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BayesianStatistics_OGimenez.R
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BayesianStatistics_OGimenez.R
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## ----setup, include = FALSE----------------------------------------------------------------------------------------
knitr::opts_chunk$set(cache = TRUE,
echo = TRUE,
message = FALSE,
warning = FALSE,
fig.height=6,
fig.width = 1.777777*6,
tidy = FALSE,
comment = NA,
highlight = TRUE,
prompt = FALSE,
crop = TRUE,
comment = "#>",
collapse = TRUE)
knitr::opts_knit$set(width = 60)
library(tidyverse)
library(reshape2)
theme_set(theme_light(base_size = 16))
make_latex_decorator <- function(output, otherwise) {
function() {
if (knitr:::is_latex_output()) output else otherwise
}
}
insert_pause <- make_latex_decorator(". . .", "\n")
insert_slide_break <- make_latex_decorator("----", "\n")
insert_inc_bullet <- make_latex_decorator("> *", "*")
insert_html_math <- make_latex_decorator("", "$$")
## ---- out.width = '13cm',out.height='7cm',fig.align='center',echo=FALSE--------------------------------------------
knitr::include_graphics('img/brace_yourself.jpeg')
## ---- out.width = '13cm',out.height='7cm',fig.align='center',echo=FALSE--------------------------------------------
knitr::include_graphics('img/books.jpeg')
## ---- fig.align = 'center', echo = FALSE---------------------------------------------------------------------------
knitr::include_graphics('img/mccarthy.png')
## ---- fig.align = 'center', echo = FALSE---------------------------------------------------------------------------
knitr::include_graphics('img/kery.png')
## ---- fig.align = 'center', echo = FALSE---------------------------------------------------------------------------
knitr::include_graphics('img/kruschke.png')
## ---- fig.align = 'center', echo = FALSE---------------------------------------------------------------------------
knitr::include_graphics('img/mcelreath.png')
## ---- fig.align = 'center', echo = FALSE---------------------------------------------------------------------------
knitr::include_graphics('img/gelmanhill.png')
## ---- echo=FALSE, fig.align='center'-------------------------------------------------------------------------------
knitr::include_graphics('img/frequentists_vs_bayesians_2x.png')
## ----collapse=TRUE-------------------------------------------------------------------------------------------------
dbinom(x=3,size=10,prob=0.1)
## ----collapse=TRUE-------------------------------------------------------------------------------------------------
dbinom(x=3,size=10,prob=0.9)
## ----collapse=TRUE-------------------------------------------------------------------------------------------------
dbinom(x=3,size=10,prob=0.25)
## ----collapse=TRUE-------------------------------------------------------------------------------------------------
dbinom(x=3,size=10,prob=0.1)
dbinom(x=3,size=10,prob=0.9)
dbinom(x=3,size=10,prob=0.25)
## ----collapse=TRUE-------------------------------------------------------------------------------------------------
lik.fun <- function(parameter){
ll <- dbinom(x=3, size=10, prob=parameter)
return(ll)
}
lik.fun(0.3)
lik.fun(0.6)
## ---- echo=FALSE---------------------------------------------------------------------------------------------------
lik.fun <- function(parameter){
ll <- dbinom(x=3, size=10, prob=parameter)
return(ll)
}
p.grid = seq(0,1,by=0.01)
lik = rep(NA,length(p.grid))
for (i in 1:length(p.grid)){
lik[i] <- lik.fun(p.grid[i])
}
plot(p.grid,lik,xlab='Probability of getting a successful breeding',ylab='Likelihood',type='l',lwd=3,cex.lab=1.5)
abline(v=0.3,lty=2,lwd=2,col='blue')
## ----collapse=TRUE-------------------------------------------------------------------------------------------------
lik.fun <- function(parameter) dbinom(x=3, size=10, prob=parameter)
# ?optimize
optimize(lik.fun,c(0,1),maximum=TRUE)
## ---- echo=FALSE---------------------------------------------------------------------------------------------------
lik.fun <- function(parameter) dbinom(x=3, size=10, prob=parameter)
