Language Reference: KALDFF Call :: SAS/IML(R) 9.22 User's Guide

Language Reference

KALDFF Call

CALL KALDFF( pred, vpred, initial, s2, data, lead, int, coef, var, intd, coefd <*>, n0 <*>, at <*>, mt <*>, qt ) ;

The KALDFF subroutine computes the one-step forecast of state vectors in an SSM by using the diffuse Kalman filter. The call estimates the conditional expectation of $\text{[math]}$ , and also estimates the initial random vector, $\text{[math]}$ , and its covariance matrix.

The input arguments to the KALDFF subroutine are as follows:

data: is a $\text{[math]}$ matrix that contains data $\text{[math]}$ .
lead: is the number of steps to forecast after the end of the data set.
int: is an $\text{[math]}$ matrix for a time-invariant fixed matrix, or a $\text{[math]}$ matrix that contains fixed matrices for the time-variant model in the transition equation and the measurement equation—that is, $\text{[math]}$ .
coef: is an $\text{[math]}$ matrix for a time-invariant coefficient, or a $\text{[math]}$ matrix that contains coefficients at each time in the transition equation and the measurement equation—that is, $\text{[math]}$ .
var: is an $\text{[math]}$ matrix for a time-invariant variance matrix for the error in the transition equation and the error in the measurement equation, or a $\text{[math]}$ matrix that contains covariance matrices for the error in the transition equation and the error in the measurement equation—that is, $\text{[math]}$ .
intd: is an $\text{[math]}$ vector that contains the intercept term in the equation for the initial state vector $\text{[math]}$ and the mean effect $\text{[math]}$ —that is, $\text{[math]}$ .
coefd: is an $\text{[math]}$ matrix that contains coefficients for the initial state $\text{[math]}$ in the equation for the initial state vector $\text{[math]}$ and the mean effect $\text{[math]}$ —that is, $\text{[math]}$ .
n0: is an optional scalar including an initial denominator. If $\text{[math]}$ , the denominator for $\text{[math]}$ is $\text{[math]}$ plus the number $\text{[math]}$ of elements of $\text{[math]}$ . If $\text{[math]}$ or $\text{[math]}$ is not specified, the denominator for $\text{[math]}$ is $\text{[math]}$ . With $\text{[math]}$ , the initial values, $\text{[math]}$ , and $\text{[math]}$ , are assumed to be known and, hence, $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ are used for input that contains the initial values. If the value of $\text{[math]}$ is negative or $\text{[math]}$ is not specified, the initial values for $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ are computed. The value of $\text{[math]}$ is updated as $\text{[math]}$ after the KALDFF call.
at: is an optional $\text{[math]}$ matrix. If $\text{[math]}$ , $\text{[math]}$ contains $\text{[math]}$ . However, only the first matrix $\text{[math]}$ is used as input. When you specify the KALDFF call, $\text{[math]}$ returns $\text{[math]}$ . If $\text{[math]}$ is negative or the matrix $\text{[math]}$ contains missing values, $\text{[math]}$ is automatically computed.
mt: is an optional $\text{[math]}$ matrix. If $\text{[math]}$ , $\text{[math]}$ contains $\text{[math]}$ . However, only the first matrix $\text{[math]}$ is used as input. If $\text{[math]}$ is negative or the matrix $\text{[math]}$ contains missing values, $\text{[math]}$ is used for output, and it contains $\text{[math]}$ . Note that the matrix $\text{[math]}$ can be used as an input matrix if either of the off-diagonal elements is not missing. The missing element $\text{[math]}$ is replaced by the nonmissing element $\text{[math]}$ .
qt: is an optional $\text{[math]}$ matrix. If $\text{[math]}$ , $\text{[math]}$ contains $\text{[math]}$ . However, only the first matrix $\text{[math]}$ is used as input. If $\text{[math]}$ is negative or the matrix $\text{[math]}$ contains missing values, $\text{[math]}$ is used for output and contains $\text{[math]}$ . The matrix $\text{[math]}$ can also be used as an input matrix if either of the off-diagonal elements is not missing since the missing element $\text{[math]}$ is replaced by the nonmissing element $\text{[math]}$ .

The KALDFF call returns the following values:

pred: is a $\text{[math]}$ matrix that contains estimated predicted state vectors $\text{[math]}$ .
vpred: is a $\text{[math]}$ matrix that contains estimated mean square errors of predicted state vectors $\text{[math]}$ .
initial: is an $\text{[math]}$ matrix that contains an estimate and its variance for initial state $\text{[math]}$ , that is, $\text{[math]}$ .
s2: is a scalar that contains the estimated variance $\text{[math]}$ .

The KALDFF call computes the one-step forecast of state vectors in an SSM by using the diffuse Kalman filter. The SSM for the diffuse Kalman filter is written

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

where $\text{[math]}$ is an $\text{[math]}$ state vector, $\text{[math]}$ is an $\text{[math]}$ observed vector, and

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

It is assumed that the noise vector $\text{[math]}$ is independent and $\text{[math]}$ is independent of the vector $\text{[math]}$ . The matrices, $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ , are assumed to be known. The KALDFF call estimates the conditional expectation of the state vector $\text{[math]}$ given the observations. The KALDFF subroutine also produces the estimates of the initial random vector $\text{[math]}$ and its covariance matrix. For $\text{[math]}$ -step forecasting where $\text{[math]}$ , the estimated conditional expectation at time $\text{[math]}$ is computed with observations given up to time $\text{[math]}$ . The estimated $\text{[math]}$ -step forecast and its estimated MSE are denoted $\text{[math]}$ and $\text{[math]}$ (for $\text{[math]}$ ). $\text{[math]}$ and $\text{[math]}$ are last-column-deleted submatrices of $\text{[math]}$ and $\text{[math]}$ , respectively. The algorithm for one-step prediction is given as follows:

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

where $\text{[math]}$ is the number of elements of $\text{[math]}$ plus $\text{[math]}$ . Unless initial values are given and $\text{[math]}$ , initial values are set as follows:

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

For $\text{[math]}$ -step forecasting where $\text{[math]}$ ,

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

If there is a missing observation, the KALDFF call computes the one-step forecast for the observation that follows the missing observation as the two-step forecast from the previous observation.

An example that uses the KALDFF call is in the documentation for the KALDFS call.

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