KFKSDS (version 1.6)

KFKSDS: Kalman Filter, Smoother and Disturbance Smoother

Description

Kalman filter, smoother and disturbance smoother.

Usage

KSDS(y, ss, kf)
KSDS.deriv(y, ss, kf)
KFKSDS.deriv.C(y, ss)
KFKSDS.deriv.steady.C(y, ss, convergence = c(0.001, 10, 1.2))

Arguments

y

a numeric time series or vector.

ss

a list containing the matrices of the state space model.

kf

a list containing the output returned by the function KF.

convergence

a numeric vector of length three to control and determine the convergence of the filter and smoother. See details below.

Value

The function KSDS returns a list containing the following elements:

ahat

smoothed state disturbance.

varhat

covariance matrix of ahat.

r

weighted sum of innovations used to obtain ahat.

N

intermediate matrix used to obtain varahat

epshat

smoothed estimate of the disturbance term in the observation equation.

vareps

error variance of epshat.

etahat

smoothed estimate of the disturbance term(s) in the state equation.

vareta

error variance of etahat.

The function KSDS-deriv returns also dahat, dvarahat, dr, dN which are the derivatives referred to the elements defined above.The functions KFKSDS.deriv.C and KFKSDS.deriv.steady.C return a list containing the elements already defined above: epshat, vareps, etahat, vareta, r, N, dr, dN, dahat and dvareps.

Details

See the details section and the section ‘state space representation’ in KF.

The iteration at which the the Kalman smoother converges is determined as the iteration where the Kalman filter converged multiplied by the factor convergence[3]. It should be equal or greater than unity. It can be omitted by setting it equal to one.

These interfaces are fully implemented in R.

In these functions, the Kalman smoother and the disturbance smoother are run in a single loop instead of running two sepate loops over the same observations.

These functions return a relatively extended output than can be used, for example, to implement an expectation-maximization algorithm.

References

Durbin, J. and Koopman, S. J. (2001). Time Series Analysis by State Space Methods. Oxford University Press.

Harvey, A. C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.

See Also

KF, KS; char2numeric in package stsm.

Examples

# See tests comparing the analytical derivatives returned by
# these functions and the numerical derivatives evaluated with
# function 'numDeriv::grad' in the folder 'KFKSDS/inst/tests'
# of the source package

# local level plus seasonal model with arbitrary parameter values
# for the 'JohnsonJohnson' time series
m <- stsm::stsm.model(model = "llm+seas", y = JohnsonJohnson, 
  pars = c("var1" = 2, "var2" = 15, "var3" = 30))
ss <- stsm::char2numeric(m)
kfd <- KF.deriv(m@y, ss)
ksds <- KSDS(m@y, ss, kfd)
da <- KSDS.deriv(m@y, ss, kfd)
db <- KFKSDS.deriv.C(m@y, ss)
# the ouput is the same but is returned with a different arrangement
dim(da$dahat)
dim(db$dahat)
for (i in seq_along(m@y))
  stopifnot(all.equal(da$dahat[i,,], db$dahat[,,i]))
dim(da$dN)
dim(db$dN)
for (i in seq_along(m@y))
  for (j in seq_len(3))
    stopifnot(all.equal(diag(da$dN[i,,,j]), db$dN[,j,i], check.attributes = FALSE))