Bayesian Statistics: Mixture Models (Assessment Week2 B Codes)

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## https://stackoverflow.com/a/36846793/3806250
options(width = 999)
knitr::opts_chunk$set(class.source = 'gradient1', class.output = 'gradient2', class.error = 'bg-danger')

1 受講生によるテストの実施：The EM algorithm for zero-inflated mixtures

課題を受ける準備はできていますか？

以下に提出のための指示が表示されます。

1.1 説明

A biologist is interest in characterizing the number of eggs laid by a particular bird species. To do this, they sample n=300n=300 nests on a site in Southern California. The observations are contained in the attached file nestsize.csv:

1.1.1 Review criteria

1.2 自分の提出物

1.3 ピアレビューディスカッション

1.3.1 ルーブリック

1.3.1.1 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', 'data.table', 'reprex', 
          'timetk', 'plyr', 'dplyr', 'stringr', 'magrittr', 
          'tdplyr', 'tidyverse', 'formattable', 
          'echarts4r', '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(cache = TRUE, warning = FALSE, 
                      message = FALSE, cache.lazy = FALSE)

rm(pkgs)

dat <- fread('data/nestsize.csv') %>% as.matrix
head(dat)

##      V1
## [1,]  4
## [2,]  0
## [3,]  0
## [4,]  0
## [5,]  9
## [6,]  3

Source : 300 x 1

1.3.1.2 Assignment

#### Example of an EM algorithm for fitting a location mixture of 2 Gaussian components
#### The algorithm is tested using simulated data

## Clear the environment and load required libraries
rm(list=ls())
#set.seed(81196)    # So that results are reproducible (same simulated data every time)
set.seed(12345)

x = read.csv("data/nestsize.csv",header=FALSE)$V1
KK         = 2

## Run the actual EM algorithm
## Initialize the parameters
w     = 0.5                       #Assign equal weight to each component to start with
lambda    = mean(x)   #Random cluster centers randomly spread over the support of the data
n = length(x)

# Plot the initial guess for the density
xx = seq(0,10,length=11)
ddeg = function(x,log=FALSE) {
  p = pmin(0.99999,pmax(0.00001,as.integer(x==0)))
  if(log==FALSE) {
    p
  } else {
    .Primitive("log")(pmax(1.0e-40,p)) 
  }
}
yy = w*ddeg(xx) + (1-w)*dpois(xx, lambda)
#par(mfrow=c(1,1),mar=c(4,4,4,4))
#plot(xx, yy, type="l", ylim=c(0, max(yy)), xlab="x", ylab="Initial density")
#points(jitter(x), jitter(rep(0,n)),col="red")

tibble(xx, yy) %>% 
  e_charts(x = xx) %>% 
  e_line(yy) %>% 
  e_datazoom(
    type = "slider", 
    toolbox = FALSE, 
    bottom = -5) %>% 
  e_tooltip() %>% 
  e_title("Exopential Regression") %>% 
  e_x_axis(xx, axisPointer = list(show = TRUE)) %>% 
  e_legend(
    orient = 'vertical', 
    type = c('scroll'), 
    right = 80)

s  = 0
sw = FALSE
QQ = -Inf
QQ.out = NULL
epsilon = 10^(-8)

##Checking convergence of the algorithm
while(!sw){
  ## E step
  v = array(0, dim=c(n,KK))
  v.logs = array(0, dim=c(n,KK))
  v.logs[,1] = log(w) + log(ddeg(x))    #Compute the log of the weights
  v.logs[,2] = log(1-w) + log(dpois(x, lambda))  #Compute the log of the weights
  for(i in 1:n){
    #v[i,] = exp(v.logs[i,])
    v[i,] = exp(v.logs[i,] - max(v.logs[i,]))/sum(exp(v.logs[i,] - max(v.logs[i,])))  #Go from logs to actual weights in a numerically stable manner
 }
  
  ## M step
  # Weights
  #w = max(1.0e-10,sum(v[,1])/(sum(v[,1])+sum(v[,2])))
  w = mean(v[,1])#sum(v[,1])/(sum(v[,1])+sum(v[,2]))
  mu = 0
  for(i in 1:n){
    mu = mu + v[i,2]*x[i]
  }
  mu = mu/sum(v[,2])
  ##Check convergence
  QQn = 0
  for(i in 1:n){
    logp1 = ddeg(x[i],log=TRUE)
    first = v[i,1]*(log(w) + logp1)
    second = v[i,2]*(log(1-w) + dpois(x, lambda,log=TRUE))
    QQn = QQn + first + second
    #cat("v[",i,",1]",v[i,1],"v[",i,",2]",v[i,2],"w",w,"logp1",logp1,"QQn",QQn,"first",first,"second",second,"\n")
  }
  epsilon_cond = abs(QQn-QQ)/abs(QQn)
  cat("QQn",QQn,"QQ",QQ,"epsilon_cond",epsilon_cond,"mu",mu,"w",w,"\n")
  if(epsilon_cond<epsilon){
    sw=TRUE
  }
  QQ = QQn
  QQ.out = c(QQ.out, QQ)
  
  s = s + 1
  print(paste(s, QQn))
  
  #Plot current estimate over data
  #layout(matrix(c(1,2,3),3,1), widths=c(1,1,1), heights=c(1,1,3))
  #par(mar=c(3.1,4.1,0.5,0.5))
  #plot(QQ.out[1:s],type="l", xlim=c(1,max(10,s)), las=1, ylab="Q", lwd=2)

  par(mar=c(5,4,1.5,0.5))
  xx = seq(0,10,length=11)
  yy = w*ddeg(xx) + (1-w)*dpois(xx, mu)
  #plot(xx, yy, type="l", ylim=c(0,1), main=paste("s =",s,"   Q =", round(QQ.out[s],4)," w=",w," mu=",mu), lwd=2, col="red", lty=2, xlab="x", ylab="Density")
  #lines(xx.true, yy.true, lwd=2)
  #points(x, rep(0,n), col="red")
  #legend(6,0.22,c("Truth","Estimate"),col=c("black","red"), lty=c(1,2))
}