plot(lik.fun,0,1,xlab="probability of success (p)",ylab="log-likelihood(p)",main="Binomial likelihood with 3 successes ot of 10 attempts",lwd=3,cex.lab=1.5, cex.axis=1.5, cex.main=1.5, cex.sub=1.5)
abline(v=0.3,h=0.26682,col='blue',lty=2,lwd=2)
## ------------------------------------------------------------------------------------------------------------------
# set seed for random numbers
set.seed(2020)
# simulate data from Normal distribution
n <- 100
height <- rnorm(n, mean=170, sd=10)
## ----echo=FALSE----------------------------------------------------------------------------------------------------
data.frame(height=height) %>%
ggplot(aes(x=height))+
geom_histogram(color="blue",fill="dodgerblue") +
labs(x = "Height", y = 'Density')
## ------------------------------------------------------------------------------------------------------------------
negloglik <- function(theta, data) {
mu <- theta[1]
sigma <- theta[2]
x <- data
-sum(dnorm(x, mean = mu, sd = sigma, log = TRUE))
}
negloglik(theta = c(150,1), height)
## ------------------------------------------------------------------------------------------------------------------
fit <- optim(par = c(1,1), fn = negloglik, data = height)
fit
## ---- echo = FALSE-------------------------------------------------------------------------------------------------
binwidth <- 1
df <- data.frame(x = height) %>%
ggplot(aes(x = height, mean = fit$par[1], sd = fit$par[2], binwidth = binwidth, n = n)) +
geom_histogram(binwidth = binwidth,
colour = "white",
fill = "cornflowerblue",
size = 0.1) +
stat_function(fun = function(x) dnorm(x, mean = fit$par[1], sd = fit$par[2]) * n * binwidth,
color = "darkred", size = 1)
df
## ---- echo=FALSE---------------------------------------------------------------------------------------------------
x <- seq(0, 1, length=200)
par(mfrow = c(2,3))
# distribution a posteriori beta
plot(x,dbeta(x, 1, 1),type='l',xlab='q',ylab='Density',main='beta(1,1)',lwd=3,col='red',ylim=c(0,1.5))
plot(x,dbeta(x, 2, 1),type='l',xlab='q',ylab='',main='beta(2,1)',lwd=3,col='red',ylim=c(0,2))
plot(x,dbeta(x, 1, 2),type='l',xlab='q',ylab='',main='beta(1,2)',lwd=3,col='red',ylim=c(0,2))
plot(x,dbeta(x, 2, 2),type='l',xlab='q',ylab='Density',main='beta(2,2)',lwd=3,col='red',ylim=c(0,1.5))
plot(x,dbeta(x, 10, 10),type='l',xlab='q',ylab='',main='beta(10,10)',lwd=3,col='red',ylim=c(0,3.5))
plot(x,dbeta(x, 0.8, 0.8),type='l',xlab='q',ylab='',main='beta(0.8,0.8)',lwd=3,col='red',ylim=c(0.5,2.5))
## ----echo=FALSE----------------------------------------------------------------------------------------------------
x <- seq(0, 1, length=200)
# distribution a posteriori beta
plot(x,dbeta(x, 20,39),type='l',xlab='',ylab='',main='',lwd=3,col='red')
# distribution a priori uniforme
points(x,dbeta(x, 1, 1),type='l',lwd=3)
## ----echo=FALSE----------------------------------------------------------------------------------------------------
x <- seq(0, 1, length=200)
# distribution a posteriori beta
plot(x,dbeta(x, 20,39),type='l',xlab='',ylab='',main='',lwd=3,col='red')
# distribution a priori uniforme
points(x,dbeta(x, 1, 1),type='l',lwd=3)
abline(v = 19/57, lwd = 3, lty = 2, col = 'blue')
text(x = 0.28, y = 0, 'MLE', col = 'blue')
## ----echo=FALSE----------------------------------------------------------------------------------------------------
x <- seq(0, 1, length=200)
# distribution a posteriori beta
plot(x,dbeta(x, .5+19,.5+57-19),type='l',xlab='',ylab='',main='',lwd=3,col='red')
# distribution a priori uniforme
points(x,dbeta(x, .5, .5),type='l',lwd=3)
## ----echo=FALSE----------------------------------------------------------------------------------------------------
x <- seq(0, 1, length=200)
# distribution a posteriori beta
plot(x,dbeta(x, 2+19,2+57-19),type='l',xlab='',ylab='',main='',lwd=3,col='red')
# distribution a priori uniforme
points(x,dbeta(x, 2, 2),type='l',lwd=3)
## ----echo=FALSE----------------------------------------------------------------------------------------------------
x <- seq(0, 1, length=200)
# distribution a posteriori beta
plot(x,dbeta(x, 20+19,1+57-19),type='l',xlab='',ylab='',main='',lwd=3,col='red')
# distribution a priori uniforme
points(x,dbeta(x, 20, 1),type='l',lwd=3)
