Abhinav Kumar - class-lab7.1
This week, we will do a brief introduction to Bayesian regression. The topic is discussed in further detail in Gelman et al., chapter 9.
A simple example of discrete predictive simulations
We can make predictions using known information.
# probability of n girls from 400 births with probability of .488
?rbinom## starting httpd help server ... donen.girls <- rbinom (1, 400, .488)
print (n.girls)## [1] 195# 1000 estimates using a loop instead
n.sims <- 1000
n.girls <- rep (NA, n.sims)
for (s in 1:n.sims){
n.girls[s] <- rbinom (1, 400, .488)
}
hist (n.girls, main="")
# equivalently
n.sims <- 1000
n.girls <- rbinom (n.sims, 400, .488)
hist (n.girls, main="")
Accounting for twins
Complicating the simulation to account for twins.
birth.type <- sample (c("fraternal twin", "identical twin", "single birth"),
size=400, replace=TRUE, prob=c(1/25, 1/300, 1 - 1/25 - 1/300))
girls <- rep (NA, 400)
for (i in 1:400){
if (birth.type[i]=="single birth"){
girls[i] <- rbinom (1, 1, .488)}
else if (birth.type[i]=="identical twin"){
girls[i] <- 2*rbinom (1, 1, .495)}
else if (birth.type[i]=="fraternal twin"){
girls[i] <- rbinom (1, 2, .495)}
}
n.girls <- sum (girls)
n.girls## [1] 199 # putting in a loop
n.sims <- 1000
n.girls <- rep (NA, n.sims)
for (s in 1:n.sims){
birth.type <- sample (c("fraternal twin", "identical twin", "single birth"),
size=400, replace=TRUE, prob=c(1/25, 1/300, 1 - 1/25 - 1/300))
girls <- rep (NA, 400)
for (i in 1:400){
if (birth.type[i]=="single birth"){
girls[i] <- rbinom (1, 1, .488)}
else if (birth.type[i]=="identical twin"){
girls[i] <- 2*rbinom (1, 1, .495)}
else if (birth.type[i]=="fraternal twin"){
girls[i] <- rbinom (1, 2, .495)}
}
n.girls[s] <- sum (girls)
}
hist (n.girls, main="")
A simple example of continuous predictive simulations for height
# simulating 10 height data points
woman <- rbinom (10, 1, .52)
?rnorm
height <- ifelse (woman==0, rnorm (10, 69.1, 2.9), rnorm (10, 64.5, 2.7))
avg.height <- mean (height)
print(avg.height)## [1] 67.31085 # simulation loop for average height
n.sims <- 1000
?rep
avg.height <- rep (NA, n.sims)
for (s in 1:n.sims){
sex <- rbinom (10, 1, .52)
height <- ifelse (sex==0, rnorm (10, 69.1, 2.9), rnorm (10, 64.5, 2.7))
avg.height[s] <- mean (height)
}
hist (avg.height, main="Average height of 10 adults") 
## Simulation using custom-made functions
Height.sim <- function (n.adults){
sex <- rbinom (n.adults, 1, .52)
height <- ifelse (sex==0, rnorm (10, 69.1, 2.9), rnorm (10, 64.5, 2.7))
return (mean(height))
}
avg.height <- replicate (1000, Height.sim (n.adults=10))
hist (avg.height, main="Average height of 10 adults")
Predictive uncertainty via an informal Bayesian approach
Chapter 4 example
library(haven)
urlfile="http://www.stat.columbia.edu/~gelman/arm/examples/earnings/heights.dta"
heights <- read_dta(url(urlfile))
View(heights)
# data cleaning
# (for simplicity) remove cases with missing data
heights$male <- 2 - heights$sex
# subset and removing missing data
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionch4 = heights %>% filter(earn>0) %>% select(earn, male, height)
summary(ch4)## earn male height
## Min. : 200 Min. :0.0000 Min. :58.00
## 1st Qu.: 10000 1st Qu.:0.0000 1st Qu.:64.00
## Median : 20000 Median :0.0000 Median :66.00
## Mean : 23155 Mean :0.4237 Mean :66.92
## 3rd Qu.: 30000 3rd Qu.:1.0000 3rd Qu.:70.00
## Max. :200000 Max. :1.0000 Max. :77.00complete=complete.cases(ch4)
table(complete)## complete
## TRUE
## 1192ch4c=subset(ch4,complete==T)
#attach(ch4c)
log.earn <- log(ch4c$earn)
earn.logmodel.3 <- lm(log.earn ~ height + male + height:male, data=ch4c)
library(texreg)## Version: 1.37.5
## Date: 2020-06-17
## Author: Philip Leifeld (University of Essex)
##
## Consider submitting praise using the praise or praise_interactive functions.
## Please cite the JSS article in your publications -- see citation("texreg").screenreg(earn.logmodel.3)##
## ========================
## Model 1
## ------------------------
## (Intercept) 8.39 ***
## (0.84)
## height 0.02
## (0.01)
## male -0.08
## (1.26)
## height:male 0.01
## (0.02)
## ------------------------
## R^2 0.09
## Adj. R^2 0.08
## Num. obs. 1192
## ========================
## *** p < 0.001; ** p < 0.01; * p < 0.05Prediction uncertainty
x.new <- data.frame(height=68, male=1)
pred.interval <- predict(earn.logmodel.3,x.new,interval="prediction",
level=.95)
print(pred.interval)## fit lwr upr
## 1 9.972807 8.241334 11.70428mean(pred.interval)## [1] 9.972807(11.70428-9.972807)/(1.96)## [1] 0.8834046print (exp (pred.interval))## fit lwr upr
## 1 21435.57 3794.597 121088.9## Constructing the predictive interval using simulation
# 1000 iterations
pred <- exp (rnorm (1000, 9.95, .88))
pred.original.scale <- rnorm (1000, 9.95, .88)
# Histograms (Figure 7.2)
par (mfrow=c(2,2))
hist (pred.original.scale, xlab="log(earnings)", main="")
hist (pred, xlab="earnings", main="")
Bayesian inference
We will take up the issue of uncertainty related to regression parameters using the Bayesian approach. Let’s first begin by loading the rstanarm package. Hopefully, this works for your local machines.
library(rstanarm)## Warning: package 'rstanarm' was built under R version 4.1.3## Loading required package: Rcpp## This is rstanarm version 2.21.1## - See https://mc-stan.org/rstanarm/articles/priors for changes to default priors!## - Default priors may change, so it's safest to specify priors, even if equivalent to the defaults.## - For execution on a local, multicore CPU with excess RAM we recommend calling## options(mc.cores = parallel::detectCores())OLS
Let’s begin by reading in the hibbs data, which contains personal income growth and electoral vote share for incumbents’ party, and then estimating an OLS model.
library(readr)
urlfile="https://raw.githubusercontent.com/avehtari/ROS-Examples/master/ElectionsEconomy/data/hibbs.dat"
hibbs=read.table(url(urlfile),header=T)
head(hibbs)# Estimate the model and save the results
m1_ols = lm(vote ~ growth, data=hibbs)
summary(m1_ols)##
## Call:
## lm(formula = vote ~ growth, data = hibbs)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.9929 -0.6674 0.2556 2.3225 5.3094
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.2476 1.6219 28.514 8.41e-14 ***
## growth 3.0605 0.6963 4.396 0.00061 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.763 on 14 degrees of freedom
## Multiple R-squared: 0.5798, Adjusted R-squared: 0.5498
## F-statistic: 19.32 on 1 and 14 DF, p-value: 0.00061# formatted regression results using screenreg and visualization of the data and predicted values
screenreg(m1_ols)##
## ======================
## Model 1
## ----------------------
## (Intercept) 46.25 ***
## (1.62)
## growth 3.06 ***
## (0.70)
## ----------------------
## R^2 0.58
## Adj. R^2 0.55
## Num. obs. 16
## ======================
## *** p < 0.001; ** p < 0.01; * p < 0.05plot(hibbs$growth, hibbs$vote, xlab="Avg recent growth in personal income (%)",
ylab="Incumbent party's vote share (%)",
main="Forecasting the election from the economy")+abline(m1_ols)
## integer(0)## Outcome: If economic growth is zero, the vote for incumbment party will be 46.25 .
## Outcome: For any +ve economic growth of 1%, the incumbment president's party vote increases by 3.1% .Bayesian regression
Bayesian regression combines MLE (which is also used in classical regression) with a penalty linked to assumptions about priors. Here, we set the priors to be default and then null (like OLS), and then estimate via MLE as opposed to a sampling approach.
How can one interpret the output? Similar to OLS output.
Coefficient estimates = Median
MAD_SD = uncertainties (like coefficient SE)
sigma = residual standard errorConsistent with standard deviations in the normal distribution, about 68% of vote outcomes will fall with +/- the residual standard error.
?stan_glm
# default priors
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## Chain 4:print(m2.br)## stan_glm
## family: gaussian [identity]
## formula: vote ~ growth
## observations: 16
## predictors: 2
## ------
## Median MAD_SD
## (Intercept) 46.3 1.6
## growth 3.1 0.7
##
## Auxiliary parameter(s):
## Median MAD_SD
## sigma 3.9 0.7
##
## ------
## * For help interpreting the printed output see ?print.stanreg
## * For info on the priors used see ?prior_summary.stanreg# equivalent to no prior information, but not MLE
m2.nopriors=stan_glm(vote ~ growth,prior_intercept=NULL,prior=NULL,
prior_aux=NULL,data=hibbs)##
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## Chain 4:print(m2.nopriors)## stan_glm
## family: gaussian [identity]
## formula: vote ~ growth
## observations: 16
## predictors: 2
## ------
## Median MAD_SD
## (Intercept) 46.3 1.7
## growth 3.1 0.7
##
## Auxiliary parameter(s):
## Median MAD_SD
## sigma 4.0 0.8
##
## ------
## * For help interpreting the printed output see ?print.stanreg
## * For info on the priors used see ?prior_summary.stanreg# equivalent to no penalty in maximum penalized likelihood estimation, or just MLE
m2.nopriorsmle=stan_glm(vote ~ growth,prior_intercept=NULL,prior=NULL,
prior_aux=NULL,algorithm="optimizing",data=hibbs)
print(m2.nopriorsmle)## stan_glm
## family: gaussian [identity]
## formula: vote ~ growth
## observations: 16
## predictors: 2
## ------
## Median MAD_SD
## (Intercept) 46.2 1.6
## growth 3.1 0.7
##
## Auxiliary parameter(s):
## Median MAD_SD
## sigma 3.9 0.8
##
## ------
## * For help interpreting the printed output see ?print.stanreg
## * For info on the priors used see ?prior_summary.stanregParameter estimates as simulation-based results
All parameter estimates represent the outcomes of posterior simulations (e.g., median).
print(m2.br)## stan_glm
## family: gaussian [identity]
## formula: vote ~ growth
## observations: 16
## predictors: 2
## ------
## Median MAD_SD
## (Intercept) 46.3 1.6
## growth 3.1 0.7
##
## Auxiliary parameter(s):
## Median MAD_SD
## sigma 3.9 0.7
##
## ------
## * For help interpreting the printed output see ?print.stanreg
## * For info on the priors used see ?prior_summary.stanreg# accessing simulation information
sims=as.matrix(m2.br)
head(sims)## parameters
## iterations (Intercept) growth sigma
## [1,] 47.37298 2.655321 4.978187
## [2,] 47.46683 2.752677 4.742360
## [3,] 47.04543 2.958429 4.507205
## [4,] 47.57576 2.449818 3.644981
## [5,] 48.51272 2.343477 4.577593
## [6,] 44.79675 3.673835 3.149167dim(sims)## [1] 4000 3# column summaries (median and scaled median absolute deviation) from simulations via the apply()
Median=apply(sims,2,median)
MAD_SD=apply(sims,2,mad)
print(cbind(Median,MAD_SD))## Median MAD_SD
## (Intercept) 46.281954 1.6268636
## growth 3.050043 0.7110332
## sigma 3.891934 0.7367086hist(sims[,2])
hist(sims[,1])
Uncertainty interval
Similar to OLS, Bayesian uncertainty assumes normally distributed error, which means that +/- 2 standard errors has approximately a 95% chance of covering the true value. Here, a 95% “uncertainty interval” = median +/- 2 MAD SD or through the simulation results. The MAD SD is akin to a standard error.
# model parameters attached to the m2.br object
names(m2.br)## [1] "coefficients" "ses" "fitted.values"
## [4] "linear.predictors" "residuals" "df.residual"
## [7] "covmat" "y" "model"
## [10] "data" "family" "offset"
## [13] "weights" "prior.weights" "contrasts"
## [16] "na.action" "formula" "terms"
## [19] "prior.info" "algorithm" "stan_summary"
## [22] "stanfit" "rstan_version" "call"
## [25] "stan_function" "compute_mean_PPD" "xlevels"# normal distribution and 95% UI
(lower=m2.br$coefficients[2]-2*m2.br$ses[2])## growth
## 1.627976(higher=m2.br$coefficients[2]+2*m2.br$ses[2])## growth
## 4.472109# picking out the values associated with .025 and .975 in simulation results
# simulation results of the parameters
quantile(sims[,"growth"], c(.025,.975))## 2.5% 97.5%
## 1.588589 4.542411Predictions and uncertainty in Bayesian regression
We can use the posterior distribution matrix to calculate uncertainty around parameter estimates.
Point predictions
One can make different kinds of predictions based on the Bayesian regression.
