\name{SS}
\alias{SS}
\alias{is.SS}
\alias{is.innov.SS}
\alias{is.nonInnov.SS}
\title{State Space Models}
\description{Construct a }
\usage{
SS(F.=NULL, G=NULL, H=NULL, K=NULL, Q=NULL, R=NULL, z0=NULL, P0=NULL, rootP0=NULL,
constants=NULL,
description=NULL, names=NULL, input.names=NULL, output.names=NULL)
is.SS(obj)
is.innov.SS(obj)
is.nonInnov.SS(obj)
}
\arguments{
\item{F.}{(nxn) state transition matrix.}
\item{H}{(pxn) output matrix.}
\item{Q}{(nxn) matrix specifying the system noise distribution.}
\item{R}{(pxp) matrix specifying the output (measurement) noise distribution.}
\item{G}{(nxp) input (control) matrix. G should be NULL if there is no input.}
\item{K}{(nxp) matrix specifying the Kalman gain.}
\item{z0}{vector indicating estimate of the state at time 0.
Set to zero if not supplied.}
\item{rootP0}{matrix indicating a square root of the initial tracking
error (e.g. chol(P0)).}
\item{P0}{matrix indicating initial tracking error P(t=1|t=0).
Set to I if rootP0 or P0 are not supplied.}
\item{constants}{NULL or a list of logical matrices with the same names as
matices above, indicating which elements should be considered constants.}
\item{description}{String. An arbitrary description.}
\item{names}{A list with elements input and output, each a vector of
strings. Arguments input.names and output.names should not be used if
argument names is used.}
\item{input.names}{
A vector of character strings indicating input variable names.
}
\item{output.names}{
A vector of character strings indicating output variable names.
}
\item{obj}{an object.}
}
\value{An SS TSmodel}
\details{
State space models have a
further sub-class: innov or non-innov, indicating an innovations form
or a non-innovations form.
The state space (SS) model is defined by:
z(t) =Fz(t-1) + Gu(t) + Qe(t)
y(t) = Hz(t) + Rw(t)
or the innovations model:
z(t) =Fz(t-1) + Gu(t) + Kw(t-1)
y(t) = Hz(t) + w(t)
Matrices are as specified above in the arguments, and
\describe{
\item{y}{is the p dimensional output data.}
\item{u}{is the m dimensional exogenous (input) data.}
\item{z}{is the n dimensional (estimated) state at time t,
E[z(t)|y(t-1), u(t)] denoted E[z(t)|t-1]. Note: In the case where
there is no input u this corresponds to what
would usually be called the predicted state - not the filtered state.
An initial value for z can
be specified as z0 and an initial one step ahead state tracking
error (for non-innovations models) as P0. In the object returned
by \code{l.ss}, \code{state} is a time series matrix corresponding to z.}
\item{z0}{An initial value for z can be specified as z0.}
\item{P0}{An initial one step ahead state tracking error (for
non-innovations models) can be specified as P0.}
\item{rootP0}{Alternatively, a square root of P0 can be specified. This can
be an upper triangular matrix so that only the required number of parameters
are used.}
\item{K, Q, R}{
For sub-class \code{innov} the Kalman gain K is specified but not Q and R.
For sub-class \code{non-innov} Q and R are specified but not the Kalman gain K.}
\item{e and w}{are typically assumed to be white noise in the
non-innovations form, in which case
the covariance of the system noise is QQ' and the covariance of
the measurement noise is RR'. The covariance of e and w can be specified
otherwise in the simulate
method \code{simulate.SS} for this class of model, but the assumption is
usually maintained when estimating models of this form (although, not by all
authors).}
}
Typically, an non-innovations form is harder to identify than an innovations
form. Non-innovations form would typically be choosen when there is
considerable theoretical or physical knowledge of the system (e.g. the
system was built from known components with measured physical values).
By default, elements in parameter matrices are treated as constants if they
are exactly 1.0 or 0.0, and as parameters otherwise. A value of 1.001 would
be treated as a parameter, and this is the easiest way to initialize an
element which is not to be treated as a constant of value 1.0. Any matrix
elements can be fixed to constants by specifying the list \code{constants}.
Matrices which are not specified in the list will be treated in the default
way. An alternative for fixing constants is the function \code{fixConstants}.
}
\references{
Anderson, B. D. O. and Moore, J. B. (1979) \emph{Optimal Filtering}.
Prentice-Hall. (note p.39,44.)
}
\seealso{
\code{\link{TSmodel}}
\code{\link{ARMA}}
\code{\link{simulate.SS}}
\code{\link{l.SS}}
\code{\link{state}}
\code{\link{smoother}}
\code{\link{fixConstants}}
}
\examples{
f <- array(c(.5,.3,.2,.4),c(2,2))
h <- array(c(1,0,0,1),c(2,2))
k <- array(c(.5,.3,.2,.4),c(2,2))
ss <- SS(F=f,G=NULL,H=h,K=k)
is.SS(ss)
ss
}
\concept{DSE}
\keyword{ts}