particle_smoother {bssm}

Particle Smoothing

Description

Function particle_smoother performs particle smoothing based on either bootstrap particle filter (Gordon et al. 1993), \psi-auxiliary particle filter (\psi-APF) (Vihola et al. 2020), extended Kalman particle filter (Van Der Merwe et al. 2001), or its version based on iterated EKF (Jazwinski, 1970). The smoothing phase is based on the filter-smoother algorithm by Kitagawa (1996).

Usage

particle_smoother(model, particles, ...)

## S3 method for class 'lineargaussian'
particle_smoother(
  model,
  particles,
  method = "psi",
  seed = sample(.Machine$integer.max, size = 1),
  ...
)

## S3 method for class 'nongaussian'
particle_smoother(
  model,
  particles,
  method = "psi",
  seed = sample(.Machine$integer.max, size = 1),
  max_iter = 100,
  conv_tol = 1e-08,
  ...
)

## S3 method for class 'ssm_nlg'
particle_smoother(
  model,
  particles,
  method = "bsf",
  seed = sample(.Machine$integer.max, size = 1),
  max_iter = 100,
  conv_tol = 1e-08,
  iekf_iter = 0,
  ...
)

## S3 method for class 'ssm_sde'
particle_smoother(
  model,
  particles,
  L,
  seed = sample(.Machine$integer.max, size = 1),
  ...
)

Arguments

model

A model object of class bssm_model.

particles

Number of particles as a positive integer. Suitable values depend on the model, the data, and the chosen algorithm. While larger values provide more accurate estimates, the run time also increases with respect to the number of particles, so it is generally a good idea to test the filter first with a small number of particles, e.g., less than 100.

...

Ignored.

method

Choice of particle filter algorithm. For Gaussian and non-Gaussian models with linear dynamics, options are "bsf" (bootstrap particle filter, default for non-linear models) and "psi" (\psi-APF, the default for other models), and for non-linear models option "ekf" (extended Kalman particle filter) is also available.

seed

Seed for the C++ RNG (positive integer).

max_iter

Maximum number of iterations used in Gaussian approximation, as a positive integer. Default is 100 (although typically only few iterations are needed).

conv_tol

Positive tolerance parameter used in Gaussian approximation. Default is 1e-8.

iekf_iter

Non-negative integer. If zero (default), first approximation for non-linear Gaussian models is obtained from extended Kalman filter. If iekf_iter > 0, iterated extended Kalman filter is used with iekf_iter iterations.

L

Positive integer defining the discretization level for SDE model.

Details

See one of the vignettes for \psi-APF in case of nonlinear models.

Value

List with samples (alpha) from the smoothing distribution and corresponding weights (weights), as well as smoothed means and covariances (alphahat and Vt) of the states and estimated log-likelihood (logLik).

References

Gordon, NJ, Salmond, DJ, Smith, AFM (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings-F, 140, 107-113. https://doi.org/10.1049/ip-f-2.1993.0015

Vihola, M, Helske, J, Franks, J. Importance sampling type estimators based on approximate marginal Markov chain Monte Carlo. Scand J Statist. 2020; 1-38. https://doi.org/10.1111/sjos.12492

Van Der Merwe, R, Doucet, A, De Freitas, N, Wan, EA (2001). The unscented particle filter. In Advances in neural information processing systems, p 584-590.

Jazwinski, A 1970. Stochastic Processes and Filtering Theory. Academic Press.

Kitagawa, G (1996). Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. Journal of Computational and Graphical Statistics, 5, 1-25. https://doi.org/10.2307/1390750

Examples

set.seed(1)
x <- cumsum(rnorm(100))
y <- rnorm(100, x)
model <- ssm_ulg(y, Z = 1, T = 1, R = 1, H = 1, P1 = 1)
system.time(out <- particle_smoother(model, particles = 1000))
# same with simulation smoother:
system.time(out2 <- sim_smoother(model, particles = 1000, 
  use_antithetic = TRUE))
ts.plot(out$alphahat, rowMeans(out2), col = 1:2)