### The data

As described before; the data are from KNMI; ROYAL NETHERLANDS METEOROLOGICAL INSTITUTE. The actual page for the data is http://www.knmi.nl/klimatologie/monv/reeksen/select_rr.html. The data selected were from 6 locations; De Bilt, Groningen, Heerde, Kerkwerve, Ter Apel and Winterswijk (Sibinkweg). The blog starts with data entered in a data.frame named all which was derived two weeks ago. The data selected is again every 5th day in January 1906-195 against January 2004-2013.

sel1 <- all[(all$year <1916 | all$year>2003)

& all$Month=='Jan'

& all$location=='DE-BILT'

& all$day %in% seq(1,31,by=5),]

sel1$decade <- factor(c('1906-1915','2004-2013')[1+(sel1$year>1950)])

datain <- list(

isCensored = 1-as.numeric(sel1$RD==0),

N = nrow(sel1),

RD = ifelse(sel1$RD==0,NA,sel1$RD/10),

period = 1+(sel1$year>1950)

)

model1 <- function() {

for ( i in 1:N ) {

isCensored[i] ~ dinterval( RD[i] , 0 )

RD[i] ~ dt( mu[period[i]] , tau[period[i]],nu )

}

for (i in 1:2) {

tau[i] <- 1/pow(sigma[i],2)

sigma[i] ~ dlnorm(sigmamu,sigmatau)

mu[i] ~ dnorm(mumu,mutau)

}

mumu ~ dnorm(0,1e-6)

mutau <- 1/pow(musigma,2)

musigma ~ dunif(0,100)

sigmamu ~ dnorm(0,1e-6)

sigmatau <- 1/pow(sigmasigma,2)

sigmasigma ~ dunif(1,100)

mudif <- mu[2]-mu[1]

sigmadif <- sigma[2]-sigma[1]

nu <- 1/nuinv

nuinv ~ dunif(0,.5)

}

params <- c('mu','sigma','mudif','sigmadif','nu')

inits <- function() {

list(mumu = rnorm(1,0),

musigma = runif(1,1,2),

sigmamu = rnorm(1,0),

sigmasigma = runif(1,1,2),

nuinv = runif(1,0,.5))

}

jagsfit <- jags(datain, model=model1, inits=inits,

parameters=params,progress.bar="gui",

n.chains=1,n.iter=10000,n.burnin=3000,n.thin=3)

jagsfit

### The model

The model is almost off the shelf. Truncation, two t-distributions with different means and standard deviations. Priors there taken from*Data Analysis Using Regression and Multilevel/Hierarchical Models*, Gelman and Hill.sel1 <- all[(all$year <1916 | all$year>2003)

& all$Month=='Jan'

& all$location=='DE-BILT'

& all$day %in% seq(1,31,by=5),]

sel1$decade <- factor(c('1906-1915','2004-2013')[1+(sel1$year>1950)])

datain <- list(

isCensored = 1-as.numeric(sel1$RD==0),

N = nrow(sel1),

RD = ifelse(sel1$RD==0,NA,sel1$RD/10),

period = 1+(sel1$year>1950)

)

model1 <- function() {

for ( i in 1:N ) {

isCensored[i] ~ dinterval( RD[i] , 0 )

RD[i] ~ dt( mu[period[i]] , tau[period[i]],nu )

}

for (i in 1:2) {

tau[i] <- 1/pow(sigma[i],2)

sigma[i] ~ dlnorm(sigmamu,sigmatau)

mu[i] ~ dnorm(mumu,mutau)

}

mumu ~ dnorm(0,1e-6)

mutau <- 1/pow(musigma,2)

musigma ~ dunif(0,100)

sigmamu ~ dnorm(0,1e-6)

sigmatau <- 1/pow(sigmasigma,2)

sigmasigma ~ dunif(1,100)

mudif <- mu[2]-mu[1]

sigmadif <- sigma[2]-sigma[1]

nu <- 1/nuinv

nuinv ~ dunif(0,.5)

}

params <- c('mu','sigma','mudif','sigmadif','nu')

inits <- function() {

list(mumu = rnorm(1,0),

musigma = runif(1,1,2),

sigmamu = rnorm(1,0),

sigmasigma = runif(1,1,2),

nuinv = runif(1,0,.5))

}

jagsfit <- jags(datain, model=model1, inits=inits,

parameters=params,progress.bar="gui",

n.chains=1,n.iter=10000,n.burnin=3000,n.thin=3)

jagsfit

Inference for Bugs model at "C:/Users/.../Local/Temp/Rtmp4ydV3G/modele0c40226a9c.txt", fit using jags,

1 chains, each with 10000 iterations (first 3000 discarded), n.thin = 3

n.sims = 2334 iterations saved

mu.vect sd.vect 2.5% 25% 50% 75% 97.5%

mu[1] 0.026 0.480 -0.925 -0.284 0.042 0.332 0.951

mu[2] 0.917 0.361 0.250 0.665 0.900 1.152 1.658

mudif 0.891 0.615 -0.231 0.449 0.880 1.315 2.109

nu 2.331 0.348 2.006 2.085 2.212 2.465 3.290

sigma[1] 2.980 0.597 2.018 2.555 2.913 3.327 4.361

sigma[2] 2.309 0.395 1.642 2.024 2.274 2.552 3.197

sigmadif -0.671 0.689 -2.108 -1.098 -0.622 -0.214 0.588

deviance 509.412 9.431 492.173 502.886 508.990 515.454 528.740

DIC info (using the rule, pD = var(deviance)/2)

pD = 44.5 and DIC = 553.9

DIC is an estimate of expected predictive error (lower deviance is better).

### Interpretation

The means shown in the first decade is very close to 0, with an increase to 1 mm in the last decade. The difference is just 0.9 mm a very small increase, with 0 just within the posterior 95% interval.

The standard deviation decreases from first to last decade. It is almost like the rain becomes more predictable. Again the 95% posterior interval of the difference does include 0.

The degrees of freedom of the t distribution is two to three, quite heavy tailed. Hence where almost half of a the days don't have rain, half of the days have a little rain (a few mm, certainly not more than 6 mm), a few dais have more rain.