## ---- out.width = '11cm',out.height='7cm',fig.align='center',echo=FALSE--------------------------------------------
knitr::include_graphics('img/falling_man.jpg')
## ---- out.width = '10cm',out.height='6cm',fig.align='center',echo=FALSE--------------------------------------------
knitr::include_graphics('img/dipper.png')
## ----include=FALSE-------------------------------------------------------------------------------------------------
# read in data
data <- as.matrix(read.table("dat/dipper.dat"))
# number of individuals
n <- dim(data)[[1]]
# number of capture occasions
K <- dim(data)[[2]]
# compute the date of first capture
e <- NULL
for (i in 1:n){
temp <- 1:K
e <- c(e,min(temp[data[i,]==1]))
}
# data
datax <- list(N=n,Years=K,obs=data,First=e)
# mark-recapture analysis for European Dippers
model <-
paste("
model
{
for (i in 1:N){
alive[i,First[i]] <- 1
for (j in (First[i]+1):Years){
alive[i,j] ~ dbern(alivep[i,j])
alivep[i,j] <- surv * alive[i,j-1]
obs[i,j] ~ dbern(sightp[i,j])
sightp[i,j] <- resight * alive[i,j]
}
}
surv~dunif(0,1)
resight~dunif(0,1)
}
")
writeLines(model,"code/CJS.txt")
# In JAGS we have to give good initial values for the latent state alive. At all occasions when an individual was observed, its state is alive = 1 for sure. In addition, if an individual was not observed at an occasion, but was alive for sure, because it was observed before and thereafter (i.e. has a capture history of e.g. {101} or {10001}), then we know that the individual was alive at all of these occasions, and thus alive = 1. Therefore, we should provide initial values of alive = 1 at these positions as well. The following function provides such initial values from the observed capture histories (from Kery and Schaub book)
known.state.cjs <- function(ch){
state <- ch
for (i in 1:dim(ch)[1]){
n1 <- min(which(ch[i,]==1))
n2 <- max(which(ch[i,]==1))
state[i,n1:n2] <- 1
state[i,n1] <- NA
}
state[state==0] <- NA
return(state)
}
Xinit <- known.state.cjs(data)
# first list of inits
init1 <- list(surv=.1,resight=.1,alive=Xinit)
# second list of inits
init2 <- list(surv=.9,resight=.9,alive=Xinit)
# specify the parameters to be monitored
parameters <- c("resight","surv")
# load R2jags
library(R2jags)
# run the MCMC analysis WITHOUT PRIOR INFORMATION
CJS.sim <-jags(data=datax, inits=list(init1,init2), parameters,n.iter=1000,model.file="code/CJS.txt",n.chains=2,n.burnin=500)
# to see the numerical results
# CJS.sim
# traceplot(CJS.sim) # diagnostic de convergence
# keep 3 first years only
data = data[,1:3]
databis = NULL
for (i in 1:nrow(data)){
# discard all non existing individuals i.e. those that were never captured
# test whether there was at least 1 detection and keep this individual if it was the case
if (sum(data[i,] == c(0,0,0))<3) databis = rbind(databis,data[i,])
}
data = databis
# number of individuals
n <- dim(data)[[1]]
# number of capture occasions
K <- dim(data)[[2]]
# compute the date of first capture
e <- NULL
for (i in 1:n){
temp <- 1:K
e <- c(e,min(temp[data[i,]==1]))
}
# data
datax <- list(N=n,Years=K,obs=data,First=e)
Xinit <- known.state.cjs(data)
# first list of inits
init1 <- list(surv=.1,resight=.1,alive=Xinit)
# second list of inits
init2 <- list(surv=.9,resight=.9,alive=Xinit)
# specify the parameters to be monitored
parameters <- c("resight","surv")
# run the MCMC analysis WITHOUT PRIOR INFORMATION
CJS.sim.wo.apriori <-jags(data=datax, inits=list(init1,init2), parameters,n.iter=1000,model.file="code/CJS.txt",n.chains=2,n.burnin=500)
# same model but with informative prior on survival
model <-
paste("
model
{
for (i in 1:N){
alive[i,First[i]] <- 1
for (j in (First[i]+1):Years){
alive[i,j] ~ dbern(alivep[i,j])
alivep[i,j] <- surv * alive[i,j-1]
obs[i,j] ~ dbern(sightp[i,j])
sightp[i,j] <- resight * alive[i,j]
}
}
surv~dnorm(0.57,187.6) # Norm(0.57,sd=0.073) ; precision = 1/var = 1/0.073^2
resight~dunif(0,1)
}
")
writeLines(model,"code/CJS2.txt")
CJS.sim.apriori <-jags(data=datax, inits=list(init1,init2), parameters,n.iter=1000,model.file="code/CJS2.txt",n.chains=2,n.burnin=500)
## ----echo=FALSE, message=FALSE, warning=FALSE----------------------------------------------------------------------
res = as.mcmc(CJS.sim.wo.apriori)
