Bayesian Statistics: Mixture Models (Assessment Week 4A with Codes)

Theme Song

---

1 Setting

1.1 SCSS Setup

# install.packages("remotes")
library('BBmisc', 'rmsfuns')
#remotes::install_github("rstudio/sass")
lib('sass')

## sass 
## TRUE

/* https://stackoverflow.com/a/66029010/3806250 */
h1 { color: #002C54; }
h2 { color: #2F496E; }
h3 { color: #375E97; }
h4 { color: #556DAC; }
h5 { color: #92AAC7; }

/* ----------------------------------------------------------------- */
/* https://gist.github.com/himynameisdave/c7a7ed14500d29e58149#file-broken-gradient-animation-less */
.hover01 {
  /* color: #FFD64D; */
  background: linear-gradient(155deg, #EDAE01 0%, #FFEB94 100%);
  transition: all 0.45s;
  &:hover{
    background: linear-gradient(155deg, #EDAE01 20%, #FFEB94 80%);
    }
  }

.hover02 {
  color: #FFD64D;
  background: linear-gradient(155deg, #002C54 0%, #4CB5F5 100%);
  transition: all 0.45s;
  &:hover{
    background: linear-gradient(155deg, #002C54 20%, #4CB5F5 80%);
    }
  }

.hover03 {
  color: #FFD64D;
  background: linear-gradient(155deg, #A10115 0%, #FF3C5C 100%);
  transition: all 0.45s;
  &:hover{
    background: linear-gradient(155deg, #A10115 20%, #FF3C5C 80%);
    }
  }

## https://stackoverflow.com/a/36846793/3806250
options(width = 999)
knitr::opts_chunk$set(class.source = 'hover01', class.output = 'hover02', class.error = 'hover03')

1.2 Setup

if(!suppressPackageStartupMessages(require('BBmisc'))) {
  install.packages('BBmisc', dependencies = TRUE, INSTALL_opts = '--no-lock')
}
suppressPackageStartupMessages(require('BBmisc'))
# suppressPackageStartupMessages(require('rmsfuns'))

pkgs <- c('devtools', 'knitr', 'kableExtra', 'tidyr', 
          'readr', 'lubridate', 'reprex', 'echarts4r', 
          'timetk', 'plyr', 'dplyr', 'stringr', 'magrittr', 
          'tdplyr', 'tidyverse', 'formattable', 
          'paletteer')

suppressAll(lib(pkgs))
# load_pkg(pkgs)

## Set the timezone but not change the datetime
Sys.setenv(TZ = 'Asia/Tokyo')
## options(knitr.table.format = 'html') will set all kableExtra tables to be 'html', otherwise need to set the parameter on every single table.
options(warn = -1, knitr.table.format = 'html')#, digits.secs = 6)

## https://stackoverflow.com/questions/39417003/long-vectors-not-supported-yet-abnor-in-rmd-but-not-in-r-script
knitr::opts_chunk$set(message = FALSE, warning = FALSE)#, 
                      #cache = TRUE, cache.lazy = FALSE)

rm(pkgs)

2 受講生によるテスト：MCMC algorithms and density estimation

課題をすぐに提出してください

課題の提出期限は、5月31日 15:59 JSTですが、可能であれば1日か2日早く提出してください。 早い段階で提出すると、他の受講生のレビューを時間内に得る可能性が高くなります。

2.1 説明

The R dataset faithful contains data on waiting time between eruptions (the column named waiting) and the duration of the eruption (the column named eruptions) for the famous Old Faithful geyser in Yellowstone National Park, Wyoming, USA.

In this case, you are asked to modify the MCMC algorithm provided in “Sample code for density estimation problems” (as opposed to the EM algorithm you used in the previous peer assignment) to provide a (Bayesian) density estimate the marginal distribution of the duration of the eruptions using a location-and-scale mixture of 2 univariate Gaussian distributions (as opposed to the location mixture of 6 univariate Gaussian distributions that we used for the galaxies dataset). Assume that the priors are \(\omega∼Beta(1,1)\), \(\mu_k∼Normal(η,\tau^2)\) and \(1/σ^2_k∼Gamma(d,q)\), where \(\eta\), \(\tau^2\), \(d\) and \(q\) are selected using an empirical Bayes approach similar to the one we used in “Sample code for density estimation problems”.

2.1.1 Review criteria

Reviewers will check whether the code has been modified correctly, and whether the density estimate you generate appears correct. Please remember that you are being asked to use a location-and-scale mixture to generate the density estimate, so the “Sample code for density estimation problems” cannot be used directly and requires some modification. Before submitting your answer, it might be useful to compare the density estimate generated by your algorithm against a kernel density estimate generated by the R function density(), and agains the answer to the previous peer assignment.

2.2 自分の提出物

2.2.1 Assignment

Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.

## Load package
#if(!suppressPackageStartupMessages(require('invgamma'))) {
  ##install.packages('invgamma', dependencies = TRUE, INSTALL_opts = '--no-lock')
#  devtools::install_github('dkahle/invgamma', force=TRUE)
#}

## these few packages have `rinvgamma` function.
#library('EDISON', 'extraDistr', 'MCMCpack')

## function in 'invgamma' will be more accurate than 'MCMCpack'
#suppressPackageStartupMessages(require('invgamma'))
suppressAll(lib('EDISON', 'extraDistr', 'MCMCpack', 'invgamma'))

### Get a "Bayesian" kernel density estimator based on the same location mixture of 6 normals
## Priors set up using an "empirical Bayes" approach
### Loading data and setting up global variables
library(MASS)
library(MCMCpack)
data(faithful)
KK      <- 2  # As asked
x       <- faithful[,1]
n       <- length(x)
aa      <- rep(1,KK)  
eta     <- mean(x)    
tau     <- sqrt(var(x))
dd      <- 2
qq      <- var(x)/KK
mu_0    <- rnorm(KK, eta, tau)
sigma_0 <- invgamma::rinvgamma(KK, dd, qq)

## Initialize the parameters
w       <- rep(1,KK)/KK
mu      <- rnorm(KK, mean(x), sd(x))
sigma   <- sd(x)/KK
cc      <- sample(1:KK, n, replace=TRUE, prob=w)

## Number of iterations of the sampler
rrr     <- 12000
burn    <- 2000

## Storing the samples
cc.out    <- array(0, dim=c(rrr, n))
w.out     <- array(0, dim=c(rrr, KK))
mu.out    <- array(0, dim=c(rrr, KK))
sigma.out <- rep(0, rrr)
logpost   <- rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v <- rep(0,KK)
    for(k in 1:KK){
      v[k] <- log(w[k]) + dnorm(x[i], mu[k], sigma, log=TRUE)  #Compute the log of the weights
    }
    v     <- exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] <- sample(1:KK, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  ## https://cran.r-project.org/web/packages/rBeta2009/rBeta2009.pdf
  ## gtools, extraDistr, MCMCpack also have rbeta and rdirichlet functions.
  w <- as.vector(rdirichlet(1, aa + tabulate(cc, nbins=KK)))
  
  # Sample the means
  for(k in 1:KK){
    nk       <- sum(cc==k)
    xsumk    <- sum(x[cc==k])
    tau2.hat <- 1/(nk/sigma^2 + 1/sigma_0[k]^2)
    mu.hat   <- tau2.hat*(xsumk/sigma^2 + mu_0[k]/sigma_0[k]^2)
    mu[k]    <- rnorm(1, mu.hat, sqrt(tau2.hat))
  }
  
  # Sample the variances
  dd.star    <- dd + n/2
  qq.star    <- qq + sum((x - mu[cc])^2)/2
  sigma      <- sqrt(1/rgamma(1, dd.star, qq.star))
  
  # Store samples
  cc.out[s,]   <- cc
  w.out[s,]    <- w
  mu.out[s,]   <- mu
  sigma.out[s] <- sigma
  for(i in 1:n){
    logpost[s] <- logpost[s] + log(w[cc[i]]) + dnorm(x[i], mu[cc[i]], sigma, log=TRUE)
  }
  logpost[s]   <- logpost[s] + log(ddirichlet(w, aa))
  for(k in 1:KK){
    logpost[s] <- logpost[s] + dnorm(mu[k], eta, tau, log=TRUE)
  }
  logpost[s]   <- logpost[s] + dgamma(1/sigma^2, dd, qq, log=TRUE) - 4*log(sigma)
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].

xx  <- seq(0,7,length.out=150)
nxx <- length(xx)

## Plot EM Density Estimate
density.EM = rep(0, nxx)
for(s in 1:nxx){
  for(k in 1:KK){
    density.EM[s] = density.EM[s] + w[k]*dnorm(xx[s], mu[k], sigma)
  }
}

#yy = density(x)
#plot(Eruptions, density.EM, col="red", lwd=2, type="l")
#plot(xx, rep(density.EM, 2), col="red", lwd=2, type="l")
#points(x, rep(0,n))

## Compute the samples of the density over a dense grid
density.mcmc <- array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
    for(k in 1:KK){
        density.mcmc[s,] <- density.mcmc[s,] + w.out[s+burn,k]*dnorm(xx,mu.out[s+burn,k],sigma.out[s+burn])
    }
}
density.mcmc.m <- apply(density.mcmc , 2, mean)

Provide the a graph of the density estimate on the interval [0,7].