What was the probability that Clinton would defeat Trump? As indicated by Gelman et al., growth prior to the 2016 election averaged 2% annually. Plugging this into our model, we get y=52.3= 46.3+3*2, but we need to include uncertainty. What is the probability of receiving at least 50% of the vote, assuming a predicted value of y at 52.3 and a standard deviation of 3.9.
print(m2.br)## stan_glm
## family: gaussian [identity]
## formula: vote ~ growth
## observations: 16
## predictors: 2
## ------
## Median MAD_SD
## (Intercept) 46.3 1.6
## growth 3.1 0.7
##
## Auxiliary parameter(s):
## Median MAD_SD
## sigma 3.9 0.7
##
## ------
## * For help interpreting the printed output see ?print.stanreg
## * For info on the priors used see ?prior_summary.stanregnames(m2.br)## [1] "coefficients" "ses" "fitted.values"
## [4] "linear.predictors" "residuals" "df.residual"
## [7] "covmat" "y" "model"
## [10] "data" "family" "offset"
## [13] "weights" "prior.weights" "contrasts"
## [16] "na.action" "formula" "terms"
## [19] "prior.info" "algorithm" "stan_summary"
## [22] "stanfit" "rstan_version" "call"
## [25] "stan_function" "compute_mean_PPD" "xlevels"# manual point prediction of growth 2
(coefs_1=coef(m2.br))## (Intercept) growth
## 46.281954 3.050043(predict1=coefs_1[1] + coefs_1[2]*2)## (Intercept)
## 52.38204# calculating the probability that
?pnorm
1-pnorm(50,mean=predict1,sd=3.9)## [1] 0.7293272# automatic calculation
new=data.frame(growth=2)
(predict2=predict(m2.br, newdata=new))## 1
## 52.39294Linear predictor with uncertainty
We can generate estimates of uncertainty with regards to the fitted line. Here, the output returns simulations where the mean=the coefficient estimate and a standard deviation can be calculated to represent uncertainty.
# linear prediction with uncertainty
(y_pred1=posterior_linpred(m2.br, newdata=new))##
## iterations 1
## [1,] 52.68362
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## [184,] 50.71196
## [185,] 51.36933
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## [3455,] 52.99221
## [3456,] 52.93789
## [3457,] 51.63605
## [3458,] 51.88225
## [3459,] 52.35103
## [3460,] 52.40855
## [3461,] 51.41683
## [3462,] 53.21700
## [3463,] 52.96008
## [3464,] 52.61973
## [3465,] 51.99641
## [3466,] 51.92933
## [3467,] 51.56700
## [3468,] 51.77826
## [3469,] 49.50911
## [3470,] 50.30041
## [3471,] 50.38347
## [3472,] 53.29508
## [3473,] 53.22956
## [3474,] 53.41759
## [3475,] 52.81728
## [3476,] 53.38211
## [3477,] 51.59547
## [3478,] 53.46798
## [3479,] 52.67159
## [3480,] 50.85109
## [3481,] 52.27683
## [3482,] 52.62532
## [3483,] 52.47024
## [3484,] 52.58464
## [3485,] 54.34626
## [3486,] 51.30133
## [3487,] 51.92588
## [3488,] 51.15641
## [3489,] 52.28304
## [3490,] 53.02223
## [3491,] 50.87819
## [3492,] 52.32443
## [3493,] 52.36145
## [3494,] 51.88026
## [3495,] 52.02574
## [3496,] 53.12327
## [3497,] 52.84307
## [3498,] 52.08520
## [3499,] 52.46818
## [3500,] 53.11423
## [3501,] 52.70161
## [3502,] 51.92780
## [3503,] 52.71606
## [3504,] 52.13838
## [3505,] 53.24373
## [3506,] 54.54268
## [3507,] 53.41334
## [3508,] 52.87973
## [3509,] 54.03485
## [3510,] 53.82168
## [3511,] 51.14855
## [3512,] 53.17277
## [3513,] 53.25689
## [3514,] 52.66458
## [3515,] 51.10076
## [3516,] 52.01373
## [3517,] 52.72539
## [3518,] 53.29360
## [3519,] 51.45406
## [3520,] 53.52804
## [3521,] 51.56427
## [3522,] 51.97182
## [3523,] 53.14861
## [3524,] 52.71683
## [3525,] 52.83630
## [3526,] 52.88083
## [3527,] 52.15012
## [3528,] 52.48664
## [3529,] 50.69430
## [3530,] 50.34671
## [3531,] 50.96160
## [3532,] 51.81726
## [3533,] 51.19485
## [3534,] 51.29485
## [3535,] 51.04012
## [3536,] 53.66602
## [3537,] 51.39220
## [3538,] 52.08215
## [3539,] 52.10073
## [3540,] 52.41868
## [3541,] 52.16969
## [3542,] 52.50543
## [3543,] 52.01493
## [3544,] 51.95433
## [3545,] 50.79876
## [3546,] 53.28688
## [3547,] 53.50570
## [3548,] 54.96161
## [3549,] 50.10638
## [3550,] 50.31200
## [3551,] 53.29525
## [3552,] 51.44094
## [3553,] 52.89995
## [3554,] 53.40785
## [3555,] 52.84942
## [3556,] 51.80559
## [3557,] 53.84495
## [3558,] 50.69822
## [3559,] 51.30185
## [3560,] 51.44040
## [3561,] 52.00240
## [3562,] 51.72457
## [3563,] 52.65673
## [3564,] 51.80965
## [3565,] 52.63798
## [3566,] 52.96021
## [3567,] 51.96044
## [3568,] 51.50527
## [3569,] 52.46306
## [3570,] 51.97589
## [3571,] 53.88414
## [3572,] 53.72526
## [3573,] 52.24908
## [3574,] 52.22641
## [3575,] 51.21916
## [3576,] 51.96760
## [3577,] 51.11654
## [3578,] 51.85604
## [3579,] 53.20212
## [3580,] 52.66250
## [3581,] 53.72258
## [3582,] 53.40402
## [3583,] 52.18472
## [3584,] 52.89514
## [3585,] 52.80174
## [3586,] 52.51587
## [3587,] 52.45305
## [3588,] 53.10415
## [3589,] 52.53994
## [3590,] 52.96110
## [3591,] 52.07646
## [3592,] 52.79412
## [3593,] 52.14314
## [3594,] 51.14212
## [3595,] 51.25781
## [3596,] 51.50160
## [3597,] 51.71480
## [3598,] 52.02043
## [3599,] 52.94915
## [3600,] 52.49200
## [3601,] 52.90006
## [3602,] 52.94847
## [3603,] 51.37479
## [3604,] 52.38931
## [3605,] 53.51737
## [3606,] 51.36160
## [3607,] 53.69738
## [3608,] 50.76202
## [3609,] 52.21952
## [3610,] 52.32289
## [3611,] 52.02616
## [3612,] 52.71011
## [3613,] 52.87229
## [3614,] 51.74690
## [3615,] 54.48741
## [3616,] 51.81415
## [3617,] 51.33866
## [3618,] 53.11057
## [3619,] 52.50313
## [3620,] 53.49093
## [3621,] 52.61503
## [3622,] 51.74299
## [3623,] 52.72064
## [3624,] 52.44360
## [3625,] 53.91102
## [3626,] 52.79976
## [3627,] 53.20390
## [3628,] 53.93240
## [3629,] 52.65646
## [3630,] 52.39714
## [3631,] 52.49176
## [3632,] 51.79352
## [3633,] 52.08369
## [3634,] 52.11212
## [3635,] 52.78161
## [3636,] 52.54523
## [3637,] 54.06487
## [3638,] 53.46477
## [3639,] 52.48162
## [3640,] 53.33816
## [3641,] 53.08052
## [3642,] 53.77932
## [3643,] 55.06696
## [3644,] 52.65938
## [3645,] 51.93046
## [3646,] 52.30463
## [3647,] 53.43888
## [3648,] 52.82730
## [3649,] 52.25920
## [3650,] 52.36637
## [3651,] 51.85643
## [3652,] 52.61849
## [3653,] 51.87703
## [3654,] 52.02660
## [3655,] 53.47351
## [3656,] 50.96025
## [3657,] 51.66078
## [3658,] 51.96793
## [3659,] 52.76046
## [3660,] 52.25326
## [3661,] 52.93205
## [3662,] 52.12366
## [3663,] 51.84901
## [3664,] 52.80206
## [3665,] 52.30201
## [3666,] 52.75768
## [3667,] 51.45723
## [3668,] 52.48620
## [3669,] 52.49515
## [3670,] 52.57375
## [3671,] 51.69973
## [3672,] 51.84706
## [3673,] 51.55868
## [3674,] 51.42989
## [3675,] 51.90982
## [3676,] 52.20506
## [3677,] 53.36990
## [3678,] 53.46195
## [3679,] 51.65327
## [3680,] 51.92813
## [3681,] 52.43505
## [3682,] 51.97182
## [3683,] 51.85084
## [3684,] 52.79064
## [3685,] 52.16793
## [3686,] 52.36433
## [3687,] 51.57443
## [3688,] 52.83381
## [3689,] 52.43274
## [3690,] 52.02471
## [3691,] 51.76746
## [3692,] 52.32784
## [3693,] 52.55010
## [3694,] 51.71134
## [3695,] 52.49154
## [3696,] 52.54458
## [3697,] 52.83884
## [3698,] 53.63326
## [3699,] 51.37370
## [3700,] 52.36091
## [3701,] 51.93844
## [3702,] 52.62401
## [3703,] 52.26842
## [3704,] 51.77105
## [3705,] 52.31454
## [3706,] 52.50516
## [3707,] 53.13574
## [3708,] 51.36548
## [3709,] 51.59242
## [3710,] 54.27071
## [3711,] 51.35525
## [3712,] 53.89268
## [3713,] 48.84423
## [3714,] 51.16296
## [3715,] 52.71390
## [3716,] 52.29235
## [3717,] 53.25426
## [3718,] 52.95119
## [3719,] 52.88826
## [3720,] 52.46717
## [3721,] 52.46340
## [3722,] 51.22154
## [3723,] 52.22738
## [3724,] 51.47375
## [3725,] 52.57209
## [3726,] 52.20057
## [3727,] 52.54964
## [3728,] 51.44547
## [3729,] 51.93900
## [3730,] 51.86925
## [3731,] 49.45700
## [3732,] 49.99622
## [3733,] 49.85707
## [3734,] 52.12108
## [3735,] 52.83180
## [3736,] 51.41828
## [3737,] 52.65940
## [3738,] 52.62965
## [3739,] 52.68869
## [3740,] 51.75993
## [3741,] 53.27191
## [3742,] 53.54464
## [3743,] 55.90753
## [3744,] 55.75723
## [3745,] 54.99226
## [3746,] 54.97493
## [3747,] 54.56772
## [3748,] 54.71758
## [3749,] 54.51432
## [3750,] 52.39452
## [3751,] 52.23218
## [3752,] 52.35725
## [3753,] 52.54853
## [3754,] 53.33668
## [3755,] 52.95544
## [3756,] 50.47197
## [3757,] 54.27855
## [3758,] 50.51254
## [3759,] 51.88113
## [3760,] 51.82160
## [3761,] 51.94196
## [3762,] 52.21126
## [3763,] 53.49813
## [3764,] 51.50512
## [3765,] 52.07854
## [3766,] 53.86560
## [3767,] 53.65467
## [3768,] 53.02473
## [3769,] 52.75839
## [3770,] 52.88680
## [3771,] 52.20539
## [3772,] 53.18648
## [3773,] 52.68104
## [3774,] 52.04005
## [3775,] 52.39881
## [3776,] 52.68264
## [3777,] 52.62078
## [3778,] 53.22617
## [3779,] 53.42627
## [3780,] 51.27516
## [3781,] 53.99502
## [3782,] 50.74799
## [3783,] 53.37938
## [3784,] 53.58563
## [3785,] 51.05333
## [3786,] 53.40475
## [3787,] 52.66714
## [3788,] 51.86839
## [3789,] 51.66151
## [3790,] 51.57005
## [3791,] 53.29841
## [3792,] 51.60273
## [3793,] 52.83555
## [3794,] 51.69257
## [3795,] 52.07766
## [3796,] 51.21841
## [3797,] 51.66793
## [3798,] 51.44348
## [3799,] 52.56870
## [3800,] 51.66727
## [3801,] 51.65505
## [3802,] 52.21793
## [3803,] 52.51668
## [3804,] 52.43013
## [3805,] 51.92522
## [3806,] 53.16930
## [3807,] 51.44336
## [3808,] 51.88152
## [3809,] 51.86550
## [3810,] 52.02850
## [3811,] 51.42333
## [3812,] 51.75278
## [3813,] 51.96925
## [3814,] 53.06410
## [3815,] 52.41905
## [3816,] 52.50538
## [3817,] 53.49614
## [3818,] 51.45583
## [3819,] 52.53628
## [3820,] 52.14117
## [3821,] 53.22063
## [3822,] 52.51317
## [3823,] 52.67839
## [3824,] 50.93803
## [3825,] 50.57411
## [3826,] 53.20153
## [3827,] 52.50993
## [3828,] 53.51443
## [3829,] 51.98625
## [3830,] 52.95788
## [3831,] 52.50710
## [3832,] 51.34958
## [3833,] 53.23948
## [3834,] 51.21448
## [3835,] 51.53212
## [3836,] 51.49381
## [3837,] 53.27355
## [3838,] 52.92585
## [3839,] 52.65439
## [3840,] 50.83692
## [3841,] 51.35791
## [3842,] 52.70401
## [3843,] 51.40107
## [3844,] 53.26791
