Time Series Analysis and Examples: Example 10.4: Diffuse Kalman Filtering

Time Series Analysis and Examples

Example 10.4: Diffuse Kalman Filtering

The nonstationary SSM is simulated to analyze the diffuse Kalman filter call KALDFF. The transition equation is generated by using the following formula:

$[ z_{1t} \ z_{2t} ] = [ 1.5 & -0.5 \ 1.0 & 0.0 ] [ z_{1t-1} \ z_{2t-1} ] + [ \eta_{1t} \ 0 ]$

where $\eta_{1t} \sim n(0,1)$ . The transition equation is nonstationary since the transition matrix

has one unit root. Here is the code:

  
    proc iml; 
       z_1 = 0; z_2 = 0; 
       do i = 1 to 30; 
          z = 1.5*z_1 - .5*z_2 + rannor(1234567); 
          z_2 = z_1; 
          z_1 = z; 
          x =  z + .8*rannor(1234578); 
          if ( i > 10 ) then y = y // x; 
       end;

The KALDFF and KALCVF calls produce one-step prediction, and the result shows that two predictions coincide after the fifth observation (Output 10.4.1). Here is the code:

  
       t = nrow(y); 
       h = { 1 0 }; 
       f = { 1.5 -.5, 1 0 }; 
       rt = .64; 
       vt = diag({1 0}); 
       ny = nrow(h); 
       nz = ncol(h); 
       nb = nz; 
       nd = nz; 
       a  = j(nz,1,0); 
       b  = j(ny,1,0); 
       int = j(ny+nz,nb,0); 
       coef = f // h; 
       var = ( vt || j(nz,ny,0) ) // 
             ( j(ny,nz,0) || rt ); 
       intd = j(nz+nb,1,0); 
       coefd = i(nz) // j(nb,nd,0); 
       at = j(t*nz,nd+1,0); 
       mt = j(t*nz,nz,0); 
       qt = j(t*(nd+1),nd+1,0); 
       n0 = -1; 
       call kaldff(kaldff_p,dvpred,initial,s2,y,0,int, 
                   coef,var,intd,coefd,n0,at,mt,qt); 
       call kalcvf(kalcvf_p,vpred,filt,vfilt,y,0,a,f,b,h,var); 
       print kalcvf_p kaldff_p;

Output 10.4.1: Diffuse Kalman Filtering

Diffuse Kalman Filtering

KALCVF_P		KALDFF_P
0	0	0	0
1.441911	0.961274	1.1214871	0.9612746
-0.882128	-0.267663	-0.882138	-0.267667
-0.723156	-0.527704	-0.723158	-0.527706
1.2964969	0.871659	1.2964968	0.8716585
-0.035692	0.1379633	-0.035692	0.1379633
-2.698135	-1.967344	-2.698135	-1.967344
-5.010039	-4.158022	-5.010039	-4.158022
-9.048134	-7.719107	-9.048134	-7.719107
-8.993153	-8.508513	-8.993153	-8.508513
-11.16619	-10.44119	-11.16619	-10.44119
-10.42932	-10.34166	-10.42932	-10.34166
-8.331091	-8.822777	-8.331091	-8.822777
-9.578258	-9.450848	-9.578258	-9.450848
-6.526855	-7.241927	-6.526855	-7.241927
-5.218651	-5.813854	-5.218651	-5.813854
-5.01855	-5.291777	-5.01855	-5.291777
-6.5699	-6.284522	-6.5699	-6.284522
-4.613301	-4.995434	-4.613301	-4.995434
-5.057926	-5.09007	-5.057926	-5.09007

The likelihood function for the diffuse Kalman filter under the finite initial covariance matrix $\sigma_\delta$ is written

$\lambda({y}) = -\frac{1}2[{y}^\char93 \log(\hat{\sigma}^2) + \sum_{t=1}^t \log(|{d}_t|)]$

where ${y}^(\char93 )$ is the dimension of the matrix $({y}^'_1, ... , {y}^'_t)^'$ . The likelihood function for the diffuse Kalman filter under the diffuse initial covariance matrix $(\sigma_\delta arrow \infty)$ is computed as $\lambda({y}) - \frac{1}2\log(|{s}|)$ , where the

matrix is the upper $n_\delta x n_\delta$ matrix of ${q}_t$ . Output 10.4.2 displays the log likelihood and the diffuse log likelihood. Here is the code:

  
       d = 0; 
       do i = 1 to t; 
          dt = h*mt[(i-1)*nz+1:i*nz,]*h` + rt; 
          d = d + log(det(dt)); 
       end; 
       s = qt[(t-1)*(nd+1)+1:t*(nd+1)-1,1:nd]; 
       log_l = -(t*log(s2) + d)/2; 
       dff_logl = log_l - log(det(s))/2; 
       print log_l dff_logl;

Output 10.4.2: Diffuse Likelihood Function

Diffuse Kalman Filtering

	LOG_L
Log L	-11.42547

	DFF_LOGL
Diffuse Log L	-9.457596

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