## Plot Bayesian estimate with pointwise credible bands along with kernel density estimate and frequentist point estimate
colscale <- c("black", "blue", "red")
yy <- density(x)
density.mcmc.lq <- replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq <- replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)

plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruption Duration", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col=colscale[1], lwd=2)
lines(xx, density.EM, col=colscale[2], lty=2, lwd=2)
lines(yy, col=colscale[3], lty=3, lwd=2)
points(x, rep(0,n))
legend(5, 0.45, c("KDE","EM","MCMC"), col=colscale[c(3,2,1)], lty=c(3,2,1), lwd=2, bty="n")

2.2.2 Marking

Is the sampler for \(c\) correct?

The full conditional for \(c_i\) is given by,

\[\begin{align} \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \end{align}\]

(This assumes that the mixture weights are parameterized as

for(i in 1:n){
  v    = rep(0,2)
  v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
  v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
  v    = exp(v - max(v))/sum(exp(v - max(v)))
  cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}

## Error in sample.int(length(x), size, replace, prob): NA in probability vector

• 0点 No
• 1点 Yes

Is the sampler for \(\omega\) correct?

The full conditional is simply

\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]

The following one line of code implements it:

w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))

• 0点 No
• 1点 Yes

Are the sampler for \(\mu\) correct?

\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]

The following code implements the sampler:

# Sample the means
for(k in 1:2){
  nk       = sum(cc==k)
  xsumk    = sum(x[cc==k])
  tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
  mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
  mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Are the samplers for \(\sigma_k\) correct?

The full conditional takes the form:

\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]

The following code implements the sampler:

# Sample the variances
for(k in 1:2){
  nk    = sum(cc==k)
  dd.star = dd + nk/2
  qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
  sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Does the code to generate the density estimate appear correct?

This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

Note that this will generate not only a point estimate, but also pointwise credible intervals.

• 0点 No
• 1点 Yes

Does the graph appear to be correct?

The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.

The full code for the MCMC algorithm follows:

x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))

## Priors set up using an "empirical Bayes" approach
aa  = rep(1,2)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/2

## Initialize the parameters
w     = 1/2
mu    = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc    = sample(1:2, n, replace=T, prob=c(1/2,1/2))

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = rep(0, rrr)
mu.out    = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v    = rep(0,2)
    v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
    v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
    v    = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
  
  # Sample the parameters of the components
  for(k in 1:2){
    # Sample the means
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
    # Sample the variances
    dd.star = dd + nk/2
    qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
  }
  
  # Store samples
  cc.out[s,]    = cc
  w.out[s]     = w
  mu.out[s,]    = mu
  sigma.out[s,] = sigma
  logpost[s] = 0
  for(i in 1:n){
    if(cc[i] == 1){
      logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
    }else{
      logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
    }
  }
  logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
  for(k in 1:2){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
  }
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

## Compute the samples of the density over a dense grid
xx  = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) + 
                               (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

• 0点 No
• 1点 Yes

2.3 ピアレビュー

2.3.1 1st Peer

2.3.1.1 Assignment

Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.

### Get a "Bayesian" kernel density estimator based on the same location mixture of 6 normals
## Priors set up using an "empirical Bayes" approach
library(MASS)
library(MCMCpack)
data(faithful)
KK = 2          # As asked
x  = faithful[,1]
aa      = rep(1,KK)  
eta     = mean(x)    
tau     = sqrt(var(x))
dd      = 2
qq      = var(x)/KK
mu_0    = rnorm(KK, eta, tau)
sigma_0 = rinvgamma(KK, dd, qq)

## Initialize the parameters
w     = rep(1,KK)/KK
mu    = rnorm(KK, mean(x), sd(x))
sigma = sd(x)/KK
cc    = sample(1:KK, n, replace=T, prob=w)

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = array(0, dim=c(rrr, KK))
mu.out    = array(0, dim=c(rrr, KK))
sigma.out = rep(0, rrr)
logpost   = rep(0, rrr)

for(s in 1:rrr){
  
  # Sample the indicators
  for(i in 1:n){
    v = rep(0, KK)
    
    for(k in 1:KK){
      v[k] = log(w[k]) + dnorm(x[i], mu[k], sigma, log = TRUE)  #Compute the log of the weights
    }
    v      = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i]  = sample(1:KK, 1, replace = TRUE, prob = v)
  }
  
  # Sample the weights
  w = as.vector(rdirichlet(1, aa + tabulate(cc, nbins=KK)))
  
  # Sample the means
  for(k in 1:KK){
    nk       = sum(cc==k)
    xsumk    = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma^2 + 1/sigma_0[k]^2)
    mu.hat   = tau2.hat*(xsumk/sigma^2 + mu_0[k]/sigma_0[k]^2)
    mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
  }
  
  # Sample the variances
  dd.star = dd + n/2
  qq.star = qq + sum((x - mu[cc])^2)/2
  sigma   = sqrt(1/rgamma(1, dd.star, qq.star))
  
  # Store samples
  cc.out[s,]   = cc
  w.out[s,]    = w
  mu.out[s,]   = mu
  sigma.out[s] = sigma
  
  for(i in 1:n){
    logpost[s] = logpost[s] + log(w[cc[i]]) + dnorm(x[i], mu[cc[i]], sigma, log = TRUE)
  }
  logpost[s] = logpost[s] + log(ddirichlet(w, aa))
  for(k in 1:KK){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = TRUE)
  }
  logpost[s] = logpost[s] + dgamma(1/sigma^2, dd, qq, log = TRUE) - 4*log(sigma)
  
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].

xx  = seq(0,7,length.out=150)
nxx = length(xx)

## Compute the samples of the density over a dense grid
density.mcmc = array(0, dim = c(rrr-burn, length(xx)))
for(s in 1:(rrr-burn)){
    for(k in 1:KK){
        density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn,k]*dnorm(xx,mu.out[s+burn,k],sigma.out[s+burn])
    }
}
density.mcmc.m = apply(density.mcmc , 2, mean)

Provide the a graph of the density estimate on the interval [0,7].

## Plot Bayesian estimate with pointwise credible bands along with kernel density estimate and frequentist point estimate
colscale = c("black", "blue", "red")
yy = density(x)
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)

plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruption Duration", ylab="Density")

polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col=colscale[1], lwd=2)
lines(xx, density.EM, col=colscale[2], lty=2, lwd=2)
lines(yy, col=colscale[3], lty=3, lwd=2)
points(x, rep(0,n))
legend(5, 0.45, c("KDE","EM","MCMC"), col=colscale[c(3,2,1)], lty=c(3,2,1), lwd=2, bty="n")

2.3.1.2 Marking

Is the sampler for \(c\) correct?

The full conditional for \(c_i\) is given by,

\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]

(This assumes that the mixture weights are parameterized as

for(i in 1:n){
  v    = rep(0,2)
  v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
  v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
  v    = exp(v - max(v))/sum(exp(v - max(v)))
  cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}

## Error in sample.int(length(x), size, replace, prob): NA in probability vector

• 0点 No
• 1点 Yes

Is the sampler for \(\omega\) correct?

The full conditional is simply

\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]

The following one line of code implements it:

w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))

• 0点 No
• 1点 Yes

Are the sampler for \(\mu\) correct?

\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]

The following code implements the sampler:

# Sample the means
for(k in 1:2){
  nk       = sum(cc==k)
  xsumk    = sum(x[cc==k])
  tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
  mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
  mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Are the samplers for \(\sigma_k\) correct?

The full conditional takes the form:

\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]

The following code implements the sampler:

# Sample the variances
for(k in 1:2){
  nk    = sum(cc==k)
  dd.star = dd + nk/2
  qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
  sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Does the code to generate the density estimate appear correct?

This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

Note that this will generate not only a point estimate, but also pointwise credible intervals.

• 0点 No
• 1点 Yes

Does the graph appear to be correct?

The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.

The full code for the MCMC algorithm follows:

x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))

## Priors set up using an "empirical Bayes" approach
aa  = rep(1,2)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/2

## Initialize the parameters
w     = 1/2
mu    = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc    = sample(1:2, n, replace=T, prob=c(1/2,1/2))

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = rep(0, rrr)
mu.out    = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v    = rep(0,2)
    v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
    v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
    v    = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
  
  # Sample the parameters of the components
  for(k in 1:2){
    # Sample the means
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
    # Sample the variances
    dd.star = dd + nk/2
    qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
  }
  
  # Store samples
  cc.out[s,]    = cc
  w.out[s]     = w
  mu.out[s,]    = mu
  sigma.out[s,] = sigma
  logpost[s] = 0
  for(i in 1:n){
    if(cc[i] == 1){
      logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
    }else{
      logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
    }
  }
  logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
  for(k in 1:2){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
  }
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

## Compute the samples of the density over a dense grid
xx  = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) + 
                               (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

• 0点 No
• 1点 Yes

2.3.2 2nd Peer

2.3.2.1 Assignment

Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.

