Example 10.4: Diffuse Kalman Filtering

Time Series Analysis and Control 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 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 F has one unit root.

   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).

      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⁽#) 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 S matrix is the upper $N_\delta x N_\delta$ matrix of Q_t. See the section "KALDFF Call" for more details on matrix notation. Output 10.4.2 displays the log likelihood and the diffuse log likelihood.

      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

	LIKL
Log L	-11.42547

	LIKL_DFF
Diffuse Log L	-9.457596

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