#' Sequential Importance Sampling with arbitrary kernel for 1-D NLSS
#'
#' Runs sequential importance sampling \strong{without} resampling on a given Non-Linear State Space models with user-specified kernel as proposal.
#'
#' This algorithm is a slightly more generic version of \code{\link{sis}}. It is not recommended and included for illustrative purposes only. This version
#' is therefore a minimalistic and only supports NLLS with univariate states. Use \code{\link{sisr}} instead.
#'
#'
#' @param nlss Non-linear state space model
#' @param y Sequence of observations. Its length T is the number of timesteps.
#' @param N Number of particles
#' @param proposal.rnd Function sampling from the proposal kernel to use
#' @param proposal.logpdf Function computing the log-pdf of the proposal kernel
#' @param initial.proposal.rnd Function sampling from the proposal kernel to use at initial timestep
#' @param initial.proposal.logpdf Function computing the log-pdf of the proposal kernel at initial timestep
#' @return A list with the following components:
#' \item{particles}{Array (T, N, D) of the sampled particles}
#' \item{logweights}{Array (T, N) of the \strong{logarithm} of the \strong{non-normalized} importance weights of the particles}
#' \item{weights}{Array (T, N) of the \strong{normalized} importance weights of the particles}
#' \item{t}{Indices 1 to T, included for ease of plotting}
#'
#' @export
#' @seealso \code{\link{sisr}}
siskernel <- function (nlss, y, N,
proposal.rnd=prior.rnd,
proposal.logpdf=prior.logpdf,
initial.proposal.rnd=initial.rnd,
initial.proposal.logpdf=initial.logpdf) {
T <- length(y);
p <- array(dim=c(T,N));
lw <- array(dim=c(T,N));
w <- array(dim=c(T,N))
particles <- initial.proposal.rnd(nlss, N=N, y[1])
if(!is.na(y[1]))
logweights <- initial.logpdf(nlss, particles) - initial.proposal.logpdf(nlss, particles, y[1], t=1) + loclike.logpdf(nlss, particles, y[1])
else
logweights <- array(0,dim=c(1,N))
weights <- normalized.exponential(logweights)
# Store the history
p[1,] <- particles; lw[1,] <- logweights; w[1,] <- weights
if (T > 1)
for (t in 2:T) {
xpast <- particles
particles <- proposal.rnd(nlss, xpast, y[t], t=t)
if (!is.na(y[t]))
logweights <- logweights + loclike.logpdf(nlss, particles, y=y[t], t=t)
logweights <- logweights +
prior.logpdf(nlss, xpast, particles, y=y[t], t=t) -
proposal.logpdf(nlss, xpast, particles, y=y[t], t=t)
weights <- normalized.exponential(logweights)
# Store the history
p[t,] <- particles; lw[t,] <- logweights; w[t,] <- weights
}
# Return the weighted sample
list(particles=p, logweights=lw, weights=w, t=1:T)
}
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