### Loading data and setting up global variables
library(MASS)
library(MCMCpack)
data(faithful)
KK = 2          # As asked
x  = faithful[,1]
n  = length(x)
set.seed(781209)

### Get a "Bayesian" kernel density estimator based on the same location mixture of 2 normals
## Priors set up using an "empirical Bayes" approach
aa  = rep(1,KK)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/KK

w     = rep(1,KK)/KK
mu    = rnorm(KK, mean(x), sd(x))
sigma = rep(sd(x)/KK, KK)
cc    = sample(1:KK, n, replace=T, prob=w)

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = array(0, dim=c(rrr, KK))
mu.out    = array(0, dim=c(rrr, KK))
sigma.out = array(0, dim=c(rrr, KK))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v = rep(0,KK)
    for(k in 1:KK){
      v[k] = log(w[k]) + dnorm(x[i], mu[k], sigma[k], log=TRUE)  #Compute the log of the weights
    }
    v = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:KK, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  w = as.vector(rdirichlet(1, aa + tabulate(cc, nbins=KK)))
  
  # Sample the means
  for(k in 1:KK){
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
  }
  
  # Sample the variances
  for(k in 1:KK) {
    dd.star = dd + sum(cc==k)/2
    sumsq = 0
    for(i in 1:n) {
      if (cc[i]==k) {
        sumsq = sumsq + ((x[i]-mu[k])^2)
      }
    }
    qq.star = qq + sumsq/2
    sigma[k] = sqrt(invgamma::rinvgamma(1, shape = dd.star, scale = qq.star))
  }
  
  # Store samples
  cc.out[s,]   = cc
  w.out[s,]    = w
  mu.out[s,]   = mu
  sigma.out[s,] = sigma
  for(i in 1:n){
    logpost[s] = logpost[s] + log(w[cc[i]]) + dnorm(x[i], mu[cc[i]], sigma[cc[i]], log=TRUE)
  }
  logpost[s] = logpost[s] + log(ddirichlet(w, aa))
  for(k in 1:KK){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log=TRUE)
    logpost[s] = logpost[s] + log(invgamma::dinvgamma(sigma[k]^2, shape = dd, scale = qq))
  }
  if(s/500==floor(s/500)){
    print(paste("s =",s, w, mu, sigma, logpost[s]))
  }
}

## [1] "s = 500 0.649085670059537 4.29911087249897 0.0257392238442076 -Inf" "s = 500 0.350914329940463 2.0443538746226 0.0628798226170892 -Inf" 
## [1] "s = 1000 0.610845911800996 4.29694654403535 0.0289124372846902 -Inf" "s = 1000 0.389154088199004 2.05483569589295 0.0624528246656942 -Inf"
## [1] "s = 1500 0.629196978893728 4.32378234378882 0.030702655328229 -Inf"  "s = 1500 0.370803021106272 2.10253788421881 0.0505840813269322 -Inf"
## [1] "s = 2000 0.627386732958307 4.37176414528319 0.0356727144879368 -Inf" "s = 2000 0.372613267041693 2.23126658610265 0.0327306141248162 -Inf"
## [1] "s = 2500 0.566046915491541 4.32940008616899 0.0319139853798889 -Inf" "s = 2500 0.433953084508459 2.12765612972159 0.0410992074845721 -Inf"
## [1] "s = 3000 0.653872666444745 4.29845212862576 0.0260588569160813 -Inf" "s = 3000 0.346127333555255 2.05535047361625 0.0702653455053356 -Inf"
## [1] "s = 3500 0.582224297034073 4.37065567611536 0.0363411341561569 -Inf" "s = 3500 0.417775702965927 2.22577997866447 0.0314457936341981 -Inf"
## [1] "s = 4000 0.665909134869133 4.325826413912 0.0340783429168065 -Inf"   "s = 4000 0.334090865130867 2.11746845867021 0.0423207327093899 -Inf"
## [1] "s = 4500 0.628317248101969 4.3480902800189 0.0312968909865187 -Inf"  "s = 4500 0.371682751898031 2.17342350345536 0.0370138146381416 -Inf"
## [1] "s = 5000 0.612880487641458 4.30026290842618 0.0291594757927606 -Inf" "s = 5000 0.387119512358542 2.05156086617752 0.0735930347626931 -Inf"
## [1] "s = 5500 0.616179295286607 4.34239234358705 0.0321352284146182 -Inf" "s = 5500 0.383820704713393 2.15375066181538 0.0439223899293136 -Inf"
## [1] "s = 6000 0.669821317256534 4.32279879041033 0.0308292952489493 -Inf" "s = 6000 0.330178682743465 2.09575927765975 0.0466482534935236 -Inf"
## [1] "s = 6500 0.609947882643349 4.31857618733397 0.0294962361277672 -Inf" "s = 6500 0.390052117356652 2.09191927003505 0.0481351039363759 -Inf"
## [1] "s = 7000 0.623659776526185 4.32523521957499 0.0339824678763501 -Inf" "s = 7000 0.376340223473815 2.10478504970728 0.0454606227039882 -Inf"
## [1] "s = 7500 0.578396886682662 4.29729261346055 0.0269529439567729 -Inf" "s = 7500 0.421603113317338 2.04775316275951 0.0681558344506739 -Inf"
## [1] "s = 8000 0.571530834052 4.32816299195062 0.030712930378534 -Inf"  "s = 8000 0.428469165948 2.10266044832463 0.0447823861370567 -Inf"
## [1] "s = 8500 0.659023347228691 4.30499580371751 0.0275301233706194 -Inf" "s = 8500 0.340976652771309 2.04572919364044 0.0629367367720857 -Inf"
## [1] "s = 9000 0.658823960112415 4.31873791415184 0.0291946972712849 -Inf" "s = 9000 0.341176039887585 2.08813246762313 0.0581863751304524 -Inf"
## [1] "s = 9500 0.664632822634927 4.30719472785145 0.0280979178249494 -Inf" "s = 9500 0.335367177365073 2.05907879149062 0.0594190710889116 -Inf"
## [1] "s = 10000 0.647014865450983 4.29982122293235 0.0291321983432132 -Inf" "s = 10000 0.352985134549017 2.05014032929293 0.062598052424088 -Inf" 
## [1] "s = 10500 0.627145742442364 4.34381801258206 0.032485618320239 -Inf"  "s = 10500 0.372854257557636 2.14705294389961 0.0409444871041369 -Inf"
## [1] "s = 11000 0.655524925803427 4.32224655771092 0.0299280249968171 -Inf" "s = 11000 0.344475074196573 2.08027405411962 0.05354583502245 -Inf"  
## [1] "s = 11500 0.616112315848646 4.29279361287758 0.0273676126462287 -Inf" "s = 11500 0.383887684151354 2.04642359476572 0.0619673516755033 -Inf"
## [1] "s = 12000 0.681278494513206 4.32460797592666 0.0288678266517917 -Inf" "s = 12000 0.318721505486794 2.1103452221058 0.0493909009877983 -Inf"

Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].

Eruptions = seq(0,7, length.out=150)
## Compute the samples of the density over a dense grid
density.mcmc = array(0, dim=c(rrr-burn,length(Eruptions)))
for(s in 1:(rrr-burn)){
  for(k in 1:KK){
    density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn,k]*dnorm(Eruptions,mu.out[s+burn,k],sigma.out[s+burn,k])
  }
}
density.mcmc.m = apply(density.mcmc , 2, mean)

## Plot Bayesian estimate with pointwise credible bands along with kernel density estimate and frequentist point estimate
colscale = c("black", "blue", "red")
#yy = density(x)
yy = density(Eruptions)
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(Eruptions, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Duration", ylab="Density")
polygon(c(Eruptions,rev(Eruptions)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(Eruptions, density.mcmc.m, col=colscale[3], lwd=2)
#lines(xx, density.EM, col=colscale[2], lty=2, lwd=2)
#lines(yy, col=colscale[3], lty=3, lwd=2)
points(x, rep(0,n))

Provide the a graph of the density estimate on the interval [0,7].

Source :

2.3.2.2 Marking

Is the sampler for \(c\) correct?

The full conditional for \(c_i\) is given by,

\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]

(This assumes that the mixture weights are parameterized as

for(i in 1:n){
  v    = rep(0,2)
  v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
  v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
  v    = exp(v - max(v))/sum(exp(v - max(v)))
  cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}

• 0点 No
• 1点 Yes

Is the sampler for \(\omega\) correct?

The full conditional is simply

\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]

The following one line of code implements it:

w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))

• 0点 No
• 1点 Yes

Are the sampler for \(\mu\) correct?

\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]

The following code implements the sampler:

# Sample the means
for(k in 1:2){
  nk       = sum(cc==k)
  xsumk    = sum(x[cc==k])
  tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
  mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
  mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Are the samplers for \(\sigma_k\) correct?

The full conditional takes the form:

\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]

The following code implements the sampler:

# Sample the variances
for(k in 1:2){
  nk    = sum(cc==k)
  dd.star = dd + nk/2
  qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
  sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Does the code to generate the density estimate appear correct?

This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

Note that this will generate not only a point estimate, but also pointwise credible intervals.

• 0点 No
• 1点 Yes

Does the graph appear to be correct?

The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.

The full code for the MCMC algorithm follows:

x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))

## Priors set up using an "empirical Bayes" approach
aa  = rep(1,2)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/2

## Initialize the parameters
w     = 1/2
mu    = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc    = sample(1:2, n, replace=T, prob=c(1/2,1/2))

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = rep(0, rrr)
mu.out    = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v    = rep(0,2)
    v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
    v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
    v    = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
  
  # Sample the parameters of the components
  for(k in 1:2){
    # Sample the means
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
    # Sample the variances
    dd.star = dd + nk/2
    qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
  }
  
  # Store samples
  cc.out[s,]    = cc
  w.out[s]     = w
  mu.out[s,]    = mu
  sigma.out[s,] = sigma
  logpost[s] = 0
  for(i in 1:n){
    if(cc[i] == 1){
      logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
    }else{
      logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
    }
  }
  logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
  for(k in 1:2){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
  }
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

## Compute the samples of the density over a dense grid
xx  = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) + 
                               (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

• 0点 No
• 1点 Yes

2.3.3 3rd Peer

2.3.3.1 Assignment

Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.

