kalmanFilter: Kalman filter

Description

The basic Kalman filter for multivariate, linear, Gaussian processes.

Usage

kalmanFilter(object, X0 = rinit(object), A, Q, C, R, tol = 1e-06)

Value

A named list containing the following elements:

object: the ‘pomp’ object
A, Q, C, R: as in the call
filter.mean: $E[X_t|y^*_1,\dots,y^*_t]$
pred.mean: $E[X_t|y^*_1,\dots,y^*_{t-1}]$
forecast: $E[Y_t|y^*_1,\dots,y^*_{t-1}]$
cond.logLik: $f(y^*_t|y^*_1,\dots,y^*_{t-1})$
logLik: $f(y^*_1,\dots,y^*_T)$

Arguments

object: a pomp object containing data;
X0: length-m vector containing initial state. This is assumed known without uncertainty.
A: $m\times m$ latent state-process transition matrix. $E[X_{t+1} | X_t] = A.X_t$.
Q: $m\times m$ latent state-process covariance matrix. $Var[X_{t+1} | X_t] = Q$
C: $n\times m$ link matrix. $E[Y_t | X_t] = C.X_t$.
R: $n\times n$ observation process covariance matrix. $Var[Y_t | X_t] = R$
tol: numeric; the tolerance to be used in computing matrix pseudoinverses via singular-value decomposition. Singular values smaller than tol are set to zero.

Details

If the latent state is $X$, the observed variable is $Y$, $X_t \in R^m$, $Y_t \in R^n$, and $$X_t \sim \mathrm{MVN}(A X_{t-1}, Q)$$ $$Y_t \sim \mathrm{MVN}(C X_t, R)$$ where $\mathrm{MVN}(M,V)$ denotes the multivariate normal distribution with mean $M$ and variance $V$. Then the Kalman filter computes the exact likelihood of $Y$ given $A$, $C$, $Q$, and $R$.

Examples

Run this code

if (require(dplyr)) {

  gompertz() -> po

  po |>
    as.data.frame() |>
    mutate(
      logY=log(Y)
    ) |>
    select(time,logY) |>
    pomp(times="time",t0=0) |>
    kalmanFilter(
      X0=c(logX=0),
      A=matrix(exp(-0.1),1,1),
      Q=matrix(0.01,1,1),
      C=matrix(1,1,1),
      R=matrix(0.01,1,1)
    ) -> kf

  po |>
    pfilter(Np=1000) -> pf

  kf$logLik
  logLik(pf) + sum(log(obs(pf)))

}