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

$y_t = b_t + H_t z_t + \epsilon_t$

where b_t is an N_y ×1 vector, H_t is an N_y ×N_z matrix, the sequence of observation noise $\epsilon_t$ is independent, z_t is an N_z ×1 state vector, and y_t is an N_y ×1 observed vector.

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

$z_{t+1} = a_t + F_t z_t + \eta_t$

where a_t is an N_z ×1 vector, F_t is an N_z ×N_z transition matrix, and the sequence of transition noise $\eta_t$ 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

$z_t = a_t + F_t z_{t-1} + \eta_t$

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, $E(\eta_t \epsilon^'_t) = 0$ . It is assumed that

$E(\eta_t \eta^'_s) & = & V_t \delta_{ts} \E(\epsilon_t \epsilon^'_s) & = & R_t \delta_{ts} \E(\eta_t \epsilon^'_s) & = & G_t \delta_{ts} \$

where

$\delta_{ts} = \{ 1 & { if } t = s \ 0 & { if } t \neq s .$

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

$y_t & = & X_t \beta + H_t z_t + \epsilon_t \z_{t+1} & = & W_t \beta + F_t z_t + \eta_t \z_0 & = & a+ A\delta \\beta & = & b+ B\delta$

where $\delta$ is a random variable with a mean of $\mu$ and a variance of $\sigma^2\Sigma$ . When $\Sigma arrow \infty$ , 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 $z_{t+1| t}$ and the filtered estimate $z_{t| t}$ , together with their covariance matrices $P_{t+1| t}$ and $P_{t| t}$ , using forward recursions. You can obtain the k-step prediction $z_{t+k| t}$ and its covariance matrix $P_{t+k| t}$ with the KALCVF call. The KALCVS call uses backward recursions to compute the smoothed estimate $z_{t| T}$ and its covariance matrix $P_{t| T}$ when there are T observations in the complete data.

The KALDFF call produces one-step prediction of the state and the unobserved random vector $\delta$ as well as their covariance matrices. The KALDFS call computes the smoothed estimate $z_{t| T}$ and its covariance matrix $P_{t| T}$ .

See Chapter 17, "Language Reference," for more information about Kalman filtering subroutines.

Kalman Filter Subroutines

Example 10.2: Kalman Filtering: Likelihood Function Evaluation

Example 10.3: Kalman Filtering: Estimating an SSM Using the EM Algorithm

Example 10.4: Diffuse Kalman Filtering