### Get a "Bayesian" kernel density estimator based on the same location mixture of 6 normals
## Priors set up using an "empirical Bayes" approach
library(MASS)
library(MCMCpack)
data(faithful)
KK = 2          # As asked
x  = faithful[,1]
aa  = rep(1,KK)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/KK
mu_0 = rnorm(KK, eta, tau)
sigma_0 = rinvgamma(KK, dd, qq)

## Initialize the parameters
w     = rep(1,KK)/KK
mu    = rnorm(KK, mean(x), sd(x))
sigma = sd(x)/KK
cc    = sample(1:KK, n, replace=T, prob=w)

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = array(0, dim=c(rrr, KK))
mu.out    = array(0, dim=c(rrr, KK))
sigma.out = rep(0, rrr)
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v = rep(0,KK)
    for(k in 1:KK){
      v[k] = log(w[k]) + dnorm(x[i], mu[k], sigma, log=TRUE)  #Compute the log of the weights
    }
    v = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:KK, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  ## https://cran.r-project.org/web/packages/rBeta2009/rBeta2009.pdf
  w = as.vector(rdirichlet(1, aa + tabulate(cc, nbins=KK)))
  
  # Sample the means
  for(k in 1:KK){
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma^2 + 1/sigma_0[k]^2)
    mu.hat  = tau2.hat*(xsumk/sigma^2 + mu_0[k]/sigma_0[k]^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
  }
  
  # Sample the variances
  dd.star = dd + n/2
  qq.star = qq + sum((x - mu[cc])^2)/2
  sigma = sqrt(1/rgamma(1, dd.star, qq.star))
  
  # Store samples
  cc.out[s,]   = cc
  w.out[s,]    = w
  mu.out[s,]   = mu
  sigma.out[s] = sigma
  for(i in 1:n){
    logpost[s] = logpost[s] + log(w[cc[i]]) + dnorm(x[i], mu[cc[i]], sigma, log=TRUE)
  }
  logpost[s] = logpost[s] + log(ddirichlet(w, aa))
  for(k in 1:KK){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log=TRUE)
  }
  logpost[s] = logpost[s] + dgamma(1/sigma^2, dd, qq, log=TRUE) - 4*log(sigma)
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].

xx  = seq(0,7,length.out=150)
nxx = length(xx)

density.EM = rep(0, nxx)
for(i in 1:nxx){
  for(k in 1:KK){
    density.EM[i] = density.EM[i] + ((k-1) - w*(-1)**k) * dnorm(xx[i], mu[k], sigma[k])
  }
}
plot(xx, density.EM, col="red", lwd=2, type="l")

## Error in plot.window(...): need finite 'ylim' values

points(x,rep(0,n))

## Compute the samples of the density over a dense grid
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
    for(k in 1:KK){
        density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn,k]*dnorm(xx,mu.out[s+burn,k],sigma.out[s+burn])
    }
}
density.mcmc.m = apply(density.mcmc , 2, mean)

Provide the a graph of the density estimate on the interval [0,7].

## Plot Bayesian estimate with pointwise credible bands along with kernel density estimate and frequentist point estimate
colscale = c("black", "blue", "red")
yy = density(x)
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruption Duration", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col=colscale[1], lwd=2)
lines(xx, density.EM, col=colscale[2], lty=2, lwd=2)
lines(yy, col=colscale[3], lty=3, lwd=2)
points(x, rep(0,n))
legend(5, 0.45, c("KDE","EM","MCMC"), col=colscale[c(3,2,1)], lty=c(3,2,1), lwd=2, bty="n")

2.3.3.2 Marking

Is the sampler for \(c\) correct?

The full conditional for \(c_i\) is given by,

\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]

(This assumes that the mixture weights are parameterized as

for(i in 1:n){
  v    = rep(0,2)
  v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
  v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
  v    = exp(v - max(v))/sum(exp(v - max(v)))
  cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}

## Error in sample.int(length(x), size, replace, prob): NA in probability vector

• 0点 No
• 1点 Yes

Is the sampler for \(\omega\) correct?

The full conditional is simply

\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]

The following one line of code implements it:

w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))

• 0点 No
• 1点 Yes

Are the sampler for \(\mu\) correct?

\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]

The following code implements the sampler:

# Sample the means
for(k in 1:2){
  nk       = sum(cc==k)
  xsumk    = sum(x[cc==k])
  tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
  mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
  mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Are the samplers for \(\sigma_k\) correct?

The full conditional takes the form:

\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]

The following code implements the sampler:

# Sample the variances
for(k in 1:2){
  nk    = sum(cc==k)
  dd.star = dd + nk/2
  qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
  sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Does the code to generate the density estimate appear correct?

This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

Note that this will generate not only a point estimate, but also pointwise credible intervals.

• 0点 No
• 1点 Yes

Does the graph appear to be correct?

The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.

The full code for the MCMC algorithm follows:

x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))

## Priors set up using an "empirical Bayes" approach
aa  = rep(1,2)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/2

## Initialize the parameters
w     = 1/2
mu    = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc    = sample(1:2, n, replace=T, prob=c(1/2,1/2))

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = rep(0, rrr)
mu.out    = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v    = rep(0,2)
    v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
    v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
    v    = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
  
  # Sample the parameters of the components
  for(k in 1:2){
    # Sample the means
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
    # Sample the variances
    dd.star = dd + nk/2
    qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
  }
  
  # Store samples
  cc.out[s,]    = cc
  w.out[s]     = w
  mu.out[s,]    = mu
  sigma.out[s,] = sigma
  logpost[s] = 0
  for(i in 1:n){
    if(cc[i] == 1){
      logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
    }else{
      logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
    }
  }
  logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
  for(k in 1:2){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
  }
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

## Compute the samples of the density over a dense grid
xx  = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) + 
                               (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

• 0点 No
• 1点 Yes

2.3.4 4th Peer

2.3.4.1 Assignment

Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.

### Loading data and setting up global variables
library(MASS)
library(MCMCpack)
data(faithful)
KK = 2          # As asked
x  = faithful[,1]
n  = length(x)
set.seed(781209)

### Get a "Bayesian" kernel density estimator based on the same location mixture of 2 normals
## Priors set up using an "empirical Bayes" approach
aa  = rep(1,KK)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/KK

w     = rep(1,KK)/KK
mu    = rnorm(KK, mean(x), sd(x))
sigma = rep(sd(x)/KK, KK)
cc    = sample(1:KK, n, replace=T, prob=w)

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = array(0, dim=c(rrr, KK))
mu.out    = array(0, dim=c(rrr, KK))
sigma.out = array(0, dim=c(rrr, KK))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v = rep(0,KK)
    for(k in 1:KK){
      v[k] = log(w[k]) + dnorm(x[i], mu[k], sigma[k], log=TRUE)  #Compute the log of the weights
    }
    v = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:KK, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  w = as.vector(rdirichlet(1, aa + tabulate(cc, nbins=KK)))
  
  # Sample the means
  for(k in 1:KK){
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
  }
  
  # Sample the variances
  for(k in 1:KK) {
    dd.star = dd + sum(cc==k)/2
    sumsq = 0
    for(i in 1:n) {
      if (cc[i]==k) {
        sumsq = sumsq + ((x[i]-mu[k])^2)
      }
    }
    qq.star = qq + sumsq/2
    sigma[k] = sqrt(MCMCpack::rinvgamma(1, shape = dd.star, scale = qq.star))
  }
  
  # Store samples
  cc.out[s,]   = cc
  w.out[s,]    = w
  mu.out[s,]   = mu
  sigma.out[s,] = sigma
  for(i in 1:n){
    logpost[s] = logpost[s] + log(w[cc[i]]) + dnorm(x[i], mu[cc[i]], sigma[cc[i]], log=TRUE)
  }
  logpost[s] = logpost[s] + log(ddirichlet(w, aa))
  for(k in 1:KK){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log=TRUE)
    logpost[s] = logpost[s] + log(MCMCpack::dinvgamma(sigma[k]^2, shape = dd, scale = qq))
  }
  if(s/500==floor(s/500)){
    print(paste("s =",s, w, mu, sigma, logpost[s]))
  }
}