## QQn -686.9775 -548.4393 -548.4393 -548.4393 -1930.059 -540.3046 -548.4393 -540.3046 -686.9775 -686.9775 -875.8969 -548.4393 -448.0969 -686.9775 -548.4393 -548.4393 -548.4393 -448.0969 -875.8969 -448.0969 -548.4393 -548.4393 -1351.956 -540.3046 -548.4393 -540.3046 -1629.858 -686.9775 -540.3046 -548.4393 -875.8969 -448.0969 -686.9775 -686.9775 -875.8969 -448.0969 -1099.334 -548.4393 -540.3046 -448.0969 -548.4393 -875.8969 -432.6534 -448.0969 -540.3046 -540.3046 -548.4393 -548.4393 -448.0969 -548.4393 -548.4393 -540.3046 -875.8969 -548.4393 -448.0969 -1351.956 -686.9775 -540.3046 -548.4393 -548.4393 -686.9775 -540.3046 -432.6534 -686.9775 -432.6534 -1351.956 -548.4393 -548.4393 -432.6534 -1099.334 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -540.3046 -540.3046 -1099.334 -548.4393 -548.4393 -432.6534 -548.4393 -448.0969 -686.9775 -540.3046 -548.4393 -686.9775 -686.9775 -548.4393 -548.4393 -432.6534 -448.0969 -686.9775 -548.4393 -432.6534 -432.6534 -432.6534 -548.4393 -686.9775 -548.4393 -548.4393 -548.4393 -540.3046 -548.4393 -686.9775 -548.4393 -548.4393 -432.6534 -540.3046 -548.4393 -540.3046 -540.3046 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -540.3046 -540.3046 -540.3046 -548.4393 -875.8969 -548.4393 -548.4393 -548.4393 -548.4393 -448.0969 -432.6534 -548.4393 -548.4393 -448.0969 -548.4393 -548.4393 -548.4393 -448.0969 -686.9775 -432.6534 -432.6534 -548.4393 -540.3046 -548.4393 -875.8969 -548.4393 -448.0969 -548.4393 -548.4393 -875.8969 -548.4393 -875.8969 -448.0969 -686.9775 -540.3046 -432.6534 -540.3046 -548.4393 -540.3046 -548.4393 -548.4393 -448.0969 -686.9775 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -686.9775 -875.8969 -448.0969 -548.4393 -432.6534 -875.8969 -548.4393 -548.4393 -548.4393 -548.4393 -432.6534 -686.9775 -686.9775 -448.0969 -432.6534 -448.0969 -548.4393 -548.4393 -540.3046 -540.3046 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -540.3046 -875.8969 -448.0969 -432.6534 -1629.858 -540.3046 -548.4393 -548.4393 -686.9775 -548.4393 -548.4393 -548.4393 -875.8969 -548.4393 -540.3046 -448.0969 -432.6534 -540.3046 -432.6534 -548.4393 -1099.334 -448.0969 -448.0969 -875.8969 -548.4393 -432.6534 -548.4393 -548.4393 -548.4393 -540.3046 -548.4393 -548.4393 -448.0969 -548.4393 -686.9775 -686.9775 -448.0969 -540.3046 -548.4393 -548.4393 -548.4393 -548.4393 -432.6534 -448.0969 -548.4393 -548.4393 -875.8969 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -875.8969 -548.4393 -548.4393 -448.0969 -686.9775 -448.0969 -432.6534 -1099.334 -540.3046 -686.9775 -1099.334 -686.9775 -448.0969 -1351.956 -875.8969 -548.4393 -432.6534 -448.0969 -448.0969 -448.0969 -432.6534 -1351.956 -540.3046 -448.0969 -875.8969 -875.8969 -548.4393 -686.9775 -548.4393 -448.0969 -448.0969 -548.4393 -432.6534 -548.4393 -686.9775 -548.4393 -875.8969 -686.9775 -686.9775 -448.0969 -548.4393 -540.3046 -548.4393 -548.4393 -540.3046 -548.4393 -686.9775 -432.6534 -548.4393 -432.6534 -548.4393 -540.3046 -686.9775 -548.4393 -548.4393 QQ -Inf epsilon_cond Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf mu 2.913898 w 0.3689199 
##   [1] "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -1930.05943861639" "1 -540.304630953612" "1 -548.439337762417" "1 -540.304630953612" "1 -686.97752505613"  "1 -686.97752505613"  "1 -875.896856220031" "1 -548.439337762417" "1 -448.096867014809" "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -448.096867014809" "1 -875.896856220031" "1 -448.096867014809" "1 -548.439337762417" "1 -548.439337762417" "1 -1351.95565178614" "1 -540.304630953612" "1 -548.439337762417" "1 -540.304630953612" "1 -1629.85796553955" "1 -686.97752505613"  "1 -540.304630953612" "1 -548.439337762417" "1 -875.896856220031" "1 -448.096867014809" "1 -686.97752505613"  "1 -686.97752505613"  "1 -875.896856220031" "1 -448.096867014809" "1 -1099.33403980972" "1 -548.439337762417" "1 -540.304630953612" "1 -448.096867014809" "1 -548.439337762417" "1 -875.896856220031" "1 -432.653392563174" "1 -448.096867014809" "1 -540.304630953612"
##  [46] "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -448.096867014809" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -875.896856220031" "1 -548.439337762417" "1 -448.096867014809" "1 -1351.95565178614" "1 -686.97752505613"  "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -686.97752505613"  "1 -540.304630953612" "1 -432.653392563174" "1 -686.97752505613"  "1 -432.653392563174" "1 -1351.95565178614" "1 -548.439337762417" "1 -548.439337762417" "1 -432.653392563174" "1 -1099.33403980972" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -540.304630953612" "1 -1099.33403980972" "1 -548.439337762417" "1 -548.439337762417" "1 -432.653392563174" "1 -548.439337762417" "1 -448.096867014809" "1 -686.97752505613"  "1 -540.304630953612" "1 -548.439337762417" "1 -686.97752505613"  "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417"
##  [91] "1 -432.653392563174" "1 -448.096867014809" "1 -686.97752505613"  "1 -548.439337762417" "1 -432.653392563174" "1 -432.653392563174" "1 -432.653392563174" "1 -548.439337762417" "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -548.439337762417" "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417" "1 -432.653392563174" "1 -540.304630953612" "1 -548.439337762417" "1 -540.304630953612" "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -540.304630953612" "1 -540.304630953612" "1 -548.439337762417" "1 -875.896856220031" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -448.096867014809" "1 -432.653392563174" "1 -548.439337762417" "1 -548.439337762417" "1 -448.096867014809" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -448.096867014809"
## [136] "1 -686.97752505613"  "1 -432.653392563174" "1 -432.653392563174" "1 -548.439337762417" "1 -540.304630953612" "1 -548.439337762417" "1 -875.896856220031" "1 -548.439337762417" "1 -448.096867014809" "1 -548.439337762417" "1 -548.439337762417" "1 -875.896856220031" "1 -548.439337762417" "1 -875.896856220031" "1 -448.096867014809" "1 -686.97752505613"  "1 -540.304630953612" "1 -432.653392563174" "1 -540.304630953612" "1 -548.439337762417" "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -448.096867014809" "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -686.97752505613"  "1 -875.896856220031" "1 -448.096867014809" "1 -548.439337762417" "1 -432.653392563174" "1 -875.896856220031" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -432.653392563174" "1 -686.97752505613"  "1 -686.97752505613" 
## [181] "1 -448.096867014809" "1 -432.653392563174" "1 -448.096867014809" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -875.896856220031" "1 -448.096867014809" "1 -432.653392563174" "1 -1629.85796553955" "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -875.896856220031" "1 -548.439337762417" "1 -540.304630953612" "1 -448.096867014809" "1 -432.653392563174" "1 -540.304630953612" "1 -432.653392563174" "1 -548.439337762417" "1 -1099.33403980972" "1 -448.096867014809" "1 -448.096867014809" "1 -875.896856220031" "1 -548.439337762417" "1 -432.653392563174" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417"
## [226] "1 -448.096867014809" "1 -548.439337762417" "1 -686.97752505613"  "1 -686.97752505613"  "1 -448.096867014809" "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -432.653392563174" "1 -448.096867014809" "1 -548.439337762417" "1 -548.439337762417" "1 -875.896856220031" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -875.896856220031" "1 -548.439337762417" "1 -548.439337762417" "1 -448.096867014809" "1 -686.97752505613"  "1 -448.096867014809" "1 -432.653392563174" "1 -1099.33403980972" "1 -540.304630953612" "1 -686.97752505613"  "1 -1099.33403980972" "1 -686.97752505613"  "1 -448.096867014809" "1 -1351.95565178614" "1 -875.896856220031" "1 -548.439337762417" "1 -432.653392563174" "1 -448.096867014809" "1 -448.096867014809" "1 -448.096867014809" "1 -432.653392563174" "1 -1351.95565178614" "1 -540.304630953612" "1 -448.096867014809" "1 -875.896856220031"
## [271] "1 -875.896856220031" "1 -548.439337762417" "1 -686.97752505613"  "1 -548.439337762417" "1 -448.096867014809" "1 -448.096867014809" "1 -548.439337762417" "1 -432.653392563174" "1 -548.439337762417" "1 -686.97752505613"  "1 -548.439337762417" "1 -875.896856220031" "1 -686.97752505613"  "1 -686.97752505613"  "1 -448.096867014809" "1 -548.439337762417" "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -548.439337762417" "1 -686.97752505613"  "1 -432.653392563174" "1 -548.439337762417" "1 -432.653392563174" "1 -548.439337762417" "1 -540.304630953612" "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417"
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## [226] "3 -456.07289329404"  "3 -563.347226269638" "3 -711.455913676417" "3 -711.455913676417" "3 -456.07289329404"  "3 -554.650557348045" "3 -563.347226269638" "3 -563.347226269638" "3 -563.347226269638" "3 -563.347226269638" "3 -439.562552176448" "3 -456.07289329404"  "3 -563.347226269638" "3 -563.347226269638" "3 -913.426176973794" "3 -563.347226269638" "3 -563.347226269638" "3 -563.347226269638" "3 -563.347226269638" "3 -563.347226269638" "3 -913.426176973794" "3 -563.347226269638" "3 -563.347226269638" "3 -456.07289329404"  "3 -711.455913676417" "3 -456.07289329404"  "3 -439.562552176448" "3 -1152.29885623858" "3 -554.650557348045" "3 -711.455913676417" "3 -1152.29885623858" "3 -711.455913676417" "3 -456.07289329404"  "3 -1422.37208363752" "3 -913.426176973794" "3 -563.347226269638" "3 -439.562552176448" "3 -456.07289329404"  "3 -456.07289329404"  "3 -456.07289329404"  "3 -439.562552176448" "3 -1422.37208363752" "3 -554.650557348045" "3 -456.07289329404"  "3 -913.426176973794"
## [271] "3 -913.426176973794" "3 -563.347226269638" "3 -711.455913676417" "3 -563.347226269638" "3 -456.07289329404"  "3 -456.07289329404"  "3 -563.347226269638" "3 -439.562552176448" "3 -563.347226269638" "3 -711.455913676417" "3 -563.347226269638" "3 -913.426176973794" "3 -711.455913676417" "3 -711.455913676417" "3 -456.07289329404"  "3 -563.347226269638" "3 -554.650557348045" "3 -563.347226269638" "3 -563.347226269638" "3 -554.650557348045" "3 -563.347226269638" "3 -711.455913676417" "3 -439.562552176448" "3 -563.347226269638" "3 -439.562552176448" "3 -563.347226269638" "3 -554.650557348045" "3 -711.455913676417" "3 -563.347226269638" "3 -563.347226269638"
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## [216] "11 -457.03480363972"  "11 -918.486366561081" "11 -565.270407011048" "11 -440.376515648916" "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -556.49580844735"  "11 -565.270407011048" "11 -565.270407011048" "11 -457.03480363972"  "11 -565.270407011048" "11 -714.706275791091" "11 -714.706275791091" "11 -457.03480363972"  "11 -556.49580844735"  "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -440.376515648916" "11 -457.03480363972"  "11 -565.270407011048" "11 -565.270407011048" "11 -918.486366561081" "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -918.486366561081" "11 -565.270407011048" "11 -565.270407011048" "11 -457.03480363972"  "11 -714.706275791091" "11 -457.03480363972"  "11 -440.376515648916" "11 -1159.49955072164" "11 -556.49580844735"  "11 -714.706275791091" "11 -1159.49955072164" "11 -714.706275791091" "11 -457.03480363972" 
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5.344276e-09 5.206111e-09 5.206111e-09 9.834959e-09 5.344276e-09 5.344276e-09 2.859153e-09 5.344276e-09 3.269111e-09 7.176401e-09 5.206111e-09 5.344276e-09 7.176401e-09 7.176401e-09 5.344276e-09 5.344276e-09 2.859153e-09 3.269111e-09 7.176401e-09 5.344276e-09 2.859153e-09 2.859153e-09 2.859153e-09 5.344276e-09 7.176401e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.206111e-09 5.344276e-09 7.176401e-09 5.344276e-09 5.344276e-09 2.859153e-09 5.206111e-09 5.344276e-09 5.206111e-09 5.206111e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.206111e-09 5.206111e-09 5.206111e-09 5.344276e-09 8.714014e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 3.269111e-09 2.859153e-09 5.344276e-09 5.344276e-09 3.269111e-09 5.344276e-09 5.344276e-09 5.344276e-09 3.269111e-09 7.176401e-09 2.859153e-09 2.859153e-09 5.344276e-09 5.206111e-09 5.344276e-09 8.714014e-09 5.344276e-09 3.269111e-09 5.344276e-09 5.344276e-09 8.714014e-09 5.344276e-09 8.714014e-09 3.269111e-09 7.176401e-09 5.206111e-09 2.859153e-09 5.206111e-09 5.344276e-09 5.206111e-09 5.344276e-09 5.344276e-09 3.269111e-09 7.176401e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 7.176401e-09 8.714014e-09 3.269111e-09 5.344276e-09 2.859153e-09 8.714014e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 2.859153e-09 7.176401e-09 7.176401e-09 3.269111e-09 2.859153e-09 3.269111e-09 5.344276e-09 5.344276e-09 5.206111e-09 5.206111e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.206111e-09 8.714014e-09 3.269111e-09 2.859153e-09 1.124658e-08 5.206111e-09 5.344276e-09 5.344276e-09 7.176401e-09 5.344276e-09 5.344276e-09 5.344276e-09 8.714014e-09 5.344276e-09 5.206111e-09 3.269111e-09 2.859153e-09 5.206111e-09 2.859153e-09 5.344276e-09 9.834959e-09 3.269111e-09 3.269111e-09 8.714014e-09 5.344276e-09 2.859153e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.206111e-09 5.344276e-09 5.344276e-09 3.269111e-09 5.344276e-09 7.176401e-09 7.176401e-09 3.269111e-09 5.206111e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 2.859153e-09 3.269111e-09 5.344276e-09 5.344276e-09 8.714014e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 8.714014e-09 5.344276e-09 5.344276e-09 3.269111e-09 7.176401e-09 3.269111e-09 2.859153e-09 9.834959e-09 5.206111e-09 7.176401e-09 9.834959e-09 7.176401e-09 3.269111e-09 1.064785e-08 8.714014e-09 5.344276e-09 2.859153e-09 3.269111e-09 3.269111e-09 3.269111e-09 2.859153e-09 1.064785e-08 5.206111e-09 3.269111e-09 8.714014e-09 8.714014e-09 5.344276e-09 7.176401e-09 5.344276e-09 3.269111e-09 3.269111e-09 5.344276e-09 2.859153e-09 5.344276e-09 7.176401e-09 5.344276e-09 8.714014e-09 7.176401e-09 7.176401e-09 3.269111e-09 5.344276e-09 5.206111e-09 5.344276e-09 5.344276e-09 5.206111e-09 5.344276e-09 7.176401e-09 2.859153e-09 5.344276e-09 2.859153e-09 5.344276e-09 5.206111e-09 7.176401e-09 5.344276e-09 5.344276e-09 mu 2.704756 w 0.3192758 
##   [1] "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -2055.57680170529" "17 -556.496457574345" "17 -565.271083871812" "17 -556.496457574345" "17 -714.707424972011" "17 -714.707424972011" "17 -918.488159827284" "17 -565.271083871812" "17 -457.035138401513" "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -457.035138401513" "17 -918.488159827284" "17 -457.035138401513" "17 -565.271083871812" "17 -565.271083871812" "17 -1431.996283094"   "17 -556.496457574345" "17 -565.271083871812" "17 -556.496457574345" "17 -1731.75987724688" "17 -714.707424972011" "17 -556.496457574345" "17 -565.271083871812" "17 -918.488159827284" "17 -457.035138401513" "17 -714.707424972011" "17 -714.707424972011" "17 -918.488159827284" "17 -457.035138401513" "17 -1159.50210575533" "17 -565.271083871812" "17 -556.496457574345" "17 -457.035138401513" "17 -565.271083871812" "17 -918.488159827284" "17 -440.376797759056"
##  [44] "17 -457.035138401513" "17 -556.496457574345" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -457.035138401513" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -918.488159827284" "17 -565.271083871812" "17 -457.035138401513" "17 -1431.996283094"   "17 -714.707424972011" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -714.707424972011" "17 -556.496457574345" "17 -440.376797759056" "17 -714.707424972011" "17 -440.376797759056" "17 -1431.996283094"   "17 -565.271083871812" "17 -565.271083871812" "17 -440.376797759056" "17 -1159.50210575533" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -556.496457574345" "17 -1159.50210575533" "17 -565.271083871812" "17 -565.271083871812" "17 -440.376797759056" "17 -565.271083871812" "17 -457.035138401513" "17 -714.707424972011" "17 -556.496457574345" "17 -565.271083871812"
##  [87] "17 -714.707424972011" "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812" "17 -440.376797759056" "17 -457.035138401513" "17 -714.707424972011" "17 -565.271083871812" "17 -440.376797759056" "17 -440.376797759056" "17 -440.376797759056" "17 -565.271083871812" "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -565.271083871812" "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812" "17 -440.376797759056" "17 -556.496457574345" "17 -565.271083871812" "17 -556.496457574345" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -556.496457574345" "17 -556.496457574345" "17 -565.271083871812" "17 -918.488159827284" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -457.035138401513" "17 -440.376797759056" "17 -565.271083871812"
## [130] "17 -565.271083871812" "17 -457.035138401513" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -457.035138401513" "17 -714.707424972011" "17 -440.376797759056" "17 -440.376797759056" "17 -565.271083871812" "17 -556.496457574345" "17 -565.271083871812" "17 -918.488159827284" "17 -565.271083871812" "17 -457.035138401513" "17 -565.271083871812" "17 -565.271083871812" "17 -918.488159827284" "17 -565.271083871812" "17 -918.488159827284" "17 -457.035138401513" "17 -714.707424972011" "17 -556.496457574345" "17 -440.376797759056" "17 -556.496457574345" "17 -565.271083871812" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -457.035138401513" "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -714.707424972011" "17 -918.488159827284" "17 -457.035138401513" "17 -565.271083871812" "17 -440.376797759056"
## [173] "17 -918.488159827284" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -440.376797759056" "17 -714.707424972011" "17 -714.707424972011" "17 -457.035138401513" "17 -440.376797759056" "17 -457.035138401513" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -918.488159827284" "17 -457.035138401513" "17 -440.376797759056" "17 -1731.75987724688" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -918.488159827284" "17 -565.271083871812" "17 -556.496457574345" "17 -457.035138401513" "17 -440.376797759056" "17 -556.496457574345" "17 -440.376797759056" "17 -565.271083871812" "17 -1159.50210575533" "17 -457.035138401513"
## [216] "17 -457.035138401513" "17 -918.488159827284" "17 -565.271083871812" "17 -440.376797759056" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -457.035138401513" "17 -565.271083871812" "17 -714.707424972011" "17 -714.707424972011" "17 -457.035138401513" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -440.376797759056" "17 -457.035138401513" "17 -565.271083871812" "17 -565.271083871812" "17 -918.488159827284" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -918.488159827284" "17 -565.271083871812" "17 -565.271083871812" "17 -457.035138401513" "17 -714.707424972011" "17 -457.035138401513" "17 -440.376797759056" "17 -1159.50210575533" "17 -556.496457574345" "17 -714.707424972011" "17 -1159.50210575533" "17 -714.707424972011" "17 -457.035138401513"
## [259] "17 -1431.996283094"   "17 -918.488159827284" "17 -565.271083871812" "17 -440.376797759056" "17 -457.035138401513" "17 -457.035138401513" "17 -457.035138401513" "17 -440.376797759056" "17 -1431.996283094"   "17 -556.496457574345" "17 -457.035138401513" "17 -918.488159827284" "17 -918.488159827284" "17 -565.271083871812" "17 -714.707424972011" "17 -565.271083871812" "17 -457.035138401513" "17 -457.035138401513" "17 -565.271083871812" "17 -440.376797759056" "17 -565.271083871812" "17 -714.707424972011" "17 -565.271083871812" "17 -918.488159827284" "17 -714.707424972011" "17 -714.707424972011" "17 -457.035138401513" "17 -565.271083871812" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -565.271083871812" "17 -714.707424972011" "17 -440.376797759056" "17 -565.271083871812" "17 -440.376797759056" "17 -565.271083871812" "17 -556.496457574345" "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812"

