Fixed-interval Smoothing

MATLAB MEX Function Reference

Fixed-interval Smoothing (kalcvs.dll)

Fixed-interval smoothing is concerned with the smoothing of a finite set of data, i.e., with obtaining ${z}_{t| T}$ for fixed T and all t in the interval t = 1, ... , T.

Syntax

[sm, vsm] = kalcvs(data, a, F, b, H, var, pred, vpred)
[sm, vsm] = kalcvs(data, a, F, b, H, var, pred, vpred, un, vun)

Description

KALCVS uses backward recursions to compute the smoothed estimate ${z}_{t| T}$ and its covariance matrix, ${P}_{t| T}$ , where T is the number of observations in the complete data set.

The inputs to the KALCVS function are as follows.

data: is a N_y×T matrix containing data ( y₁, ... , y_T)'.
a: is an N_z×1 vector for a time-invariant input vector in the transition equation, or a N_z×T vector containing input vectors in the transition equation.
F: is an N_z×N_z matrix for a time-invariant transition matrix in the transition equation, or a N_z×N_z×T matrix containing T transition matrices.
b: is an N_y×1 vector for a time-invariant input vector in the measurement equation, or a N_y×T vector containing input vectors in the measurement equation.
H: is an N_y×N_z matrix for a time-invariant measurement matrix in the measurement equation, or a N_y×N_z×T matrix containing T time variant H_t matrices in the measurement equation.
var: is an (N_y+N_z)×(N_y+N_z) covariance matrix for the errors in the transition and the measurement equations, or a (N_y+N_z)×(N_y + N_z)×T matrix containing covariance matrices in the transition equation and measurement equation noises, that is, $(\eta^'_t, \epsilon^'_t)^'$ .
pred: is a N_z×T matrix containing one-step forecasts $({z}_{1|}, ... , {z}_{T| T-1})^'$ .
vpred: is a N_z×N_z×T matrix containing mean square error matrices of predicted state vectors $({P}_{1|}, ... , {P}_{T| T-1})^'$ .
un: is an optional N_z×1 vector containing u_T. The returned value is u₀.
vun: is an optional N_z×N_z matrix containing U_T. The returned value is U₀.

The KALCVS function returns the following values:

sm: is a N_z×T matrix containing smoothed state vectors $({z}_{1| T}, ... , {z}_{T| T})^'$ .
vsm: is a N_z×N_z×T matrix containing covariance matrices of smoothed state vectors $({P}_{1| T}, ... , {P}_{T| T})^'$ .

Algorithm

When the Kalman filtering is performed using KALCVF function, the KALCVS function computes smoothed state vectors and their covariance matrices. The fixed-interval smoothing state vector at time t is obtained by the conditional expectation given all observations.

The smoothing algorithm uses one-step forecasts and their covariance matrices, which are obtained using KALCVF function. For notation, ${z}_{t| T}$ is the smoothed value of the state vector z_t, and the mean square error matrix is denoted ${P}_{t| T}$ . For smoothing,

$\hat{\epsilon}_t & = & {y}_t - {b}_t - {H}_t {z}_{t| t-1} \ {D}_t & = & {H}_t ... ...{t-1} \ {P}_{t| T} & = & {P}_{t| t-1} - {P}_{t| t-1} {U}_{t-1} {P}_{t| t-1}$

where t = T, T-1, ... , 1. The initial values are u_T = 0 and U_T = 0.

When the SSM is specified using the alternative transition equation

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

the fixed-interval smoothing is performed using the following backward recursions:

$\hat{\epsilon}_t & = & {y}_t - {b}_t - {H}_t {z}_{t| t-1} \ {D}_t & = & {H}_t ... ...{t-1} \ {P}_{t| T} & = & {P}_{t| t-1} - {P}_{t| t-1} {U}_{t-1} {P}_{t| t-1}$

where it is assumed that G_t = 0.

You can use the KALCVS function regardless of the specification of the transition equation when G_t = 0. Harvey (1991) gives the following fixed-interval smoothing formula, which produces the same smoothed value:

${z}_{t| T} & = & {z}_{t| t} + {P}^*_t ({z}_{t+1| T} - {z}_{t+1| t}) \ {P}_{t| T} & = & {P}_{t| t} + {P}^*_t ({P}_{t+1| T} - {P}_{t+1| t}) {P}^{*'_t$

where

${P}^*_t = {P}_{t| t} {F}^'_t {P}^-_{t+1| t}$

under the shifted transition equation, but

${P}^*_t = {P}_{t| t} {F}^'_{t+1} {P}_{t+1| t}^-$

under the alternative transition equation.

The KALCVS function is accompanied by the KALCVF function, as shown in the following code. Note that you do not need to specify UN and VUN.

[logl, pred, vpred] = kalcvf(y, 0, a, F, b, H, var);
[sm, vsm] = kalcvs(y, a, F, b, H, var, pred, vpred);

You can also compute the smoothed estimate and its covariance matrix on an observation-by-observation basis. When the SSM is time invariant, the following example performs smoothing. In this situation, you should initialize UN and VUN as matrices of value 0.

[logl, pred, vpred] = kalcvf(y, 0, a, F, b, H, var);
n = size(y,2);
nz = size(F,1);
un = zeros(nz,1);
vun = zeros(nz,nz);
sm = zeros(nz,n);
vsm = zeros(nz,nz,n);
for i=n:-1:1
	y_i = y(:,i);
	pred_i  = pred(:,i);
	vpred_i = vpred(:,:,i);
	[sm_i, vsm_i] = kalcvs(y_i, a, F, b, H, var, pred_i, vpred_i, un, vun);
	sm(:,i)  = sm_i;
	vsm(:,:,i) = vsm_i;
end

See also

KALCVF performs covariance filtering and prediction

Getting Started with State Space Models

Kalman Filtering Example 1: Likelihood Function Evaluation

Kalman Filtering Example 2: Estimating an SSM Using the EM Algorithm

References

[1] Harvey, A.C., Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge: Cambridge University Press, 1991.

[2] Anderson, B.D.O., and J.B. Moore, Optimal Filtering, Englewood Cliffs, NJ: Prentice-Hall, 1979.

[3] Hamilton, J.D., Time Series Analysis, Princeton, 1994.

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