Language Reference: KALCVF Call :: SAS/IML(R) 9.2 User's Guide

Language Reference

KALCVF Call

computes the one-step prediction ${z}_{t+1| t}$ and the filtered estimate ${z}_{t| t}$ , as well as their covariance matrices. The call uses forward recursions, and you can also use it to obtain -step estimates.

CALL KALCVF( pred, vpred, filt, vfilt, data, lead, , , , ,

var <, z0, vz0>);

The inputs to the KALCVF subroutine are as follows:

data: is a matrix containing data $({y}_1, ... , {y}_t)^'$ .
lead: is the number of steps to forecast after the end of the data.
: is an vector for a time-invariant input vector in the transition equation, or a $(t+{lead})n_z x 1$ vector containing input vectors in the transition equation.
: is an matrix for a time-invariant transition matrix in the transition equation, or a $(t+{lead})n_z x n_z$ matrix containing transition matrices in the transition equation.
: is an vector for a time-invariant input vector in the measurement equation, or a $(t+{lead})n_y x 1$ vector containing input vectors in the measurement equation.
: is an matrix for a time-invariant measurement matrix in the measurement equation, or a $(t+{lead})n_y x n_z$ matrix containing measurement matrices in the measurement equation.
var: is an matrix for a time-invariant variance matrix for the error in the transition equation and the error in the measurement equation, or a $(t+{lead})(n_y + n_z) x (n_y + n_z)$ matrix containing variance matrices for the error in the transition equation and the error in the measurement equation - that is, $(\eta^'_t, \epsilon^'_t)^'$ .
: is an optional initial state vector ${z}^'_{1|}$ .
: is an optional covariance matrix of an initial state vector ${p}_{1|}$ .

The KALCVF call returns the following values:

pred: is a $(t+{lead}) x n_z$ matrix containing one-step predicted state vectors $({z}_{1|}, ... , {z}_{t+1| t}, {z}_{t+2| t}, ... , {z}_{t+{lead}| t})^'$ .
vpred: is a $(t+{lead})n_z x n_z$ matrix containing mean square errors of predicted state vectors $({p}_{1|}, ... , {p}_{t+1| t}, {p}_{t+2| t}, ... , {p}_{t+{lead}| t})^'$ .
filt: is a matrix containing filtered state vectors $({z}_{1| 1}, ... , {z}_{t| t})^'$ .
vfilt: is a matrix containing mean square errors of filtered state vectors $({p}_{1| 1}, ... , {p}_{t| t})^'$ .

The KALCVF call computes the conditional expectation of the state vector ${z}_t$ given the observations, assuming that the mean and the variance of the initial state vector are known. The filtered value is the conditional expectation of the state vector ${z}_t$ given the observations up to time

. For

-step forecasting where $k\gt$ , the conditional expectation at time

is computed given observations up to

. For notation, ${v}_t$ and ${{r}}_t$ are variances of $\eta_t$ and ${\epsilon}_t$ , respectively, and ${g}_t$ is a covariance of $\eta_t$ and ${\epsilon}_t$ . ${a}^-$ stands for the generalized inverse of

. The filtered value and its covariance matrix are denoted ${z}_{t| t}$ and ${p}_{t| t}$ , respectively. For $k\gt$ , ${z}_{t+k| t}$ and ${p}_{t+k| t}$ stand for the

-step forecast of ${z}_{t+k}$ and its mean square error. The Kalman filtering algorithm for one-step prediction and filtering is given as follows:

$\hat{\epsilon}_t & = & {y}_t - {b}_t - {h}_t {z}_{t| t-1} \ {d}_t & = & {h}_t ... ... {p}_{t+1| t} & = & {f}_t {p}_{t| t-1}{f}^'_t + {v}_t - {k}_t {d}_t {k}^'_t$

