Welcome State space notation Functions > kal.fil.mul.can
Last update Jul 10, 2004

DESCRIPTION:

Extended Kalman filter for multinomial data with k categories assuming canonical link and linear Gaussian latent process. The specific model is
Observation equation:
(yt | ht)
~
Mn(nt,pt),
Link function:
log(\fracpitpkt)
=
lit,
i
=
1,,k-1,
Signal:
lt,
=
FtTqt,
System equation:
qt
=
Gtqt-1 + wt,
    wt
~
Np(0,Wt),
Initial prior:
q0
~
Np(m0,C0).

USAGE:

      kal.fil.bin.can(ssm)

REQUIRED ARGUMENTS:

ssm:
Object of class ssm (state space model).
The following attributes must be specified in ssm:
Yt:
n × k matrix containing the multinomial observations arranged row-wise.
nt:
1 × n vector containing the number of trials at each time-point.
Should be nt <- apply(Yt,1,sum).
Ft:
Function returning a p × (k-1) design matrix.
Gt:
Function returning a p × p evolution transfer matrix.
Wt:
Function returning a p × p evolution variance matrix.
m0:
p × 1 vector (prior mean).
C0:
p × p matrix (prior variance).

OPTIONAL ARGUMENTS:

The following attributes of the ssm object might be needed:
Xt:
Matrix containing the covariates needed in Ft, Gt and/or Wt. This matrix must have n rows, such that row t contains the covariates needed for generating the system matrices at time t.
psi:
Parameter vector must be supplied if it is needed in the calculation of the system matrices.
The following attributes of the ssm object are optional:
fam:
Tacitly assumed to have value multinomial.
link:
Tacitly assumed to have value canonical.
m.start:
Used to specify the Taylor expansion points. These are not meant to be set by the user. Applying the Kalman smoother on a ssm object that has been filtered will set the m.start attribute to the smoothed states. This is needed in the iterated extended Kalman filter and smoother.

VALUE:

Returns an object of class ssm with the same attributes as the object in the call, but with attribute filtered updated with:
mt:
Posterior means mt given the observations up to and including time t. These are arranged row-wise in a matrix.
Ct:
Posterior variances Ct given the observations up to and including time t. These are arranged in a 3-dimensional array such that Ct is placed in filtered$Ct[,,t].
llh:
Log-likelihood of of the approximating Gaussian state space model.
Rt:
Prior variances for the states arranged in 3-dimensional array such that Rt is placed in filtered$Rt[,,t]. The reason for returning these is that they are needed in the Kalman smoother.

SIDE EFFECTS:

None

DETAILS:

The filter works by sequentially approximating the multinomial observation density with a Gaussian density based on a Taylor expansion around the values supplied in the attribute m.start of the ssm object. If m.start is not supplied, the Taylor expansion will be around the one-step forecast mean of the observation.

REFERENCES:

Durbin & Koopman, (2002), Time Series Analysis by State Space Models, Oxford Statistical Science Series.
Fahrmeir & Tutz, (1994), Multivariate Statistical Modelling Based on Generalized Linear Models, Springer Series in Statistics.
Klein, (2003), State Space Models for Exponential Family Data, Ph.D. Thesis, Department of Statistics, University of Southern Denmark.

EXAMPLES:

# Specify a state space model
ss <- ssm(Ft = function(i,x,phi)  
                       {t(cbind(1,diag(5)))},
          Gt = function(i,x,phi)  
                       {diag(c(0.95,rep(1,5)))},
          Wt = function(i,x,phi) 
                       {diag(c(0.1,rep(0.0010,5)))},
          m0 = c(0,-0.2,-0.1,0,0.1,0.2),
          C0 = diag(10,6),
          nt = rep(20,50),
          fam = "multinomial",
          link = "canonical")

# Simulate observations
ss <-simulate.ssm(ss, n=50)

# Apply the extended Kalman filter
ss <- kal.fil.mul.can(ss)

# Extract category probabilities (i=1,...,5)
# response function
h <- function(lambda) exp(lambda)/(1+sum(exp(lambda)))
props <- t(apply(ss$filtered$mt[,2:6]
                 +ss$filtered$mt[,1],1,h))
sim.props <- t(apply(ss$simulated$lambda,1,h))

# Plot the simulated and filtered signal 
# on the probability scale one at a time
for (i in 1:5)
{
plot(props[,i] , ylim=c(0,0.6), type="l")
lines(sim.props[,i],lty=2)
legend(0,0.6,
       legend=c("Filtered signals on probability scale",
                "Simulated signals on probability scale"),
       lty=c(1,2))
title(paste("Results from kal.fil.mul.can, 
signal component",i))
}