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 Time Series Analysis and Control Examples

## Example 10.3: Kalman Filtering: 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 (1989) for details of this data set. Since this EM algorithm uses filtering and smoothing, you can learn how to use the KALCVF and KALCVS calls to analyze the data. Consider the bivariate SSM:

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 and covariance . For estimation, the matrix is fixed as
while the mean vector is updated by the smoothing procedure such that .Note that this estimation requires an extra smoothing step since the usual smoothing procedure does not produce .

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

where the index i represents the current iteration number, and
It is necessary to compute the value of recursively such that
where and the initial value is derived using the formula
Note that the initial value of the state vector is updated for each iteration
The objective function value is computed as -2l in the IML module LIK. The log-likelihood function is written
where .

The iteration history is shown in Output 10.3.1. As shown in Output 10.3.2, 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 (1989, p. 469). Finally, multistep forecasts of yt are computed using the KALCVF call.

   call kalcvf(pred,vpred,filt,vfilt,y,15,a,f,b,h,var,z0,vz0);


The predicted values of the state vector zt and their standard errors are shown in Output 10.3.3.

   title 'SSM Estimation using EM Algorithm';
data one;
input y1 y2 @@;
datalines;
/*. . . data lines omitted . . .*/
;

proc iml;
start lik(y,pred,vpred,h,rt);
n = nrow(y);
nz = ncol(h);
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 + rt;
sum1 = sum1 + log(det(ft));
sum2 = sum2 + et_i*inv(ft)*et_i;
end;
return(sum1+sum2);
finish;

start main;
use one;
read all into y var {y1 y2};
t = nrow(y);
ny = ncol(y);
nz = ny;
f = i(nz);
h = i(ny);

/*-- observation noise variance is diagonal --*/
rt = 1e-5#i(ny);

/*-- transition noise variance --*/
vt = .1#i(nz);
a = j(nz,1,0);
b = j(ny,1,0);
myu = j(nz,1,0);
sigma = .1#i(nz);
converge = 0;
do iter = 1 to 100 while( converge = 0 );

         /*--- construct big cov matrix --*/
var = ( vt || j(nz,ny,0) ) //
( j(ny,nz,0) || rt );

/*-- initial values are changed --*/
z0  = myu * f;
vz0 = f * sigma * f + vt;

/*-- filtering to get one-step prediction and filtered value --*/
call kalcvf(pred,vpred,filt,vfilt,y,0,a,f,b,h,var,z0,vz0);

/*-- smoothing using one-step prediction values --*/
call kalcvs(sm,vsm,y,a,f,b,h,var,pred,vpred);

/*-- compute likelihood values --*/
logl = lik(y,pred,vpred,h,rt);

/*-- store old parameters and function values --*/
myu0 = myu;
f0 = f;
vt0 = vt;
rt0 = rt;
logl0 = logl;
itermat = itermat // ( iter || logl0 || shape(f0,1) || myu0 );

/*-- obtain P*(t) to get P_T_0 and Z_T_0   --*/
/*-- these values are not usually needed   --*/
/*-- See Harvey (1989 p154) or Shumway (1988, p177) --*/
jt1 = sigma * f * inv(vpred[1:nz,]);
p_t_0  = sigma + jt1*(vsm[1:nz,] - vpred[1:nz,])*jt1;
z_t_0  = myu + jt1*(sm[1,] - pred[1,]);
p_t1_t  = vpred[(t-1)*nz+1:t*nz,];
p_t1_t1 = vfilt[(t-2)*nz+1:(t-1)*nz,];
kt = p_t1_t*h*inv(h*p_t1_t*h+rt);


         /*-- obtain P_T_TT1. See Shumway (1988, p180) --*/
p_t_ii1 = (i(nz)-kt*h)*f*p_t1_t1;
st0 = vsm[(t-1)*nz+1:t*nz,] + 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-1)*nz+1:t*nz,]*h;
do i = t to 2 by -1;
p_i1_i1 = vfilt[(i-2)*nz+1:(i-1)*nz,];
p_i1_i  = vpred[(i-1)*nz+1:i*nz,];
jt1 = p_i1_i1 * f * inv(p_i1_i);
p_i1_i  = vpred[(i-2)*nz+1:(i-1)*nz,];
if ( i > 2 ) then
p_i2_i2 = vfilt[(i-3)*nz+1:(i-2)*nz,];
else
p_i2_i2 = sigma;
jt2 = p_i2_i2 * f * inv(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-2)*nz+1:(i-1)*nz,];
sm1 = sm[i-1,];
st0 = st0 + ( temp + sm1 * sm1 );
if ( i > 2 ) then
st1 = st1 + ( p_t_ii1 + sm1 * sm[i-2,]);
else st1 = st1 + ( p_t_ii1 + sm1 * z_t_0);
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 * inv(st00);
vt = (st0 - st1 * inv(st00) * st1)/t;
rt = cov / t;