## [3845,] 52.48525
## [3846,] 52.06205
## [3847,] 51.69932
## [3848,] 53.13377
## [3849,] 51.94070
## [3850,] 52.11086
## [3851,] 51.77590
## [3852,] 51.23356
## [3853,] 51.31389
## [3854,] 52.94328
## [3855,] 49.84882
## [3856,] 53.49575
## [3857,] 51.14937
## [3858,] 54.35516
## [3859,] 54.09892
## [3860,] 52.47105
## [3861,] 52.90663
## [3862,] 53.02941
## [3863,] 52.82974
## [3864,] 52.85610
## [3865,] 52.56015
## [3866,] 52.68434
## [3867,] 52.36232
## [3868,] 52.18701
## [3869,] 51.94071
## [3870,] 51.63001
## [3871,] 53.83146
## [3872,] 50.20037
## [3873,] 54.29828
## [3874,] 52.20601
## [3875,] 51.61299
## [3876,] 53.77476
## [3877,] 52.27936
## [3878,] 54.58837
## [3879,] 54.70501
## [3880,] 52.27975
## [3881,] 51.98762
## [3882,] 50.63754
## [3883,] 53.20989
## [3884,] 53.97433
## [3885,] 52.49458
## [3886,] 52.61676
## [3887,] 52.48194
## [3888,] 53.39295
## [3889,] 52.06787
## [3890,] 52.29481
## [3891,] 52.53268
## [3892,] 51.90285
## [3893,] 53.53895
## [3894,] 52.88744
## [3895,] 52.54292
## [3896,] 52.09690
## [3897,] 51.62917
## [3898,] 51.31048
## [3899,] 51.73731
## [3900,] 52.29247
## [3901,] 52.65906
## [3902,] 52.36907
## [3903,] 52.96351
## [3904,] 51.81076
## [3905,] 52.94990
## [3906,] 51.64870
## [3907,] 52.98614
## [3908,] 52.02119
## [3909,] 52.35077
## [3910,] 51.21220
## [3911,] 51.78476
## [3912,] 51.08883
## [3913,] 52.54869
## [3914,] 52.44782
## [3915,] 51.88221
## [3916,] 52.02617
## [3917,] 52.64491
## [3918,] 52.26423
## [3919,] 52.71199
## [3920,] 51.98716
## [3921,] 52.12526
## [3922,] 51.68626
## [3923,] 50.77180
## [3924,] 52.18623
## [3925,] 50.48421
## [3926,] 50.69143
## [3927,] 51.90569
## [3928,] 52.54721
## [3929,] 51.44153
## [3930,] 52.62363
## [3931,] 52.02598
## [3932,] 52.27694
## [3933,] 51.67729
## [3934,] 52.83620
## [3935,] 52.27166
## [3936,] 53.80151
## [3937,] 52.55600
## [3938,] 52.39422
## [3939,] 52.48726
## [3940,] 52.42831
## [3941,] 52.93481
## [3942,] 51.13587
## [3943,] 53.39810
## [3944,] 49.89263
## [3945,] 49.73819
## [3946,] 54.29075
## [3947,] 51.64182
## [3948,] 53.01864
## [3949,] 51.66297
## [3950,] 51.61986
## [3951,] 51.58735
## [3952,] 53.14199
## [3953,] 51.89710
## [3954,] 53.82722
## [3955,] 51.65209
## [3956,] 51.38849
## [3957,] 49.66788
## [3958,] 51.80471
## [3959,] 53.43819
## [3960,] 51.16972
## [3961,] 52.35120
## [3962,] 51.65501
## [3963,] 51.16199
## [3964,] 52.70489
## [3965,] 51.95024
## [3966,] 53.01212
## [3967,] 52.87585
## [3968,] 53.99199
## [3969,] 51.70175
## [3970,] 52.70976
## [3971,] 53.50272
## [3972,] 50.89784
## [3973,] 51.38565
## [3974,] 51.07108
## [3975,] 51.07957
## [3976,] 53.58180
## [3977,] 51.34324
## [3978,] 50.98654
## [3979,] 51.55426
## [3980,] 53.00574
## [3981,] 51.86014
## [3982,] 50.89987
## [3983,] 51.31304
## [3984,] 52.34095
## [3985,] 52.58707
## [3986,] 52.99104
## [3987,] 53.05569
## [3988,] 53.68138
## [3989,] 52.87023
## [3990,] 51.71759
## [3991,] 52.17011
## [3992,] 51.84673
## [3993,] 51.90960
## [3994,] 52.76526
## [3995,] 54.62258
## [3996,] 53.45097
## [3997,] 53.23753
## [3998,] 52.98033
## [3999,] 51.76064
## [4000,] 52.65380hist(y_pred1)
# manually calculating it
head(sims)## parameters
## iterations (Intercept) growth sigma
## [1,] 47.37298 2.655321 4.978187
## [2,] 47.46683 2.752677 4.742360
## [3,] 47.04543 2.958429 4.507205
## [4,] 47.57576 2.449818 3.644981
## [5,] 48.51272 2.343477 4.577593
## [6,] 44.79675 3.673835 3.149167a=sims[,1]
b=sims[,2]
(y_pred2=a+b*2)## [1] 52.68362 52.97218 52.96229 52.47539 53.19967 52.14442 50.71921 52.66878
## [9] 50.85714 52.18429 52.39691 52.78913 51.69621 52.93418 52.28870 52.35676
## [17] 52.33167 52.71242 53.54627 51.34035 52.97566 52.06719 53.83511 50.55492
## [25] 49.62704 54.41625 53.09712 52.23700 52.39791 51.66447 52.61487 52.51493
## [33] 52.78430 52.39485 53.38523 53.19981 53.83205 54.43044 50.54545 53.98743
## [41] 50.60060 53.65485 50.97469 50.37916 50.16137 50.32514 51.52620 51.54998
## [49] 53.21906 51.92946 51.99001 54.43769 50.55368 50.01388 52.93131 52.73121
## [57] 52.38350 53.06871 52.48428 54.16200 50.66236 52.09231 52.32538 51.99930
## [65] 51.49005 50.85918 51.70370 52.14362 52.06710 52.81782 52.32419 52.61743
## [73] 53.40422 54.18104 52.31393 52.05987 52.49631 51.84431 50.93628 51.90898
## [81] 52.50629 51.18643 53.81747 52.32434 51.99933 52.31804 51.58908 51.58516
## [89] 53.24429 52.84391 52.95079 51.62006 53.20122 51.69839 52.43982 51.49315
## [97] 51.26969 52.42913 53.94172 52.19517 52.31080 50.88375 53.36042 51.59873
## [105] 51.90490 53.14814 51.47926 52.96795 52.91709 52.34106 52.20869 52.98321
## [113] 52.81158 53.04151 51.18904 50.70350 51.15050 51.07173 52.17741 51.27084
## [121] 52.95890 52.49674 52.03204 52.66952 51.96054 52.50907 51.70293 51.71085
## [129] 51.60608 53.10731 53.10731 51.56520 52.34943 52.54114 52.21730 52.24096
## [137] 51.66241 51.34721 52.08278 52.60492 52.34026 52.59605 52.70509 51.15378
## [145] 53.51540 52.20970 52.28856 51.36834 51.46790 52.45454 51.64665 51.43447
## [153] 51.88043 53.07892 50.33380 50.20640 51.89671 53.63936 52.96810 51.22024
## [161] 53.77692 50.63570 51.51625 52.87817 52.77540 51.68343 52.76945 51.54538
## [169] 52.09503 52.54576 52.10733 51.61746 51.24666 52.03996 50.66273 54.74783
## [177] 54.83356 54.52328 54.59878 54.20905 55.98072 56.13471 54.42625 50.71196
## [185] 51.36933 55.91892 55.98987 49.07386 54.37519 50.76038 54.45041 54.45688
## [193] 54.51977 53.15372 51.46585 51.45180 50.80861 52.10762 51.80146 52.10719
## [201] 51.30946 51.91328 52.19536 52.80846 51.20482 54.29109 54.59084 53.15773
## [209] 53.01827 52.55889 53.29362 53.62729 53.36534 54.58437 53.92669 51.44721
## [217] 53.28643 50.96200 52.96465 51.17502 52.93424 51.28487 51.24978 52.80815
## [225] 50.42383 51.58023 53.42837 51.99109 52.12437 54.27086 51.72509 54.05864
## [233] 51.08184 51.21516 50.87930 53.32303 51.99192 51.05966 50.77037 52.65657
## [241] 52.72447 52.13060 51.12260 49.82711 49.98218 53.01145 52.89215 52.36656
## [249] 52.20680 53.08813 51.63420 52.93042 53.34878 53.63177 52.20670 51.31085
## [257] 49.23035 51.79102 52.03905 51.71984 50.81558 53.48001 51.47305 52.66141
## [265] 51.78139 50.88300 50.78778 52.94944 51.74840 52.76543 51.85137 52.60711
## [273] 51.52440 51.51415 53.39378 51.89427 50.77455 51.41352 51.50447 51.54673
## [281] 50.39464 52.41797 52.75940 51.51904 52.82755 52.04965 51.53034 52.61316
## [289] 52.43566 51.60957 53.41127 51.93962 52.67875 52.09053 50.75772 54.32625
## [297] 53.62947 53.14243 52.87548 52.03493 50.53305 52.60820 52.76987 52.81755
## [305] 52.68138 52.43305 52.50210 53.47743 52.05024 52.69584 51.68906 53.15658
## [313] 52.77141 51.99321 51.02970 53.41132 51.19271 52.63147 52.05607 53.02546
## [321] 51.90960 50.29731 54.69162 48.87554 49.54034 51.32957 52.22249 51.16000
## [329] 53.76691 51.05551 52.93229 51.84306 52.77192 51.08826 52.98312 52.74930
## [337] 50.81056 50.57572 49.52098 55.22749 49.64310 50.08785 53.07046 51.37795
## [345] 52.41392 52.04405 53.77973 52.27774 53.13195 52.40813 51.67847 52.40611
## [353] 53.20499 51.18451 51.38588 54.26401 53.57205 52.81130 52.73229 52.60077
## [361] 51.99164 53.04812 51.02051 53.85840 51.52027 52.72914 53.68738 53.72360
## [369] 50.99994 51.58275 51.73111 52.11110 51.14168 52.01078 54.57916 54.17610
## [377] 54.23700 54.77563 55.40577 54.54965 54.65089 53.74103 52.86250 51.57199
## [385] 51.64477 54.50106 54.70578 54.96076 54.89924 50.66910 51.05586 53.58765
## [393] 51.44963 52.51681 54.16196 52.95742 50.52932 51.04464 51.48824 51.45205
## [401] 52.90644 51.83164 55.38458 53.49401 52.86675 51.25030 53.82247 53.88311
## [409] 50.83053 53.22933 54.66067 55.49107 53.27452 52.89430 52.50672 52.99704
## [417] 51.62466 52.46648 50.36170 50.32899 50.26138 52.20256 50.70594 54.50627
## [425] 51.92043 53.59538 51.35890 53.39575 53.28104 53.46709 53.00343 51.25565
## [433] 53.07850 52.97846 51.52928 52.14054 53.30658 50.97464 51.29696 51.11542
## [441] 52.35517 52.44623 52.59192 52.71586 52.29028 51.72653 51.77420 51.74165
## [449] 53.20665 52.04560 52.68250 52.82397 51.64696 51.94551 52.18880 52.07820
## [457] 51.58374 52.22457 53.12765 51.23820 51.96973 53.30429 50.68143 54.02258
## [465] 50.84473 53.59200 52.59317 52.86005 52.59356 53.17138 52.81942 51.16841
## [473] 52.07012 52.39824 52.20852 52.26085 51.68696 50.97048 51.51692 52.01256
## [481] 51.71856 51.11912 51.06227 51.14108 53.80583 54.02431 52.35295 54.11457
## [489] 53.53546 52.12628 52.78503 52.51527 51.77494 51.90069 52.96135 53.25398
## [497] 53.02506 53.38055 51.69338 52.28055 52.52685 53.79419 52.65841 52.65841
## [505] 51.47223 53.39260 51.40275 50.94030 51.69097 53.31214 55.46025 51.04367
## [513] 51.59042 52.44369 51.97217 52.90380 53.22425 51.28704 52.75867 52.69454
## [521] 51.94156 51.80822 52.68476 53.06947 52.30008 52.34810 52.32214 53.60582
## [529] 51.09539 51.73646 51.01828 51.00201 52.93775 51.40542 52.24836 52.98759
## [537] 51.69267 53.18736 52.12724 51.83155 52.54320 51.98591 51.40385 53.20243
## [545] 53.88078 53.73121 50.94505 54.04960 53.23754 54.25385 53.38320 52.89571
## [553] 52.64732 52.44920 52.49920 53.15466 52.57515 51.84220 53.44683 51.03735
## [561] 51.84142 51.48619 51.91240 52.62641 51.89992 52.87130 52.82257 50.58608
## [569] 53.00924 51.49632 53.32664 51.35374 51.12751 52.50267 52.05328 52.08334
## [577] 53.77994 51.10722 50.93283 53.43909 51.34049 53.02914 52.19953 53.60763
## [585] 54.40586 53.78201 53.71229 51.28569 53.91284 54.20096 51.53076 51.07737
## [593] 52.53032 52.10909 51.67298 52.23201 51.44492 52.83623 53.63292 52.54368
## [601] 51.81884 52.77453 51.74236 52.32980 53.55721 52.33413 52.32256 52.72165
## [609] 54.15260 51.04884 54.07956 51.97021 52.17435 52.95466 52.84791 52.43562
## [617] 52.35452 52.39469 53.42802 51.78877 52.07307 53.88865 53.23520 51.64995
## [625] 53.18896 53.80250 52.67030 51.55728 51.30820 51.43960 53.81788 