res = rbind(res[[1]],res[[2]])
#head(res)
res2 = as.mcmc(CJS.sim.apriori)
res2 = rbind(res2[[1]],res2[[2]])
#head(res2)
plot(density(res2[,'surv']),xlab='survival',ylab='probability density',col='red',lwd=4,main='',xlim=c(0.2,1))
lines(density(res[,'surv']),xlab='survival',ylab='probability density',col='blue',lwd=4,main='')
legend('topleft',lwd=2,legend=c('with prior info','without prior info'),col=c('red','blue'))
## ----echo = TRUE---------------------------------------------------------------------------------------------------
(alpha <- ( (1 - 0.57)/(0.073*0.073) - (1/0.57) )*0.57^2)
(beta <- alpha * ( (1/0.57) - 1))
## ----eval = FALSE, echo = TRUE-------------------------------------------------------------------------------------
## alpha <- ( (1 - 0.57)/(0.073*0.073) - (1/0.57) )*0.57^2
## beta <- alpha * ( (1/0.57) - 1)
## n <- 10000
## samp <- rbeta(n, alpha, beta)
## (mu <- mean(samp))
## (sigma <- sqrt(var(samp)))
## ----echo=1, fig.height=3, fig.width=3, fig.align='left'-----------------------------------------------------------
plot(density(rnorm(1000, 0, 1000)),
main="", xlab="Height (m)")
## ----echo=1, fig.height=3, fig.width=3, fig.align='left'-----------------------------------------------------------
plot(density(rnorm(1000, 2, 0.5)),
main="", xlab="Height (m)")
## ----echo=1, fig.height=3, fig.width=3, fig.align='left'-----------------------------------------------------------
plot(density(plogis(rnorm(1000,0,10)),
from = 0, to = 1), main='', xlab='survival')
## ----echo=1, fig.height=3, fig.width=3, fig.align='left'-----------------------------------------------------------
plot(density(plogis(rnorm(1000,0,1.5)),
from = 0, to = 1), main='', xlab='survival')
## ------------------------------------------------------------------------------------------------------------------
y <- 19 # nb of success
n <- 57 # nb of attempts
## ------------------------------------------------------------------------------------------------------------------
a <- 1; b <- 1; p <- seq(0,1,.002)
plot(p, dbeta(p,a,b), type='l', lwd=3)
## ------------------------------------------------------------------------------------------------------------------
numerator <- function(p) dbinom(y,n,p)*dbeta(p,a,b)
## ------------------------------------------------------------------------------------------------------------------
denominator <- integrate(numerator,0,1)$value
## ------------------------------------------------------------------------------------------------------------------
plot(p, numerator(p)/denominator,type="l", lwd=3, col="green", lty=2)
## ----eval = FALSE--------------------------------------------------------------------------------------------------
## lines(p, dbeta(p,y+a,n-y+b), col='darkred', lwd=3)
## ----echo = FALSE--------------------------------------------------------------------------------------------------
plot(p, numerator(p)/denominator,type="l", lwd=3, col="green", lty=2)
lines(p, dbeta(p,y+a,n-y+b), col='darkred', lwd=3)
## ----eval = FALSE--------------------------------------------------------------------------------------------------
## lines(p, dbeta(p,a,b), col='darkblue', lwd=3)
## ----echo = FALSE--------------------------------------------------------------------------------------------------
plot(p, numerator(p)/denominator,type="l", lwd=3, col="green", lty=2)
lines(p, dbeta(p,y+a,n-y+b), col='darkred', lwd=3)
lines(p, dbeta(p,a,b), col='darkblue', lwd=3)
## ---- out.width = '9cm',out.height='3cm',fig.align='center',echo=FALSE---------------------------------------------
knitr::include_graphics('img/metropolis.png')
## ---- out.width = '11cm',out.height='7cm',fig.align='center',echo=FALSE--------------------------------------------
knitr::include_graphics('img/maniac.png')
## ------------------------------------------------------------------------------------------------------------------
pd <- function(x){
values <- c(5, 10, 4, 4, 20, 20, 12, 5)
ifelse(x %in% 1:length(values), values[x], 0)
}
prob_dist <- data.frame(x = 1:8, prob = pd(1:8))
prob_dist
