dev.nm.stat.timeseries.linear.multivariate.stationaryprocess

Class MultivariateForecastOneStep

java.lang.Object

dev.nm.stat.timeseries.linear.multivariate.stationaryprocess.MultivariateForecastOneStep

public class MultivariateForecastOneStep
extends Object

The innovation algorithm is an efficient way to obtain a one step least square linear predictor for a multivariate linear time series with known auto-covariance and these properties (not limited to ARMA processes):

• \(\{x_{t}\}\) can be non-stationary.
• \(E(x_{t}) = 0\) for all t.

See Also:

• "P. J. Brockwell and R. A. Davis, "Proposition. 5.2.2. Chapter 5. Prediction of Stationary Processes," in Time Series: Theory and Methods, Springer, 2006."
• "P. J. Brockwell and R. A. Davis, "Proposition. 11.4.2. Chapter 11.4 Best Linear Predictors of Second Order Random Vectors," in Time Series: Theory and Methods, Springer, 2006."

Constructor Summary

Constructors

Constructor and Description
MultivariateForecastOneStep(MultivariateIntTimeTimeSeries Xt, MultivariateAutoCovarianceFunction K) Construct an instance of InnovationAlgorithm for a multivariate time series with known auto-covariance structure.

Method Summary

Modifier and Type	Method and Description
ImmutableMatrix	covariance(int n) Get the covariance matrix for prediction errors for \(\hat{x}_{n+1}\), made at time n.
ImmutableMatrix	theta(int i, int j) Get the coefficients of the linear predictor.
ImmutableVector	xHat(int n) Get the one-step prediction \(\hat{X}_{n+1} = P_{\mathfrak{S_n}}X_{n+1}\), made at time n.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

MultivariateForecastOneStep

public MultivariateForecastOneStep(MultivariateIntTimeTimeSeries Xt,
                                   MultivariateAutoCovarianceFunction K)

Construct an instance of InnovationAlgorithm for a multivariate time series with known auto-covariance structure.

Parameters:

Xt - an m-dimensional time series, length t

K - auto-covariance function K(i, j) = E(Xi * Xj'), a m x m matrix

Method Detail

xHat

public ImmutableVector xHat(int n)

Get the one-step prediction \(\hat{X}_{n+1} = P_{\mathfrak{S_n}}X_{n+1}\), made at time n.

Parameters:

n - time, ranging from 0 to T, the end of observation time

Returns:

the one-step prediction \(\hat{X}_{n+1}\)

theta

public ImmutableMatrix theta(int i,
                             int j)

Get the coefficients of the linear predictor.

Parameters:

i - i, ranging from 1 to t

j - j, ranging from 1 to t

Returns:

Θ[i][j]

covariance

public ImmutableMatrix covariance(int n)

Get the covariance matrix for prediction errors for \(\hat{x}_{n+1}\), made at time n.

Parameters:

n - time, ranging from 0 to T, the end of observation time

Returns:

the covariance matrix for prediction errors at time n