#Plot final estimate over data
#layout(matrix(c(1,2,3),3,1), widths=c(1,1,1), heights=c(1,1,3))
#plot(QQ.out[1:s],type="l", xlim=c(1,max(10,s)), las=1, ylab="Q", lwd=2)
Qplot <- tibble(Index = 1:length(QQ.out[1:s]), Value = QQ.out[1:s])
Qplot %>% 
  e_charts(x = Index) %>% 
  e_line(Value) %>% 
  e_datazoom(
    type = "slider", 
    toolbox = FALSE, 
    bottom = -5) %>% 
  e_tooltip() %>% 
  e_title("Exopential Regression") %>% 
  e_x_axis(Index, axisPointer = list(show = TRUE)) %>% 
  e_legend(
    orient = 'vertical', 
    type = c('scroll'), 
    right = 80)

##par(mar=c(5,4,1.5,0.5))
xx = seq(0,10,length=11)
yy = w*ddeg(xx) + (1-w)*dpois(xx, mu)
#plot(xx, yy, type="l", ylim=c(0,1), main=paste("s =",s,"   Q =", round(QQ.out[s],4)," w=",w," mu=",mu), lwd=2, col="red", lty=2, xlab="x", ylab="Density")
#points(x, rep(0,n), col="red")
#legend(6,0.22,c("Truth","Estimate"),col=c("black","red"), lty=c(1,2), bty="n")
tibble(xx, yy) %>% 
  e_charts(x = xx) %>% 
  e_line(yy) %>% 
  e_datazoom(
    type = "slider", 
    toolbox = FALSE, 
    bottom = -5) %>% 
  e_tooltip() %>% 
  e_title("Exopential Regression") %>% 
  e_x_axis(xx, axisPointer = list(show = TRUE)) %>% 
  e_legend(
    orient = 'vertical', 
    type = c('scroll'), 
    right = 80)

paste('w =', w); paste('mu =', mu)

## [1] "w = 0.319275757851501"

## [1] "mu = 2.7047563178536"

paste('QQ =', QQ); paste('QQn =', QQn)

##   [1] "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -2055.57680170529" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -714.707424972011" "QQ = -714.707424972011" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -918.488159827284" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -1431.996283094"   "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -1731.75987724688" "QQ = -714.707424972011" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -714.707424972011" "QQ = -918.488159827284" "QQ = -457.035138401513" "QQ = -1159.50210575533" "QQ = -565.271083871812" "QQ = -556.496457574345"
##  [40] "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -440.376797759056" "QQ = -457.035138401513" "QQ = -556.496457574345" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -1431.996283094"   "QQ = -714.707424972011" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -556.496457574345" "QQ = -440.376797759056" "QQ = -714.707424972011" "QQ = -440.376797759056" "QQ = -1431.996283094"   "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -1159.50210575533" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -556.496457574345" "QQ = -1159.50210575533"
##  [79] "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -440.376797759056" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812"
## [118] "QQ = -556.496457574345" "QQ = -556.496457574345" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -440.376797759056" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -556.496457574345" "QQ = -440.376797759056" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -556.496457574345"
## [157] "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -918.488159827284" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -714.707424972011" "QQ = -714.707424972011" "QQ = -457.035138401513" "QQ = -440.376797759056" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -918.488159827284"
## [196] "QQ = -457.035138401513" "QQ = -440.376797759056" "QQ = -1731.75987724688" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -457.035138401513" "QQ = -440.376797759056" "QQ = -556.496457574345" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -1159.50210575533" "QQ = -457.035138401513" "QQ = -457.035138401513" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -714.707424972011" "QQ = -457.035138401513" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812"
## [235] "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -457.035138401513" "QQ = -440.376797759056" "QQ = -1159.50210575533" "QQ = -556.496457574345" "QQ = -714.707424972011" "QQ = -1159.50210575533" "QQ = -714.707424972011" "QQ = -457.035138401513" "QQ = -1431.996283094"   "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -457.035138401513" "QQ = -457.035138401513" "QQ = -457.035138401513" "QQ = -440.376797759056" "QQ = -1431.996283094"   "QQ = -556.496457574345" "QQ = -457.035138401513" "QQ = -918.488159827284" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -714.707424972011"
## [274] "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -714.707424972011" "QQ = -714.707424972011" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812"