## ----echo=FALSE----------------------------------------------------------------------------------------------------
prob_dist %>%
ggplot(aes(x = x, y = prob)) +
geom_col(width = 0.3) +
labs(x = 'x', y = 'Probability')
## ------------------------------------------------------------------------------------------------------------------
random_walk <- function(pd, start, num_steps){
y <- rep(0, num_steps)
current <- start
for (j in 1:num_steps){
candidate <- current + sample(c(-1, 1), 1)
prob <- pd(candidate) / pd(current)
if (runif(1) < prob) current <- candidate
y[j] <- current
}
return(y)
}
## ------------------------------------------------------------------------------------------------------------------
out <- random_walk(pd, 4, 10000)
head(out)
tail(out)
## ----echo = FALSE--------------------------------------------------------------------------------------------------
sim <- data.frame(out) %>%
group_by(out) %>%
summarize(N = n(), prob = N / 10000) %>%
select(out, prob) %>%
add_column(Type = 'Simulated')
truth <- data.frame(out = 1:8, prob = pd(1:8)/sum(pd(1:8)), Type = 'Actual')
bind_rows(sim,truth) %>%
ggplot() +
aes(x = out, y = prob, fill = Type) +
geom_col(width = 0.5, position = "dodge") +
labs(x = 'x', y = 'Probability')
## ---- out.width = '9cm',out.height='5cm',fig.align='center',echo=FALSE---------------------------------------------
knitr::include_graphics('img/plummer.png')
## ---- out.width = '10cm',out.height='8cm',fig.align='center',echo=FALSE--------------------------------------------
knitr::include_graphics('img/stork_world.png')
## ------------------------------------------------------------------------------------------------------------------
nbchicks <- c(151,105,73,107,113,87,77,108,118,122,112,120,122,89,69,71,
53,41,53,31,35,14,18)
nbpairs <- c(173,164,103,113,122,112,98,121,132,136,133,137,145,117,90,80,
67,54,58,39,42,23,23)
temp <- c(15.1,13.3,15.3,13.3,14.6,15.6,13.1,13.1,15.0,11.7,15.3,14.4,14.4,
12.7,11.7,11.9,15.9,13.4,14.0,13.9,12.9,15.1,13.0)
rain <- c(67,52,88,61,32,36,72,43,92,32,86,28,57,55,66,26,28,96,48,90,86,
78,87)
datax <- list(N = 23, nbchicks = nbchicks, nbpairs = nbpairs,
temp = (temp - mean(temp))/sd(temp),
rain = (rain - mean(rain))/sd(rain))
## ---- echo=TRUE, eval=FALSE----------------------------------------------------------------------------------------
## {
## # Likelihood
## for( i in 1 : N){
## nbchicks[i] ~ dbin(p[i],nbpairs[i])
## logit(p[i]) <- a + b.temp * temp[i] + b.rain * rain[i]
## }
## # ...
## ---- echo=TRUE, eval=FALSE----------------------------------------------------------------------------------------
## # Priors
## a ~ dnorm(0,0.001)
## b.temp ~ dnorm(0,0.001)
## b.rain ~ dnorm(0,0.001)
## }
## ----message=FALSE, warning=FALSE----------------------------------------------------------------------------------
model <-
paste("
model
{
for( i in 1 : N)
{
nbchicks[i] ~ dbin(p[i],nbpairs[i])
logit(p[i]) <- a + b.temp * temp[i] + b.rain * rain[i]
}
a ~ dnorm(0,0.001)
b.temp ~ dnorm(0,0.001)
b.rain ~ dnorm(0,0.001)
}
")
writeLines(model,"code/logistic.txt")
## ---- message=FALSE, warning=FALSE, eval = FALSE-------------------------------------------------------------------
## logistic <- function() {
## for( i in 1 : N)
## {
## nbchicks[i] ~ dbin(p[i],nbpairs[i])
## logit(p[i]) <- a + b.temp * temp[i] + b.rain * rain[i]
## }
##
## # priors for regression parameters
## a ~ dnorm(0,0.001)
## b.temp ~ dnorm(0,0.001)
## b.rain ~ dnorm(0,0.001)
## }
## ----message=FALSE, warning=FALSE----------------------------------------------------------------------------------
# list of lists of initial values (one for each MCMC chain)
init1 <- list(a = -0.5, b.temp = -0.5, b.rain = -0.5)
init2 <- list(a = 0.5, b.temp = 0.5, b.rain = 0.5)
inits <- list(init1,init2)
# specify parameters that need to be estimated
parameters <- c("a","b.temp","b.rain")
# specify nb iterations for burn-in and final inference
nb.burnin <- 1000
nb.iterations <- 2000
