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

MATLAB MEX Function Reference

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

This example estimates the normal SSM of the mink-muskrat data using the EM algorithm. The mink-muskrat series are detrended. Refer to Harvey (1991) for details of this data set. Since this EM algorithm uses filtering and smoothing, you can learn how to use the KALCVF and KALCVS functions to analyze the data. Consider the bivariate SSM:

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

where H is a 2 ×2 identity matrix, the observation noise has a time invariant covariance matrix R, and the covariance matrix of the transition equation is also assumed to be time invariant. The initial state z₀ has mean $\mu$ and covariance $\Sigma$ . For estimation, the $\Sigma$ matrix is fixed as

$[ 0.1 & 0.0 \ 0.0 & 0.1 ]$

while the mean vector $\mu$ is updated by the smoothing procedure such that $\hat{\mu} = {z}_{0| T}$ . Note that this estimation requires an extra smoothing step since the usual smoothing procedure does not produce ${z}_{T|}$ .

The EM algorithm maximizes the expected log-likelihood function given the current parameter estimates. In practice, the log-likelihood function of the normal SSM is evaluated while the parameters are updated using the M-step of the EM maximization

${F}^{i+1} & = & {S}_t(1)[{S}_{t-1}(0)]^{-1} \ {V}^{i+1} & = & \frac{1}T ( {S}_t... ...y}_t - {H}{z}_{t| T})^' + {H}{P}_{t| T} {H}^' ] \ \mu^{i+1} & = & {z}_{0| T}$

where the index i represents the current iteration number, and

${S}_t(0) & = & \sum_{t=1}^T ({P}_{t| T} + {z}_{t| T} {z}^'_{t| T}), \ {S}_t(1) & = & \sum_{t=1}^T ({P}_{t,t-1| T} + {z}_{t| T} {z}^'_{t-1| T})$

It is necessary to compute the value of ${P}_{t,t-1| T}$ recursively such that

${P}_{t-1,t-2| T} = {P}_{t-1| t-1} {P}^{*'_{t-2} + {P}^*_{t-1} ({P}_{t,t-1| T} - {F}{P}_{t-1| t-1}) {P}^{*'_{t-2}$

where ${P}^*_t = {P}_{t| t}{F}^'{P}^-_{t+1| t}$ and the initial value ${P}_{T,T-1| T}$ is derived using the formula

${P}_{T,T-1| T} = [ {I}- {P}_{t| t-1} {H}^' ({H}{P}_{t| t-1} {H}^' + {{R}}) {H} ] {F}{P}_{T-1| T-1}$

Note that the initial value of the state vector is updated for each iteration

${z}_{1|} & = & {F}\mu^i \ {P}_{1|} & = & {F}^i\Sigma {F}^{i' + {V}^i$

The objective function value is computed as -2l in the IML module LIK. The log-likelihood function is written

$\ell = -\frac{1}2 \sum_{t=1}^T \log(|{C}_t|) - \frac{1}2 \sum_{t=1}^T ({y}_t - {H}{z}_{t| t-1}) {C}^{-1}_t ({y}_t - {H}{z}_{t| t-1})^'$

where ${C}_t = {H}{P}_{t| t-1}{H}^' + {{R}}$ .

The iteration history is shown in Output 1. As shown in Output 2, the eigenvalues of F are within the unit circle, which indicates that the SSM is stationary. However, the muskrat series (Y1) is reported to be difference stationary. The estimated parameters are almost identical to those of the VAR(1) estimates. Refer to Harvey (1991, p. 469). Finally, multistep forecasts of y_t are computed using the KALCVF function.

[logl, pred, vpred] = kalcvf(y, 15, a, F, b, H, var, z0, vz0);

The predicted values of the state vector z_t and their standard errors are shown in Output 3.