##   [1] "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -2055.57680170529" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -714.707424972011" "QQn = -714.707424972011" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -918.488159827284" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -1431.996283094"   "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -1731.75987724688" "QQn = -714.707424972011" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -714.707424972011" "QQn = -918.488159827284" "QQn = -457.035138401513" "QQn = -1159.50210575533" "QQn = -565.271083871812"
##  [39] "QQn = -556.496457574345" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -440.376797759056" "QQn = -457.035138401513" "QQn = -556.496457574345" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -1431.996283094"   "QQn = -714.707424972011" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -556.496457574345" "QQn = -440.376797759056" "QQn = -714.707424972011" "QQn = -440.376797759056" "QQn = -1431.996283094"   "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -1159.50210575533" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345"
##  [77] "QQn = -556.496457574345" "QQn = -1159.50210575533" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -440.376797759056" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812"
## [115] "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -556.496457574345" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -440.376797759056" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -556.496457574345"
## [153] "QQn = -440.376797759056" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -918.488159827284" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -714.707424972011" "QQn = -714.707424972011" "QQn = -457.035138401513" "QQn = -440.376797759056" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812"
## [191] "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -918.488159827284" "QQn = -457.035138401513" "QQn = -440.376797759056" "QQn = -1731.75987724688" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -457.035138401513" "QQn = -440.376797759056" "QQn = -556.496457574345" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -1159.50210575533" "QQn = -457.035138401513" "QQn = -457.035138401513" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -714.707424972011"
## [229] "QQn = -714.707424972011" "QQn = -457.035138401513" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -457.035138401513" "QQn = -440.376797759056" "QQn = -1159.50210575533" "QQn = -556.496457574345" "QQn = -714.707424972011" "QQn = -1159.50210575533" "QQn = -714.707424972011" "QQn = -457.035138401513" "QQn = -1431.996283094"   "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -457.035138401513" "QQn = -457.035138401513" "QQn = -457.035138401513" "QQn = -440.376797759056"
## [267] "QQn = -1431.996283094"   "QQn = -556.496457574345" "QQn = -457.035138401513" "QQn = -918.488159827284" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -714.707424972011" "QQn = -714.707424972011" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812"

Reference from tutorial : Sample Code for EM Example 2

1.3.2 ルーブリック

Are the initial values reasonable?

In this problem there are only two unknown parameters: the weight of the mixture (\(\omega\)), and the mean of the Poisson component (\(\lambda\)). As long as the parameters are in the right range (i.e., \(0 < \omega < 1\) and \(\lambda > 0\)), you should give credit for this prompt.

An example of a default choice that would likely work well in most cases would be something like:

w      = 0.5
lambda = mean(x)

## Error in mean(x): object 'x' not found

• 0点 No

• 1点 Yes

Are the observation-specific weights \(v_{i,k}\) computed correctly (E step)?

Note that two cases need to be considered. If the observation is a zero, then in can come from either the point mass or the Poisson distribution. On the other hand, if the observation is different from zero, it can only come from the Poisson component. Hence:

\[v_{i,1}\propto \left\{\begin{matrix} \omega & x_i=0 \\ 0 & otherwise \\ \end{matrix}\right.\]

and

\[v_{i,2}\propto \left\{\begin{matrix} (1-\omega)\frac{e^{-\lambda}\lambda^{x_i} }{x!} & x_i=0 \\ 1 & otherwise \\ \end{matrix}\right.\]

Hence, the code for the E step could look something like

## E step
v = array(0, dim=c(n,2))

## Error in array(0, dim = c(n, 2)): 'list' object cannot be coerced to type 'integer'

for(i in 1:n){
  if(x[i]==0){
    v[i,1] = w    
    v[i,2] = (1-w)*dpois(x[i], lambda)  
    v[i,]  = v[i,]/sum(v[i,])
  }else{
    v[i,1] = 0
    v[i,2] = 1
  }
}

## Error in 1:n: NA/NaN argument

However, be open-minded when reviewing this code, as there are many ways to accomplish the same thing in R.

• 0点 No

• 1点 Yes

Are the formulas for the maximum of the \(Q\) functions correct (M step)?

Again, when computing the \(Q\) you need to consider two cases. In particular, recall that the \(Q\) function can be written as a sum over the observations:

\[Q\left(w, \lambda \mid \hat{w}^{(t)}, \hat{\lambda}^{(t)} \right) = \\ \sum_{i=1}^{n} Q_i\left(w, \lambda \mid \hat{w}^{(t)}, \hat{\lambda}^{(t)} , x_i \right)\]

where

\[Q_i\left(w, \lambda \mid \hat{w}^{(t)}, \hat{\lambda}^{(t)} , x_i \right) = \\ \begin{cases} v_{i,1} \log w + v_{i,2} \left[ \log (1-w) -\lambda + x_i \log \lambda - \log x_i! \right] & x_i=0 \\ v_{i,2} \left[ \log (1-w) -\lambda + x_i \log \lambda - \log x! \right] & \text{otherwise} \end{cases} \] Computing the derivatives and setting them to zero yields

\(\hat{\omega}^(t+1) = \frac{1}{n}\sum_{i=1}^{n}v_{i,1}\) and \(\hat{\lambda}^{(t+1)} = \frac{\sum_{i=1}^{n} x_{i}}{\sum_{i=1}^{n} v_{i,2}}\).

Hence, the code for this section might look something like:

w = mean(v[,1])

## Error in mean(v[, 1]): object 'v' not found

lambda = sum(x)/sum(v[,2])

## Error in eval(expr, envir, enclos): object 'x' not found

• 0点 No
• 1点 Yes

Is the converge check correct?

The only change required in the sample code has to do with the calculation of the Q function. As discussed in the previous prompt, this reduces to

\[Q\left(w, \lambda \mid \hat{w}^{(t)}, \hat{\lambda}^{(t)} \right) = \\ \sum_{i=1}^{n} Q_i\left(w, \lambda \mid \hat{w}^{(t)}, \hat{\lambda}^{(t)} , x_i \right)\]

where

\[Q_i\left(w, \lambda \mid \hat{w}^{(t)}, \hat{\lambda}^{(t)} , x_i \right) = \\ \begin{cases} v_{i,1} \log w + v_{i,2} \left[ \log (1-w) -\lambda + x_i \log \lambda - \log x_i! \right] & x_i=0 \\ v_{i,2} \left[ \log (1-w) -\lambda + x_i \log \lambda - \log x! \right] & \text{otherwise} \end{cases} \]

Hence, the code for this prompt might look something like:

##Check convergence
QQn = 0
for(i in 1:n){
  if(x[i]==0){
    QQn = QQn + v[i,1]*log(w) + v[i,2]*(log(1-w) + dpois(x[i], lambda, log=TRUE))
  }else{
    QQn = QQn + v[i,2]*(log(1-w) + dpois(x[i], lambda, log=TRUE))
  }
}

## Error in 1:n: NA/NaN argument

if(abs(QQn-QQ)/abs(QQn)<epsilon){
  sw=TRUE
}

## Error in eval(expr, envir, enclos): object 'QQ' not found

QQ = QQn

• 0点 No

• 1点 Yes

Provide your maximum likelihood estimates \(\hat{\omega}\) and \(\hat{\lambda}\).

\(w_hat = 0.3192758\) \(lambda_hat = 2.704756\)

Are the estimates of the parameters generated by the algorithm correct?

The point estimates, rounded to two decimal places, are \(\hat{\lambda} = 3.07\) and \(\hat{w} = 0.40\).

• 0点 No

• 1点 Yes

1.4 レビュー

1.4.1 1st Peer

#### Example of an EM algorithm for fitting a location mixture of 2 Gaussian components
#### The algorithm is tested using simulated data

## Clear the environment and load required libraries
rm(list=ls())
#set.seed(81196)    # So that results are reproducible (same simulated data every time)
set.seed(12345)

x = read.csv("data/nestsize.csv",header=FALSE)$V1
KK         = 2

## Run the actual EM algorithm
## Initialize the parameters
w     = 0.5                       #Assign equal weight to each component to start with
lambda    = mean(x)   #Random cluster centers randomly spread over the support of the data
n = length(x)

# Plot the initial guess for the density
xx = seq(0,10,length=11)
ddeg = function(x,log=FALSE) {
  p = pmin(0.99999,pmax(0.00001,as.integer(x==0)))
  if(log==FALSE) {
    p
  } else {
    .Primitive("log")(pmax(1.0e-40,p)) 
  }
}
yy = w*ddeg(xx) + (1-w)*dpois(xx, lambda)
par(mfrow=c(1,1),mar=c(4,4,4,4))
plot(xx, yy, type="l", ylim=c(0, max(yy)), xlab="x", ylab="Initial density")
points(jitter(x), jitter(rep(0,n)),col="red")

s  = 0
sw = FALSE
QQ = -Inf
QQ.out = NULL
epsilon = 10^(-8)

##Checking convergence of the algorithm
while(!sw){
  ## E step
  v = array(0, dim=c(n,KK))
  v.logs = array(0, dim=c(n,KK))
  v.logs[,1] = log(w) + log(ddeg(x))    #Compute the log of the weights
  v.logs[,2] = log(1-w) + log(dpois(x, lambda))  #Compute the log of the weights
  for(i in 1:n){
    #v[i,] = exp(v.logs[i,])
    v[i,] = exp(v.logs[i,] - max(v.logs[i,]))/sum(exp(v.logs[i,] - max(v.logs[i,])))  #Go from logs to actual weights in a numerically stable manner
  }
  