## ----eval = FALSE--------------------------------------------------------------------------------------------------
## # load R2jags
## library(R2jags)
## # run Jags
## storks <- jags(data = datax,
## inits = inits,
## parameters.to.save = parameters,
## model.file = "code/logistic.txt",
## # model.file = logistic, # if a function was written
## n.chains = 2,
## n.iter = nb.iterations,
## n.burnin = nb.burnin)
## storks
## ----echo=FALSE, message=FALSE, warning=FALSE----------------------------------------------------------------------
library(R2jags)
storks <- jags(data = datax,
inits = inits,
parameters.to.save = parameters,
model.file = "code/logistic.txt",
n.chains = 2,
n.iter = nb.iterations,
n.burnin = nb.burnin)
storks
## ---- echo=FALSE---------------------------------------------------------------------------------------------------
knitr::include_graphics('img/acf.png')
## ----message=FALSE, warning=FALSE----------------------------------------------------------------------------------
traceplot(storks,mfrow = c(1, 2), varname = c('b.rain','b.temp'), ask = FALSE)
## ----message=FALSE, warning=FALSE----------------------------------------------------------------------------------
autocorr.plot(as.mcmc(storks),ask = FALSE)
## ---- out.width = '11cm',out.height='7cm',fig.align='center',echo=FALSE--------------------------------------------
knitr::include_graphics('img/mcmc.png')
## ------------------------------------------------------------------------------------------------------------------
storks
## ----message=FALSE, warning=FALSE----------------------------------------------------------------------------------
res <- as.mcmc(storks) # convert outputs in a list
res <- rbind(res[[1]],res[[2]]) # put two MCMC lists on top of each other
head(res)
## ----message=FALSE, warning=FALSE----------------------------------------------------------------------------------
# probability that the effect of rainfall is negative
mean(res[,'b.rain'] < 0)
## ----message=FALSE, warning=FALSE----------------------------------------------------------------------------------
# probability that the effect of temperature is negative
mean(res[,'b.temp'] < 0)
## ----message=FALSE, warning=FALSE----------------------------------------------------------------------------------
quantile(res[,'b.rain'],c(0.025,0.975))
## ----message=FALSE, warning=FALSE----------------------------------------------------------------------------------
quantile(res[,'b.temp'],c(0.025,0.975))
## ----echo=FALSE, message=FALSE, warning=FALSE----------------------------------------------------------------------
par(mfrow=c(1,2))
plot(density(res[,'b.rain']),xlab="",ylab="", main="Rainfall",lwd=3)
abline(v=0,col='red',lwd=2)
plot(density(res[,'b.temp']),xlab="",ylab="", main="Temperature",lwd=3)
abline(v=0,col='red',lwd=2)
## ------------------------------------------------------------------------------------------------------------------
stupid_pd <- res[,'b.rain']^2 + cos(res[,'b.temp'])
head(stupid_pd)
## ------------------------------------------------------------------------------------------------------------------
plot(density(stupid_pd), xlab = '', main = '', lwd = 3)
## ----include=FALSE-------------------------------------------------------------------------------------------------
model <-
paste("
model
{
for( i in 1 : N)
{
nbchicks[i] ~ dbin(p[i],nbpairs[i])
logit(p[i]) <- a + b.temp * temp[i] + b.rain * rain[i]
}
# priors for regression parameters
a ~ dnorm(0,0.001)
b.temp ~ dnorm(0,0.001)
b.rain ~ dnorm(0,0.001)
}
")
writeLines(model,"code/logistic.txt")
init1 <- list(a = -0.5, b.temp = -0.5, b.rain = -0.5)
init2 <- list(a = 0.5, b.temp = 0.5, b.rain = 0.5)
inits <- list(init1,init2)
parameters <- c("a","b.temp","b.rain")
nb.burnin <- 1000
nb.iterations <- 2000
storks <- jags(data = datax,
inits = inits,
parameters.to.save = parameters,
model.file = "code/logistic.txt",
n.chains = 2,
n.iter = nb.iterations,
n.burnin = nb.burnin)
## ------------------------------------------------------------------------------------------------------------------
# calculate wAIC with JAGS
# https://sourceforge.net/p/mcmc-jags/discussion/610036/thread/8211df61/#ea5c
samples <- jags.samples(storks$model,c("WAIC","deviance"), type = "mean",
n.iter = 2000,
n.burnin = 1000,
n.thin = 1)