## [1] "s = 500 0.65972914904991 4.28469003268875 0.405452035141612 -284.691001291928" "s = 500 0.34027085095009 1.99006962679139 0.250995183146609 -284.691001291928"
## [1] "s = 1000 0.595942847037826 4.27800551958405 0.377348480705164 -286.969076980901" "s = 1000 0.404057152962174 2.03894151043637 0.30982681142742 -286.969076980901" 
## [1] "s = 1500 0.647284731246712 4.32792289508716 0.42383087208717 -283.984683957035"  "s = 1500 0.352715268753288 2.06264939688377 0.293003644635685 -283.984683957035"
## [1] "s = 2000 0.681875974292433 4.31964666156271 0.411719777867089 -284.392728508546" "s = 2000 0.318124025707567 2.05510718985465 0.295371255576304 -284.392728508546"
## [1] "s = 2500 0.601103182730055 4.24400160743064 0.446830400212041 -285.459906307434" "s = 2500 0.398896817269945 2.03587612253421 0.241473527538476 -285.459906307434"
## [1] "s = 3000 0.661235314614079 4.28417774604423 0.399650525878003 -285.992706414487" "s = 3000 0.338764685385921 2.07827156671072 0.297205073536538 -285.992706414487"
## [1] "s = 3500 0.644527213106916 4.28812055333749 0.441452138427281 -283.120726285281" "s = 3500 0.355472786893084 2.01146104149005 0.258057420679898 -283.120726285281"
## [1] "s = 4000 0.687810647492561 4.2844578377926 0.458980223432612 -287.498421765689"  "s = 4000 0.312189352507439 2.07489936081685 0.259843371397385 -287.498421765689"
## [1] "s = 4500 0.675398367857913 4.23091315341658 0.410829138479542 -285.115602530381" "s = 4500 0.324601632142087 2.00548178399042 0.263106146522126 -285.115602530381"
## [1] "s = 5000 0.623677983258687 4.2986070783669 0.460990184040201 -287.679382635602"  "s = 5000 0.376322016741313 2.04273694828362 0.296539486159871 -287.679382635602"
## [1] "s = 5500 0.650641729955276 4.30297831389609 0.410588171771523 -284.091323207027" "s = 5500 0.349358270044724 2.05603614578286 0.314091813936972 -284.091323207027"
## [1] "s = 6000 0.691002769510701 4.30695193759895 0.43643062048752 -285.25059244604"  "s = 6000 0.308997230489299 2.00724366068866 0.258042136715897 -285.25059244604"
## [1] "s = 6500 0.639653531193993 4.30228067466188 0.466022578492273 -289.071680298941" "s = 6500 0.360346468806007 2.0370347870673 0.223458414339796 -289.071680298941" 
## [1] "s = 7000 0.657266286407271 4.25015359390783 0.408697090783594 -285.609121794953" "s = 7000 0.342733713592729 1.98352245342408 0.281123375854885 -285.609121794953"
## [1] "s = 7500 0.582150245729404 4.27752884035096 0.405335835898845 -288.834567817749" "s = 7500 0.417849754270596 2.03280535197236 0.302512079952417 -288.834567817749"
## [1] "s = 8000 0.597745067402946 4.30797847743645 0.426676105386858 -286.886008869559" "s = 8000 0.402254932597054 1.97410330661178 0.266695259035665 -286.886008869559"
## [1] "s = 8500 0.67048977917052 4.34923962430531 0.440934482727733 -287.976277284725" "s = 8500 0.32951022082948 1.99450886009298 0.253385375719033 -287.976277284725"
## [1] "s = 9000 0.676526152024463 4.32975227643208 0.424164941801625 -286.420409681512" "s = 9000 0.323473847975537 2.03912276672083 0.304347017719118 -286.420409681512"
## [1] "s = 9500 0.675252958080913 4.32408885533291 0.42428978538769 -284.557683174045"  "s = 9500 0.324747041919087 2.01338154879308 0.269521567887484 -284.557683174045"
## [1] "s = 10000 0.657398814367743 4.2933157987876 0.458707092652969 -283.811172985985"  "s = 10000 0.342601185632257 2.03282618126385 0.248515730931108 -283.811172985985"
## [1] "s = 10500 0.662058917060533 4.31483111471995 0.427552521301718 -284.108178852953" "s = 10500 0.337941082939467 2.00950256527503 0.279659447389632 -284.108178852953"
## [1] "s = 11000 0.669552336247041 4.36928639963658 0.427369327130037 -288.518332324442" "s = 11000 0.330447663752959 1.97992561769347 0.296908732685882 -288.518332324442"
## [1] "s = 11500 0.627693164995977 4.20937955935309 0.437031029444333 -287.287695169602" "s = 11500 0.372306835004023 2.0025029877848 0.248798427816352 -287.287695169602" 
## [1] "s = 12000 0.706166055902923 4.32329970006521 0.423366944818915 -291.385728914099" "s = 12000 0.293833944097077 2.09369936916931 0.272464988230699 -291.385728914099"

Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].

Eruptions = seq(0,7, length.out=150)
## Compute the samples of the density over a dense grid
density.mcmc = array(0, dim=c(rrr-burn,length(Eruptions)))
for(s in 1:(rrr-burn)){
  for(k in 1:KK){
    density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn,k]*dnorm(Eruptions,mu.out[s+burn,k],sigma.out[s+burn,k])
  }
}
density.mcmc.m = apply(density.mcmc , 2, mean)

## Plot Bayesian estimate with pointwise credible bands along with kernel density estimate and frequentist point estimate
colscale = c("black", "blue", "red")
#yy = density(x)
yy = density(Eruptions)
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(Eruptions, density.mcmc.m, type="n", ylim=c(0,max(density.mcmc.uq)), xlab="Duration", ylab="Density")
polygon(c(Eruptions,rev(Eruptions)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(Eruptions, density.mcmc.m, col=colscale[3], lwd=2)
#lines(xx, density.EM, col=colscale[2], lty=2, lwd=2)
#lines(yy, col=colscale[3], lty=3, lwd=2)
points(x, rep(0,n))

Provide the a graph of the density estimate on the interval [0,7].

2.3.4.2 Marking

Is the sampler for \(c\) correct?

The full conditional for \(c_i\) is given by,

\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]

(This assumes that the mixture weights are parameterized as

for(i in 1:n){
  v    = rep(0,2)
  v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
  v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
  v    = exp(v - max(v))/sum(exp(v - max(v)))
  cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}

• 0点 No
• 1点 Yes

Is the sampler for \(\omega\) correct?

The full conditional is simply

\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]

The following one line of code implements it:

w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))

• 0点 No
• 1点 Yes

Are the sampler for \(\mu\) correct?

\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]

The following code implements the sampler:

# Sample the means
for(k in 1:2){
  nk       = sum(cc==k)
  xsumk    = sum(x[cc==k])
  tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
  mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
  mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Are the samplers for \(\sigma_k\) correct?

The full conditional takes the form:

\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]

The following code implements the sampler:

# Sample the variances
for(k in 1:2){
  nk    = sum(cc==k)
  dd.star = dd + nk/2
  qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
  sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Does the code to generate the density estimate appear correct?

This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

Note that this will generate not only a point estimate, but also pointwise credible intervals.

• 0点 No
• 1点 Yes

Does the graph appear to be correct?

The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.

The full code for the MCMC algorithm follows:

x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))

## Priors set up using an "empirical Bayes" approach
aa  = rep(1,2)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/2

## Initialize the parameters
w     = 1/2
mu    = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc    = sample(1:2, n, replace=T, prob=c(1/2,1/2))

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = rep(0, rrr)
mu.out    = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v    = rep(0,2)
    v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
    v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
    v    = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
  
  # Sample the parameters of the components
  for(k in 1:2){
    # Sample the means
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
    # Sample the variances
    dd.star = dd + nk/2
    qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
  }
  
  # Store samples
  cc.out[s,]    = cc
  w.out[s]     = w
  mu.out[s,]    = mu
  sigma.out[s,] = sigma
  logpost[s] = 0
  for(i in 1:n){
    if(cc[i] == 1){
      logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
    }else{
      logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
    }
  }
  logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
  for(k in 1:2){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
  }
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

## Compute the samples of the density over a dense grid
xx  = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) + 
                               (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

• 0点 No
• 1点 Yes

2.3.5 5th Peer

2.3.5.1 Assignment

Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.

x  = faithful$eruptions
KK = 2          # Based on the description of the dataset
n  = length(x)

## Initialize the parameters
w     = rep(1,KK)/KK
mu    = rnorm(KK, mean(x), sd(x))
sigma = rep(1,KK)*sd(x)/KK
cc    = sample(1:KK, n, replace=T, prob=w)

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = array(0, dim=c(rrr, KK))
mu.out    = array(0, dim=c(rrr, KK))
sigma.out = array(0, dim=c(rrr,KK))
logpost   = rep(0, rrr)

#prior
eta = mean(x)
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/KK

for(s in 1:rrr){   # Sample the indicators
  
  for(i in 1:n){
    v       = rep(0,KK)
    
    for(k in 1:KK){
      v[k]  = log(w[k]) + dnorm(x[i], mu[k], sigma[k], log=TRUE)  #Compute the log of the weights     }
      v     = exp(v - max(v))/sum(exp(v - max(v)))
      cc[i] = sample(1:KK, 1, replace=TRUE, prob=v)
      
      # Sample the weights
      w[1]  = rbeta(1,1+sum(cc==1),1+sum(cc==2))
      w[2]  = 1-w[1]
      
      # Sample the means
      for(k in 1:KK){
        nk       = sum(cc==k)
        xsumk    = sum(x[cc==k])
        tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
        mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
        mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
      }
      # Sample the variances
      dd.star = dd + n/2
      for (k in 1:KK) {
        qq.star  = qq + sum((x[cc==k] - mu[k])^2)/2
        sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
      }
      
      # Store samples
      cc.out[s,]    = cc
      w.out[s,]     = w
      mu.out[s,]    = mu
      sigma.out[s,] = sigma
      
      for(i in 1:n){
        logpost[s] = logpost[s] + log(w[cc[i]]) + dnorm(x[i], mu[cc[i]], sigma[cc[i]], log=TRUE)
      }
      
      logpost[s] = logpost[s] + dbeta(w[1],1,1,log = T)
      
      for(k in 1:KK){
        logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log=TRUE)+dgamma(1/sigma[k]^2, dd, qq, log=TRUE) - 4*log(sigma[k])
        }
      if(s/500==floor(s/500)){
          print(paste("s =",s))
      }
    }
  }
}

## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"

Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].

xx  = seq(0,7,length.out=150)
nxx = length(xx)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))

for(s in 1:(rrr-burn)){
  for(k in 1:KK){
    density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn,k]*dnorm(xx,mu.out[s+burn,k],sigma.out[s+burn,k])
  }
}

density.mcmc.m  = replace_na(apply(density.mcmc , 2, mean, na.rm=TRUE), 0)
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)

plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")

lines(xx, density.mcmc.m, col="red", lwd=2)
points(x, rep(0,n))

Provide the a graph of the density estimate on the interval [0,7].

2.3.5.2 Marking

Is the sampler for \(c\) correct?