  ## M step
  # Weights
  #w = max(1.0e-10,sum(v[,1])/(sum(v[,1])+sum(v[,2])))
  w = mean(v[,1])#sum(v[,1])/(sum(v[,1])+sum(v[,2]))
  mu = 0
  for(i in 1:n){
    mu    = mu + v[i,2]*x[i]
  }
  mu=mu/sum(v[,2])
  ##Check convergence
  QQn = 0
  for(i in 1:n){
    logp1 = ddeg(x[i],log=TRUE)
    first = v[i,1]*(log(w) + logp1)
    second = v[i,2]*(log(1-w) + dpois(x, lambda,log=TRUE))
    QQn = QQn + first + second
    #cat("v[",i,",1]",v[i,1],"v[",i,",2]",v[i,2],"w",w,"logp1",logp1,"QQn",QQn,"first",first,"second",second,"\n")
  }
  epsilon_cond = abs(QQn-QQ)/abs(QQn)
  cat("QQn",QQn,"QQ",QQ,"epsilon_cond",epsilon_cond,"mu",mu,"w",w,"\n")
  if(epsilon_cond<epsilon){
    sw=TRUE
  }
  QQ = QQn
  QQ.out = c(QQ.out, QQ)
  
  s = s + 1
  print(paste(s, QQn))
  
  #Plot current estimate over data
  layout(matrix(c(1,2,3),3,1), widths=c(1,1,1), heights=c(1,1,3))
  par(mar=c(3.1,4.1,0.5,0.5))
  plot(QQ.out[1:s],type="l", xlim=c(1,max(10,s)), las=1, ylab="Q", lwd=2)

  par(mar=c(5,4,1.5,0.5))
  xx = seq(0,10,length=11)
  yy = w*ddeg(xx) + (1-w)*dpois(xx, mu)
  plot(xx, yy, type="l", ylim=c(0,1), main=paste("s =",s,"   Q =", round(QQ.out[s],4)," w=",w," mu=",mu), lwd=2, col="red", lty=2, xlab="x", ylab="Density")
  #lines(xx.true, yy.true, lwd=2)
  points(x, rep(0,n), col="red")
  legend(6,0.22,c("Truth","Estimate"),col=c("black","red"), lty=c(1,2))
}

## QQn -686.9775 -548.4393 -548.4393 -548.4393 -1930.059 -540.3046 -548.4393 -540.3046 -686.9775 -686.9775 -875.8969 -548.4393 -448.0969 -686.9775 -548.4393 -548.4393 -548.4393 -448.0969 -875.8969 -448.0969 -548.4393 -548.4393 -1351.956 -540.3046 -548.4393 -540.3046 -1629.858 -686.9775 -540.3046 -548.4393 -875.8969 -448.0969 -686.9775 -686.9775 -875.8969 -448.0969 -1099.334 -548.4393 -540.3046 -448.0969 -548.4393 -875.8969 -432.6534 -448.0969 -540.3046 -540.3046 -548.4393 -548.4393 -448.0969 -548.4393 -548.4393 -540.3046 -875.8969 -548.4393 -448.0969 -1351.956 -686.9775 -540.3046 -548.4393 -548.4393 -686.9775 -540.3046 -432.6534 -686.9775 -432.6534 -1351.956 -548.4393 -548.4393 -432.6534 -1099.334 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -540.3046 -540.3046 -1099.334 -548.4393 -548.4393 -432.6534 -548.4393 -448.0969 -686.9775 -540.3046 -548.4393 -686.9775 -686.9775 -548.4393 -548.4393 -432.6534 -448.0969 -686.9775 -548.4393 -432.6534 -432.6534 -432.6534 -548.4393 -686.9775 -548.4393 -548.4393 -548.4393 -540.3046 -548.4393 -686.9775 -548.4393 -548.4393 -432.6534 -540.3046 -548.4393 -540.3046 -540.3046 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -540.3046 -540.3046 -540.3046 -548.4393 -875.8969 -548.4393 -548.4393 -548.4393 -548.4393 -448.0969 -432.6534 -548.4393 -548.4393 -448.0969 -548.4393 -548.4393 -548.4393 -448.0969 -686.9775 -432.6534 -432.6534 -548.4393 -540.3046 -548.4393 -875.8969 -548.4393 -448.0969 -548.4393 -548.4393 -875.8969 -548.4393 -875.8969 -448.0969 -686.9775 -540.3046 -432.6534 -540.3046 -548.4393 -540.3046 -548.4393 -548.4393 -448.0969 -686.9775 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -686.9775 -875.8969 -448.0969 -548.4393 -432.6534 -875.8969 -548.4393 -548.4393 -548.4393 -548.4393 -432.6534 -686.9775 -686.9775 -448.0969 -432.6534 -448.0969 -548.4393 -548.4393 -540.3046 -540.3046 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -540.3046 -875.8969 -448.0969 -432.6534 -1629.858 -540.3046 -548.4393 -548.4393 -686.9775 -548.4393 -548.4393 -548.4393 -875.8969 -548.4393 -540.3046 -448.0969 -432.6534 -540.3046 -432.6534 -548.4393 -1099.334 -448.0969 -448.0969 -875.8969 -548.4393 -432.6534 -548.4393 -548.4393 -548.4393 -540.3046 -548.4393 -548.4393 -448.0969 -548.4393 -686.9775 -686.9775 -448.0969 -540.3046 -548.4393 -548.4393 -548.4393 -548.4393 -432.6534 -448.0969 -548.4393 -548.4393 -875.8969 -548.4393 -548.4393 -548.4393 -548.4393 -548.4393 -875.8969 -548.4393 -548.4393 -448.0969 -686.9775 -448.0969 -432.6534 -1099.334 -540.3046 -686.9775 -1099.334 -686.9775 -448.0969 -1351.956 -875.8969 -548.4393 -432.6534 -448.0969 -448.0969 -448.0969 -432.6534 -1351.956 -540.3046 -448.0969 -875.8969 -875.8969 -548.4393 -686.9775 -548.4393 -448.0969 -448.0969 -548.4393 -432.6534 -548.4393 -686.9775 -548.4393 -875.8969 -686.9775 -686.9775 -448.0969 -548.4393 -540.3046 -548.4393 -548.4393 -540.3046 -548.4393 -686.9775 -432.6534 -548.4393 -432.6534 -548.4393 -540.3046 -686.9775 -548.4393 -548.4393 QQ -Inf epsilon_cond Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf mu 2.913898 w 0.3689199 
##   [1] "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -1930.05943861639" "1 -540.304630953612" "1 -548.439337762417" "1 -540.304630953612" "1 -686.97752505613"  "1 -686.97752505613"  "1 -875.896856220031" "1 -548.439337762417" "1 -448.096867014809" "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -448.096867014809" "1 -875.896856220031" "1 -448.096867014809" "1 -548.439337762417" "1 -548.439337762417" "1 -1351.95565178614" "1 -540.304630953612" "1 -548.439337762417" "1 -540.304630953612" "1 -1629.85796553955" "1 -686.97752505613"  "1 -540.304630953612" "1 -548.439337762417" "1 -875.896856220031" "1 -448.096867014809" "1 -686.97752505613"  "1 -686.97752505613"  "1 -875.896856220031" "1 -448.096867014809" "1 -1099.33403980972" "1 -548.439337762417" "1 -540.304630953612" "1 -448.096867014809" "1 -548.439337762417" "1 -875.896856220031" "1 -432.653392563174" "1 -448.096867014809" "1 -540.304630953612"
##  [46] "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -448.096867014809" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -875.896856220031" "1 -548.439337762417" "1 -448.096867014809" "1 -1351.95565178614" "1 -686.97752505613"  "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -686.97752505613"  "1 -540.304630953612" "1 -432.653392563174" "1 -686.97752505613"  "1 -432.653392563174" "1 -1351.95565178614" "1 -548.439337762417" "1 -548.439337762417" "1 -432.653392563174" "1 -1099.33403980972" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -540.304630953612" "1 -1099.33403980972" "1 -548.439337762417" "1 -548.439337762417" "1 -432.653392563174" "1 -548.439337762417" "1 -448.096867014809" "1 -686.97752505613"  "1 -540.304630953612" "1 -548.439337762417" "1 -686.97752505613"  "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417"
##  [91] "1 -432.653392563174" "1 -448.096867014809" "1 -686.97752505613"  "1 -548.439337762417" "1 -432.653392563174" "1 -432.653392563174" "1 -432.653392563174" "1 -548.439337762417" "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -548.439337762417" "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417" "1 -432.653392563174" "1 -540.304630953612" "1 -548.439337762417" "1 -540.304630953612" "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -540.304630953612" "1 -540.304630953612" "1 -548.439337762417" "1 -875.896856220031" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -448.096867014809" "1 -432.653392563174" "1 -548.439337762417" "1 -548.439337762417" "1 -448.096867014809" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -448.096867014809"
## [136] "1 -686.97752505613"  "1 -432.653392563174" "1 -432.653392563174" "1 -548.439337762417" "1 -540.304630953612" "1 -548.439337762417" "1 -875.896856220031" "1 -548.439337762417" "1 -448.096867014809" "1 -548.439337762417" "1 -548.439337762417" "1 -875.896856220031" "1 -548.439337762417" "1 -875.896856220031" "1 -448.096867014809" "1 -686.97752505613"  "1 -540.304630953612" "1 -432.653392563174" "1 -540.304630953612" "1 -548.439337762417" "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -448.096867014809" "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -686.97752505613"  "1 -875.896856220031" "1 -448.096867014809" "1 -548.439337762417" "1 -432.653392563174" "1 -875.896856220031" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -432.653392563174" "1 -686.97752505613"  "1 -686.97752505613" 
## [181] "1 -448.096867014809" "1 -432.653392563174" "1 -448.096867014809" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -875.896856220031" "1 -448.096867014809" "1 -432.653392563174" "1 -1629.85796553955" "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -875.896856220031" "1 -548.439337762417" "1 -540.304630953612" "1 -448.096867014809" "1 -432.653392563174" "1 -540.304630953612" "1 -432.653392563174" "1 -548.439337762417" "1 -1099.33403980972" "1 -448.096867014809" "1 -448.096867014809" "1 -875.896856220031" "1 -548.439337762417" "1 -432.653392563174" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417"
## [226] "1 -448.096867014809" "1 -548.439337762417" "1 -686.97752505613"  "1 -686.97752505613"  "1 -448.096867014809" "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -432.653392563174" "1 -448.096867014809" "1 -548.439337762417" "1 -548.439337762417" "1 -875.896856220031" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -548.439337762417" "1 -875.896856220031" "1 -548.439337762417" "1 -548.439337762417" "1 -448.096867014809" "1 -686.97752505613"  "1 -448.096867014809" "1 -432.653392563174" "1 -1099.33403980972" "1 -540.304630953612" "1 -686.97752505613"  "1 -1099.33403980972" "1 -686.97752505613"  "1 -448.096867014809" "1 -1351.95565178614" "1 -875.896856220031" "1 -548.439337762417" "1 -432.653392563174" "1 -448.096867014809" "1 -448.096867014809" "1 -448.096867014809" "1 -432.653392563174" "1 -1351.95565178614" "1 -540.304630953612" "1 -448.096867014809" "1 -875.896856220031"
## [271] "1 -875.896856220031" "1 -548.439337762417" "1 -686.97752505613"  "1 -548.439337762417" "1 -448.096867014809" "1 -448.096867014809" "1 -548.439337762417" "1 -432.653392563174" "1 -548.439337762417" "1 -686.97752505613"  "1 -548.439337762417" "1 -875.896856220031" "1 -686.97752505613"  "1 -686.97752505613"  "1 -448.096867014809" "1 -548.439337762417" "1 -540.304630953612" "1 -548.439337762417" "1 -548.439337762417" "1 -540.304630953612" "1 -548.439337762417" "1 -686.97752505613"  "1 -432.653392563174" "1 -548.439337762417" "1 -432.653392563174" "1 -548.439337762417" "1 -540.304630953612" "1 -686.97752505613"  "1 -548.439337762417" "1 -548.439337762417"

