com.imsl.stat
Class KalmanFilter

java.lang.Object
  com.imsl.stat.KalmanFilter

All Implemented Interfaces:

Cloneable, Serializable

public class KalmanFilter

extends Object

implements Serializable, Cloneable

Performs Kalman filtering and evaluates the likelihood function for the state-space model.

Class KalmanFilter is based on a recursive algorithm given by Kalman (1960), which has come to be known as the Kalman filter. The underlying model is known as the state-space model. The model is specified stage by stage where the stages generally correspond to time points at which the observations become available. KalmanFilter avoids many of the computations and storage requirements that would be necessary if one were to process all the data at the end of each stage in order to estimate the state vector. This is accomplished by using previous computations and retaining in storage only those items essential for processing of future observations.

The notation used here follows that of Sallas and Harville (1981). Let (input in y using method update) be the vector of observations that become available at time k. The subscript k is used here rather than t, which is more customary in time series, to emphasize that the model is expressed in stages and that these stages need not correspond to equally spaced time points. In fact, they need not correspond to time points of any kind. The observation equation for the state-space model is

Here, (input in z using method update) is an known matrix and is the state vector. The state vector is allowed to change with time in accordance with the state equation

starting with .

The change in the state vector from time k to k + 1 is explained in part by the transition matrix (the identity matrix by default, or optionally using method setTransitionMatrix), which is assumed known. It is assumed that the q-dimensional are independently distributed multivariate normal with mean vector 0 and variance-covariance matrix , that the -dimensional are independently distributed multivariate normal with mean vector 0 and variance-covariance matrix , and that the and are independent of each other. Here, is the mean of and is assumed known, is an unknown positive scalar. (input in Q) and (input in R) are assumed known.

Denote the estimator of the realization of the state vector given the observations by

By definition, the mean squared error matrix for

is

At the time of the k-th invocation, we have

and

, which were computed from the k-1-st invocation, input in b and covb, respectively. During the k-th invocation, KalmanFilter computes the filtered estimate

along with . These quantities are given by the update equations:

where

and where

Here, (stored in getPredictionError) is the one-step-ahead prediction error, and is the variance-covariance matrix for . is obtained from method getCovV. The "start-up values" needed on the first invocation of KalmanFilter are

and input via b and covb, respectively. Computations for the k-th invocation are completed by KalmanFilter computing the one-step-ahead estimate

along with given by the prediction equations:

If both the filtered estimates and one-step-ahead estimates are needed by the user at each time point, KalmanFilter can be used twice for each time point-first without methods SetTransitionMatrix and setQ to produce

and , and second without method update to produce

and (Without methods SetTransitionMatrix and setQ, the prediction equations are skipped. Without method update, the update equations are skipped.).

Often, one desires the estimate of the state vector more than one-step-ahead, i.e., an estimate of

is needed where . At time j, KalmanFilter is invoked with method update to compute

Subsequent invocations of KalmanFilter without method update can compute

Computations for

and assume the variance-covariance matrices of the errors in the observation equation and state equation are known up to an unknown positive scalar multiplier, . The maximum likelihood estimate of based on the observations , is given by

where

N and SS are input arguments rank and SumofSquares. Updated values are obtained from methods getRank and getSumofSquares

If is known, the and can be input as the variance-covariance matrices exactly. The earlier discussion is then simplified by letting .

In practice, the matrices , , and are generally not completely known. They may be known functions of an unknown parameter vector . In this case, KalmanFilter can be used in conjunction with an optimization class (see MinUnconMultiVar, JMSL Math package), to obtain a maximum likelihood estimate of . The natural logarithm of the likelihood function for differs by no more than an additive constant from

(Harvey 1981, page 14, equation 2.21).

Here,

(input in logDeterminant, updated by getLogDeterminant) is the natural logarithm of the determinant of V where is the variance-covariance matrix of the observations.

Minimization of over all and produces maximum likelihood estimates. Equivalently, minimization of where

produces maximum likelihood estimates

Minimization of instead of , reduces the dimension of the minimization problem by one. The two optimization problems are equivalent since

minimizes for all , consequently,

can be substituted for in to give a function that differs by no more than an additive constant from .

The earlier discussion assumed to be nonsingular. If is singular, a modification for singular distributions described by Rao (1973, pages 527-528) is used. The necessary changes in the preceding discussion are as follows:

1. Replace by a generalized inverse.
2. Replace by the product of the nonzero eigenvalues of .
3. Replace N by

Maximum likelihood estimation of parameters in the Kalman filter is discussed by Sallas and Harville (1988) and Harvey (1981, pages 111-113).

See Also:

Example, Serialized Form

Constructor Summary

KalmanFilter(double[] b, double[] covb, int rank, double sumOfSquaress, double logDeterminant)
Constructor for KalmanFilter.