52.22101
## [633] 51.65836 51.52834 52.01140 52.16445 51.21230 51.07860 53.55981 54.30689
## [641] 50.27949 53.90248 50.31610 50.66691 53.03194 51.47247 53.19976 51.23923
## [649] 53.24628 51.56270 51.63565 53.51943 53.49614 53.00960 53.03856 51.71694
## [657] 51.62361 52.19669 53.98500 53.20964 54.27782 52.28746 52.56906 52.22910
## [665] 52.75555 52.89098 52.45868 51.64266 53.52066 51.94661 52.16665 50.91113
## [673] 52.74955 54.13561 51.99433 51.86477 51.13906 51.33253 52.63272 52.65716
## [681] 52.77157 51.99312 52.25321 52.49698 52.63001 50.90887 53.10279 52.31538
## [689] 52.44317 52.35872 52.65029 52.56364 52.35488 52.38634 52.84352 53.99540
## [697] 51.67946 54.00270 54.06579 54.23099 54.62046 51.16236 53.05021 53.96019
## [705] 51.81084 52.46682 52.14236 54.16232 52.01782 51.93364 52.76478 51.29444
## [713] 53.98588 53.03227 50.76511 51.95838 52.49259 53.05053 52.77539 52.05603
## [721] 52.85649 52.77911 51.42749 52.11291 52.78209 52.50210 52.86070 52.55034
## [729] 52.04585 52.91281 51.52949 51.91700 52.49440 52.64817 52.76530 52.02521
## [737] 51.70350 52.34054 52.20654 53.09138 53.09603 51.38099 51.69055 52.53047
## [745] 52.53047 52.59098 53.07435 53.77600 52.65842 52.44575 53.73272 52.64239
## [753] 52.04019 50.80886 53.82537 53.70410 52.13311 52.32436 53.67237 53.27990
## [761] 53.03947 51.31732 53.78675 53.77617 52.42516 52.14514 52.56671 51.27120
## [769] 51.42093 50.05227 49.46340 49.98219 53.96689 50.25424 51.21938 52.83397
## [777] 51.89894 52.61193 51.65950 52.93363 51.10600 52.96223 52.52058 51.58948
## [785] 53.28813 52.00779 54.05196 50.33174 53.09851 51.64526 54.26573 50.72705
## [793] 53.67485 51.24505 52.16124 51.78709 51.89274 54.10182 51.65279 52.17096
## [801] 52.56010 53.87513 53.68112 52.93016 54.27710 50.89826 52.65964 53.48001
## [809] 54.14228 50.65001 52.14006 53.61125 53.69727 52.59316 52.05669 52.81610
## [817] 51.46719 52.39360 50.89309 51.27217 52.16820 52.10078 52.96585 52.71755
## [825] 52.04052 50.65021 51.66643 50.96055 54.06754 53.07548 52.97481 53.13085
## [833] 52.59559 53.94426 53.33580 53.09498 52.47399 52.92136 51.02033 53.73350
## [841] 50.79883 53.60215 51.77615 51.09318 53.23331 51.36738 51.28168 53.13750
## [849] 50.20527 52.40539 51.82537 51.67869 49.98547 49.85613 50.63568 51.27983
## [857] 51.51756 50.96937 51.85998 53.50641 54.18312 54.29617 50.61606 52.58777
## [865] 52.62380 52.02035 52.81115 51.74859 54.29570 54.21003 50.40278 52.69003
## [873] 53.78249 53.66252 52.52925 52.13542 53.97704 53.15867 52.06100 52.69548
## [881] 52.34795 52.26787 54.05341 53.11118 52.95053 51.88168 51.97475 51.19465
## [889] 52.12371 52.74325 53.28859 53.55922 52.98090 52.76316 51.19760 53.06747
## [897] 52.34729 52.44669 52.01227 51.50418 53.37118 53.44932 53.88440 51.84299
## [905] 53.36776 51.47530 53.90570 53.59701 52.99948 51.84699 51.67890 50.97641
## [913] 51.80187 51.92403 51.35702 51.92552 53.02349 51.42984 51.49890 52.62686
## [921] 52.59544 53.01221 52.21294 52.87000 52.02228 53.03544 53.18069 51.80497
## [929] 52.28406 52.13522 52.13522 52.64895 52.10379 51.53013 53.16324 51.59033
## [937] 51.82351 52.94055 52.57270 51.60223 51.34008 51.38864 51.36953 52.61917
## [945] 50.94166 50.71236 52.07483 54.24177 53.13923 53.34918 52.83444 52.29913
## [953] 53.52193 53.55206 51.15668 53.64349 53.08595 52.37352 51.94321 52.15079
## [961] 52.06730 52.29456 52.60735 53.27950 52.40139 51.43860 53.13430 52.85529
## [969] 51.19530 51.32708 52.26341 52.73891 52.20255 52.70778 52.00397 52.70368
## [977] 51.95146 51.54572 51.95323 51.41504 53.24149 51.60605 52.00237 52.86054
## [985] 51.91964 52.25689 52.06029 52.31481 53.70026 53.37907 51.48149 52.20980
## [993] 50.95489 51.79947 52.13026 52.61794 51.86340 51.84802 51.00242 54.09154
## [1001] 53.30504 51.43433 52.06187 52.33145 51.70745 50.66167 50.44078 52.46670
## [1009] 53.26858 51.57921 53.08563 52.88047 52.29818 52.17076 52.08007 53.01174
## [1017] 51.80435 50.79464 55.31908 55.14641 54.12986 51.88241 52.13074 52.30363
## [1025] 51.18103 51.92404 52.35817 51.44065 53.75312 49.87863 50.86016 52.63027
## [1033] 51.05428 52.27681 53.22070 52.01687 52.31723 52.73172 53.06975 53.21213
## [1041] 53.11903 54.59970 52.37008 52.23247 52.23689 53.73175 52.90104 54.14352
## [1049] 52.19975 53.11890 51.82727 53.05569 51.62604 53.57126 50.61437 53.39030
## [1057] 52.39538 53.46248 51.35947 54.07272 51.74837 51.79546 53.42738 52.84307
## [1065] 51.73499 52.89211 53.50323 52.38890 51.70380 52.06684 53.70227 52.27087
## [1073] 52.86769 51.31390 52.34005 52.68032 54.01265 50.32179 50.90404 53.31085
## [1081] 53.83000 52.84352 52.58870 52.90239 51.00081 51.58046 55.27356 52.40263
## [1089] 50.97987 52.88844 51.83914 53.55254 51.30043 53.68571 52.65120 52.22784
## [1097] 52.73109 51.35145 53.19991 53.90707 50.86018 51.99017 52.04347 52.86648
## [1105] 52.80285 53.09255 53.27682 52.46762 53.27644 51.85332 50.74248 52.03197
## [1113] 52.46884 52.34382 51.80021 51.21122 53.59286 53.38114 51.37341 51.14509
## [1121] 53.49409 51.68662 53.54498 51.99543 51.48201 52.80204 52.75705 52.03745
## [1129] 53.08168 52.92508 52.80324 53.04988 52.18437 52.99823 52.61854 52.28639
## [1137] 51.64452 50.21700 54.27254 52.66855 53.08696 51.84964 51.86808 54.15169
## [1145] 52.93455 53.19766 50.55343 53.22201 51.82435 54.42462 51.17297 50.96686
## [1153] 50.35190 53.08350 51.84702 52.85594 52.84935 51.87789 52.36838 52.82820
## [1161] 53.79140 53.22457 52.01383 51.34529 51.97341 51.70694 51.42518 53.34865
## [1169] 53.23560 54.18554 52.13324 53.54756 51.91956 51.00860 51.56560 51.49203
## [1177] 52.75917 52.70597 52.37529 51.54688 50.47299 53.49112 52.72292 51.80927
## [1185] 51.14524 53.90130 53.01012 53.72169 51.99354 52.42496 53.16482 51.28871
## [1193] 52.47017 52.76652 51.74329 52.58496 52.85766 52.98454 51.06132 53.27802
## [1201] 51.49340 51.30958 52.46064 50.11080 53.94909 54.04256 51.24923 53.93033
## [1209] 51.77830 53.79263 53.44875 53.80585 50.25072 50.38347 51.36613 52.71201
## [1217] 52.77472 53.08737 51.66534 53.55189 52.59608 52.11235 52.77335 52.95066
## [1225] 52.73611 52.57749 50.82737 50.28203 50.77247 53.02128 53.37228 53.41510
## [1233] 51.30473 52.17484 52.54248 51.98740 54.56216 52.82324 52.80185 52.26516
## [1241] 53.11490 51.59284 53.55030 51.14371 53.21226 51.36321 52.84908 52.54025
## [1249] 50.92466 51.64411 53.16592 51.96298 52.98540 52.30758 52.17981 52.97588
## [1257] 53.58773 51.76514 51.65927 51.76180 52.02722 52.68533 52.40575 52.02697
## [1265] 50.59122 52.79058 53.67073 51.72141 53.08168 52.44066 52.07121 52.22378
## [1273] 52.45130 52.34657 52.08389 51.11996 52.16184 52.47400 52.36934 52.15791
## [1281] 52.38678 52.38407 50.11280 49.88845 51.33692 51.98524 50.77266 53.77146
## [1289] 50.71673 53.32501 52.02975 52.90495 52.40587 53.77491 50.67963 51.08577
## [1297] 52.05260 52.32757 51.82346 51.26347 51.39626 53.96895 51.56224 51.34917
## [1305] 51.98785 50.31913 51.89994 53.32235 54.85735 52.48078 50.62000 54.11102
## [1313] 53.83347 53.76381 52.09524 53.78847 51.84218 52.45539 52.28014 52.71158
## [1321] 52.27787 53.08051 53.21037 51.55014 52.18601 51.88263 52.65856 52.03184
## [1329] 52.47247 52.14318 52.40658 52.03796 52.30377 52.37657 52.97983 52.34263
## [1337] 51.17326 52.14855 53.15521 51.77535 52.77899 52.58150 52.11577 51.60696
## [1345] 52.04970 52.10891 51.97738 50.86928 52.92497 52.85700 53.22138 52.10691
## [1353] 53.76482 54.02786 53.25251 52.64218 52.92418 53.79650 50.87111 53.98804
## [1361] 50.87198 51.07503 50.71188 51.29777 51.17938 53.99505 54.52606 53.00291
## [1369] 51.24403 53.88506 53.99709 53.67468 54.73608 51.43554 52.70294 53.67698
## [1377] 52.48299 52.70139 52.75136 52.77939 51.99886 53.26359 54.05706 53.80477
## [1385] 53.45081 53.30413 53.46887 51.17736 52.65023 52.55133 52.96236 52.14102
## [1393] 53.10619 52.50800 51.90837 52.33659 50.76999 50.51343 50.28048 53.50149
## [1401] 53.10522 51.30150 52.66479 53.25073 50.76770 54.40581 51.24106 52.72548
## [1409] 51.78457 52.96218 53.95637 53.55086 50.43840 51.77471 52.60528 52.07400
## [1417] 50.72404 50.86503 55.13447 49.92684 53.73612 52.47829 52.04079 52.36575
## [1425] 51.34847 52.55263 51.72116 52.40928 52.75445 52.01301 51.64546 52.62281
## [1433] 52.19442 52.57594 51.83078 51.88997 52.78024 52.58461 52.95983 51.54393
## [1441] 52.81499 51.28933 51.40771 51.79656 51.94491 53.06684 51.34070 51.03039
## [1449] 53.69804 52.24163 51.92113 51.27456 52.46622 52.77918 51.48996 51.81342
## [1457] 52.21410 52.12637 52.44302 52.23878 51.90705 53.63564 52.40631 53.18840
## [1465] 51.79656 52.99205 51.44323 53.35035 52.55853 53.22823 53.74539 52.98753
## [1473] 51.65232 52.42374 51.75407 53.06087 51.05078 52.67933 52.18318 54.13156
## [1481] 50.39253 52.92557 52.20334 53.40916 55.04713 53.56838 52.82107 50.49530
## [1489] 53.47183 52.82413 52.68259 53.45675 53.39283 53.17663 51.63241 51.14987
## [1497] 53.29454 51.78256 53.43101 52.80118 52.03095 52.64464 52.18631 53.35985
## [1505] 51.77306 51.23178 52.67207 53.84526 50.90002 51.73320 52.95679 52.45191
## [1513] 50.80962 55.84277 50.10649 50.63423 51.60696 54.37534 55.01504 55.24402
## [1521] 51.70942 51.73584 51.97162 52.94103 51.88011 52.52258 54.02623 50.74545
## [1529] 54.21058 52.64279 51.22006 51.97222 52.27595 51.75366 51.65647 51.61073
## [1537] 51.02823 51.85241 53.11303 51.62782 52.05998 51.95029 52.00250 52.95218
## [1545] 52.31776 52.16404 52.33430 51.43698 52.05306 53.23578 51.35296 51.29632
## [1553] 53.18050 52.49943 51.61153 51.86181 51.92130 52.87034 51.05858 53.33019
## [1561] 53.02742 52.77618 53.64465 50.95536 52.35634 54.30426 51.95992 50.57166
## [1569] 53.21386 53.18804 51.44285 53.59059 53.72639 50.67843 53.35213 53.15335
## [1577] 51.30527 52.84261 50.70170 50.85516 50.63977 50.86637 52.12794 52.39923
## [1585] 52.35256 52.48148 53.54461 51.23379 52.43064 53.35667 53.68937 53.32773
## [1593] 53.17743 51.89094 