## QQn -705.5679 -559.8223 -559.8223 -559.8223 -2013.321 -551.2644 -559.8223 -551.2644 -705.5679 -705.5679 -904.3157 -559.8223 -454.2595 -705.5679 -559.8223 -559.8223 -559.8223 -454.2595 -904.3157 -454.2595 -559.8223 -559.8223 -1405.141 -551.2644 -559.8223 -551.2644 -1697.502 -705.5679 -551.2644 -559.8223 -904.3157 -454.2595 -705.5679 -705.5679 -904.3157 -454.2595 -1139.377 -559.8223 -551.2644 -454.2595 -559.8223 -904.3157 -438.0126 -454.2595 -551.2644 -551.2644 -559.8223 -559.8223 -454.2595 -559.8223 -559.8223 -551.2644 -904.3157 -559.8223 -454.2595 -1405.141 -705.5679 -551.2644 -559.8223 -559.8223 -705.5679 -551.2644 -438.0126 -705.5679 -438.0126 -1405.141 -559.8223 -559.8223 -438.0126 -1139.377 -559.8223 -559.8223 -559.8223 -559.8223 -559.8223 -551.2644 -551.2644 -1139.377 -559.8223 -559.8223 -438.0126 -559.8223 -454.2595 -705.5679 -551.2644 -559.8223 -705.5679 -705.5679 -559.8223 -559.8223 -438.0126 -454.2595 -705.5679 -559.8223 -438.0126 -438.0126 -438.0126 -559.8223 -705.5679 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## [271] "3 -913.426176973794" "3 -563.347226269638" "3 -711.455913676417" "3 -563.347226269638" "3 -456.07289329404"  "3 -456.07289329404"  "3 -563.347226269638" "3 -439.562552176448" "3 -563.347226269638" "3 -711.455913676417" "3 -563.347226269638" "3 -913.426176973794" "3 -711.455913676417" "3 -711.455913676417" "3 -456.07289329404"  "3 -563.347226269638" "3 -554.650557348045" "3 -563.347226269638" "3 -563.347226269638" "3 -554.650557348045" "3 -563.347226269638" "3 -711.455913676417" "3 -439.562552176448" "3 -563.347226269638" "3 -439.562552176448" "3 -563.347226269638" "3 -554.650557348045" "3 -711.455913676417" "3 -563.347226269638" "3 -563.347226269638"

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## [271] "6 -918.23404590802"  "6 -565.175135814838" "6 -714.544561328685" "6 -565.175135814838" "6 -456.987656947124" "6 -456.987656947124" "6 -565.175135814838" "6 -440.336775685858" "6 -565.175135814838" "6 -714.544561328685" "6 -565.175135814838" "6 -918.23404590802"  "6 -714.544561328685" "6 -714.544561328685" "6 -456.987656947124" "6 -565.175135814838" "6 -556.40443867717"  "6 -565.175135814838" "6 -565.175135814838" "6 -556.40443867717"  "6 -565.175135814838" "6 -714.544561328685" "6 -440.336775685858" "6 -565.175135814838" "6 -440.336775685858" "6 -565.175135814838" "6 -556.40443867717"  "6 -714.544561328685" "6 -565.175135814838" "6 -565.175135814838"

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## [226] "9 -457.03270809622"  "9 -565.266170121427" "9 -714.699082444371" "9 -714.699082444371" "9 -457.03270809622"  "9 -556.491745155435" "9 -565.266170121427" "9 -565.266170121427" "9 -565.266170121427" "9 -565.266170121427" "9 -440.374749674974" "9 -457.03270809622"  "9 -565.266170121427" "9 -565.266170121427" "9 -918.475141604677" "9 -565.266170121427" "9 -565.266170121427" "9 -565.266170121427" "9 -565.266170121427" "9 -565.266170121427" "9 -918.475141604677" "9 -565.266170121427" "9 -565.266170121427" "9 -457.03270809622"  "9 -714.699082444371" "9 -457.03270809622"  "9 -440.374749674974" "9 -1159.48355753158" "9 -556.491745155435" "9 -714.699082444371" "9 -1159.48355753158" "9 -714.699082444371" "9 -457.03270809622"  "9 -1431.97148256356" "9 -918.475141604677" "9 -565.266170121427" "9 -440.374749674974" "9 -457.03270809622"  "9 -457.03270809622"  "9 -457.03270809622"  "9 -440.374749674974" "9 -1431.97148256356" "9 -556.491745155435" "9 -457.03270809622"  "9 -918.475141604677"
## [271] "9 -918.475141604677" "9 -565.266170121427" "9 -714.699082444371" "9 -565.266170121427" "9 -457.03270809622"  "9 -457.03270809622"  "9 -565.266170121427" "9 -440.374749674974" "9 -565.266170121427" "9 -714.699082444371" "9 -565.266170121427" "9 -918.475141604677" "9 -714.699082444371" "9 -714.699082444371" "9 -457.03270809622"  "9 -565.266170121427" "9 -556.491745155435" "9 -565.266170121427" "9 -565.266170121427" "9 -556.491745155435" "9 -565.266170121427" "9 -714.699082444371" "9 -440.374749674974" "9 -565.266170121427" "9 -440.374749674974" "9 -565.266170121427" "9 -556.491745155435" "9 -714.699082444371" "9 -565.266170121427" "9 -565.266170121427"

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##  [44] "10 -457.034235967656" "10 -556.494707702432" "10 -556.494707702432" "10 -565.269259237649" "10 -565.269259237649" "10 -457.034235967656" "10 -565.269259237649" "10 -565.269259237649" "10 -556.494707702432" "10 -918.483325685568" "10 -565.269259237649" "10 -457.034235967656" "10 -1431.98707372049" "10 -714.704327098731" "10 -556.494707702432" "10 -565.269259237649" "10 -565.269259237649" "10 -714.704327098731" "10 -556.494707702432" "10 -440.376037258885" "10 -714.704327098731" "10 -440.376037258885" "10 -1431.98707372049" "10 -565.269259237649" "10 -565.269259237649" "10 -440.376037258885" "10 -1159.49521810787" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -556.494707702432" "10 -556.494707702432" "10 -1159.49521810787" "10 -565.269259237649" "10 -565.269259237649" "10 -440.376037258885" "10 -565.269259237649" "10 -457.034235967656" "10 -714.704327098731" "10 -556.494707702432" "10 -565.269259237649"
##  [87] "10 -714.704327098731" "10 -714.704327098731" "10 -565.269259237649" "10 -565.269259237649" "10 -440.376037258885" "10 -457.034235967656" "10 -714.704327098731" "10 -565.269259237649" "10 -440.376037258885" "10 -440.376037258885" "10 -440.376037258885" "10 -565.269259237649" "10 -714.704327098731" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -556.494707702432" "10 -565.269259237649" "10 -714.704327098731" "10 -565.269259237649" "10 -565.269259237649" "10 -440.376037258885" "10 -556.494707702432" "10 -565.269259237649" "10 -556.494707702432" "10 -556.494707702432" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -556.494707702432" "10 -556.494707702432" "10 -556.494707702432" "10 -565.269259237649" "10 -918.483325685568" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -457.034235967656" "10 -440.376037258885" "10 -565.269259237649"
## [130] "10 -565.269259237649" "10 -457.034235967656" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -457.034235967656" "10 -714.704327098731" "10 -440.376037258885" "10 -440.376037258885" "10 -565.269259237649" "10 -556.494707702432" "10 -565.269259237649" "10 -918.483325685568" "10 -565.269259237649" "10 -457.034235967656" "10 -565.269259237649" "10 -565.269259237649" "10 -918.483325685568" "10 -565.269259237649" "10 -918.483325685568" "10 -457.034235967656" "10 -714.704327098731" "10 -556.494707702432" "10 -440.376037258885" "10 -556.494707702432" "10 -565.269259237649" "10 -556.494707702432" "10 -565.269259237649" "10 -565.269259237649" "10 -457.034235967656" "10 -714.704327098731" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -714.704327098731" "10 -918.483325685568" "10 -457.034235967656" "10 -565.269259237649" "10 -440.376037258885"
## [173] "10 -918.483325685568" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -440.376037258885" "10 -714.704327098731" "10 -714.704327098731" "10 -457.034235967656" "10 -440.376037258885" "10 -457.034235967656" "10 -565.269259237649" "10 -565.269259237649" "10 -556.494707702432" "10 -556.494707702432" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -556.494707702432" "10 -918.483325685568" "10 -457.034235967656" "10 -440.376037258885" "10 -1731.74811380431" "10 -556.494707702432" "10 -565.269259237649" "10 -565.269259237649" "10 -714.704327098731" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -918.483325685568" "10 -565.269259237649" "10 -556.494707702432" "10 -457.034235967656" "10 -440.376037258885" "10 -556.494707702432" "10 -440.376037258885" "10 -565.269259237649" "10 -1159.49521810787" "10 -457.034235967656"
## [216] "10 -457.034235967656" "10 -918.483325685568" "10 -565.269259237649" "10 -440.376037258885" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -556.494707702432" "10 -565.269259237649" "10 -565.269259237649" "10 -457.034235967656" "10 -565.269259237649" "10 -714.704327098731" "10 -714.704327098731" "10 -457.034235967656" "10 -556.494707702432" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -440.376037258885" "10 -457.034235967656" "10 -565.269259237649" "10 -565.269259237649" "10 -918.483325685568" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -565.269259237649" "10 -918.483325685568" "10 -565.269259237649" "10 -565.269259237649" "10 -457.034235967656" "10 -714.704327098731" "10 -457.034235967656" "10 -440.376037258885" "10 -1159.49521810787" "10 -556.494707702432" "10 -714.704327098731" "10 -1159.49521810787" "10 -714.704327098731" "10 -457.034235967656"
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##  [44] "11 -457.03480363972"  "11 -556.49580844735"  "11 -556.49580844735"  "11 -565.270407011048" "11 -565.270407011048" "11 -457.03480363972"  "11 -565.270407011048" "11 -565.270407011048" "11 -556.49580844735"  "11 -918.486366561081" "11 -565.270407011048" "11 -457.03480363972"  "11 -1431.99286679396" "11 -714.706275791091" "11 -556.49580844735"  "11 -565.270407011048" "11 -565.270407011048" "11 -714.706275791091" "11 -556.49580844735"  "11 -440.376515648916" "11 -714.706275791091" "11 -440.376515648916" "11 -1431.99286679396" "11 -565.270407011048" "11 -565.270407011048" "11 -440.376515648916" "11 -1159.49955072164" "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -556.49580844735"  "11 -556.49580844735"  "11 -1159.49955072164" "11 -565.270407011048" "11 -565.270407011048" "11 -440.376515648916" "11 -565.270407011048" "11 -457.03480363972"  "11 -714.706275791091" "11 -556.49580844735"  "11 -565.270407011048"
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## [216] "11 -457.03480363972"  "11 -918.486366561081" "11 -565.270407011048" "11 -440.376515648916" "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -556.49580844735"  "11 -565.270407011048" "11 -565.270407011048" "11 -457.03480363972"  "11 -565.270407011048" "11 -714.706275791091" "11 -714.706275791091" "11 -457.03480363972"  "11 -556.49580844735"  "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -440.376515648916" "11 -457.03480363972"  "11 -565.270407011048" "11 -565.270407011048" "11 -918.486366561081" "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -565.270407011048" "11 -918.486366561081" "11 -565.270407011048" "11 -565.270407011048" "11 -457.03480363972"  "11 -714.706275791091" "11 -457.03480363972"  "11 -440.376515648916" "11 -1159.49955072164" "11 -556.49580844735"  "11 -714.706275791091" "11 -1159.49955072164" "11 -714.706275791091" "11 -457.03480363972" 
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## [259] "12 -1431.99501934055" "12 -918.487496464788" "12 -565.27083348883"  "12 -440.376693401808" "12 -457.035014567419" "12 -457.035014567419" "12 -457.035014567419" "12 -440.376693401808" "12 -1431.99501934055" "12 -556.49621745061"  "12 -457.035014567419" "12 -918.487496464788" "12 -918.487496464788" "12 -565.27083348883"  "12 -714.706999868857" "12 -565.27083348883"  "12 -457.035014567419" "12 -457.035014567419" "12 -565.27083348883"  "12 -440.376693401808" "12 -565.27083348883"  "12 -714.706999868857" "12 -565.27083348883"  "12 -918.487496464788" "12 -714.706999868857" "12 -714.706999868857" "12 -457.035014567419" "12 -565.27083348883"  "12 -556.49621745061"  "12 -565.27083348883"  "12 -565.27083348883"  "12 -556.49621745061"  "12 -565.27083348883"  "12 -714.706999868857" "12 -440.376693401808" "12 -565.27083348883"  "12 -440.376693401808" "12 -565.27083348883"  "12 -556.49621745061"  "12 -714.706999868857" "12 -565.27083348883"  "12 -565.27083348883"

