SS: Representation of Gaussian State Space Model
In ClausDethlefsen/sspir: State Space Models in R

Description Usage Arguments Details Value Author(s) See Also Examples

Creates an SS-object describing a Gaussian state space model.

SS(y = NA, x = list(x = NA),
   Fmat = function(tt,x,phi) { return(matrix(1)) },
   Gmat = function(tt,x,phi) { return(matrix(1)) },
   Vmat = function(tt,x,phi) { return(matrix(phi[1])) },
   Wmat = function(tt,x,phi) { return(matrix(phi[2])) },
   m0 = matrix(0),
   C0 = matrix(100),
   phi = c(1,1))

## S3 method for class 'SS'
C0(ssm) 
## S3 method for class 'SS'
m0(ssm) 
## S3 method for class 'SS'
Fmat(ssm)
## S3 method for class 'SS'
Gmat(ssm)
## S3 method for class 'SS'
Vmat(ssm)
## S3 method for class 'SS'
Wmat(ssm)
## S3 method for class 'SS'
phi(ssm) 
## S3 replacement method for class 'SS'
C0(ssm) <- value
## S3 replacement method for class 'SS'
m0(ssm) <- value
## S3 replacement method for class 'SS'
Fmat(ssm) <- value
## S3 replacement method for class 'SS'
Gmat(ssm) <- value
## S3 replacement method for class 'SS'
Vmat(ssm) <- value
## S3 replacement method for class 'SS'
Wmat(ssm) <- value
## S3 replacement method for class 'SS'
phi(ssm) <- value

`y`	a matrix giving a multivariate time series of observations. The observation at time `tt` is `y[tt,]`. The dimension of `y` is n \times d. Preferably, `y` is a `ts` object.
`x`	a list of entities (eg. covariates) passed as argument to the functions `Fmat`, `Gmat`, `Vmat`, and `Wmat`.
`Fmat`	a function depending on the parameter-vector `phi`, covariates `x` and returns the p \times d design matrix at time `tt`.
`Gmat`	a function depending on the parameter-vector `phi`, covariates `x` and returns the p \times p evolution matrix at time `tt`.
`Vmat`	a function depending on the parameter-vector `phi`, covariates `x` and returns the d \times d (positive definit) variance matrix at time `tt`. The matrix `Vmat` may be specified as a constant matrix not depending on `phi` coded with `matrix`.
`Wmat`	a function depending on the parameter-vector `phi`, covariates `x` and returns the p \times p (positive semidefinite) evolution variance matrix at time `tt`. The matrix `Wmat` may be specified as a constant matrix not depending on `phi` coded with `matrix`.
`m0`	a 1 \times p matrix giving the initial state.
`C0`	a p \times p variance matrix giving the variance matrix of the initial state.
`phi`	a (hyper) parameter vector passed as argument to the functions `Fmat`, `Gmat`, `Vmat`, and `Wmat`.
`ssm`	an object of class `SS`.
`value`	an object to be assigned to the element of the state space model.

The state space model is given by

Y_t = F_t^T * θ_t + v_t, v_t ~ N(0,V_t)

θ_t = G_t * θ_{t-1} + w_t, w_t ~ N(0,W_t)

for t=1,...,n. The matrices F_t, G_t, V_t, and W_t may depend on a parameter vector φ. The initialization is given as

θ_0 ~ N(m_0,C_0).

An object of class SS, which is a list with the following components

`y`	as input.
`x`	as input.
`Fmat`	as input.
`Gmat`	as input.
`Vmat`	as input.
`Wmat`	as input.
`m0`	as input.
`C0`	as input.
`phi`	as input.
`n`	the number of time points
`d`	the dimension of each observation.
`p`	the dimension of the state vector at each timepoint.
`ytilde`	adjusted observations for use in the extended Kalman filter, see `extended`.
`iteration`	an integer giving the number of iterations used in the extended Kalman filter, see `extended`.
`m`	after Kalman filtering (or smoothing), holds the conditional mean of the state vectors given the observations up till time t (filtering) or all observations (smoothing). This is organised in a n \times p dimensional matrix holding m_t (m_t^*) in rows. Is returned as a `ts` object.
`C`	after Kalman filtering (or smoothing), holds the conditional variance of the state vectors given the observations up til time t (filtering) or all observations (smoothing). This is organised in a list holding the p \times p dimensional matrices C_t (C_t^*).
`mu`	after Kalman smoothing, holds the conditional mean of the signal (μ_t=F_t^\top θ_t) given all observations. This is organised in a n \times d dimensional matrix holding μ_t in rows.
`loglik`	the log-likelihood value after Kalman filtering.

Claus Dethlefsen, Søren Lundbye-Christensen and Anette Luther Christensen

ssm for a glm-like interface of specifying models, kfilter for Kalman filter and smoother for Kalman smoother.

data(kurit)  ## West & Harrison, page 40
m1 <- SS(y=kurit,
         Fmat=function(tt,x,phi) return(matrix(1)),
         Gmat=function(tt,x,phi) return(matrix(1)),
         Wmat=function(tt,x,phi) return(matrix(5)), ## Alternatively Wmat=matrix(5)
         Vmat=function(tt,x,phi) return(matrix(100)), ## Alternatively Vmat=matrix(100)
         m0=matrix(130),C0=matrix(400)
         )

plot(m1$y)
m1.f <- kfilter(m1)
m1.s <- smoother(m1.f)
lines(m1.f$m,lty=2,col=2)
lines(m1.s$m,lty=2,col=2)

## make a model with an intervention at time 10
m2 <- m1
Wmat(m2) <- function(tt,x,phi) {
  if (tt==10) return(matrix(900))
  else return(matrix(5))
}

m2.f <- kfilter(m2)
m2.s <- smoother(m2.f)
lines(m2.f$m,lty=2,col=4)
lines(m2.s$m,lty=2,col=4)

## Use 'ssm' to construct an SS skeleton
phi.start <- StructTS(log10(UKgas),type="BSM")$coef[c(4,1,2,3)]
gasmodel <- ssm( log10(UKgas) ~ -1+
                 tvar(polytime(time,1))+
                 tvar(sumseason(time,12)),
                 phi=phi.start)

m0(gasmodel)
C0(gasmodel)
phi(gasmodel)

fit <- getFit(gasmodel)
plot( fit$m[,1:3]  )