For Kalman Filter Durbin/Koopman give at p67 equations 4.13:
v = y - Z a, F = Z P Z' + H, K = T P Z' / F + H,
a[t+1] = T a + K v, P[t+1] = T P L' + R Q R'
for P1 = 0, Q=0, T=Z=R=1 that reduces to:
v = y - a, F = H, K = H, a[t+1] = a + K v, P[t+1] = 0
(equivalent to exponential moving average, Durbin/Koopman p49)
for vector y = c(1,2,3,4,5), H = 0.66 manual calculations
using the equations above give a = c(1,1.66,2.55,3.51,4.50).
KalmanRun with these parameters gives res$states = (1,1,1,1,1).
Looking into the code of arima.c we have at line 109 an equivalent of
a[t+1] = anew + Pnew * resid0 / gain
where gain = mod$h = H (by line 97), resid0 = y-a = v (by lines 94-96)
Since Pnew = 0, then a[t+1] = a, which explains why the computation
returns res$states = c(1,1,1,1,1).
The help file says "'states', the contemporaneous state
estimates",
which I assumed to mean 'a' in the equations above. But that
assumption does not agree with the numerical results. It also
does not agree with the coding(?) as a[t+1] = a + K v differs
substantially from a[t+1] = anew + Pnew * resid0 / gain.
So, what does 'states' contain? Where did I go wrong?
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