## ------------------------------------------------------------------------------------------------------------------
samples$p_waic <- samples$WAIC
samples$waic <- samples$deviance + samples$p_waic
tmp <- sapply(samples, sum)
waic <- round(c(waic = tmp[["waic"]], p_waic = tmp[["p_waic"]]),1)
waic
## ------------------------------------------------------------------------------------------------------------------
# model specification
model <-
paste("
model
{
for( i in 1 : N)
{
nbchicks[i] ~ dbin(p[i],nbpairs[i])
logit(p[i]) <- a + b * cov[i]
}
# priors for regression parameters
a ~ dnorm(0,0.001)
b ~ dnorm(0,0.001)
}
")
writeLines(model,"code/logtemp.txt")
## ------------------------------------------------------------------------------------------------------------------
# list of lists of initial values (one for each MCMC chain)
init1 <- list(a = -0.5, b = -0.5)
init2 <- list(a = 0.5, b = 0.5)
inits <- list(init1,init2)
# specify parameters that need to be estimated
parameters <- c("a","b")
# specify nb iterations for burn-in and final inference
nb.burnin <- 1000
nb.iterations <- 2000
# read in data
datax <- list(N = 23, nbchicks = nbchicks, nbpairs = nbpairs,
cov = (temp - mean(temp))/sd(temp))
## ----message=FALSE, warning=FALSE----------------------------------------------------------------------------------
# load R2jags to run Jags through R
storks_temp <- jags(data = datax,
inits = inits,
parameters.to.save = parameters,
model.file = "code/logtemp.txt",
n.chains = 2,
n.iter = nb.iterations,
n.burnin = nb.burnin)
## ------------------------------------------------------------------------------------------------------------------
# compute WAIC
samples <- jags.samples(storks_temp$model,c("WAIC","deviance"), type = "mean",
n.iter = 2000,
n.burnin = 1000,
n.thin = 1)
samples$p_waic <- samples$WAIC
samples$waic <- samples$deviance + samples$p_waic
tmp <- sapply(samples, sum)
waic_temp <- round(c(waic = tmp[["waic"]], p_waic = tmp[["p_waic"]]),1)
## ----message=FALSE, warning=FALSE, paged.print=FALSE---------------------------------------------------------------
# read in data
datax <- list(N = 23, nbchicks = nbchicks, nbpairs = nbpairs,
cov = (rain - mean(rain))/sd(rain))
## ----message=FALSE, warning=FALSE, paged.print=FALSE---------------------------------------------------------------
# load R2jags to run Jags through R
storks_temp <- jags(data = datax,
inits = inits,
parameters.to.save = parameters,
model.file = "code/logtemp.txt",
n.chains = 2,
n.iter = nb.iterations,
n.burnin = nb.burnin)
## ----message=FALSE, warning=FALSE, paged.print=FALSE---------------------------------------------------------------
# compute WAIC
samples <- jags.samples(storks_temp$model,c("WAIC","deviance"), type = "mean",
n.iter = 2000,
n.burnin = 1000,
n.thin = 1)
samples$p_waic <- samples$WAIC
samples$waic <- samples$deviance + samples$p_waic
tmp <- sapply(samples, sum)
waic_rain <- round(c(waic = tmp[["waic"]], p_waic = tmp[["p_waic"]]),1)
## ------------------------------------------------------------------------------------------------------------------
# model specification
model <-
paste("
model
{
for( i in 1 : N)
{
nbchicks[i] ~ dbin(p[i],nbpairs[i])
logit(p[i]) <- a
}
# priors for regression parameters
a ~ dnorm(0,0.001)
}
")
writeLines(model,"code/lognull.txt")
## ----message=FALSE, warning=FALSE, paged.print=FALSE---------------------------------------------------------------
# list of lists of initial values (one for each MCMC chain)
init1 <- list(a = -0.5)
init2 <- list(a = 0.5)
inits <- list(init1,init2)
# specify parameters that need to be estimated
parameters <- c("a")
# specify nb iterations for burn-in and final inference
nb.burnin <- 1000
nb.iterations <- 2000
# read in data
datax <- list(N = 23, nbchicks = nbchicks, nbpairs = nbpairs)
## ----message=FALSE, warning=FALSE, paged.print=FALSE---------------------------------------------------------------
# load R2jags to run Jags through R
storks_temp <- jags(data = datax,
inits = inits,
parameters.to.save = parameters,
model.file = "code/lognull.txt",
n.chains = 2,
n.iter = nb.iterations,
n.burnin = nb.burnin)