52.16065 52.21125 54.45029 53.31699 53.43497 52.71693
## [1601] 54.23049 53.62329 51.38294 52.50425 52.74965 52.56973 53.01076 54.70987
## [1609] 52.79952 53.73592 50.98543 52.74520 51.65214 52.82198 52.00159 53.47764
## [1617] 51.52921 51.93999 52.49519 52.58172 52.61586 52.11348 52.99072 52.71342
## [1625] 53.69263 52.74063 52.01965 51.90708 54.44858 51.52774 52.88079 54.89411
## [1633] 54.74142 53.90538 53.61680 53.38320 52.10368 51.99996 54.01224 50.63687
## [1641] 51.92700 53.27323 53.32920 54.87269 50.89891 53.36662 52.43923 51.87045
## [1649] 52.38955 53.57925 53.95084 53.72413 51.31104 50.64392 52.66119 52.86002
## [1657] 53.67637 50.16328 54.59232 52.33963 51.31854 53.74510 53.72570 50.60910
## [1665] 52.05376 52.03406 52.87160 52.12290 52.19613 51.28312 51.96680 52.40214
## [1673] 52.55549 50.73826 51.74863 52.29411 52.95263 53.89969 53.66318 52.44509
## [1681] 52.20240 52.48984 53.24276 51.41245 50.87606 50.06042 53.17927 53.44858
## [1689] 53.92992 54.10579 52.13191 52.34806 51.33802 50.49848 50.37483 54.14459
## [1697] 54.15169 50.40167 53.36802 54.46213 50.88392 53.75005 51.65850 51.34555
## [1705] 52.70687 53.12383 52.35963 52.49850 52.55970 53.72239 53.33084 53.07582
## [1713] 51.96317 53.05063 52.25322 52.22799 51.98510 51.49180 51.18952 51.70867
## [1721] 51.45325 51.25887 51.16877 51.37393 52.70131 52.54761 52.87785 54.31566
## [1729] 52.49828 52.51703 52.37534 53.19014 51.11096 51.42784 51.53342 52.96438
## [1737] 52.30970 52.21404 51.30940 53.27976 51.78442 53.16962 52.78590 53.15280
## [1745] 51.88479 52.43386 51.46232 50.94817 51.23017 51.65397 51.75976 53.03485
## [1753] 51.56739 50.93919 51.54230 49.37382 51.85217 52.10867 51.98244 52.79475
## [1761] 51.13256 53.04838 52.71365 52.17992 52.47521 51.92045 53.13864 50.03563
## [1769] 50.81212 54.10994 52.98082 53.58132 50.51188 49.82990 49.00853 49.75233
## [1777] 52.79208 52.85789 53.53796 51.31149 51.69462 52.08469 52.70972 52.90168
## [1785] 51.91297 50.51147 52.62126 53.36405 52.62080 52.60787 51.90523 52.18111
## [1793] 51.45505 50.33850 50.48945 53.35696 52.30714 52.19082 51.44680 52.56948
## [1801] 52.29409 52.78959 50.74040 51.74503 52.50632 52.59443 52.74152 51.97732
## [1809] 52.15254 52.29689 52.36578 53.14231 51.87984 52.81809 52.46762 52.45534
## [1817] 53.26469 52.56718 52.55477 52.55556 54.47414 50.50408 53.77323 50.38248
## [1825] 51.80579 50.78354 53.33904 52.56686 50.18770 51.25759 53.41140 52.70324
## [1833] 53.17441 52.29683 51.93567 52.42739 51.89743 52.96953 51.71474 52.94440
## [1841] 52.16676 53.55294 51.21100 51.85671 53.15820 52.30341 51.71893 53.02675
## [1849] 52.75109 52.34455 52.23389 52.80743 51.10068 52.65379 53.04305 53.83555
## [1857] 50.98865 52.45375 52.64224 51.57529 50.60520 50.54341 54.15161 54.02428
## [1865] 53.80698 52.17256 52.28059 53.41215 52.86606 52.01458 52.55731 52.15952
## [1873] 52.77294 51.81326 51.51503 51.78721 53.58769 52.52568 52.79254 52.64969
## [1881] 52.33660 52.59223 51.89880 51.10424 52.25900 52.73145 53.31236 53.56303
## [1889] 52.64302 52.79432 52.12301 51.92063 51.54334 53.62428 52.76789 52.03664
## [1897] 52.03664 52.41590 53.27655 51.24790 51.39170 53.11129 51.45765 53.07988
## [1905] 52.62631 52.91227 51.63968 51.21269 52.73816 51.82330 52.08888 53.31979
## [1913] 53.31899 51.80805 53.35896 53.47698 54.30979 53.46867 52.81880 54.04988
## [1921] 53.34282 51.49575 51.61082 51.97067 53.28329 52.88754 51.50957 52.75948
## [1929] 51.65169 52.55938 51.43558 53.25426 52.11264 53.28402 53.09210 52.73019
## [1937] 52.93760 52.23956 50.52014 51.81398 54.86280 53.00307 51.64456 51.53098
## [1945] 53.12281 52.12762 53.62275 51.26370 52.62516 51.69591 52.46932 51.42486
## [1953] 51.32750 52.36801 51.56649 52.91991 51.82294 52.69510 51.88619 53.37004
## [1961] 52.93018 51.72816 52.72000 51.66347 52.53990 51.57329 52.78220 51.28152
## [1969] 54.45360 52.29347 51.43554 52.78873 52.86895 51.53076 50.60396 52.72255
## [1977] 54.42520 54.63544 53.12702 52.70026 52.45351 51.85731 51.51416 52.45196
## [1985] 51.96865 51.81975 54.56934 53.85993 53.10646 53.77298 54.04944 49.68183
## [1993] 49.37178 50.86843 50.85651 51.89310 52.23958 52.01741 53.45272 52.12451
## [2001] 52.44674 52.32642 51.92398 52.97159 52.75828 52.78878 53.17683 51.60960
## [2009] 51.08293 51.15655 51.70171 52.70397 51.95388 51.56773 53.08336 51.75272
## [2017] 53.05526 52.08085 53.20425 51.57945 54.25630 54.11540 52.94106 51.70391
## [2025] 51.00487 50.26264 55.58397 55.46016 55.65119 54.82684 51.11389 52.50342
## [2033] 53.42753 51.66044 51.85844 53.13335 51.83507 52.45243 53.06581 53.60871
## [2041] 52.42648 53.07253 52.38484 51.27327 52.37805 52.25996 52.59216 52.95014
## [2049] 53.46520 52.31262 52.42138 54.14104 54.05884 53.94960 54.22853 50.94852
## [2057] 52.72734 51.73746 51.61040 53.89156 51.98547 53.08616 51.84793 53.02314
## [2065] 51.59787 51.16503 51.44702 53.19754 50.82140 50.18397 50.89158 52.40573
## [2073] 52.38990 52.52729 52.36197 52.10030 52.36122 52.53873 52.84798 53.22779
## [2081] 53.50510 52.99763 52.65185 53.35667 50.52132 51.55796 52.20219 52.72089
## [2089] 52.66922 51.14609 51.80934 51.70232 53.07306 52.60568 51.59040 51.33011
## [2097] 53.74452 51.97822 51.77626 53.82563 51.45532 52.14666 51.49559 51.42276
## [2105] 50.84549 51.32926 54.57182 51.64136 52.67149 51.93805 53.04964 53.66232
## [2113] 53.57155 53.80456 53.14874 53.78734 53.42181 51.53580 52.04105 51.69767
## [2121] 51.20123 52.55566 50.82433 51.28422 53.15645 52.46699 52.70897 52.23519
## [2129] 53.41546 51.53436 52.09813 51.93040 50.75927 53.74264 53.63581 51.66620
## [2137] 53.08621 51.92784 52.57917 53.04751 53.21481 52.56566 52.53663 51.98772
## [2145] 52.79121 51.77930 51.49744 52.74470 52.61701 52.11432 51.81921 51.28922
## [2153] 52.36658 52.35175 51.97963 51.65717 52.39565 52.36625 50.60958 53.92175
## [2161] 53.95528 53.62677 54.91773 50.66485 54.44654 50.75380 53.70143 52.85619
## [2169] 52.17058 51.67668 53.33078 51.45822 52.10944 52.22137 52.21551 51.44482
## [2177] 52.16026 52.89794 51.78373 53.52734 50.54640 50.70432 52.06442 52.77606
## [2185] 52.12677 51.62452 51.78496 52.35610 52.65293 52.38617 52.25473 52.61234
## [2193] 53.62348 51.36996 51.61970 53.16970 52.36589 52.55110 52.98780 53.16499
## [2201] 51.81105 51.91497 51.85227 53.42024 50.65267 50.49370 52.29133 52.37896
## [2209] 50.82028 52.15353 53.64346 52.30345 51.97344 54.34641 50.89703 51.76517
## [2217] 51.76616 51.59323 50.09040 54.67910 54.38490 53.07243 53.80461 52.90935
## [2225] 52.70186 52.01280 52.67903 52.47070 52.38634 51.44742 53.07110 51.40042
## [2233] 51.64295 52.34046 53.01506 52.99984 52.17541 51.29644 53.17715 51.76772
## [2241] 52.48210 51.45346 53.61478 51.83420 52.38648 49.96902 49.81139 50.31517
## [2249] 54.56558 51.79606 49.18121 53.24055 51.65269 51.91889 51.89209 51.48784
## [2257] 53.48677 51.33940 53.04500 53.13000 52.99016 52.49800 52.37728 52.50379
## [2265] 51.90991 52.58938 51.86179 52.82640 52.14482 51.45805 53.47805 54.16309
## [2273] 54.64768 54.59145 54.57291 54.62966 51.37746 54.23002 53.41330 49.40565
## [2281] 50.58946 53.06697 51.36468 52.71417 51.91225 52.75028 52.62389 52.73499
## [2289] 51.37724 53.34951 53.66403 54.23337 53.89924 51.45214 52.70905 51.37874
## [2297] 53.07530 53.07530 52.70337 51.27667 51.75857 52.56158 51.63189 53.39390
## [2305] 50.83419 52.49825 51.74431 52.63136 51.78386 53.04767 52.73565 53.31587
## [2313] 52.00949 51.79211 52.75555 56.64204 54.80974 54.34104 53.61844 53.27883
## [2321] 53.24704 52.60579 51.97714 51.62433 52.50086 53.80705 53.59164 53.36349
## [2329] 50.54201 52.18274 51.05407 53.10831 53.07036 51.64103 52.75593 52.30061
## [2337] 53.07946 51.62625 52.22363 53.20633 51.63533 52.51140 51.39600 51.49761
## [2345] 50.89817 53.24448 52.09253 52.07376 50.60978 54.18339 51.07421 50.71871
## [2353] 52.60082 52.28919 51.47403 52.23421 50.63117 53.76991 53.09268 51.07967
## [2361] 52.64528 53.87092 54.13743 52.04891 51.32491 51.21906 52.07369 51.66054
## [2369] 52.58031 53.68388 53.26936 52.47889 52.55038 52.67617 51.54231 52.37647
## [2377] 52.22205 52.77814 51.84240 53.95376 52.39796 52.73777 52.53754 52.32049
## [2385] 52.73229 52.21718 52.25456 51.68992 54.42313 51.89242 50.82988 51.00528
## [2393] 50.33071 51.85590 51.25419 53.38544 53.26691 51.95597 53.03947 53.06746
## [2401] 53.41293 54.14785 54.24105 53.40654 51.79868 52.03931 54.02786 52.21212
## [2409] 53.58027 51.60914 51.47758 51.53714 53.34293 53.89015 52.52455 53.10188
## [2417] 51.17184 51.20537 51.46159 53.63667 49.95073 51.15336 53.01444 53.74764
## [2425] 53.96630 51.20916 53.90098 51.96383 54.98046 53.07725 51.22656 52.36103
## [2433] 52.64472 52.33695 52.17325 52.54516 52.33048 53.41150 52.19026 51.81031
## [2441] 53.56290 53.67474 52.07466 52.75234 52.16794 52.49150 51.56264 52.60539
## [2449] 52.63825 52.63410 52.21382 52.35588 51.66686 52.44681 52.06053 51.54399
## [2457] 53.96993 50.58001 50.08501 53.36188 54.66700 52.19981 53.44797 51.26029
## [2465] 52.35095 51.68983 52.58984 52.52493 52.67749 52.18158 52.88570 51.85470
## [2473] 51.81376 50.22876 50.19736 53.32928 52.76665 51.51043 52.31415 51.43993
## [2481] 53.87648 53.40395 52.58292 53.54298 51.19498 51.63469 53.07416 53.12818
## [2489] 51.64529 51.92567 52.43950 51.19425 52.58264 52.74064 52.06913 53.20194
## [2497] 52.47850 54.39691 51.34268 53.04072 52.25305 51.86462 53.11357 53.05467
## [2505] 51.79822 52.98411 53.69054 52.32163 52.71794 52.62524 52.45472 52.57121
## [2513] 52.38008 52.69353 52.00159 51.65993 52.83574 53.92239 50.51580 50.48732
## [2521] 51.86849 51.74667 52.91668 51.93251 51.62038 52.51300 52.35927 52.59581
## [2529] 51.96527 52.83598 52.37792 54.05099 55.08343 51.58830 52.01373 52.67754
## [2537] 52.98554 52.44503 52.58601 52.39250 52.20334 53.49118 53.98299 52.68448
## [2545] 51.47410 51.46375 53.03769 52.68798 52.43338 52.41259 53.31746 52.04072
## [2553] 52.02608 