% The mink-muskrat data set
y = [  0.10609     0.16794
      -0.16852     0.06242
      -0.23700    -0.13344
      -0.18022    -0.50616
       0.18094    -0.37943
       0.65983    -0.40132
       0.65235     0.08789
       0.21594     0.23877
      -0.11515     0.40043
      -0.00067     0.37758
      -0.00387     0.55735
      -0.25202     0.34444
      -0.65011    -0.02749
      -0.53646    -0.41519
      -0.08462     0.02591
      -0.05640    -0.11348
       0.26630     0.20544
       0.03641     0.16331
      -0.26030    -0.01498
      -0.03995     0.09657
       0.33612     0.31096
      -0.11672     0.30681
      -0.69775    -0.69351
      -0.07569    -0.56212
       0.36149    -0.36799
       0.42341    -0.24725
       0.26721     0.04478
      -0.00363     0.21637
       0.08333     0.30188
      -0.22480     0.29493
      -0.13728     0.35463
      -0.12698     0.05490
      -0.18770    -0.52573
       0.34741    -0.49541
       0.54947    -0.26250
       0.57423    -0.21936
       0.57493    -0.12012
       0.28188     0.63556
      -0.58438     0.27067
      -0.50236     0.10386
      -0.60766     0.36748
      -1.04784    -0.33493
      -0.68857    -0.46525
      -0.11450    -0.63648
       0.22005    -0.26335
       0.36533     0.07017
      -0.00151    -0.04977
       0.03740    -0.02411
       0.22438     0.30790
      -0.16196     0.41050
      -0.12862     0.34929
       0.08448    -0.14995
       0.17945    -0.03320
       0.37502     0.02953
       0.95727     0.24090
       0.86188     0.41096
       0.39464     0.24157
       0.53794     0.29385
       0.13054     0.39336
      -0.39138    -0.00323
      -1.23825    -0.56953
      -0.66286    -0.72363  ]';
% mean adjust series
t = size(y,2);
Ny = size(y,1);
Nz = Ny;
F = eye(Nz);
H = eye(Ny);
% observation noise variance is diagonal
Rt = 1e-5*eye(Ny);
% transition noise variance
Vt = .1*eye(Nz);
a = zeros(Nz,1);
b = zeros(Ny,1);
myu = zeros(Nz,1);
sigma = .1*eye(Nz);
itermat = [];
for iter=1:100
   % construct big cov matrix
   var = [Vt zeros(Nz,Ny); zeros(Ny,Nz) Rt];

   % initial values are changed
   z0 = F*myu;
   vz0 = F*sigma*F'+Vt;

   % filtering to get one-step prediction and filtered value
   [logl, pred, vpred, filt, vfilt] = kalcvf(y, 0, a, F, b, H, var, z0, vz0);
   logl = (-2*logl-Ny*log(2*pi))*t;

   % smoothing using one-step prediction values
   [sm, vsm] = kalcvs(y, a, F, b, H, var, pred, vpred);

   % store old parameters and function values
   myu0 = myu;
   F0 = F;
   Vt0 = Vt;
   Rt0 = Rt;
   logl0 = logl;
   itermat = [itermat; iter logl0 F0(:)' myu0'];

   % obtain P*(t) to get P_T_0 and Z_T_0
   % these values are not usually needed
   % See Harvey (1991 p154) or Shumway (1988, p177)
   jt1 = sigma*F'/vpred(:,:,1);
   p_t_0  = sigma+jt1*(vsm(:,:,1)-vpred(:,:,1))*jt1';
   z_t_0  = myu+jt1*(sm(:,1)-pred(:,1));
   p_t1_t = vpred(:,:,t);
   p_t1_t1 = vfilt(:,:,t-1);
   Kt = p_t1_t*H'/(H*p_t1_t*H'+Rt);

   % obtain P_T_TT1. See Shumway (1988, p180)
   p_t_ii1 = (eye(Nz)-Kt*H)*F*p_t1_t1;
   st0 = vsm(:,:,t)+sm(:,t)*sm(:,t)';
   st1 = p_t_ii1+sm(:,t)*sm(:,t-1)';
   st00 = p_t_0+z_t_0*z_t_0';
   cov = (y(:,t)-H*sm(:,t))*(y(:,t)-H*sm(:,t))'+H*vsm(:,:,t)*H';
   for i=t:-1:2
      p_i1_i1 = vfilt(:,:,i-1);
      p_i1_i  = vpred(:,:,i);
      jt1 = p_i1_i1*F'/p_i1_i;
      p_i1_i  = vpred(:,:,i-1);
      if (i>2)
         p_i2_i2 = vfilt(:,:,i-2);
      else
         p_i2_i2 = sigma;
      end
      jt2 = p_i2_i2*F'/p_i1_i;
      p_t_i1i2 = p_i1_i1*jt2'+jt1*(p_t_ii1-F*p_i1_i1)*jt2';
      p_t_ii1 = p_t_i1i2;
      temp = vsm(:,:,i-1);
      sm1 = sm(:,i-1);
      st0 = st0+(temp+sm1*sm1');
      if (i>2)
         st1 = st1+(p_t_ii1+sm1*sm(:,i-2)');
      else
         st1 = st1+(p_t_ii1+sm1*z_t_0');
      end
      st00 = st00+(temp+sm1*sm1');
      cov = cov+H*temp*H'+(y(:,i-1)-H*sm1)*(y(:,i-1)-H*sm1)';
   end

   % M-step: update the parameters
   myu = z_t_0;
   F = st1/st00;
   Vt = (st0-F*st1')/t;
   Rt = cov/t;

