Detrend Example

library(knitr)
library(nimble)
library(earlywarning)
library(ggplot2)
library(tidyr)
opts_chunk$set(dev='png', fig.width=5, fig.height=5, results='hide')

some sample data from earlywarning:

set.seed(123)
data(ibms)
plot(ibm_critical)
raw <- as.data.frame(ibm_critical)
names(raw) <- "x"

Rather than explicitly modeling the trend element predicted by the linearization, let us simply remove it:

N <- length(raw$x)
raw$t <- 1:N
detrend <- loess(x ~ t, raw)
data <- data.frame(x = detrend$residuals/sqrt(var(detrend$residuals)))
qplot(raw$t, data$x, geom='line')

LSN version

Modify the LSN model to explicitly model the changing parameter as a hidden, stochastic variable

lsn <- nimbleCode({
   theta ~ dunif(-100.0, 100.0)
   sigma_x ~ dunif(1e-10, 100.0)
   sigma_y ~ dunif(1e-10, 100.0)
       m ~ dunif(-1e2, 1e2)
    x[1] ~ dunif(-100, 100)
    y[1] ~ dunif(-100, 100)

  for(i in 1:(N-1)){
    mu_x[i] <- x[i] + y[i] * (theta - x[i]) 
    x[i+1] ~ dnorm(mu_x[i], sd = sigma_x) 
    mu_y[i] <- y[i] + m * t[i]
    y[i+1] ~ dnorm(mu_y[i], sd = sigma_y) 
  }
})

Constants in the model definition are the length of the dataset, \(N\) and the time points of the sample. Note we’ve made time explicit, we’ll assume uniform spacing here.

constants <- list(N = N, t = raw$t)

Initial values for the parameters

inits <- list(theta = 6, m = 0, sigma_x = 1, sigma_y = 1, y = rep(1,N))

and here we go:

Rmodel <- nimbleModel(code = lsn, 
                      constants = constants, 
                      data = data, 
                      inits = inits)
Cmodel <- compileNimble(Rmodel)

mcmcspec <- configureMCMC(Rmodel, print=TRUE,thin=2e2)
Rmcmc <- buildMCMC(mcmcspec)
Cmcmc <- compileNimble(Rmcmc, project = Cmodel)

Cmcmc$run(1e6)

NULL

and examine results

samples <- as.data.frame(as.matrix(Cmcmc$mvSamples))
dim(samples)

[1] 5000   84

samples <- samples[,1:4]
long <- gather(samples)

apply(samples, 2, mean)

            m       sigma_x       sigma_y         theta 
 0.0003790592  1.0792385676  0.1920288851 -0.0150533955 

ggplot(long) + 
  geom_line(aes(seq_along(value), value)) + 
    facet_wrap(~key, scale='free')

ggplot(long) + 
    geom_density(aes(value)) + 
    facet_wrap(~key, scale='free')

sessionInfo()

R version 3.1.2 (2014-10-31)
Platform: x86_64-pc-linux-gnu (64-bit)

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] methods   stats     graphics  grDevices utils     datasets 
[7] base     

other attached packages:
[1] tidyr_0.2.0        ggplot2_1.0.0      earlywarning_0.0-1
[4] nimble_0.3         yaml_2.1.13        knitr_1.9         

loaded via a namespace (and not attached):
 [1] codetools_0.2-10 colorspace_1.2-4 deSolve_1.11    
 [4] digest_0.6.8     evaluate_0.5.5   formatR_1.0     
 [7] grid_3.1.2       gtable_0.1.2     igraph_0.7.1    
[10] labeling_0.3     MASS_7.3-39      mnormt_1.5-1    
[13] munsell_0.4.2    parallel_3.1.2   plyr_1.8.1      
[16] proto_0.3-10     psych_1.5.1      Rcpp_0.11.4     
[19] reshape2_1.4.1   scales_0.2.4     stringr_0.6.2   
[22] tools_3.1.2