## ----message=FALSE, warning=FALSE, paged.print=FALSE---------------------------------------------------------------
# compute WAIC
samples <- jags.samples(storks_temp$model,c("WAIC","deviance"), type = "mean",
n.iter = 2000,
n.burnin = 1000,
n.thin = 1)
samples$p_waic <- samples$WAIC
samples$waic <- samples$deviance + samples$p_waic
tmp <- sapply(samples, sum)
waic_null <- round(c(waic = tmp[["waic"]], p_waic = tmp[["p_waic"]]),1)
## ------------------------------------------------------------------------------------------------------------------
data.frame(model = c('both_covariates', 'temp', 'rain', 'none'),
waic = c(waic[1],waic_temp[1],waic_rain[1],waic_null[1]),
p_waic = c(waic[2],waic_temp[2],waic_rain[2],waic_null[2])) %>%
arrange(waic)
## ---- out.width = '8cm',out.height='6cm',fig.align='center',echo=FALSE---------------------------------------------
knitr::include_graphics('img/pdm.png')
## ---- message=FALSE, warning=FALSE, include=FALSE------------------------------------------------------------------
# On lit le jeu de données à analyser et on le nettoie
VMG <- read.table("dat/VMG.csv", header=TRUE, dec= ".", sep =";")
# crée crée un vecteur contenant le nb de graines (en log)
y <- round(VMG$NGrTotest)
# crée un vecteur contenant la biomass
x <- VMG$Vm
# crée un vecteur contenant le nom des espèces
Sp <- VMG$Sp
# crée un vecteur contenant le numéro des espèces
species <- as.numeric(Sp)
# nombre d'espèces
nbspecies <- length(levels(Sp)) # ou bien length(unique(species))
# nombre de mesures
n <- length(y)
## ---- echo=FALSE, message=FALSE, warning=FALSE,fig.align='center',out.width = '12cm',out.height='7cm'--------------
# graphical representation
library(lattice)
dat <- data.frame(Biomass=x,nb_grain=y,Species=Sp)
xyplot(log(nb_grain) ~ Biomass | Species,data=dat,xlab='Biomass',ylab='Number of grains (log transformed)')
## ------------------------------------------------------------------------------------------------------------------
# read in data
VMG <- read.table("dat/VMG.csv", header=TRUE, dec= ".", sep =";")
# nb of seeds (log)
y <- VMG$NGrTotest
# biomass
x <- VMG$Vm
x <- (x - mean(x))/sd(x)
# species name
Sp <- VMG$Sp
# species label
species <- as.numeric(Sp)
# species name
nbspecies <- length(levels(Sp))
# total nb of measurements
n <- length(y)
## ----message=FALSE, warning=FALSE----------------------------------------------------------------------------------
model <-
paste("
model{
for(i in 1:n){
y[i] ~ dnorm(mu[i],tau.y)
mu[i] <- a + b*x[i]
}
tau.y <- 1/(sigma.y*sigma.y)
sigma.y ~ dunif(0,100)
a ~ dnorm(0,0.001)
b ~ dnorm(0,0.001)
}
")
writeLines(model,"code/completepooling.bug")
## ------------------------------------------------------------------------------------------------------------------
# data
allom.data <- list(y=y,n=n,x=x)
# initial values
init1 <- list(a=rnorm(1), b=rnorm(1),sigma.y=runif(1))
init2 <- list(a=rnorm(1), b=rnorm(1),sigma.y=runif(1))
inits <- list(init1,init2)
# parameters to be estimated
allom.parameters <- c("a", "b", "sigma.y")
## ------------------------------------------------------------------------------------------------------------------
allom.1 <- R2jags::jags(allom.data,inits,allom.parameters,
n.iter = 2500,model.file="code/completepooling.bug",
n.chains = 2, n.burn = 1000)
## ------------------------------------------------------------------------------------------------------------------
allom.1
## ----echo=FALSE, message=FALSE, warning=FALSE----------------------------------------------------------------------
library(lattice)
xyplot(y ~ x | Sp,
xlab = "Biomass", ylab = "Number of seeds",main="complete pooling",
panel = function(x, y) {
panel.xyplot(x, y)
panel.abline(a=c(1991,1677),col='red',lwd=3)
})
## ---- fig.align='center',echo=FALSE--------------------------------------------------------------------------------
knitr::include_graphics('img/varyingint.png')
## ----message=FALSE, warning=FALSE----------------------------------------------------------------------------------
model <- paste("
model {
for (i in 1:n){
y[i] ~ dnorm (mu[i], tau.y)
mu[i] <- a[species[i]] + b *x[i]}
tau.y <- pow(sigma.y, -2)
sigma.y ~ dunif (0, 100)
for (j in 1:nbspecies){ a[j] ~ dnorm (mu.a, tau.a)}