   % check convergence
   if norm((myu-myu0)./(myu0+1e-6),inf) < 1e-2 && ...
      norm((F-F0)./(F0+1e-6),inf) < 1e-2 && ...
      norm((Vt-Vt0)./(Vt0+1e-6),inf) < 1e-2 && ...
      norm((Rt-Rt0)./(Rt0+1e-6),inf) < 1e-2 && ...
      abs((logl-logl0)/(logl0+1e-6)) < 1e-3
      break
   end
end

fprintf('\n   SSM Estimation using EM Algorithm\n');
fprintf('\n   Iter  -2*log L      F11       F21       F12       F22      MYU11     MYU22\n');
fprintf('%6d   % .3f   % .4f   % .4f   % .4f   % .4f   % .4f   % .4f\n',itermat');
eval = eig(F0);
fprintf('\n   Real         Imag         MOD\n');
eval = [real(eval) imag(eval) abs(eval)];
fprintf('% .7f   % .7f   % .7f\n',eval');
var = [Vt zeros(Nz,Ny); zeros(Ny,Nz) Rt];

% initial values are changed
z0  = F*myu;
vz0 = F*sigma*F'+Vt;
itermat=[];

% multistep prediction
[logl, pred, vpred] = kalcvf(y, 15, a, F, b, H, var, z0, vz0);
for i=1:15
   itermat = [itermat; i pred(:,t+i)' sqrt(diag(vpred(:,:,t+i)))'];
end
fprintf('\n  n-Step    Z1_T_n       Z2_T_n        SE_Z1        SE_Z2\n');
fprintf('%6d    % .7f   % .7f   % .7f   % .7f\n',itermat');
fprintf('\n');

Output 1: Iteration History

   SSM Estimation using EM Algorithm

   Iter  -2*log L      F11       F21       F12       F22      MYU11     MYU22
     1   -154.010    1.0000    0.0000    0.0000    1.0000    0.0000    0.0000
     2   -237.962    0.7952    0.3263   -0.6473    0.5143    0.0530    0.0840
     3   -238.083    0.7967    0.3259   -0.6514    0.5142    0.1372    0.0977
     4   -238.126    0.7966    0.3259   -0.6517    0.5139    0.1853    0.1159
     5   -238.143    0.7964    0.3257   -0.6519    0.5138    0.2143    0.1304
     6   -238.151    0.7963    0.3255   -0.6520    0.5136    0.2324    0.1405
     7   -238.153    0.7962    0.3254   -0.6520    0.5135    0.2438    0.1473
     8   -238.155    0.7962    0.3253   -0.6521    0.5135    0.2511    0.1518
     9   -238.155    0.7962    0.3253   -0.6521    0.5134    0.2558    0.1546
    10   -238.155    0.7961    0.3253   -0.6521    0.5134    0.2588    0.1565

Output 2: Eigenvalues of F Matrix

   Real         Imag         MOD
 0.6547534    0.4383170    0.7879237
 0.6547534   -0.4383170    0.7879237

Output 3: Multistep Prediction

  n-Step    Z1_T_n       Z2_T_n        SE_Z1        SE_Z2
     1    -0.0557920   -0.5870493    0.2437666    0.2370740
     2     0.3384325   -0.3195046    0.3140478    0.2906620
     3     0.4778022   -0.0539491    0.3669731    0.3104052
     4     0.4155731    0.1276996    0.4021048    0.3218256
     5     0.2475671    0.2007098    0.4196990    0.3319293
     6     0.0661993    0.1835492    0.4268943    0.3396153
     7    -0.0670005    0.1157541    0.4307520    0.3438409
     8    -0.1288308    0.0376316    0.4341532    0.3456312
     9    -0.1271071   -0.0225813    0.4369411    0.3465325
    10    -0.0864664   -0.0529307    0.4385978    0.3473038
    11    -0.0343187   -0.0552930    0.4393282    0.3479612
    12     0.0087379   -0.0395458    0.4396666    0.3483717
    13     0.0327466   -0.0174589    0.4399360    0.3485586
    14     0.0374564    0.0016876    0.4401753    0.3486415
    15     0.0287193    0.0130482    0.4403350    0.3487034

See also

KALCVF performs covariance filtering and prediction

KALCVS performs fixed-interval smoothing

Getting Started with State Space Models

Kalman Filtering Example 1: Likelihood Function Evaluation

References

[1] Harvey, A.C., Forecasting, structural time series models and the Kalman filter, Cambridge: Cambridge University Press, 1991.

[2] Shumway, R.H., Applied Statistical Time Series Analysis, Englewood Cliffs, NJ: Prentice-Hall, 1988.

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