This example shows how to construct and conduct inference on a state space model using particle filtering algorithms. `nimble`

currently has versions of the bootstrap filter, the auxiliary particle filter, the ensemble Kalman filter, and the Liu and West filter implemented. Additionally, particle MCMC samplers are available and can be specified for both univariate and multivariate parameters.

# Model Creation

Assume is the latent state and is the observation at time for . We define our state space model as

with initial states

and prior distributions

where denotes a normal distribution with mean and standard deviation , and is a shifted, scaled -distribution with center parameter , scale parameter , and degrees of freedom.

We specify and build our state space model below, using time points:

## load the nimble library and set seed library('nimble') set.seed(1) ## define the model stateSpaceCode <- nimbleCode({ a ~ dunif(-0.9999, 0.9999) b ~ dnorm(0, sd = 1000) sigPN ~ dunif(1e-04, 1) sigOE ~ dunif(1e-04, 1) x[1] ~ dnorm(b/(1 - a), sd = sigPN/sqrt((1-a*a))) y[1] ~ dt(mu = x[1], sigma = sigOE, df = 5) for (i in 2:t) { x[i] ~ dnorm(a * x[i - 1] + b, sd = sigPN) y[i] ~ dt(mu = x[i], sigma = sigOE, df = 5) } }) ## define data, constants, and initial values data <- list( y = c(0.213, 1.025, 0.314, 0.521, 0.895, 1.74, 0.078, 0.474, 0.656, 0.802) ) constants <- list( t = 10 ) inits <- list( a = 0, b = .5, sigPN = .1, sigOE = .05 ) ## build the model stateSpaceModel <- nimbleModel(stateSpaceCode, data = data, constants = constants, inits = inits, check = FALSE)

# Construct and run a bootstrap filter

We next construct a bootstrap filter to conduct inference on the latent states of our state space model. Note that the bootstrap filter, along with the auxiliary particle filter and the ensemble Kalman filter, treat the top-level parameters `a, b, sigPN`

, and `sigOE`

as fixed. Therefore, the bootstrap filter below will proceed as though `a = 0, b = .5, sigPN = .1`

, and `sigOE = .05`

, which are the initial values that were assigned to the top-level parameters.

The bootstrap filter takes as arguments the name of the model and the name of the latent state variable within the model. The filter can also take a control list that can be used to fine-tune the algorithm’s configuration.

## build bootstrap filter and compile model and filter bootstrapFilter <- buildBootstrapFilter(stateSpaceModel, nodes = 'x') compiledList <- compileNimble(stateSpaceModel, bootstrapFilter)

## run compiled filter with 10,000 particles. ## note that the bootstrap filter returns an estimate of the log-likelihood of the model. compiledList$bootstrapFilter$run(10000)

## [1] -28.13009

Particle filtering algorithms in `nimble`

store weighted samples of the filtering distribution of the latent states in the `mvSamples`

modelValues object. Equally weighted samples are stored in the `mvEWSamples`

object. By default, `nimble`

only stores samples from the final time point.

## extract equally weighted posterior samples of x[10] and create a histogram posteriorSamples <- as.matrix(compiledList$bootstrapFilter$mvEWSamples) hist(posteriorSamples)

The auxiliary particle filter and ensemble Kalman filter can be constructed and run in the same manner as the bootstrap filter.

# Conduct inference on top-level parameters using particle MCMC

Particle MCMC can be used to conduct inference on the posterior distribution of both the latent states and any top-level parameters of interest in a state space model. The particle marginal Metropolis-Hastings sampler can be specified to jointly sample the `a, b, sigPN`

, and `sigOE`

top level parameters within `nimble`

‘s MCMC framework as follows:

## create MCMC specification for the state space model stateSpaceMCMCconf <- configureMCMC(stateSpaceModel, nodes = NULL) ## add a block pMCMC sampler for a, b, sigPN, and sigOE stateSpaceMCMCconf$addSampler(target = c('a', 'b', 'sigPN', 'sigOE'), type = 'RW_PF_block', control = list(latents = 'x')) ## build and compile pMCMC sampler stateSpaceMCMC <- buildMCMC(stateSpaceMCMCconf) compiledList <- compileNimble(stateSpaceModel, stateSpaceMCMC, resetFunctions = TRUE)

## run compiled sampler for 5000 iterations compiledList$stateSpaceMCMC$run(5000)

## |-------------|-------------|-------------|-------------| ## |-------------------------------------------------------|

## NULL

## create trace plots for each parameter library('coda')

par(mfrow = c(2,2)) posteriorSamps <- as.mcmc(as.matrix(compiledList$stateSpaceMCMC$mvSamples)) traceplot(posteriorSamps[,'a'], ylab = 'a') traceplot(posteriorSamps[,'b'], ylab = 'b') traceplot(posteriorSamps[,'sigPN'], ylab = 'sigPN') traceplot(posteriorSamps[,'sigOE'], ylab = 'sigOE')

The above `RW_PF_block`

sampler uses a multivariate normal proposal distribution to sample vectors of top-level parameters. To sample a scalar top-level parameter, use the `RW_PF`

sampler instead.