/*-- check convergence --*/
if ( max(abs((myu - myu0)/(myu0+1e-6))) < 1e-2 &
max(abs((f - f0)/(f0+1e-6))) < 1e-2 &
max(abs((vt - vt0)/(vt0+1e-6))) < 1e-2 &
max(abs((rt - rt0)/(rt0+1e-6))) < 1e-2 &
abs((logl-logl0)/(logl0+1e-6)) < 1e-3 ) then
converge = 1;
end;


      reset noname;
colnm = {'Iter' '-2*log L' 'F11' 'F12' 'F21' 'F22'
'MYU11' 'MYU22'};
print itermat[colname=colnm format=8.4];
eval = teigval(f0);
colnm = {'Real' 'Imag' 'MOD'};
eval = eval || sqrt((eval#eval)[,+]);
print eval[colname=colnm];
var = ( vt || j(nz,ny,0) ) //
( j(ny,nz,0) || rt );

/*-- initial values are changed --*/
z0  = myu * f;
vz0 = f * sigma * f + vt;
free itermat;

/*-- multistep prediction --*/
call kalcvf(pred,vpred,filt,vfilt,y,15,a,f,b,h,var,z0,vz0);
do i = 1 to 15;
itermat = itermat // ( i || pred[t+i,] ||
sqrt(vecdiag(vpred[(t+i-1)*nz+1:(t+i)*nz,])) );
end;
colnm = {'n-Step' 'Z1_T_n' 'Z2_T_n' 'SE_Z1' 'SE_Z2'};
print itermat[colname=colnm];
finish;
run;
`

Output 10.3.1: Iteration History

 SSM Estimation using EM Algorithm

 Iter -2*log L F11 F12 F21 F22 MYU11 MYU22 1.0000 -154.010 1.0000 0.0000 0.0000 1.0000 0.0000 0.0000 2.0000 -237.962 0.7952 -0.6473 0.3263 0.5143 0.0530 0.0840 3.0000 -238.083 0.7967 -0.6514 0.3259 0.5142 0.1372 0.0977 4.0000 -238.126 0.7966 -0.6517 0.3259 0.5139 0.1853 0.1159 5.0000 -238.143 0.7964 -0.6519 0.3257 0.5138 0.2143 0.1304 6.0000 -238.151 0.7963 -0.6520 0.3255 0.5136 0.2324 0.1405 7.0000 -238.153 0.7962 -0.6520 0.3254 0.5135 0.2438 0.1473 8.0000 -238.155 0.7962 -0.6521 0.3253 0.5135 0.2511 0.1518 9.0000 -238.155 0.7962 -0.6521 0.3253 0.5134 0.2558 0.1546 10.0000 -238.155 0.7961 -0.6521 0.3253 0.5134 0.2588 0.1565

Output 10.3.2: Eigenvalues of F Matrix

 Real Imag MOD 0.6547534 0.438317 0.7879237 0.6547534 -0.438317 0.7879237

Output 10.3.3: Multistep Prediction

 n-Step Z1_T_n Z2_T_n SE_Z1 SE_Z2 1 -0.055792 -0.587049 0.2437666 0.237074 2 0.3384325 -0.319505 0.3140478 0.290662 3 0.4778022 -0.053949 0.3669731 0.3104052 4 0.4155731 0.1276996 0.4021048 0.3218256 5 0.2475671 0.2007098 0.419699 0.3319293 6 0.0661993 0.1835492 0.4268943 0.3396153 7 -0.067001 0.1157541 0.430752 0.3438409 8 -0.128831 0.0376316 0.4341532 0.3456312 9 -0.127107 -0.022581 0.4369411 0.3465325 10 -0.086466 -0.052931 0.4385978 0.3473038 11 -0.034319 -0.055293 0.4393282 0.3479612 12 0.0087379 -0.039546 0.4396666 0.3483717 13 0.0327466 -0.017459 0.439936 0.3485586 14 0.0374564 0.0016876 0.4401753 0.3486415 15 0.0287193 0.0130482 0.440335 0.3487034

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