54.49652 50.92328 53.20618 53.42674 53.31062 53.20127 52.82019
## [2561] 52.51621 52.16868 52.11334 53.12605 51.81760 52.20262 53.10456 51.33691
## [2569] 52.88705 50.85553 49.57695 52.93044 53.28198 53.07741 52.09338 52.02436
## [2577] 51.78465 51.81894 52.45037 52.69768 52.96711 51.78326 53.19954 49.98855
## [2585] 49.32242 50.45992 53.70601 53.29808 52.79696 52.92049 51.60241 52.83265
## [2593] 52.89534 52.25794 52.13232 52.01953 52.92633 52.77594 51.86033 52.77940
## [2601] 51.30207 52.41395 52.05316 52.12023 51.67703 51.57541 51.16163 53.46538
## [2609] 51.14149 53.26118 52.44561 51.09834 52.23573 52.42977 51.46533 52.16496
## [2617] 52.10432 54.55807 54.01252 53.18824 53.12058 52.83444 51.70185 51.96725
## [2625] 51.97928 52.92367 51.73171 52.50735 50.97893 52.57409 52.53106 53.81855
## [2633] 54.26959 51.01918 51.70235 51.68187 51.79612 53.10142 53.44062 53.81748
## [2641] 52.80791 52.00195 52.55001 51.88100 52.58857 51.55406 53.49954 51.23982
## [2649] 53.80886 50.60862 54.08144 50.89642 53.73363 54.12784 53.77877 51.34244
## [2657] 51.94935 52.01009 52.27096 53.32160 52.55286 52.11717 52.31121 51.71085
## [2665] 51.58521 51.96261 52.53837 50.85826 50.52898 50.55272 50.35976 52.72603
## [2673] 53.12663 53.48841 51.39317 53.12726 54.10307 53.57190 52.60269 52.64771
## [2681] 52.45604 51.99608 52.13292 52.67395 53.20497 52.42212 52.60612 54.90748
## [2689] 52.50134 53.47346 54.01914 53.06863 52.82152 52.13197 51.24087 51.25631
## [2697] 52.52263 51.52183 54.44810 53.98800 53.03516 53.06637 53.03738 53.93368
## [2705] 51.87144 53.49111 50.86690 51.68657 50.62497 51.12863 53.20269 51.44758
## [2713] 50.68767 50.08576 54.55948 54.20570 52.54987 52.84018 52.75021 52.03263
## [2721] 52.59070 52.85253 52.51693 51.77712 51.85519 52.19798 52.39799 52.56475
## [2729] 54.36474 51.64642 52.36329 51.67536 51.51297 53.20106 54.65718 54.04687
## [2737] 51.41365 50.49215 52.31007 52.50283 52.70623 52.51419 53.16955 53.18661
## [2745] 53.81103 53.05763 52.67964 52.86569 52.68670 51.85545 53.12190 52.49105
## [2753] 51.74355 51.38816 54.40396 53.08715 52.19156 54.38847 50.88165 51.16246
## [2761] 53.49732 52.09512 52.28054 52.54787 53.98731 52.65740 52.00358 51.41597
## [2769] 51.89850 51.70253 52.45419 52.53812 52.15931 52.41684 51.48394 53.44415
## [2777] 51.14839 51.40059 51.32966 50.44603 52.52179 52.23501 52.63109 51.74617
## [2785] 52.57238 52.01781 52.18971 51.84846 51.87538 53.36202 52.69795 50.89504
## [2793] 51.09966 51.96595 52.02172 50.67569 50.58968 53.73878 51.98816 50.29924
## [2801] 51.49148 52.51961 53.14214 53.54410 52.74863 51.76143 52.45224 51.00405
## [2809] 52.66138 51.89351 50.10963 51.78525 53.43353 54.53257 54.54644 51.34469
## [2817] 53.15982 53.07499 51.14727 52.94853 52.47864 50.60527 55.03887 52.02219
## [2825] 53.97655 53.00031 52.35801 54.37961 54.41643 53.82009 53.71584 52.58412
## [2833] 51.65893 52.76229 51.47697 52.42644 52.28174 52.66543 52.25334 52.47686
## [2841] 52.73134 51.98733 51.02319 51.18340 53.01538 52.55691 52.17856 52.57288
## [2849] 52.15097 52.56428 53.08029 53.16887 53.61555 51.10243 52.11497 52.62691
## [2857] 51.70482 51.63059 52.83048 53.65945 53.58289 51.39547 51.61114 53.26331
## [2865] 52.06941 51.49302 52.05675 52.50986 52.50074 52.13062 51.14797 53.51775
## [2873] 53.70734 53.02558 50.87444 50.89299 52.23105 52.76660 52.14423 52.31622
## [2881] 52.76355 52.63235 52.00847 53.73628 51.52369 53.88348 53.89357 53.64801
## [2889] 52.62552 52.53964 52.01942 53.16675 53.34893 54.40985 54.43078 52.81318
## [2897] 52.57162 52.70145 52.08138 52.56610 52.29777 51.90235 51.65559 53.04637
## [2905] 51.20895 51.53339 51.21770 52.52347 52.30608 53.10559 51.96180 51.84840
## [2913] 51.41006 52.05956 52.93645 53.49328 52.83332 53.08620 52.79532 53.38766
## [2921] 51.09713 53.68156 52.07686 53.08360 52.38409 51.91157 52.21985 52.20797
## [2929] 53.30161 52.28909 55.03095 55.22216 49.91279 55.97236 50.71901 53.93547
## [2937] 53.81497 51.95259 53.47165 51.38927 52.11619 52.94000 51.99272 51.79834
## [2945] 53.01293 53.29556 49.42018 53.81064 53.48132 53.48952 52.22460 51.92590
## [2953] 52.19466 52.91240 52.22518 51.76154 53.31799 52.18523 51.46291 51.85456
## [2961] 52.61363 52.75159 50.43335 52.88908 52.85328 52.31558 52.01917 51.86418
## [2969] 53.16405 52.49688 52.35239 53.21605 51.56837 52.11971 52.87829 50.31799
## [2977] 53.04959 52.19430 52.28227 54.92783 51.52423 51.92232 52.41668 51.16728
## [2985] 52.19153 52.32702 51.61292 52.78842 51.88267 52.44530 52.56185 52.23631
## [2993] 51.71627 53.08997 54.13565 54.00574 51.65544 51.96856 51.52583 52.25975
## [3001] 52.61067 53.15067 51.92571 53.07326 52.15769 51.56567 52.03460 52.64892
## [3009] 51.47059 52.26261 51.60804 53.06112 51.95750 51.87974 53.37584 52.77525
## [3017] 52.62624 51.92519 52.37118 51.44903 51.19962 51.73253 53.26320 50.01226
## [3025] 51.75511 53.57798 52.13248 52.03074 52.93217 51.26712 52.73448 54.64563
## [3033] 51.62292 52.94084 51.62486 51.50366 51.96168 51.64632 53.01855 51.77867
## [3041] 52.92446 53.03359 53.75125 50.83459 53.65910 51.49720 54.09440 53.23926
## [3049] 53.45489 51.15654 51.21756 52.57705 51.95215 52.11918 52.29886 54.27042
## [3057] 52.54449 52.29751 52.61030 51.40155 53.85571 53.35806 53.93426 53.43794
## [3065] 52.99057 51.69380 51.54756 52.97643 52.74971 52.22919 53.13141 53.03075
## [3073] 52.91302 52.96236 53.33885 53.61579 52.83077 52.40641 52.07209 52.05561
## [3081] 51.99543 52.26309 53.29230 52.07774 50.98349 52.92681 50.96013 52.44488
## [3089] 52.23606 53.42383 53.33050 51.71851 52.06073 51.44758 53.12264 51.21030
## [3097] 50.48375 51.68733 51.53353 54.95334 54.82220 50.27175 50.67535 52.59252
## [3105] 51.79195 52.49462 52.03595 51.57758 52.97345 52.82054 53.19804 52.51285
## [3113] 52.23021 52.54780 52.40674 52.64185 54.45294 51.74144 51.91279 52.35458
## [3121] 51.68021 52.28611 51.82899 52.86593 53.70348 51.09843 52.20848 52.12569
## [3129] 52.24266 52.83541 51.99913 52.64272 51.29443 53.46131 51.50941 53.11585
## [3137] 50.55398 51.81821 52.63664 53.87628 52.76221 51.18681 52.02984 51.83936
## [3145] 52.90170 52.95452 52.00112 52.77511 51.71848 51.47928 52.19541 51.88825
## [3153] 52.65276 52.34762 53.09203 51.85372 54.28826 53.04523 52.01896 51.81136
## [3161] 51.11533 53.31926 53.10313 53.39913 51.57990 51.23669 54.47151 54.76783
## [3169] 52.83319 50.45331 51.04900 50.89806 51.01439 52.19720 53.55556 51.68171
## [3177] 52.17254 52.50153 51.34372 51.67264 53.39120 52.48568 52.19326 52.37753
## [3185] 51.59401 53.06837 52.77937 52.85183 52.63543 53.62324 52.94672 51.37279
## [3193] 52.20309 50.45584 54.60949 54.00385 54.55138 51.44084 52.53360 51.34316
## [3201] 52.38107 51.01664 51.28576 52.26950 50.76476 50.95027 54.29505 54.37518
## [3209] 53.51732 52.74917 52.44276 52.57209 52.02357 53.29549 52.47712 52.98208
## [3217] 52.47973 52.79195 52.47359 52.28219 52.14905 53.49149 52.28829 52.49240
## [3225] 52.89606 52.00423 53.22093 52.87419 53.44428 50.16614 52.07527 53.12244
## [3233] 53.15774 54.15456 54.62227 54.15175 50.97354 51.44094 52.47786 52.06906
## [3241] 53.31334 52.37789 51.81111 52.28736 52.68442 52.06800 53.35735 51.63183
## [3249] 53.31392 51.77295 51.56339 51.32149 51.38279 51.51643 52.05631 52.09830
## [3257] 53.23655 53.24348 51.58124 52.13388 52.16584 53.37745 52.24659 52.68954
## [3265] 50.67696 51.18440 53.20032 53.18142 50.45537 53.06160 53.73356 54.02046
## [3273] 52.12681 54.92915 54.46472 50.66488 53.77286 53.77885 52.73874 52.25005
## [3281] 52.20355 52.40273 53.10894 51.57006 52.91433 51.08815 53.30198 51.84368
## [3289] 53.21341 53.06576 52.82466 51.43746 51.80612 52.82436 53.52375 52.24927
## [3297] 54.93255 52.10849 52.82277 52.43310 53.92378 54.54061 54.81615 53.51327
## [3305] 51.95670 52.37361 52.54064 52.77732 51.79449 52.15283 52.23603 51.36831
## [3313] 53.32780 52.82584 52.27326 51.20304 53.49840 53.31766 52.90414 51.91142
## [3321] 51.92442 52.59374 52.05671 52.26918 52.50104 52.49912 52.87300 51.93293
## [3329] 52.87819 52.33585 52.43746 52.43430 52.62235 51.98346 51.71272 51.89083
## [3337] 53.18858 50.69060 50.52857 53.40309 53.55552 52.87311 54.68107 51.63851
## [3345] 53.45372 52.21417 52.66293 52.71309 52.05695 52.93282 52.13028 51.91396
## [3353] 50.89315 54.35665 54.29393 52.49887 52.81413 52.20644 54.35515 51.94453
## [3361] 52.26166 52.27000 52.55183 52.72860 51.58538 50.48242 51.15676 50.95769
## [3369] 50.62759 52.33276 50.60868 50.53595 52.26188 52.52047 52.63959 52.11869
## [3377] 51.37060 52.88047 50.64192 51.65133 51.42950 51.51171 51.72328 50.54645
## [3385] 54.14800 51.82642 51.97611 51.96483 51.85667 52.75649 52.26266 52.61682
## [3393] 51.98960 53.19206 52.50015 53.05163 52.12706 52.65029 51.83272 53.04626
## [3401] 52.12150 52.81244 52.56443 52.45186 53.23509 52.77943 51.05626 51.61889
## [3409] 52.00553 53.63039 51.51769 52.86272 52.85140 52.24424 52.52240 52.57236
## [3417] 52.72844 52.67353 53.08410 51.44843 53.13452 51.21432 51.76150 53.17471
## [3425] 52.58446 51.35292 51.27607 53.46232 52.05662 52.88803 52.54521 50.86415
## [3433] 51.45675 53.09295 52.39618 51.82666 51.86727 52.70642 53.16226 52.98153
## [3441] 51.34529 50.79990 53.51669 51.03196 51.36320 53.59538 52.17157 53.21947
## [3449] 52.48220 52.03198 53.06816 52.97945 52.68635 52.02436 52.99221 52.93789
## [3457] 51.63605 51.88225 52.35103 52.40855 51.41683 53.21700 52.96008 52.61973
## [3465] 51.99641 51.92933 51.56700 51.77826 49.50911 50.30041 50.38347 53.29508
## [3473] 53.22956 53.41759 52.81728 53.38211 51.59547 53.46798 52.67159 50.85109
## [3481] 52.27683 52.62532 52.47024 52.58464 54.34626 51.30133 51.92588 51.15641
## [3489] 52.28304 53.02223 50.87819 52.32443 52.36145 51.88026 52.02574 53.12327
## [3497] 52.84307 52.08520 52.46818 53.11423 52.70161 51.92780 52.71606 52.13838
## [3505] 53.24373 54.54268 53.41334 52.87973 54.03485 53.82168 51.14855 53.17277
## [3513] 53.25689 