The full conditional for \(c_i\) is given by,

\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]

(This assumes that the mixture weights are parameterized as

for(i in 1:n){
  v    = rep(0,2)
  v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
  v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
  v    = exp(v - max(v))/sum(exp(v - max(v)))
  cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}

• 0点 No
• 1点 Yes

Is the sampler for \(\omega\) correct?

The full conditional is simply

\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]

The following one line of code implements it:

w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))

• 0点 No
• 1点 Yes

Are the sampler for \(\mu\) correct?

\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]

The following code implements the sampler:

# Sample the means
for(k in 1:2){
  nk       = sum(cc==k)
  xsumk    = sum(x[cc==k])
  tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
  mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
  mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Are the samplers for \(\sigma_k\) correct?

The full conditional takes the form:

\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]

The following code implements the sampler:

# Sample the variances
for(k in 1:2){
  nk    = sum(cc==k)
  dd.star = dd + nk/2
  qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
  sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Does the code to generate the density estimate appear correct?

This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

Note that this will generate not only a point estimate, but also pointwise credible intervals.

• 0点 No
• 1点 Yes

Does the graph appear to be correct?

The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.

The full code for the MCMC algorithm follows:

x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))

## Priors set up using an "empirical Bayes" approach
aa  = rep(1,2)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/2

## Initialize the parameters
w     = 1/2
mu    = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc    = sample(1:2, n, replace=T, prob=c(1/2,1/2))

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = rep(0, rrr)
mu.out    = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v    = rep(0,2)
    v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
    v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
    v    = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
  
  # Sample the parameters of the components
  for(k in 1:2){
    # Sample the means
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
    # Sample the variances
    dd.star = dd + nk/2
    qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
  }
  
  # Store samples
  cc.out[s,]    = cc
  w.out[s]     = w
  mu.out[s,]    = mu
  sigma.out[s,] = sigma
  logpost[s] = 0
  for(i in 1:n){
    if(cc[i] == 1){
      logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
    }else{
      logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
    }
  }
  logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
  for(k in 1:2){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
  }
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

## Compute the samples of the density over a dense grid
xx  = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) + 
                               (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

• 0点 No
• 1点 Yes

2.3.6 6th Peer

2.3.6.1 Assignment

Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.

### Loading data and setting up global variables

library(MASS)
library(MCMCpack)
library(invgamma)

data(faithful)
x <- faithful$waiting
hist(x)

n = dim(faithful)[1]

## Priors set up using an "empirical Bayes" approach

KK = 2
aa  = 1
eta = 1
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/KK

## Initialize the parameters
w     = rep(1,KK)/KK
mu    = rnorm(KK, mean(x), sd(x))
sigma = rinvgamma(KK, dd, qq)
cc    = sample(1:KK, n, replace=T, prob=w)

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = array(0, dim=c(rrr, KK))
mu.out    = array(0, dim=c(rrr, KK))
sigma.out = array(0, dim=c(rrr, KK))
logpost   = rep(0, rrr)


for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v = rep(0,KK)
    for(k in 1:KK){
      v[k] = log(w[k]) + dnorm(x[i], mu[k], sigma[k], log=TRUE)  #Compute the log of the weights
                  }
      v = exp(v - max(v))/sum(exp(v - max(v)))
                  
      cc[i] = sample(1:KK, 1, replace=TRUE, prob=v)
                }
  
  # Sample the weights
  w[1] = rbeta(1, sum(cc == 1),sum(cc == 2) )
  w[2] = 1 - w[1] 
  
  # Sample the means
  for(k in 1:KK){
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
  }

  # Sample the variances
  for(k in 1:KK){
    nk    = sum(cc==k)
    dd.star = dd + nk/2
    qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
  }
  # Store samples
  cc.out[s,]   = cc
  w.out[s,]    = w
  mu.out[s,]   = mu
  sigma.out[s,] = sigma
  for(i in 1:n){
    logpost[s] = logpost[s] + log(w[cc[i]]) + dnorm(x[i], mu[cc[i]], sigma[cc[i]], log=TRUE)
  }
  
  for(k in 1:KK){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log=TRUE)
    logpost[s] = logpost[s] + dgamma(1/sigma[k]^2, dd, qq, log=TRUE) - 4*log(sigma[k])
    logpost[s] = logpost[s] + dbeta(w[k], aa, aa, log = TRUE)
               }
  
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].

colscale = c("blue", "red")
yy = density(x)
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Velocity", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col=colscale[1], lwd=2)

lines(yy, col=colscale[2], lty=3, lwd=2)
points(x, runif(n, min = 0, max = .005), col=colscale[cc])
legend(90, 0.04, c("KDE","MCMC"), col=colscale[c(2,1)], lty=c(2,1), lwd=2, bty="n")

Provide the a graph of the density estimate on the interval [0,7].

copy and paste doesn't work here

2.3.6.2 Marking

Is the sampler for \(c\) correct?

The full conditional for \(c_i\) is given by,

\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]

(This assumes that the mixture weights are parameterized as

for(i in 1:n){
  v    = rep(0,2)
  v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
  v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
  v    = exp(v - max(v))/sum(exp(v - max(v)))
  cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}

• 0点 No
• 1点 Yes

Is the sampler for \(\omega\) correct?

The full conditional is simply

\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]

The following one line of code implements it:

w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))

• 0点 No
• 1点 Yes

Are the sampler for \(\mu\) correct?

\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]

The following code implements the sampler:

# Sample the means
for(k in 1:2){
  nk       = sum(cc==k)
  xsumk    = sum(x[cc==k])
  tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
  mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
  mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Are the samplers for \(\sigma_k\) correct?

The full conditional takes the form:

\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]

The following code implements the sampler:

# Sample the variances
for(k in 1:2){
  nk    = sum(cc==k)
  dd.star = dd + nk/2
  qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
  sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Does the code to generate the density estimate appear correct?

This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

Note that this will generate not only a point estimate, but also pointwise credible intervals.

• 0点 No
• 1点 Yes

Does the graph appear to be correct?

The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.

The full code for the MCMC algorithm follows:

x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))

## Priors set up using an "empirical Bayes" approach
aa  = rep(1,2)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/2

## Initialize the parameters
w     = 1/2
mu    = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc    = sample(1:2, n, replace=T, prob=c(1/2,1/2))

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = rep(0, rrr)
mu.out    = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v    = rep(0,2)
    v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
    v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
    v    = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
  
  # Sample the parameters of the components
  for(k in 1:2){
    # Sample the means
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
    # Sample the variances
    dd.star = dd + nk/2
    qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
  }
  
  # Store samples
  cc.out[s,]    = cc
  w.out[s]     = w
  mu.out[s,]    = mu
  sigma.out[s,] = sigma
  logpost[s] = 0
  for(i in 1:n){
    if(cc[i] == 1){
      logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
    }else{
      logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
    }
  }
  logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
  for(k in 1:2){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
  }
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

## Compute the samples of the density over a dense grid
xx  = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) + 
                               (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

• 0点 No
• 1点 Yes

2.3.7 7th Peer

2.3.7.1 Assignment

Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.

rm(list=ls())
library(MCMCpack)

###### Setup data
x  = faithful$eruptions
n  = length(x)
KK = 2


###### MCMC algorithm to fit the location-and-scale mixture of 2 Gaussians

## Priors set up 
aa  = rep(1,KK)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/2


## Initialize
w     = 0.5
mu    = rnorm(KK, mean(x), sd(x))
sigma = rep(sd(x)/2,KK)
cc    = sample(1:KK, n, replace=T, prob=c(w,1-w))

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = rep(0, rrr)
mu.out    = array(0, dim=c(rrr, KK))
sigma.out = array(0, dim=c(rrr, KK))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v = rep(0,KK)
    v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
    v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
    v = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:KK, 1, replace=TRUE, prob=v)
  }
  
  
  # Sample the weights
  w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
  
  # Sample the means
  for(k in 1:KK){
    # Sample the means
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
    
  }
  
  # Sample the variances
  for(k in 1:KK){
    dd.star = dd +sum(cc==k)/2
    qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(rinvgamma(1, dd.star, qq.star))
  }
  # Store samples
  cc.out[s,]   = cc
  w.out[s]    = w
  mu.out[s,]   = mu
  sigma.out[s,] = sigma
  logpost[s] = 0
  for(i in 1:n){
    if(cc[i]==1){
      logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
    }else{
      logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
    }
  }
  logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2])
  for(k in 1:KK){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T)
    
  }
  for (k in 1:KK){
    logpost[s] = logpost[s] + dgamma(1/sigma[k]^2, dd, qq,log = T) - 4*log(sigma[k])
  }

  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

xx  = seq(0,7,length=150)

## Compute the samples of the density over a dense grid
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) + 
    (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
  
}
density.mcmc.m = apply(density.mcmc , 2, mean)

colscale = c("black", "blue")
yy = density(x)

density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruption", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)

points(x,rep(0,n))

Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].

xx  = seq(0,7,length.out=150)

## Compute the samples of the density over a dense grid
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) + 
    (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
  
}
density.mcmc.m = apply(density.mcmc , 2, mean)

colscale = c("black", "blue")
yy = density(x)

density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruption", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)

points(x,rep(0,n))

Provide the a graph of the density estimate on the interval [0,7].

xx  = seq(0,7,length.out=150)

## Compute the samples of the density over a dense grid
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) + 
    (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
  
}
density.mcmc.m = apply(density.mcmc , 2, mean)

colscale = c("black", "blue")
yy = density(x)

density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruption", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)

points(x,rep(0,n))

2.3.7.2 Marking

Is the sampler for \(c\) correct?