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7.176401e-09 3.269111e-09 5.206111e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 2.859153e-09 3.269111e-09 5.344276e-09 5.344276e-09 8.714014e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 5.344276e-09 8.714014e-09 5.344276e-09 5.344276e-09 3.269111e-09 7.176401e-09 3.269111e-09 2.859153e-09 9.834959e-09 5.206111e-09 7.176401e-09 9.834959e-09 7.176401e-09 3.269111e-09 1.064785e-08 8.714014e-09 5.344276e-09 2.859153e-09 3.269111e-09 3.269111e-09 3.269111e-09 2.859153e-09 1.064785e-08 5.206111e-09 3.269111e-09 8.714014e-09 8.714014e-09 5.344276e-09 7.176401e-09 5.344276e-09 3.269111e-09 3.269111e-09 5.344276e-09 2.859153e-09 5.344276e-09 7.176401e-09 5.344276e-09 8.714014e-09 7.176401e-09 7.176401e-09 3.269111e-09 5.344276e-09 5.206111e-09 5.344276e-09 5.344276e-09 5.206111e-09 5.344276e-09 7.176401e-09 2.859153e-09 5.344276e-09 2.859153e-09 5.344276e-09 5.206111e-09 7.176401e-09 5.344276e-09 5.344276e-09 mu 2.704756 w 0.3192758 
##   [1] "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -2055.57680170529" "17 -556.496457574345" "17 -565.271083871812" "17 -556.496457574345" "17 -714.707424972011" "17 -714.707424972011" "17 -918.488159827284" "17 -565.271083871812" "17 -457.035138401513" "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -457.035138401513" "17 -918.488159827284" "17 -457.035138401513" "17 -565.271083871812" "17 -565.271083871812" "17 -1431.996283094"   "17 -556.496457574345" "17 -565.271083871812" "17 -556.496457574345" "17 -1731.75987724688" "17 -714.707424972011" "17 -556.496457574345" "17 -565.271083871812" "17 -918.488159827284" "17 -457.035138401513" "17 -714.707424972011" "17 -714.707424972011" "17 -918.488159827284" "17 -457.035138401513" "17 -1159.50210575533" "17 -565.271083871812" "17 -556.496457574345" "17 -457.035138401513" "17 -565.271083871812" "17 -918.488159827284" "17 -440.376797759056"
##  [44] "17 -457.035138401513" "17 -556.496457574345" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -457.035138401513" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -918.488159827284" "17 -565.271083871812" "17 -457.035138401513" "17 -1431.996283094"   "17 -714.707424972011" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -714.707424972011" "17 -556.496457574345" "17 -440.376797759056" "17 -714.707424972011" "17 -440.376797759056" "17 -1431.996283094"   "17 -565.271083871812" "17 -565.271083871812" "17 -440.376797759056" "17 -1159.50210575533" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -556.496457574345" "17 -1159.50210575533" "17 -565.271083871812" "17 -565.271083871812" "17 -440.376797759056" "17 -565.271083871812" "17 -457.035138401513" "17 -714.707424972011" "17 -556.496457574345" "17 -565.271083871812"
##  [87] "17 -714.707424972011" "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812" "17 -440.376797759056" "17 -457.035138401513" "17 -714.707424972011" "17 -565.271083871812" "17 -440.376797759056" "17 -440.376797759056" "17 -440.376797759056" "17 -565.271083871812" "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -565.271083871812" "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812" "17 -440.376797759056" "17 -556.496457574345" "17 -565.271083871812" "17 -556.496457574345" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -556.496457574345" "17 -556.496457574345" "17 -565.271083871812" "17 -918.488159827284" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -457.035138401513" "17 -440.376797759056" "17 -565.271083871812"
## [130] "17 -565.271083871812" "17 -457.035138401513" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -457.035138401513" "17 -714.707424972011" "17 -440.376797759056" "17 -440.376797759056" "17 -565.271083871812" "17 -556.496457574345" "17 -565.271083871812" "17 -918.488159827284" "17 -565.271083871812" "17 -457.035138401513" "17 -565.271083871812" "17 -565.271083871812" "17 -918.488159827284" "17 -565.271083871812" "17 -918.488159827284" "17 -457.035138401513" "17 -714.707424972011" "17 -556.496457574345" "17 -440.376797759056" "17 -556.496457574345" "17 -565.271083871812" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -457.035138401513" "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -714.707424972011" "17 -918.488159827284" "17 -457.035138401513" "17 -565.271083871812" "17 -440.376797759056"
## [173] "17 -918.488159827284" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -440.376797759056" "17 -714.707424972011" "17 -714.707424972011" "17 -457.035138401513" "17 -440.376797759056" "17 -457.035138401513" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -918.488159827284" "17 -457.035138401513" "17 -440.376797759056" "17 -1731.75987724688" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -918.488159827284" "17 -565.271083871812" "17 -556.496457574345" "17 -457.035138401513" "17 -440.376797759056" "17 -556.496457574345" "17 -440.376797759056" "17 -565.271083871812" "17 -1159.50210575533" "17 -457.035138401513"
## [216] "17 -457.035138401513" "17 -918.488159827284" "17 -565.271083871812" "17 -440.376797759056" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -457.035138401513" "17 -565.271083871812" "17 -714.707424972011" "17 -714.707424972011" "17 -457.035138401513" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -440.376797759056" "17 -457.035138401513" "17 -565.271083871812" "17 -565.271083871812" "17 -918.488159827284" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -565.271083871812" "17 -918.488159827284" "17 -565.271083871812" "17 -565.271083871812" "17 -457.035138401513" "17 -714.707424972011" "17 -457.035138401513" "17 -440.376797759056" "17 -1159.50210575533" "17 -556.496457574345" "17 -714.707424972011" "17 -1159.50210575533" "17 -714.707424972011" "17 -457.035138401513"
## [259] "17 -1431.996283094"   "17 -918.488159827284" "17 -565.271083871812" "17 -440.376797759056" "17 -457.035138401513" "17 -457.035138401513" "17 -457.035138401513" "17 -440.376797759056" "17 -1431.996283094"   "17 -556.496457574345" "17 -457.035138401513" "17 -918.488159827284" "17 -918.488159827284" "17 -565.271083871812" "17 -714.707424972011" "17 -565.271083871812" "17 -457.035138401513" "17 -457.035138401513" "17 -565.271083871812" "17 -440.376797759056" "17 -565.271083871812" "17 -714.707424972011" "17 -565.271083871812" "17 -918.488159827284" "17 -714.707424972011" "17 -714.707424972011" "17 -457.035138401513" "17 -565.271083871812" "17 -556.496457574345" "17 -565.271083871812" "17 -565.271083871812" "17 -556.496457574345" "17 -565.271083871812" "17 -714.707424972011" "17 -440.376797759056" "17 -565.271083871812" "17 -440.376797759056" "17 -565.271083871812" "17 -556.496457574345" "17 -714.707424972011" "17 -565.271083871812" "17 -565.271083871812"

