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 Time Series Analysis and Control Examples

# Kalman Filter Subroutines

This section describes a collection of Kalman filtering and smoothing subroutines for time series analysis; immediately following are three examples using Kalman filtering subroutines. The state space model is a method for analyzing a wide range of time series models. When the time series is represented by the state space model (SSM), the Kalman filter is used for filtering, prediction, and smoothing of the state vector. The state space model is composed of the measurement and transition equations. The measurement (or observation) equation can be written

where bt is an Ny ×1 vector, Ht is an Ny ×Nz matrix, the sequence of observation noise is independent, zt is an Nz ×1 state vector, and yt is an Ny ×1 observed vector.

The transition (or state) equation is denoted as a first-order Markov process of the state vector.

where at is an Nz ×1 vector, Ft is an Nz ×Nz transition matrix, and the sequence of transition noise is independent. This equation is often called a shifted transition equation because the state vector is shifted forward one time period. The transition equation can also be denoted using an alternative specification
There is no real difference between the shifted transition equation and this alternative equation if the observation noise and transition equation noise are uncorrelated, that is, . It is assumed that
where
De Jong (1991a) proposed a diffuse Kalman filter that can handle an arbitrarily large initial state covariance matrix. The diffuse initial state assumption is reasonable if you encounter the case of parameter uncertainty or SSM nonstationarity. The SSM of the diffuse Kalman filter is written
where is a random variable with a mean of and a variance of . When , the SSM is said to be diffuse.

The following IML Kalman filter calls are supported:

KALCVF
performs covariance filtering and prediction

KALCVS
performs fixed-interval smoothing

KALDFF
performs diffuse covariance filtering and prediction

KALDFS
performs diffuse fixed-interval smoothing
The KALCVF call computes the one-step prediction and the filtered estimate , together with their covariance matrices and , using forward recursions. You can obtain the k-step prediction and its covariance matrix with the KALCVF call. The KALCVS call uses backward recursions to compute the smoothed estimate and its covariance matrix when there are T observations in the complete data.

The KALDFF call produces one-step prediction of the state and the unobserved random vector as well as their covariance matrices. The KALDFS call computes the smoothed estimate and its covariance matrix .