kalmanFilter

Purpose

Data filtering algorithm.

Format

kout = kalmanFilter(y, Z, d, H, T, c, R, Q, a_0, p_0)

Parameters:

• yt (px1 matrix) – numerical vector of times series data.

• Z (pxM matrix) – design matrix.

• d (px1 matrix) – observation intercept.

• H (pxp matrix) – observation disturbance covariance.

• T (mxm matrix) – transition matrix.

• c (px1 matrix) – state intercept.

• R (mxr matrix) – selection matrix.

• Q (rxr matrix) – state disturbance covariance matrix.

• a_0 (mx1 matrix) – initial prior state mean.

• p_0 (mxm matrix) – initial prior covariance matrix.

Returns:

kout (struct) –

Instance of a kalmanOut structure. The following members of kout are referencing within this routine:

kout.filtered_state	Matrix, pxT, filtered states.
kout.filtered_state_cov	Array, Txpxp, filtered state covariances.
kout.predicted_state	Matrix, px(T+1), predicted states.
kout.predicted_state_cov	Array, Txpxp, predicted state covariances.
kout.forecast	Matrix, pxT, forecasts.
kout.forecast_error	Matrix, pxT, forecast error.
kout.forecast_error_cov	Array, Txpxp, forecast error covariances.
kout.loglikelihood	Matrix, px(T+1), computed loglikelihood.

Example

new;
cls;
library tsmt;

// Load data
y = loadd( getGAUSSHome() $+ "pkgs/tsmt/examples/nile.dat");

// Set up state space matrices
sigma_e = 15099;
sigma_n = 1469.1;

// Linear level model
Z = 1;
H = sigma_e;
T = 1;
Q = sigma_n;
R = 1;
d = 0;
c = 0;

// Initial state variables
a_0 = 0;
p_0 = 10e7;

struct kalmanResult rslt;
rslt = kalmanFilter(y', Z, d, H, T, c, R, Q, a_0, p_0);

Remarks

The kalman filter inputs are based on the state space representation:

• \(y_t = d_t + Z_t\alpha_t + \epsilon_t\)

• \(\epsilon_t \sim N(0, H_t)\)

• \(\alpha_{t+1} = c_t + T_t\alpha_t + R\eta_t\)

• \(\eta_t \sim N(0, Q_t)\)

where the \(y_t\) equation is known as the observation or measurement equation, and the \(\alpha_t+1\) equation is the transition equation.

Library

tsmt

Source

kalman_filter.src