52.66458 51.10076 52.01373 52.72539 53.29360 51.45406 53.52804
## [3521] 51.56427 51.97182 53.14861 52.71683 52.83630 52.88083 52.15012 52.48664
## [3529] 50.69430 50.34671 50.96160 51.81726 51.19485 51.29485 51.04012 53.66602
## [3537] 51.39220 52.08215 52.10073 52.41868 52.16969 52.50543 52.01493 51.95433
## [3545] 50.79876 53.28688 53.50570 54.96161 50.10638 50.31200 53.29525 51.44094
## [3553] 52.89995 53.40785 52.84942 51.80559 53.84495 50.69822 51.30185 51.44040
## [3561] 52.00240 51.72457 52.65673 51.80965 52.63798 52.96021 51.96044 51.50527
## [3569] 52.46306 51.97589 53.88414 53.72526 52.24908 52.22641 51.21916 51.96760
## [3577] 51.11654 51.85604 53.20212 52.66250 53.72258 53.40402 52.18472 52.89514
## [3585] 52.80174 52.51587 52.45305 53.10415 52.53994 52.96110 52.07646 52.79412
## [3593] 52.14314 51.14212 51.25781 51.50160 51.71480 52.02043 52.94915 52.49200
## [3601] 52.90006 52.94847 51.37479 52.38931 53.51737 51.36160 53.69738 50.76202
## [3609] 52.21952 52.32289 52.02616 52.71011 52.87229 51.74690 54.48741 51.81415
## [3617] 51.33866 53.11057 52.50313 53.49093 52.61503 51.74299 52.72064 52.44360
## [3625] 53.91102 52.79976 53.20390 53.93240 52.65646 52.39714 52.49176 51.79352
## [3633] 52.08369 52.11212 52.78161 52.54523 54.06487 53.46477 52.48162 53.33816
## [3641] 53.08052 53.77932 55.06696 52.65938 51.93046 52.30463 53.43888 52.82730
## [3649] 52.25920 52.36637 51.85643 52.61849 51.87703 52.02660 53.47351 50.96025
## [3657] 51.66078 51.96793 52.76046 52.25326 52.93205 52.12366 51.84901 52.80206
## [3665] 52.30201 52.75768 51.45723 52.48620 52.49515 52.57375 51.69973 51.84706
## [3673] 51.55868 51.42989 51.90982 52.20506 53.36990 53.46195 51.65327 51.92813
## [3681] 52.43505 51.97182 51.85084 52.79064 52.16793 52.36433 51.57443 52.83381
## [3689] 52.43274 52.02471 51.76746 52.32784 52.55010 51.71134 52.49154 52.54458
## [3697] 52.83884 53.63326 51.37370 52.36091 51.93844 52.62401 52.26842 51.77105
## [3705] 52.31454 52.50516 53.13574 51.36548 51.59242 54.27071 51.35525 53.89268
## [3713] 48.84423 51.16296 52.71390 52.29235 53.25426 52.95119 52.88826 52.46717
## [3721] 52.46340 51.22154 52.22738 51.47375 52.57209 52.20057 52.54964 51.44547
## [3729] 51.93900 51.86925 49.45700 49.99622 49.85707 52.12108 52.83180 51.41828
## [3737] 52.65940 52.62965 52.68869 51.75993 53.27191 53.54464 55.90753 55.75723
## [3745] 54.99226 54.97493 54.56772 54.71758 54.51432 52.39452 52.23218 52.35725
## [3753] 52.54853 53.33668 52.95544 50.47197 54.27855 50.51254 51.88113 51.82160
## [3761] 51.94196 52.21126 53.49813 51.50512 52.07854 53.86560 53.65467 53.02473
## [3769] 52.75839 52.88680 52.20539 53.18648 52.68104 52.04005 52.39881 52.68264
## [3777] 52.62078 53.22617 53.42627 51.27516 53.99502 50.74799 53.37938 53.58563
## [3785] 51.05333 53.40475 52.66714 51.86839 51.66151 51.57005 53.29841 51.60273
## [3793] 52.83555 51.69257 52.07766 51.21841 51.66793 51.44348 52.56870 51.66727
## [3801] 51.65505 52.21793 52.51668 52.43013 51.92522 53.16930 51.44336 51.88152
## [3809] 51.86550 52.02850 51.42333 51.75278 51.96925 53.06410 52.41905 52.50538
## [3817] 53.49614 51.45583 52.53628 52.14117 53.22063 52.51317 52.67839 50.93803
## [3825] 50.57411 53.20153 52.50993 53.51443 51.98625 52.95788 52.50710 51.34958
## [3833] 53.23948 51.21448 51.53212 51.49381 53.27355 52.92585 52.65439 50.83692
## [3841] 51.35791 52.70401 51.40107 53.26791 52.48525 52.06205 51.69932 53.13377
## [3849] 51.94070 52.11086 51.77590 51.23356 51.31389 52.94328 49.84882 53.49575
## [3857] 51.14937 54.35516 54.09892 52.47105 52.90663 53.02941 52.82974 52.85610
## [3865] 52.56015 52.68434 52.36232 52.18701 51.94071 51.63001 53.83146 50.20037
## [3873] 54.29828 52.20601 51.61299 53.77476 52.27936 54.58837 54.70501 52.27975
## [3881] 51.98762 50.63754 53.20989 53.97433 52.49458 52.61676 52.48194 53.39295
## [3889] 52.06787 52.29481 52.53268 51.90285 53.53895 52.88744 52.54292 52.09690
## [3897] 51.62917 51.31048 51.73731 52.29247 52.65906 52.36907 52.96351 51.81076
## [3905] 52.94990 51.64870 52.98614 52.02119 52.35077 51.21220 51.78476 51.08883
## [3913] 52.54869 52.44782 51.88221 52.02617 52.64491 52.26423 52.71199 51.98716
## [3921] 52.12526 51.68626 50.77180 52.18623 50.48421 50.69143 51.90569 52.54721
## [3929] 51.44153 52.62363 52.02598 52.27694 51.67729 52.83620 52.27166 53.80151
## [3937] 52.55600 52.39422 52.48726 52.42831 52.93481 51.13587 53.39810 49.89263
## [3945] 49.73819 54.29075 51.64182 53.01864 51.66297 51.61986 51.58735 53.14199
## [3953] 51.89710 53.82722 51.65209 51.38849 49.66788 51.80471 53.43819 51.16972
## [3961] 52.35120 51.65501 51.16199 52.70489 51.95024 53.01212 52.87585 53.99199
## [3969] 51.70175 52.70976 53.50272 50.89784 51.38565 51.07108 51.07957 53.58180
## [3977] 51.34324 50.98654 51.55426 53.00574 51.86014 50.89987 51.31304 52.34095
## [3985] 52.58707 52.99104 53.05569 53.68138 52.87023 51.71759 52.17011 51.84673
## [3993] 51.90960 52.76526 54.62258 53.45097 53.23753 52.98033 51.76064 52.65380# basically linear prediction produces fitted values for each simulation result
cor(y_pred1,y_pred2)## [,1]
## 1 1Predictive distribution for a new observation
Here, we are now adding in uncertainty in a single election by adding a value of mean=0 and sd=sigma.
(posterior_pred1=posterior_predict(m2.br, newdata=new))## 1
## [1,] 50.96065
## [2,] 50.35337
## [3,] 58.87595
## [4,] 49.38105
## [5,] 54.83616
## [6,] 49.65421
## [7,] 44.28994
## [8,] 51.45624
## [9,] 47.39575
## [10,] 47.16994
## [11,] 54.26816
## [12,] 52.81943
## [13,] 52.29613
## [14,] 51.70085
## [15,] 41.46142
## [16,] 49.11509
## [17,] 54.01553
## [18,] 48.57151
## [19,] 58.48928
## [20,] 53.35477
## [21,] 50.58437
## [22,] 54.10488
## [23,] 55.92965
## [24,] 51.32881
## [25,] 43.55036
## [26,] 43.29596
## [27,] 45.63803
## [28,] 51.59491
## [29,] 53.29279
## [30,] 59.55529
## [31,] 47.56998
## [32,] 49.50482
## [33,] 48.78346
## [34,] 58.79080
## [35,] 47.29877
## [36,] 54.16449
## [37,] 56.46847
## [38,] 57.34339
## [39,] 43.27467
## [40,] 52.87921
## [41,] 48.25036
## [42,] 61.70903
## [43,] 49.92521
## [44,] 54.60785
## [45,] 42.86291
## [46,] 50.25535
## [47,] 60.12165
## [48,] 48.54691
## [49,] 51.12708
## [50,] 51.80988
## [51,] 54.71113
## [52,] 52.59278
## [53,] 48.52961
## [54,] 53.14651
## [55,] 50.44164
## [56,] 53.86552
## [57,] 49.25912
## [58,] 50.52377
## [59,] 55.79780
## [60,] 56.89631
## [61,] 47.65458
## [62,] 50.56659
## [63,] 56.50811
## [64,] 55.14322
## [65,] 43.56308
## [66,] 49.84377
## [67,] 50.37073
## [68,] 50.78481
## [69,] 47.66227
## [70,] 49.61500
## [71,] 54.90045
## [72,] 57.18562
## [73,] 53.41090
## [74,] 55.63837
## [75,] 46.97456
## [76,] 48.38342
## [77,] 51.60917
## [78,] 47.09057
## [79,] 53.24919
## [80,] 55.76064
## [81,] 51.77567
## [82,] 55.04211
## [83,] 55.67823
## [84,] 50.17935
## [85,] 53.88636
## [86,] 50.29675
## [87,] 47.27953
## [88,] 53.34483
## [89,] 58.24613
## [90,] 61.75237
## [91,] 49.33683
## [92,] 43.68146
## [93,] 54.33308
## [94,] 50.26591
## [95,] 43.30919
## [96,] 55.37234
## [97,] 55.58156
## [98,] 52.92824
## [99,] 56.77914
## [100,] 48.99330
## [101,] 51.33949
## [102,] 54.06399
## [103,] 55.61457
## [104,] 53.76620
## [105,] 51.98819
## [106,] 55.44116
## [107,] 53.66358
## [108,] 50.40191
## [109,] 54.94827
## [110,] 60.37403
## [111,] 54.31226
## [112,] 56.61589
## [113,] 55.46577
## [114,] 53.26087
## [115,] 48.21029
## [116,] 52.55784
## [117,] 49.08497
## [118,] 42.73817
## [119,] 50.09969
## [120,] 58.34233
## [121,] 51.99545
## [122,] 47.58682
## [123,] 42.66087
## [124,] 50.89741
## [125,] 47.05693
## [126,] 53.06742
## [127,] 57.15152
## [128,] 53.52583
## [129,] 51.98398
## [130,] 48.52282
## [131,] 56.49247
## [132,] 52.08973
## [133,] 49.78976
## [134,] 50.05054
## [135,] 44.34830
## [136,] 56.31718
## [137,] 51.97801
## [138,] 54.61116
## [139,] 55.57225
## [140,] 50.11569
## [141,] 53.17528
## [142,] 59.17657
## [143,] 46.02471
## [144,] 50.59563
## [145,] 58.09447
## [146,] 56.60453
## [147,] 48.43094
## [148,] 50.34960
## [149,] 54.83424
## [150,] 42.60526
## [151,] 48.83674
## [152,] 52.64811
## [153,] 51.97591
## [154,] 50.97054
## [155,] 49.76525
## [156,] 48.57458
## [157,] 56.73926
## [158,] 54.81150
## [159,] 62.53182
## [160,] 55.55062
## [161,] 61.21726
## [162,] 48.04074
## [163,] 49.21597
## [164,] 51.67825
## [165,] 54.38659
## [166,] 51.58071
## [167,] 52.09576
## [168,] 51.01969
## [169,] 47.97253
## [170,] 51.69902
## [171,] 58.36310
## [172,] 49.43281
## [173,] 53.87797
## [174,] 51.39904
## [175,] 41.46807
## [176,] 48.06152
## [177,] 57.72260
## [178,] 56.40064
## [179,] 53.82199
## [180,] 45.93076
## [181,] 66.09676
## [182,] 53.69219
## [183,] 54.37718
## [184,] 49.11171
## [185,] 52.07405
## [186,] 50.61569
## [187,] 57.88667
## [188,] 48.18094
## [189,] 62.82511
## [190,] 49.32220
## [191,] 52.66252
## [192,] 51.64733
## [193,] 51.07891
## [194,] 57.10231
## [195,] 47.50583
## [196,] 51.66991
## [197,] 54.48535
## [198,] 56.94167
## [199,] 45.87206
## [200,] 55.16743
## [201,] 56.53069
## [202,] 46.09942
## [203,] 52.73118
## [204,] 53.40104
## [205,] 53.88230
## [206,] 54.95105
## [207,] 46.46414
## [208,] 54.93148
## [209,] 56.02490
## [210,] 54.10006
## [211,] 55.71723
## [212,] 50.08613
## [213,] 55.56858
## [214,] 49.00822
## [215,] 52.01486
## [216,] 49.23988
## [217,] 51.37253
## [218,] 47.04216
## [219,] 58.97109
## [220,] 54.31614
## [221,] 53.78821
## [222,] 51.32095
## [223,] 50.79429
## [224,] 49.31258
## [225,] 47.87441
## [226,] 58.76128
## [227,] 52.68285
## [228,] 49.61896
## [229,] 52.22855
## [230,] 55.49713
## [231,] 49.07765
## [232,] 52.19547
## [233,] 47.63510
## [234,] 47.76498
## [235,] 57.54605
## [236,] 56.03648
## [237,] 46.81346
## [238,] 59.08784
## [239,] 47.50140
## [240,] 53.70059
## [241,] 56.20446
## [242,] 64.18380
## [243,] 54.61380
## [244,] 44.77775
