Saturday, July 27, 2013

A JAGS calculation on pattern of rain January 1906-1915 against 2003-2012

Two weeks ago I showed rain data from six stations in Netherlands years 1906 till now. Last week I showed that frequency of days with and without rain differed between December 1906-1915 and December 2003-2012. This week I am considering the same data as t-distributed with a left truncation at 0 mm rain. This can be modeled very easy in JAGS. This model gives the posterior distribution of the amount of rain moved 1 mm of rain upwards and half a mm narrower, but both these intervals include 0.

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.

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.

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