Method Summary

void	filter() Performs Kalman filtering and evaluates the likelihood function for the state-space model.
double[]	getCovB() Returns the mean squared error matrix for b divided by sigma squared.
double[][]	getCovV() Returns the variance-covariance matrix of v divided by sigma squared.
double	getLogDeterminant() Returns the natural log of the product of the nonzero eigenvalues of P where P * sigma2 is the variance-covariance matrix of the observations.
double[]	getPredictionError() Returns the one-step-ahead prediction error.
int	getRank() Returns the rank of the variance-covariance matrix for all the observations.
double[]	getStateVector() Returns the estimated state vector at time k + 1 given the observations through time k.
double	getSumOfSquares() Returns the generalized sum of squares.
void	setQ(double[][] q) Sets the Q matrix.
void	setTolerance(double tolerance) Sets the tolerance used in determining linear dependence.
void	setTransitionMatrix(double[][] t) Sets the transition matrix.
void	update(double[] y, double[][] z, double[][] r) Performs computation of the update equations.

Methods inherited from class java.lang.Object

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

Constructor Detail

KalmanFilter

public KalmanFilter(double[] b,
                    double[] covb,
                    int rank,
                    double sumOfSquaress,
                    double logDeterminant)

Constructor for KalmanFilter.

Parameters:

b - A double array containing the estimated state vector. b is the estimated state vector at time k given the observations through time k-1.

covb - A double array of size b.length by b.length such that covb * is the mean squared error matrix for b.

rank - An int scalar containing the rank of the variance-covariance matrix for all the observations.

sumOfSquaress - A double scalar containing the generalized sum of squares.

logDeterminant - A double scalar containing the natural log of the product of the nonzero eigenvalues of P where P * is the variance-covariance matrix of the observations.

Throws:

IllegalArgumentException - is thrown if the dimensions of b, and covb are not consistent.

Method Detail

filter

public final void filter()

Performs Kalman filtering and evaluates the likelihood function for the state-space model.

getCovB

public double[] getCovB()

Returns the mean squared error matrix for b divided by sigma squared.

Returns:

a double array of size b.length by b.length such that covb * is the mean squared error matrix for b.

getCovV

public double[][] getCovV()

Returns the variance-covariance matrix of v divided by sigma squared.

Returns:

a double matrix containing a y.length by y.length matrix such that covv * is the variance-covariance matrix of the one-step-ahead prediction error, getPredictionError.

getLogDeterminant

public double getLogDeterminant()

Returns the natural log of the product of the nonzero eigenvalues of P where P * sigma² is the variance-covariance matrix of the observations.

Returns:

a double scalar containing the natural log of the product of the nonzero eigenvalues of P where P * is the variance-covariance matrix of the observations. In the usual case when P is nonsingular, logDeterminant is the natural log of the determinant of P.

getPredictionError

public double[] getPredictionError()

Returns the one-step-ahead prediction error.

Returns:

a double array of size y.length containing the one-step-ahead prediction error.

getRank

public int getRank()

Returns the rank of the variance-covariance matrix for all the observations.

Returns:

An int scalar containing the rank of the variance-covariance matrix for all the observations.

getStateVector

public double[] getStateVector()

Returns the estimated state vector at time k + 1 given the observations through time k.

Returns:

a double array containing the estimated state vector at time k + 1 given the observations through time k.

getSumOfSquares

public double getSumOfSquares()

Returns the generalized sum of squares.

Returns:

a double scalar containing the generalized sum of squares. The estimate of is given by sumOfSquares / rank.

setQ

public void setQ(double[][] q)

Sets the Q matrix.

Parameters:

q - A double matrix containing the b.length by b.length matrix such that q * is the variance-covariance matrix of the error vector in the state equation. Default: There is no error term in the state equation.

setTolerance

public void setTolerance(double tolerance)

Sets the tolerance used in determining linear dependence.

Parameters:

tolerance - A double scalar containing the tolerance used in determining linear dependence. Default: tolerance = 100.0*2.2204460492503131e-16.

setTransitionMatrix

public void setTransitionMatrix(double[][] t)

Sets the transition matrix.

Parameters:

t - A double matrix containing the b.length by b.length transition matrix in the state equation. Default: t = identity matrix

update

public void update(double[] y,
                   double[][] z,
                   double[][] r)

Performs computation of the update equations.

Parameters:

y - A double array containing the observations.

z - A double matrix containing the y.length by b.length matrix relating the observations to the state vector in the observation equation.

r - A double matrix containing the y.length by y.length matrix such that r * is the variance-covariance matrix of errors in the observation equation. is a positive unknown scalar. Only elements in the upper triangle of r are referenced.