And for

-step forecasting for $k\gt 1$ ,

${z}_{t+k| t} & = & {a}_{t+k-1} + {f}_{t+k-1} {z}_{t+k-1| t} \ {p}_{t+k| t} & = & {f}_{t+k-1} {p}_{t+k-1| t} {f}^'_{t+k-1} + {v}_{t+k-1}$

When you use the alternative transition equation

${z}_t = {a}_t + {f}_t{z}_{t-1} + \eta_t$

the forward recursion algorithm is written

$\hat{\epsilon}_t & = & {y}_t - {b}_t - {h}_t{z}_{t| t-1} \ {d}_t & = & {h}_t {... ... t} & = & {f}_{t+1} {p}_{t| t-1}{f}^'_{t+1} + {v}_{t+1} - {k}_t {d}_t {k}^'_t$

And for

-step forecasting $(k\gt 1)$ ,

${z}_{t+k| t} & = & {a}_{t+k} + {f}_{t+k}{z}_{t+k-1| t} \ {p}_{t+k| t} & = & {f}_{t+k} {p}_{t+k-1| t} {f}^'_{t+k} + {v}_{t+k}$

You can use the KALCVF call when you specify the alternative transition equation and ${g}_t = 0$ .

The initial state vector and its covariance matrix of the time invariant Kalman filters are computed under the stationarity condition

${z}_{1|} & = & ({i}- {f})^- {a}\ {p}_{1|} & = & ({i}- {f}\otimes {f})^- {vec}({v})$

where

and

are the time-invariant transition matrix and the covariance matrix of transition equation noise, and vec

is an

column vector that is constructed by the stacking

columns of matrix

. Note that all eigenvalues of the matrix

are inside the unit circle when the SSM is stationary. When the preceding formula cannot be applied, the initial state vector estimate ${z}_{1|}$ is set to ${a}_1$ and its covariance matrix ${p}_{1|}$ is given by

I. Optionally, you can specify initial values.

The KALCVF call accepts missing values in observations. If there is a missing observation, the filtered state vector for the missing observation is given by the one-step forecast.

The following program gives an example of the KALCVF call:

  
    q=2; 
    p=2; 
    n=10; 
    lead=3; 
  
    total=n+lead; 
  
    seed = 25735; 
    x=round(10*normal(j(n,p,seed)))/10; 
    f=round(10*normal(j(q*total,q,seed)))/10; 
    a=round(10*normal(j(total*q,1,seed)))/10; 
    h=round(10*normal(j(p*total,q,seed)))/10; 
    b=round(10*normal(j(p*total,1,seed)))/10; 
    do i = 1 to total; 
    temp=round(10*normal(j(p+q,p+q,seed)))/10; 
    var=var//(temp*temp`); 
    end; 
  
    call kalcvf(pred,vpred,filt,vfilt,x,lead,a,f,b,h,var); 
  
    /* default initial state and covariance */ 
    call kalcvs(sm,vsm,x,a,f,b,h,var,pred,vpred); 
    print sm [format=9.4] vsm [format=9.4];

This program produces the following output:

  
                SM                 VSM 
  
           -1.5236   -0.1000    1.5813   -0.4779 
            0.3058   -0.1131   -0.4779    0.3963 
           -0.2593    0.2496    2.4629    0.2426 
           -0.5533    0.0332    0.2426    0.0944 
           -0.5813    0.1251    0.2023   -0.0228 
           -0.3017    0.7480   -0.0228    0.5799 
            1.1333   -0.2144    0.8615   -0.7653 
            1.5193   -0.6237   -0.7653    1.2334 
           -0.6641   -0.7770    1.0836    0.8706 
            0.5994    2.3333    0.8706    1.5252 
                                0.3677    0.2510 
                                0.2510    0.2051 
                                0.3243   -0.4093 
                               -0.4093    1.2287 
                                0.1736   -0.0712 
                               -0.0712    0.9048 
                                1.3153    0.8748 
                                0.8748    1.6575 
                                8.6650    0.1841 
                                0.1841    4.4770

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