Example 10.2: Kalman Filtering: Likelihood Function Evaluation

Time Series Analysis and Control Examples

Example 10.2: Kalman Filtering: Likelihood Function Evaluation

In this example, the log likelihood function of the SSM is computed using prediction error decomposition. The annual real GNP series, y_t, can be decomposed as

$y_t = \mu_t + \epsilon_t$

where $\mu_t$ is a trend component and $\epsilon_t$ is a white noise error with $\epsilon_t \sim (0, \sigma^2_\epsilon)$ .Refer to Nelson and Plosser (1982) for more details on these data. The trend component is assumed to be generated from the following stochastic equations:

$\mu_t & = & \mu_{t-1} + \beta_{t-1} + \eta_{1t} \\beta_t & = & \beta_{t-1} + \eta_{2t}$

where $\eta_{1t}$ and $\eta_{2t}$ are independent white noise disturbances with $\eta_{1t} \sim (0, \sigma^2_{\eta_1})$ and $\eta_{2t} \sim (0, \sigma^2_{\eta_2})$ .

It is straightforward to construct the SSM of the real GNP series.

$y_t & = & H{z}_t + \epsilon_t \z_t & = & F{z}_{t-1} + \eta_t$

where

$H& = & (1, 0) \ F& = & [ 1 & 1 \ 0 & 1 ] \ z_t & = & (\mu_t, \beta_t)^' \\... ...ma^2_\epsilon & 0 & 0 \ 0 & \sigma^2_{\eta 1} & 0 \ 0 & 0 & \sigma^2_{\eta 2} ]$

When the observation noise $\epsilon_t$ is normally distributed, the average log likelihood function of the SSM is

$\ell & = & \frac{1}T \sum_{t=1}^T \ell_t \ \ell_t & = & -\frac{N_y}2\log(2\pi) ... ...ac{1}2\log(|{C}_t|) - \frac{1}2 \hat{\epsilon}^'_t C^{-1}_t \hat{\epsilon}_t$

where C_t is the mean square error matrix of the prediction error $\hat{\epsilon}_t$ , such that $C_t = H{P}_{t| t-1} H^' + R_t$ .

The LIK module computes the average log likelihood function. First, the average log likelihood function is computed using the default initial values: Z0=0 and VZ0=10⁶I. The second call of module LIK produces the average log likelihood function with the given initial conditions: Z0=0 and VZ0=10^-3I. You can notice a sizable difference between the uncertain initial condition (VZ0=10⁶I) and the almost deterministic initial condition (VZ0=10^-3I) in Output 10.2.1.

Finally, the first 15 observations of one-step predictions, filtered values, and real GNP series are produced under the moderate initial condition (VZ0=10I). The data are the annual real GNP for the years 1909 to 1969.

   title 'Likelihood Evaluation of SSM';
   title2 'DATA: Annual Real GNP 1909-1969';
   data gnp;
      input y @@;
   datalines;
   116.8 120.1 123.2 130.2 131.4 125.6 124.5 134.3
   135.2 151.8 146.4 139.0 127.8 147.0 165.9 165.5
   179.4 190.0 189.8 190.9 203.6 183.5 169.3 144.2
   141.5 154.3 169.5 193.0 203.2 192.9 209.4 227.2
   263.7 297.8 337.1 361.3 355.2 312.6 309.9 323.7
   324.1 355.3 383.4 395.1 412.8 406.0 438.0 446.1
   452.5 447.3 475.9 487.7 497.2 529.8 551.0 581.1
   617.8 658.1 675.2 706.6 724.7
   ;

   proc iml;
   start lik(y,a,b,f,h,var,z0,vz0);
       nz = nrow(f);
       n = nrow(y);
       k = ncol(y);
       const = k*log(8*atan(1));
       if ( sum(z0 = .) | sum(vz0 = .) ) then
          call kalcvf(pred,vpred,filt,vfilt,y,0,a,f,b,h,var);
       else
          call kalcvf(pred,vpred,filt,vfilt,y,0,a,f,b,h,var,z0,vz0);
       et = y - pred*h`;
       sum1 = 0;
       sum2 = 0;
       do i = 1 to n;
          vpred_i = vpred[(i-1)*nz+1:i*nz,];
          et_i = et[i,];
          ft = h*vpred_i*h` + var[nz+1:nz+k,nz+1:nz+k];
          sum1 = sum1 + log(det(ft));
          sum2 = sum2 + et_i*inv(ft)*et_i`;
       end;
       return(-const-.5*(sum1+sum2)/n);
   finish;

   start main;
      use gnp;
      read all var {y};

      f = {1 1, 0 1};
      h = {1 0};
      a = j(nrow(f),1,0);
      b = j(nrow(h),1,0);
      var = diag(j(1,nrow(f)+ncol(y),1e-3));
      /*-- initial values are computed --*/
      z0 = j(1,nrow(f),.);
      vz0 = j(nrow(f),nrow(f),.);
      logl = lik(y,a,b,f,h,var,z0,vz0);
      print 'No initial values are given', logl;
      /*-- initial values are given --*/
      z0 = j(1,nrow(f),0);
      vz0 = 1e-3#i(nrow(f));
      logl = lik(y,a,b,f,h,var,z0,vz0);
      print 'Initial values are given', logl;
      z0 = j(1,nrow(f),0);
      vz0 = 10#i(nrow(f));
      call kalcvf(pred,vpred,filt,vfilt,y,1,a,f,b,h,var,z0,vz0);
      print y pred filt;
   finish;
   run;

Output 10.2.1: Average Log Likelihood of SSM

Likelihood Evaluation of SSM

DATA: Annual Real GNP 1909-1969

LOGL
-26314.66

LOGL
-91884.41

Output 10.2.2 shows the observed data, the predicted state vectors, and the filtered state vectors for the first 16 observations.

Output 10.2.2: Filtering and One-Step Prediction

Y	PRED		FILT
116.8	0	0	116.78832	0
120.1	116.78832	0	120.09967	3.3106857
123.2	123.41035	3.3106857	123.22338	3.1938303
130.2	126.41721	3.1938303	129.59203	4.8825531
131.4	134.47459	4.8825531	131.93806	3.5758561
125.6	135.51391	3.5758561	127.36247	-0.610017
124.5	126.75246	-0.610017	124.90123	-1.560708
134.3	123.34052	-1.560708	132.34754	3.0651076
135.2	135.41265	3.0651076	135.23788	2.9753526
151.8	138.21324	2.9753526	149.37947	8.7100967
146.4	158.08957	8.7100967	148.48254	3.7761324
139	152.25867	3.7761324	141.36208	-1.82012
127.8	139.54196	-1.82012	129.89187	-6.776195
147	123.11568	-6.776195	142.74492	3.3049584
165.9	146.04988	3.3049584	162.36363	11.683345
165.5	174.04698	11.683345	167.02267	8.075817

Chapter Contents
Previous
Next
Top