The full conditional for \(c_i\) is given by,

\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]

(This assumes that the mixture weights are parameterized as

for(i in 1:n){
  v    = rep(0,2)
  v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
  v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
  v    = exp(v - max(v))/sum(exp(v - max(v)))
  cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}

• 0点 No
• 1点 Yes

Is the sampler for \(\omega\) correct?

The full conditional is simply

\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]

The following one line of code implements it:

w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))

• 0点 No
• 1点 Yes

Are the sampler for \(\mu\) correct?

\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]

The following code implements the sampler:

# Sample the means
for(k in 1:2){
  nk       = sum(cc==k)
  xsumk    = sum(x[cc==k])
  tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
  mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
  mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Are the samplers for \(\sigma_k\) correct?

The full conditional takes the form:

\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]

The following code implements the sampler:

# Sample the variances
for(k in 1:2){
  nk    = sum(cc==k)
  dd.star = dd + nk/2
  qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
  sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Does the code to generate the density estimate appear correct?

This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

Note that this will generate not only a point estimate, but also pointwise credible intervals.

• 0点 No
• 1点 Yes

Does the graph appear to be correct?

The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.

The full code for the MCMC algorithm follows:

x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))

## Priors set up using an "empirical Bayes" approach
aa  = rep(1,2)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/2

## Initialize the parameters
w     = 1/2
mu    = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc    = sample(1:2, n, replace=T, prob=c(1/2,1/2))

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = rep(0, rrr)
mu.out    = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v    = rep(0,2)
    v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
    v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
    v    = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
  
  # Sample the parameters of the components
  for(k in 1:2){
    # Sample the means
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
    # Sample the variances
    dd.star = dd + nk/2
    qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
  }
  
  # Store samples
  cc.out[s,]    = cc
  w.out[s]     = w
  mu.out[s,]    = mu
  sigma.out[s,] = sigma
  logpost[s] = 0
  for(i in 1:n){
    if(cc[i] == 1){
      logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
    }else{
      logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
    }
  }
  logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
  for(k in 1:2){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
  }
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

## Compute the samples of the density over a dense grid
xx  = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) + 
                               (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

• 0点 No
• 1点 Yes

2.3.8 8th Peer

2.3.8.1 Assignment

Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.

### Get a "Bayesian" kernel density estimator based on the same location-scale mixture of 2 normals

## Priors set up using an "empirical Bayes" approach
aa  = rep(1,KK)
eta = mean(x)
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/KK

## Initialize the parameters
w     = 1/2
mu    = rnorm(KK,mean(x), sd(x))
sigma = rep(sd(x)/KK,KK)
cc    = sample(1, n, replace=T, prob=w)

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0,dim=c(rrr, n))
w.out     = array(0,dim=c(rrr, KK))
mu.out    = array(0,dim=c(rrr, KK))
sigma.out = array(0, dim=c(rrr, KK))
logpost   = rep(0,rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v     = rep(0,KK)
    v[1]  = log(w) +dnorm(x[i], mu[1], sigma[1], log=TRUE)
    v[2]  = log(1-w) +dnorm(x[i], mu[2], sigma[2], log=TRUE)
    v     = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
  }
  
  # Sample the means
  w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
  
  # Sample the means
  for(k in 1:KK){
    nk       = sum(cc==k)
    xsumk    = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
  }
  
  # Sample the variances
  for(k in 1:KK){
    nk       = sum(cc==k)
    xsumk    = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
  }
  
  # Sample the variances
  cc.out[s,]    = cc
  w.out[s,]     = w
  mu.out[s,]    = mu
  sigma.out[s,] = sigma
}

Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].

xx  = seq(0,7,length.out=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] =density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1])+    
(1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}

for(s in 1:(rrr-burn)){
  density.mcmc[s,] =density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1])+    
(1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}

density.mcmc.m = apply(density.mcmc , 2, mean)

# Plot Bayesian estimate with pointwise credible bands along with kernel density estimate 
colscale = c("black", "blue")
yy       = density(x)
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)

plot(xx, density.mcmc.m,
type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Duration",ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq,rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col=colscale[1], lwd=2)
lines(yy, col=colscale[2], lty=2, lwd=2)
points(x, rep(0,n))

Provide the a graph of the density estimate on the interval [0,7].

2.3.8.2 Marking

Is the sampler for \(c\) correct?

The full conditional for \(c_i\) is given by,

\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]

(This assumes that the mixture weights are parameterized as

for(i in 1:n){
  v    = rep(0,2)
  v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
  v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
  v    = exp(v - max(v))/sum(exp(v - max(v)))
  cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}

• 0点 No
• 1点 Yes

Is the sampler for \(\omega\) correct?

The full conditional is simply

\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]

The following one line of code implements it:

w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))

• 0点 No
• 1点 Yes

Are the sampler for \(\mu\) correct?

\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]

The following code implements the sampler:

# Sample the means
for(k in 1:2){
  nk       = sum(cc==k)
  xsumk    = sum(x[cc==k])
  tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
  mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
  mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Are the samplers for \(\sigma_k\) correct?

The full conditional takes the form:

\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]

The following code implements the sampler:

# Sample the variances
for(k in 1:2){
  nk    = sum(cc==k)
  dd.star = dd + nk/2
  qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
  sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Does the code to generate the density estimate appear correct?

This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

Note that this will generate not only a point estimate, but also pointwise credible intervals.

• 0点 No
• 1点 Yes

Does the graph appear to be correct?

The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.

The full code for the MCMC algorithm follows:

x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))

## Priors set up using an "empirical Bayes" approach
aa  = rep(1,2)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/2

## Initialize the parameters
w     = 1/2
mu    = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc    = sample(1:2, n, replace=T, prob=c(1/2,1/2))

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = rep(0, rrr)
mu.out    = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v    = rep(0,2)
    v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
    v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
    v    = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
  
  # Sample the parameters of the components
  for(k in 1:2){
    # Sample the means
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
    # Sample the variances
    dd.star = dd + nk/2
    qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
  }
  
  # Store samples
  cc.out[s,]    = cc
  w.out[s]     = w
  mu.out[s,]    = mu
  sigma.out[s,] = sigma
  logpost[s] = 0
  for(i in 1:n){
    if(cc[i] == 1){
      logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
    }else{
      logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
    }
  }
  logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
  for(k in 1:2){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
  }
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

## Compute the samples of the density over a dense grid
xx  = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) + 
                               (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

• 0点 No
• 1点 Yes

2.3.9 9th Peer

2.3.9.1 Assignment

Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.

library(MASS)
library(MCMCpack)
data(faithful)

KK = 2          # Based on the description of the dataset
x  = faithful$eruptions
n  = length(x)
set.seed(781209)

### First, compute the "Maximum Likelihood" density estimate associated with a location mixture of 6 Gaussian distributions using the EM algorithm

## Initialize the parameters
w     = rep(1,KK)/KK
mu    = rnorm(KK, mean(x), sd(x))
sigma = rep(sd(x)/KK,KK)
cc    = sample(1:KK, n, replace=T, prob=w)

epsilon = 0.000001
s       = 0
sw      = FALSE
KL      = -Inf
KL.out  = NULL

aa  = rep(1,KK)
eta = mean(x)
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/KK

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = array(0, dim=c(rrr, KK))
mu.out    = array(0, dim=c(rrr, KK))
sigma.out = array(0, dim=c(rrr, KK))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v     = rep(0,KK)
    v[1]  = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
    v[2]  = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
    v     = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:KK, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
  
  # Sample the means
  for(k in 1:KK){
    nk       = sum(cc==k)
    xsumk    = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
  }
  
  # Sample the variances
  for(k in 1:2){
    nk       = sum(cc==k)
    dd.star  = dd + nk/2
    qq.star  = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
  }
  
  # Store samples
  cc.out[s,]    = cc
  w.out[s,]     = w
  mu.out[s,]    = mu
  sigma.out[s,] = sigma
}

Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].

xx  = seq(0,7,length.out=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) +     (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}

density.mcmc.m  = apply(density.mcmc, 2, mean)
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)

Provide the a graph of the density estimate on the interval [0,7].

2.3.9.2 Marking

Is the sampler for \(c\) correct?

The full conditional for \(c_i\) is given by,

\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]

(This assumes that the mixture weights are parameterized as

for(i in 1:n){
  v    = rep(0,2)
  v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
  v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
  v    = exp(v - max(v))/sum(exp(v - max(v)))
  cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}

• 0点 No
• 1点 Yes

Is the sampler for \(\omega\) correct?

The full conditional is simply

\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]

The following one line of code implements it:

w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))

• 0点 No
• 1点 Yes

Are the sampler for \(\mu\) correct?

\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]

The following code implements the sampler:

# Sample the means
for(k in 1:2){
  nk       = sum(cc==k)
  xsumk    = sum(x[cc==k])
  tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
  mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
  mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Are the samplers for \(\sigma_k\) correct?

The full conditional takes the form:

\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]

The following code implements the sampler:

# Sample the variances
for(k in 1:2){
  nk    = sum(cc==k)
  dd.star = dd + nk/2
  qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
  sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Does the code to generate the density estimate appear correct?

This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

Note that this will generate not only a point estimate, but also pointwise credible intervals.

• 0点 No
• 1点 Yes

Does the graph appear to be correct?

The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.

