astsa (version 1.1)

Kfilter2: Kalman Filter - Model may be time varying or have inputs or correlated errors

Description

Returns the filtered values in Property 6.5 on page 354 for the state-space model, (6.97) -- (6.99). In addition, the script returns the evaluation of the likelihood at the given parameter values and the innovation sequence. For demonstrations, see Example 6.11 on page 356 and Example 6.13 on page 361.

Usage

Kfilter2(num, y, A, mu0, Sigma0, Phi, Ups, Gam, Theta, cQ, cR, 
          S, input)

Arguments

num

number of observations

y

data matrix, vector or time series

A

time-varying observation matrix, an array with dim = c(q,p,n)

mu0

initial state mean

Sigma0

initial state covariance matrix

Phi

state transition matrix

Ups

state input matrix; use Ups = 0 if not needed

Gam

observation input matrix; use Gam = 0 if not needed

Theta

state error pre-matrix

cQ

Cholesky decomposition of state error covariance matrix Q -- see details below

cR

Cholesky-type decomposition of state error covariance matrix R -- see details below

S

covariance-type matrix of state and observation errors

input

matrix or vector of inputs having the same row dimension as y; use input = 0 if not needed

Value

• xp
  one-step-ahead prediction of the state
• Pp
  mean square prediction error
• xf
  filter value of the state
• Pf
  mean square filter error
• like
  the negative of the log likelihood
• innov
  innovation series
• sig
  innovation covariances
• K
  last value of the gain, needed for smoothing

Details

Practically, the script only requires that Q or R may be reconstructed as t(cQ)%*%(cQ) or t(cR)%*%(cR), respectively.

References

http://www.stat.pitt.edu/stoffer/tsa3/ See also http://www.stat.pitt.edu/stoffer/tsa3/chap6.htm for an explanation of the difference between levels 0, 1, and 2.