#Plot final estimate over data
layout(matrix(c(1,2,3),3,1), widths=c(1,1,1), heights=c(1,1,3))

plot(QQ.out[1:s],type="l", xlim=c(1,max(10,s)), las=1, ylab="Q", lwd=2)

par(mar=c(5,4,1.5,0.5))
xx = seq(0,10,length=11)
yy = w*ddeg(xx) + (1-w)*dpois(xx, mu)
plot(xx, yy, type="l", ylim=c(0,1), main=paste("s =",s,"   Q =", round(QQ.out[s],4)," w=",w," mu=",mu), lwd=2, col="red", lty=2, xlab="x", ylab="Density")
points(x, rep(0,n), col="red")
legend(6,0.22,c("Truth","Estimate"),col=c("black","red"), lty=c(1,2), bty="n")

paste('w =', w); paste('mu =', mu)

## [1] "w = 0.319275757851501"

## [1] "mu = 2.7047563178536"

paste('QQ =', QQ); paste('QQn =', QQn)

##   [1] "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -2055.57680170529" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -714.707424972011" "QQ = -714.707424972011" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -918.488159827284" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -1431.996283094"   "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -1731.75987724688" "QQ = -714.707424972011" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -714.707424972011" "QQ = -918.488159827284" "QQ = -457.035138401513" "QQ = -1159.50210575533" "QQ = -565.271083871812" "QQ = -556.496457574345"
##  [40] "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -440.376797759056" "QQ = -457.035138401513" "QQ = -556.496457574345" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -1431.996283094"   "QQ = -714.707424972011" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -556.496457574345" "QQ = -440.376797759056" "QQ = -714.707424972011" "QQ = -440.376797759056" "QQ = -1431.996283094"   "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -1159.50210575533" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -556.496457574345" "QQ = -1159.50210575533"
##  [79] "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -440.376797759056" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812"
## [118] "QQ = -556.496457574345" "QQ = -556.496457574345" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -440.376797759056" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -556.496457574345" "QQ = -440.376797759056" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -556.496457574345"
## [157] "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -918.488159827284" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -714.707424972011" "QQ = -714.707424972011" "QQ = -457.035138401513" "QQ = -440.376797759056" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -918.488159827284"
## [196] "QQ = -457.035138401513" "QQ = -440.376797759056" "QQ = -1731.75987724688" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -457.035138401513" "QQ = -440.376797759056" "QQ = -556.496457574345" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -1159.50210575533" "QQ = -457.035138401513" "QQ = -457.035138401513" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -714.707424972011" "QQ = -457.035138401513" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812"
## [235] "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -714.707424972011" "QQ = -457.035138401513" "QQ = -440.376797759056" "QQ = -1159.50210575533" "QQ = -556.496457574345" "QQ = -714.707424972011" "QQ = -1159.50210575533" "QQ = -714.707424972011" "QQ = -457.035138401513" "QQ = -1431.996283094"   "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -457.035138401513" "QQ = -457.035138401513" "QQ = -457.035138401513" "QQ = -440.376797759056" "QQ = -1431.996283094"   "QQ = -556.496457574345" "QQ = -457.035138401513" "QQ = -918.488159827284" "QQ = -918.488159827284" "QQ = -565.271083871812" "QQ = -714.707424972011"
## [274] "QQ = -565.271083871812" "QQ = -457.035138401513" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -918.488159827284" "QQ = -714.707424972011" "QQ = -714.707424972011" "QQ = -457.035138401513" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -565.271083871812" "QQ = -714.707424972011" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -440.376797759056" "QQ = -565.271083871812" "QQ = -556.496457574345" "QQ = -714.707424972011" "QQ = -565.271083871812" "QQ = -565.271083871812"

##   [1] "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -2055.57680170529" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -714.707424972011" "QQn = -714.707424972011" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -918.488159827284" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -1431.996283094"   "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -1731.75987724688" "QQn = -714.707424972011" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -714.707424972011" "QQn = -918.488159827284" "QQn = -457.035138401513" "QQn = -1159.50210575533" "QQn = -565.271083871812"
##  [39] "QQn = -556.496457574345" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -440.376797759056" "QQn = -457.035138401513" "QQn = -556.496457574345" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -1431.996283094"   "QQn = -714.707424972011" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -556.496457574345" "QQn = -440.376797759056" "QQn = -714.707424972011" "QQn = -440.376797759056" "QQn = -1431.996283094"   "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -1159.50210575533" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345"
##  [77] "QQn = -556.496457574345" "QQn = -1159.50210575533" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -440.376797759056" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812"
## [115] "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -556.496457574345" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -440.376797759056" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -556.496457574345"
## [153] "QQn = -440.376797759056" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -918.488159827284" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -714.707424972011" "QQn = -714.707424972011" "QQn = -457.035138401513" "QQn = -440.376797759056" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812"
## [191] "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -918.488159827284" "QQn = -457.035138401513" "QQn = -440.376797759056" "QQn = -1731.75987724688" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -457.035138401513" "QQn = -440.376797759056" "QQn = -556.496457574345" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -1159.50210575533" "QQn = -457.035138401513" "QQn = -457.035138401513" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -714.707424972011"
## [229] "QQn = -714.707424972011" "QQn = -457.035138401513" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -714.707424972011" "QQn = -457.035138401513" "QQn = -440.376797759056" "QQn = -1159.50210575533" "QQn = -556.496457574345" "QQn = -714.707424972011" "QQn = -1159.50210575533" "QQn = -714.707424972011" "QQn = -457.035138401513" "QQn = -1431.996283094"   "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -457.035138401513" "QQn = -457.035138401513" "QQn = -457.035138401513" "QQn = -440.376797759056"
## [267] "QQn = -1431.996283094"   "QQn = -556.496457574345" "QQn = -457.035138401513" "QQn = -918.488159827284" "QQn = -918.488159827284" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -457.035138401513" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -918.488159827284" "QQn = -714.707424972011" "QQn = -714.707424972011" "QQn = -457.035138401513" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -565.271083871812" "QQn = -714.707424972011" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -440.376797759056" "QQn = -565.271083871812" "QQn = -556.496457574345" "QQn = -714.707424972011" "QQn = -565.271083871812" "QQn = -565.271083871812"

1.4.2 2nd Peer

Provide the EM algorithm to fit a zero-inflated Poisson distribution.

w     = 1/2  #Assign equal weight to each component to start with
maxv = 1 - 1e-9
mu = c(0, 4) # mean of uniform and poisson dist respectively
KK = 2

## E step
v = array(0, dim=c(n,KK))
v[,1] = log(w) + dunif(x , max = maxv, log = T)
v[,2] = log(1-w) + dpois(x, mu[2], log=TRUE)
for(i in 1:n){
  v[i,] = exp(v[i,] - max(v[i,]))/sum(exp(v[i,] - max(v[i,])))
}

w = mean(v[,1])
mu = rep(0, KK)
for(k in 2:KK) { # can ignore calculation of mu[1]
  for(i in 1:n){
    mu[k]    = mu[k] + v[i,k]*x[i]
  }
  mu[k] = mu[k]/sum(v[,k])
}

##Check convergence
QQn = 0
for(i in 1:n){
  comp1 = log(w) + dunif(x[i] , max = maxv, log = T)
  if (comp1 < -1e9) {comp1 = 0} # turn negative inf to zero
  QQn = QQn + v[i,1]*(comp1) + v[i,2]*(log(1-w) + dpois(x[i], mu[2], log=TRUE))
}
if(abs(QQn-QQ)/abs(QQn)<epsilon){sw=TRUE}
QQ = QQn

paste('w =', w); paste('mu =', mu)

## [1] "w = 0.41899255042371"

## [1] "mu = 0"                "mu = 3.17265008336402"

paste('QQ =', QQ); paste('QQn =', QQn)

## [1] "QQ = -536.950554571082"

## [1] "QQn = -536.950554571082"

Provide your maximum likelihood estimates \(\hat{\omega}\) and \(\hat{\lambda}\).

w = 0.3986 or 0.40 m = 3.065 or 3.07

1.4.3 3rd Peer

Provide the EM algorithm to fit a zero-inflated Poisson distribution.

w = 0.5

v = array(0, dim=c(n,2))

for(i in 1:n){
  if(x[i]==0){
    v[i,1] = w
    v[i,2] = (1-w)*dpois(x[i], lambda)
    v[i,]  = v[i,]/sum(v[i,])
    
  }else{
    v[i,1] = 0
    v[i,2] = 1
  }
}

w = mean(v[,1])
lambda = sum(x)/sum(v[,2])
QQn = 0

for(i in 1:n){
  if(x[i]==0){
    QQn = QQn + v[i,1]*log(w) + v[i,2]*(log(1-w) + dpois(x[i], lambda, log=TRUE))
    
  }else{
    QQn = QQn + v[i,2]*(log(1-w) + dpois(x[i], lambda, log=TRUE))
  }
}

if(abs(QQn-QQ)/abs(QQn)<epsilon){
  sw=TRUE
}

paste('w =', w); paste('lambda =', lambda)

## [1] "w = 0.368359448807043"

## [1] "lambda = 2.91832645933181"

paste('QQ =', QQ); paste('QQn =', QQn)

## [1] "QQ = -536.950554571082"

## [1] "QQn = -576.591134395629"

Provide your maximum likelihood estimates \(\hat{\omega}\) and \(\hat{\lambda}\).

MLE for w = 0.3986 ~ 0.4 MLE for lambda = 3.0651 ~ 3.07

1.5 ディスカッション

2 Appendix

2.1 Blooper

2.2 Documenting File Creation

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

• File creation date: 2021-05-12
• File latest updated date: 2021-05-12
• R version 4.0.5 (2021-03-31)
• 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 4.0.5 (2021-03-31)	sysname	Linux
os	Ubuntu 20.04.2 LTS	release	5.8.0-52-generic
system	x86_64, linux-gnu	version	#59~20.04.1-Ubuntu SMP Fri Apr 30 16:10:51 UTC 2021
ui	X11	nodename	Scibrokes-Trading
language	en	machine	x86_64
collate	C	login	englianhu
ctype	en_US.UTF-8	user	englianhu
tz	Asia/Tokyo	effective_user	englianhu
date	2021-05-12	Current time	2021-05-12 17:28:29 JST🗾

2.3 Reference

• Bayesian Statistics: Mixture Models (Assessment Week2 A Codes)

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