The full code for the MCMC algorithm follows:

x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))

## Priors set up using an "empirical Bayes" approach
aa  = rep(1,2)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/2

## Initialize the parameters
w     = 1/2
mu    = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc    = sample(1:2, n, replace=T, prob=c(1/2,1/2))

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = rep(0, rrr)
mu.out    = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v    = rep(0,2)
    v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
    v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
    v    = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
  
  # Sample the parameters of the components
  for(k in 1:2){
    # Sample the means
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
    # Sample the variances
    dd.star = dd + nk/2
    qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
  }
  
  # Store samples
  cc.out[s,]    = cc
  w.out[s]     = w
  mu.out[s,]    = mu
  sigma.out[s,] = sigma
  logpost[s] = 0
  for(i in 1:n){
    if(cc[i] == 1){
      logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
    }else{
      logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
    }
  }
  logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
  for(k in 1:2){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
  }
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

## Compute the samples of the density over a dense grid
xx  = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) + 
                               (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

• 0点 No
• 1点 Yes

2.3.10 10th Peer

2.3.10.1 Assignment

Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.

### Get a "Bayesian" kernel density estimator based on the same location mixture of 2 normals
data("faithful")
x  = faithful$eruptions
n  = length(x)
set.seed(781209)

## Priors set up using an "empirical Bayes" approach
KK = 2          
aa  = rep(1,KK)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/KK
## Initialize the parameters
w     = rep(1,KK)/KK
mu    = c(mean(x)-sd(x), mean(x)+sd(x))
sigma = rep(sd(x)/4,2)
cc    = sample(1:KK, n, replace=T, prob=w)

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = rep(0, rrr)
mu.out    = array(0, dim=c(rrr, KK))
sigma.out = array(0, dim=c(rrr, KK))
logpost   = rep(0, rrr)
tau.out = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v = rep(0,KK)
    v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
    v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
    v = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:KK, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  #w = as.vector(rdirichlet(1, aa + tabulate(cc, nbins=KK)))
  #w = rbeta(1, 1+sum(cc==1), 1+sum(cc==2))
  w = rbeta(2, 1 + sum(cc==1), 1 + sum(cc==2))
  
  # Sample the means
  for(k in 1:KK){
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
  }

  # Sample the variances
  for(k in 1:KK){
    nk    = sum(cc==k)
    dd.star = dd + nk/2
    qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
  }

  # Store samples
  cc.out[s,]   = cc
  w.out[s]    = w
  mu.out[s,]   = mu
  tau.out[s] = tau
  sigma.out[s,] = sigma
  logpost[s] = 0

  for(i in 1:n){
    if(cc[i]==1){
      logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
    }else{
      logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[1], sigma[2], log=TRUE)
    }
  }
  logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=T)
  #logpost[s] = logpost[s] + log(ddirichlet(w, aa), log=T)
  for(k in 1:KK){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log=TRUE)
    logpost[s] = logpost[s] + dgamma(1/sigma[k]^2, dd, qq, log=TRUE) - 4*log(sigma[k])
  }

  
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].

## Compute the samples of the density over a dense grid
xx  = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  for(k in 1:KK){
    if (k==1) {
      density.mcmc[s,] = density.mcmc[s,] + 
        w.out[s+burn]*dnorm(xx,mu.out[s+burn,k],sigma.out[s+burn,k])
    } else {
      density.mcmc[s,] = density.mcmc[s,] + 
        1-w.out[s+burn]*dnorm(xx,mu.out[s+burn,k],sigma.out[s+burn,k])
    }
  }
}
density.mcmc.m = apply(density.mcmc , 2, mean)

Provide the a graph of the density estimate on the interval [0,7].

2.3.10.2 Marking

Is the sampler for \(c\) correct?

The full conditional for \(c_i\) is given by,

\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]

(This assumes that the mixture weights are parameterized as

for(i in 1:n){
  v    = rep(0,2)
  v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
  v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
  v    = exp(v - max(v))/sum(exp(v - max(v)))
  cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}

• 0点 No
• 1点 Yes

Is the sampler for \(\omega\) correct?

The full conditional is simply

\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]

The following one line of code implements it:

w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))

• 0点 No
• 1点 Yes

Are the sampler for \(\mu\) correct?

\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]

The following code implements the sampler:

# Sample the means
for(k in 1:2){
  nk       = sum(cc==k)
  xsumk    = sum(x[cc==k])
  tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
  mu.hat   = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
  mu[k]    = rnorm(1, mu.hat, sqrt(tau2.hat))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Are the samplers for \(\sigma_k\) correct?

The full conditional takes the form:

\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]

The following code implements the sampler:

# Sample the variances
for(k in 1:2){
  nk    = sum(cc==k)
  dd.star = dd + nk/2
  qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
  sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}

Remember, however, that there are various ways in which the implementation could proceed.

• 0点 No
• 1点 Yes

Does the code to generate the density estimate appear correct?

This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

Note that this will generate not only a point estimate, but also pointwise credible intervals.

• 0点 No
• 1点 Yes

Does the graph appear to be correct?

The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.

The full code for the MCMC algorithm follows:

x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))

## Priors set up using an "empirical Bayes" approach
aa  = rep(1,2)  
eta = mean(x)    
tau = sqrt(var(x))
dd  = 2
qq  = var(x)/2

## Initialize the parameters
w     = 1/2
mu    = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc    = sample(1:2, n, replace=T, prob=c(1/2,1/2))

## Number of iterations of the sampler
rrr   = 12000
burn  = 2000

## Storing the samples
cc.out    = array(0, dim=c(rrr, n))
w.out     = rep(0, rrr)
mu.out    = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost   = rep(0, rrr)

for(s in 1:rrr){
  # Sample the indicators
  for(i in 1:n){
    v    = rep(0,2)
    v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)  #Compute the log of the weights
    v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)  #Compute the log of the weights
    v    = exp(v - max(v))/sum(exp(v - max(v)))
    cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
  }
  
  # Sample the weights
  w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
  
  # Sample the parameters of the components
  for(k in 1:2){
    # Sample the means
    nk    = sum(cc==k)
    xsumk = sum(x[cc==k])
    tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
    mu.hat  = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
    mu[k]   = rnorm(1, mu.hat, sqrt(tau2.hat))
    # Sample the variances
    dd.star = dd + nk/2
    qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
    sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
  }
  
  # Store samples
  cc.out[s,]    = cc
  w.out[s]     = w
  mu.out[s,]    = mu
  sigma.out[s,] = sigma
  logpost[s] = 0
  for(i in 1:n){
    if(cc[i] == 1){
      logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
    }else{
      logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
    }
  }
  logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
  for(k in 1:2){
    logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
  }
  if(s/500==floor(s/500)){
    print(paste("s =",s))
  }
}

## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"

## Compute the samples of the density over a dense grid
xx  = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
  density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) + 
                               (1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)

## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))

• 0点 No
• 1点 Yes

2.4 ディスカッション

3 Appendix

3.1 Blooper

3.2 Documenting File Creation

It’s useful to record some information about how your file was created.

• File creation date: 2021-05-21
• File latest updated date: 2021-06-02
• R version 4.1.0 (2021-05-18)
• rmarkdown package version: 2.8
• File version: 1.0.0
• Author Profile: ®γσ, Eng Lian Hu
• GitHub: Source Code
• Additional session information:

suppressMessages(require('dplyr', quietly = TRUE))
suppressMessages(require('magrittr', quietly = TRUE))
suppressMessages(require('formattable', quietly = TRUE))
suppressMessages(require('knitr', quietly = TRUE))
suppressMessages(require('kableExtra', quietly = TRUE))

sys1 <- devtools::session_info()$platform %>% 
  unlist %>% data.frame(Category = names(.), session_info = .)
rownames(sys1) <- NULL

sys2 <- data.frame(Sys.info()) %>% 
  dplyr::mutate(Category = rownames(.)) %>% .[2:1]
names(sys2)[2] <- c('Sys.info')
rownames(sys2) <- NULL

if (nrow(sys1) == 9 & nrow(sys2) == 8) {
  sys2 %<>% rbind(., data.frame(
  Category = 'Current time', 
  Sys.info = paste(as.character(lubridate::now('Asia/Tokyo')), 'JST🗾')))
} else {
  sys1 %<>% rbind(., data.frame(
  Category = 'Current time', 
  session_info = paste(as.character(lubridate::now('Asia/Tokyo')), 'JST🗾')))
}

sys <- cbind(sys1, sys2) %>% 
  kbl(caption = 'Additional session information:') %>% 
  kable_styling(bootstrap_options = c('striped', 'hover', 'condensed', 'responsive')) %>% 
  row_spec(0, background = 'DimGrey', color = 'yellow') %>% 
  column_spec(1, background = 'CornflowerBlue', color = 'red') %>% 
  column_spec(2, background = 'grey', color = 'black') %>% 
  column_spec(3, background = 'CornflowerBlue', color = 'blue') %>% 
  column_spec(4, background = 'grey', color = 'white') %>% 
  row_spec(9, bold = T, color = 'yellow', background = '#D7261E')

rm(sys1, sys2)
sys

Additional session information:

Category	session_info	Category	Sys.info
version	R version 4.1.0 (2021-05-18)	sysname	Linux
os	Ubuntu 20.04.2 LTS	release	5.8.0-54-generic
system	x86_64, linux-gnu	version	#61~20.04.1-Ubuntu SMP Thu May 13 00:05:49 UTC 2021
ui	X11	nodename	Scibrokes-Trading
language	en	machine	x86_64
collate	en_US.UTF-8	login	englianhu
ctype	en_US.UTF-8	user	englianhu
tz	Asia/Tokyo	effective_user	englianhu
date	2021-06-02	Current time	2021-06-02 15:16:21 JST🗾

3.3 Reference

• FR:MathJax scale to fit container #2135