## [245,] 53.60408
## [246,] 53.89280
## [247,] 50.50286
## [248,] 49.12505
## [249,] 58.17159
## [250,] 51.17103
## [251,] 54.92036
## [252,] 60.61110
## [253,] 48.75211
## [254,] 56.47748
## [255,] 50.36488
## [256,] 50.16348
## [257,] 45.21470
## [258,] 57.63888
## [259,] 47.65992
## [260,] 47.20805
## [261,] 53.12654
## [262,] 43.39542
## [263,] 44.58958
## [264,] 57.45664
## [265,] 53.05129
## [266,] 58.96903
## [267,] 49.51244
## [268,] 50.25836
## [269,] 56.22881
## [270,] 51.34562
## [271,] 55.56076
## [272,] 55.63321
## [273,] 48.82265
## [274,] 53.51002
## [275,] 48.50124
## [276,] 54.67274
## [277,] 49.88973
## [278,] 45.21720
## [279,] 53.06154
## [280,] 50.38196
## [281,] 55.81168
## [282,] 57.93463
## [283,] 59.22644
## [284,] 48.71894
## [285,] 49.65197
## [286,] 48.03493
## [287,] 51.66794
## [288,] 46.56783
## [289,] 48.52260
## [290,] 56.91339
## [291,] 54.32204
## [292,] 51.01732
## [293,] 50.13080
## [294,] 56.17243
## [295,] 49.73029
## [296,] 52.81920
## [297,] 43.96457
## [298,] 55.58015
## [299,] 61.44450
## [300,] 46.32209
## [301,] 52.24680
## [302,] 58.61048
## [303,] 48.66584
## [304,] 52.29731
## [305,] 57.49810
## [306,] 47.86072
## [307,] 55.31084
## [308,] 55.56116
## [309,] 55.63237
## [310,] 56.01536
## [311,] 51.06723
## [312,] 49.66112
## [313,] 51.63325
## [314,] 50.95710
## [315,] 48.50893
## [316,] 51.11819
## [317,] 47.81789
## [318,] 50.03389
## [319,] 49.37875
## [320,] 55.75291
## [321,] 57.65965
## [322,] 50.13615
## [323,] 48.33709
## [324,] 59.15521
## [325,] 54.96073
## [326,] 52.23386
## [327,] 63.87248
## [328,] 58.34469
## [329,] 50.04731
## [330,] 59.86884
## [331,] 54.67431
## [332,] 52.03258
## [333,] 50.43072
## [334,] 56.38959
## [335,] 54.55602
## [336,] 57.69182
## [337,] 49.63690
## [338,] 54.64637
## [339,] 52.30781
## [340,] 54.53363
## [341,] 56.61726
## [342,] 46.63725
## [343,] 62.70088
## [344,] 50.74298
## [345,] 51.02369
## [346,] 54.24946
## [347,] 54.09478
## [348,] 51.20193
## [349,] 57.64809
## [350,] 56.74557
## [351,] 59.47145
## [352,] 52.30890
## [353,] 51.34239
## [354,] 56.26221
## [355,] 54.47065
## [356,] 53.60169
## [357,] 55.95954
## [358,] 52.98171
## [359,] 53.64169
## [360,] 54.76781
## [361,] 52.23780
## [362,] 50.67034
## [363,] 57.90213
## [364,] 62.30588
## [365,] 52.03353
## [366,] 55.93005
## [367,] 53.16605
## [368,] 55.94195
## [369,] 54.57294
## [370,] 46.32880
## [371,] 50.54683
## [372,] 51.45162
## [373,] 49.77748
## [374,] 50.94102
## [375,] 55.04848
## [376,] 50.74753
## [377,] 54.96080
## [378,] 53.67903
## [379,] 66.56857
## [380,] 52.53360
## [381,] 60.11771
## [382,] 54.51628
## [383,] 50.21924
## [384,] 51.22749
## [385,] 50.35927
## [386,] 57.68184
## [387,] 57.30127
## [388,] 57.78499
## [389,] 46.26869
## [390,] 49.75492
## [391,] 60.95950
## [392,] 47.65178
## [393,] 50.16032
## [394,] 52.15218
## [395,] 59.54532
## [396,] 55.05433
## [397,] 54.52746
## [398,] 59.78503
## [399,] 53.59976
## [400,] 51.80510
## [401,] 53.15822
## [402,] 49.50605
## [403,] 57.79739
## [404,] 57.94724
## [405,] 52.52415
## [406,] 51.86551
## [407,] 52.34237
## [408,] 59.50418
## [409,] 52.34510
## [410,] 55.15986
## [411,] 58.38383
## [412,] 63.07253
## [413,] 59.68485
## [414,] 53.57808
## [415,] 54.95150
## [416,] 53.93880
## [417,] 48.66751
## [418,] 45.20211
## [419,] 36.32946
## [420,] 51.95536
## [421,] 51.98405
## [422,] 45.18702
## [423,] 55.88845
## [424,] 54.25935
## [425,] 50.26156
## [426,] 52.80042
## [427,] 54.12339
## [428,] 51.41019
## [429,] 55.13139
## [430,] 53.02085
## [431,] 49.91599
## [432,] 46.09037
## [433,] 52.33715
## [434,] 51.41972
## [435,] 54.83036
## [436,] 37.88244
## [437,] 51.58163
## [438,] 53.82110
## [439,] 46.41427
## [440,] 51.24424
## [441,] 50.30891
## [442,] 53.25014
## [443,] 55.21476
## [444,] 50.95142
## [445,] 47.77683
## [446,] 48.62935
## [447,] 50.58475
## [448,] 52.41849
## [449,] 51.57105
## [450,] 43.78641
## [451,] 54.44826
## [452,] 57.11611
## [453,] 54.41064
## [454,] 46.97574
## [455,] 56.33086
## [456,] 53.08923
## [457,] 56.56247
## [458,] 56.45169
## [459,] 49.35834
## [460,] 50.34698
## [461,] 56.60936
## [462,] 56.04435
## [463,] 49.95696
## [464,] 55.99382
## [465,] 53.16777
## [466,] 53.62193
## [467,] 66.83160
## [468,] 49.86346
## [469,] 57.14362
## [470,] 58.33859
## [471,] 46.44207
## [472,] 53.55376
## [473,] 50.08183
## [474,] 52.67471
## [475,] 46.44371
## [476,] 52.24518
## [477,] 54.68744
## [478,] 48.52167
## [479,] 53.63840
## [480,] 56.92443
## [481,] 49.26645
## [482,] 56.14852
## [483,] 46.61508
## [484,] 51.21166
## [485,] 52.50263
## [486,] 49.82538
## [487,] 57.89223
## [488,] 54.49954
## [489,] 52.27107
## [490,] 48.66449
## [491,] 55.72164
## [492,] 49.02485
## [493,] 65.06329
## [494,] 47.11733
## [495,] 57.31324
## [496,] 50.02593
## [497,] 58.05800
## [498,] 53.55050
## [499,] 50.43871
## [500,] 54.89951
## [501,] 51.80924
## [502,] 52.43650
## [503,] 50.97346
## [504,] 51.27205
## [505,] 52.75873
## [506,] 55.13854
## [507,] 50.65282
## [508,] 53.62371
## [509,] 48.89956
## [510,] 53.11176
## [511,] 50.79720
## [512,] 46.86948
## [513,] 48.43426
## [514,] 60.29431
## [515,] 49.71835
## [516,] 55.83666
## [517,] 54.66818
## [518,] 51.03210
## [519,] 54.42275
## [520,] 54.42427
## [521,] 53.87262
## [522,] 52.00974
## [523,] 54.37377
## [524,] 52.39366
## [525,] 50.29073
## [526,] 55.80625
## [527,] 46.72580
## [528,] 49.88455
## [529,] 45.52076
## [530,] 53.97507
## [531,] 54.38180
## [532,] 53.61877
## [533,] 59.24855
## [534,] 47.14376
## [535,] 51.27488
## [536,] 53.02905
## [537,] 51.63463
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## [3955,] 50.62204
## [3956,] 47.43792
## [3957,] 53.84999
## [3958,] 46.19494
## [3959,] 53.81066
## [3960,] 55.84656
## [3961,] 50.25036
## [3962,] 60.04034
## [3963,] 58.76388
## [3964,] 45.49625
## [3965,] 50.59964
## [3966,] 49.37135
## [3967,] 55.07514
## [3968,] 54.67887
## [3969,] 44.05603
## [3970,] 52.52426
## [3971,] 52.04038
## [3972,] 55.32122
## [3973,] 48.11344
## [3974,] 51.66774
## [3975,] 52.48582
## [3976,] 52.79768
## [3977,] 52.46698
## [3978,] 50.67611
## [3979,] 50.36107
## [3980,] 49.63353
## [3981,] 42.60380
## [3982,] 55.97620
## [3983,] 51.18735
## [3984,] 53.84722
## [3985,] 62.02335
## [3986,] 55.12555
## [3987,] 55.10515
## [3988,] 47.42927
## [3989,] 54.47079
## [3990,] 47.43001
## [3991,] 52.37343
## [3992,] 53.76628
## [3993,] 57.44717
## [3994,] 54.57658
## [3995,] 53.47336
## [3996,] 57.92703
## [3997,] 60.37472
## [3998,] 55.52771
## [3999,] 54.34897
## [4000,] 48.85390
## attr(,"class")
## [1] "ppd" "matrix" "array"# manual
n_sims=nrow(sims)
n_sims## [1] 4000sigma=sims[,3]
y_pred3=(a+b*2)+rnorm(n_sims,0,sigma)
hist(y_pred3)
cor(y_pred1,y_pred3)## [,1]
## 1 0.234395cor(y_pred3,posterior_pred1)## 1
## [1,] 0.04795836Including priors
- Uniform prior distribution is noninformative.
- Default informative priors.
- Weakly informative priors
# uniform
priors.none = stan_glm(vote ~ growth, data=hibbs, refresh=0,
prior_intercept = NULL, prior = NULL, prior_aux = NULL)
print(priors.none)## stan_glm
## family: gaussian [identity]
## formula: vote ~ growth
## observations: 16
## predictors: 2
## ------
## Median MAD_SD
## (Intercept) 46.2 1.7
## growth 3.1 0.7
##
## Auxiliary parameter(s):
## Median MAD_SD
## sigma 4.0 0.8
##
## ------
## * For help interpreting the printed output see ?print.stanreg
## * For info on the priors used see ?prior_summary.stanreg# default prior distribution
priors.default = stan_glm(vote ~ growth, data=hibbs, refresh=0)
print(priors.default)## stan_glm
## family: gaussian [identity]
## formula: vote ~ growth
## observations: 16
## predictors: 2
## ------
## Median MAD_SD
## (Intercept) 46.2 1.6
## growth 3.0 0.7
##
## Auxiliary parameter(s):
## Median MAD_SD
## sigma 3.9 0.8
##
## ------
## * For help interpreting the printed output see ?print.stanreg
## * For info on the priors used see ?prior_summary.stanregprior_summary(priors.default)## Priors for model 'priors.default'
## ------
## Intercept (after predictors centered)
## Specified prior:
## ~ normal(location = 52, scale = 2.5)
## Adjusted prior:
## ~ normal(location = 52, scale = 14)
##
## Coefficients
## Specified prior:
## ~ normal(location = 0, scale = 2.5)
## Adjusted prior:
## ~ normal(location = 0, scale = 10)
##
## Auxiliary (sigma)
## Specified prior:
## ~ exponential(rate = 1)
## Adjusted prior:
## ~ exponential(rate = 0.18)
## ------
## See help('prior_summary.stanreg') for more details# manual (shows slight differences)
sd_x=sd(hibbs$growth)
sd_y=sd(hibbs$vote)
mean_y=mean(hibbs$vote)
mean_y## [1] 52.055priors.default.manual = stan_glm(vote ~ growth, data=hibbs, refresh=0,
prior_intercept=normal(mean_y,2.5*sd_y), prior=normal(0,2.5*sd_y/sd_x),
prior_aux = exponential(1/sd_y))
print(priors.default.manual)## stan_glm
## family: gaussian [identity]
## formula: vote ~ growth
## observations: 16
## predictors: 2
## ------
## Median MAD_SD
## (Intercept) 46.3 1.6
## growth 3.0 0.7
##
## Auxiliary parameter(s):
## Median MAD_SD
## sigma 3.9 0.7
##
## ------
## * For help interpreting the printed output see ?print.stanreg
## * For info on the priors used see ?prior_summary.stanreg# weakly informative
# let's say incumbents have strong advantage and growth rates matter a lot
priors.inform = stan_glm(vote ~ growth, data=hibbs, refresh=0,
prior_intercept=normal(45,2.5*sd_y), prior=normal(6,2.5*sd_y/sd_x),
prior_aux = exponential(1/sd_y))
print(priors.inform)## stan_glm
## family: gaussian [identity]
## formula: vote ~ growth
## observations: 16
## predictors: 2
## ------
## Median MAD_SD
## (Intercept) 46.2 1.7
## growth 3.1 0.7
##
## Auxiliary parameter(s):
## Median MAD_SD
## sigma 3.9 0.8
##
## ------
## * For help interpreting the printed output see ?print.stanreg
## * For info on the priors used see ?prior_summary.stanreg