Description Usage Arguments Details Value References See Also Examples
These functions run the iterative equations of the Kalman smoother for a state space model upon the output from the Kalman filter.
1 2 
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 Kalman filter

See the details section and the section ‘state space representation’
in KF
.
Missing observations are allowed.
The input kf
passed to KS.deriv
must contain the
derivative terms related to the filter that are returned by
KF.deriv
or KF.deriv.C
.
When the Kalman filter was found to convergence at some iteration,
i.e., kf$convit
is not null, these functions use steady state values for
N
and varahat
in the intermediate iterations of the smoother.
For example, if the filter converged at iteration 15 in a series of length n,
the equations of the smoother are run for the first iterations from observation n
to n15; then the steady state values are used until there are 15 iterations
remaining. In the last iterations, from observation 15 to 1 the equations of
the smoother are evaluated again.
In practice, if the disturbance smoother is to be run as well, using the functions
described in KFKSDS
will be slightly more efficient.
A list containing the following elements:
ahat 
smoothed state vector. 
varhat 
covariance matrix of 
r 
weighted sum of innovations used to obtain 
N 
intermediate matrix used to obtain 
The function KS.deriv
returns also the derivatives referred to each
of the elements defined above, named respectively dahat
, dvarahat
,
dr
and dN
.
Durbin, J. and Koopman, S. J. (2001). Time Series Analysis by State Space Methods. Oxford University Press.
Harvey, A. C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.
KF
, KSDS
;
char2numeric
in package stsm.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44  # local level plus seasonal model with arbitrary parameter values
# for the 'JohnsonJohnson' time series
m < stsm::stsm.model(model = "llm+seas", y = JohnsonJohnson,
pars = c("var1" = 2, "var2" = 15, "var3" = 30))
ss < stsm::char2numeric(m)
kf < KF(m@y, ss)
ks < KS(m@y, ss, kf)
plot(ks$ahat[,1:2], main = "smoothed state vector")
kfd < KF.deriv(m@y, ss)
ksd < KS.deriv(m@y, ss, kfd)
all.equal(ks$ahat, ksd$ahat)
# extended output is required if 'KF.deriv.C' is used to obtain
# the necessary elements from the filter, set return.all = TRUE
kfdc < KF.deriv.C(m@y, ss, return.all = TRUE)
ksd < KS.deriv(m@y, ss, kfdc)
all.equal(ks$ahat, ksd$ahat)
# compare analytical and numerical derivatives
# yield same results up to a tolerance error
fcn < function(x, model, type, i)
{
m < stsm::set.pars(model, x)
ss < stsm::char2numeric(m)
kf < KF(m@y, ss)
ks < KS(m@y, ss, kf)
switch(type, "ahat" = sum(ks$ahat[,i]), "r" = sum(ks$r[,i]))
}
dahat < numDeriv::grad(func = fcn, x = m@pars, model = m, type = "ahat", i = 1)
all.equal(dahat, colSums(ksd$dahat[,1,]))
dahat < numDeriv::grad(func = fcn, x = m@pars, model = m, type = "ahat", i = 2)
all.equal(dahat, colSums(ksd$dahat[,2,]))
dahat < numDeriv::grad(func = fcn, x = m@pars, model = m, type = "ahat", i = 3)
all.equal(dahat, colSums(ksd$dahat[,3,]))
dr < numDeriv::grad(func = fcn, x = m@pars, model = m, type = "r", i = 1)
all.equal(dr, colSums(ksd$dr[,1,]), check.attributes = FALSE)
dr < numDeriv::grad(func = fcn, x = m@pars, model = m, type = "r", i = 2)
all.equal(dr, colSums(ksd$dr[,2,]), check.attributes = FALSE)
dr < numDeriv::grad(func = fcn, x = m@pars, model = m, type = "r", i = 3)
all.equal(dr, colSums(ksd$dr[,3,]), check.attributes = FALSE)

Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.