One-Dimensional Missing-Data APES via Expectation Maximization - Spectral Analysis of Signals: The Missing Data Case - page 47

We stress again that the MAPES approach is based on a surrogate like-
lihood function that is not the true likelihood of the data snapshots. However,
such surrogate likelihood functions (for instance, based on false uncorrelatedness
or Gaussian assumptions) are known to lead to satisfactory fitting criteria, under
fairly reasonable conditions (see, e.g., [42, 49]). Furthermore, it can be shown that
the EM algorithm applied to such a surrogate likelihood function (which is a valid
probability distribution function) still has the key property in (5.5) to monotonically
increase the function at each iteration.

5.5.2 MAPES-EM1 Versus MAPES-EM2
Because at each iteration and at each frequency of interest

, MAPES-EM2 esti-
mates the missing samples only once (for all data snapshots), it has a lower com-
putational complexity than MAPES-EM1, which estimates the missing samples
separately for each data snapshot.
It is also interesting to observe that MAPES-EM1 makes the assump-
tion that the snapshots

ω

y l

are independent when formulating the surrogate data
likelihood function, and it maintains this assumption when estimating the miss-
ing data—hence a “consistent” ignoring of the overlapping. On the other hand,
MAPES-EM2 makes the same assumption when formulating the surrogate data
likelihood function, but in a somewhat “inconsistent” manner it observes the over-
lapping when estimating the missing data. This suggests that MAPES-EM2, which
estimates fewer unknowns than MAPES-EM1, may not necessarily have a (much)
better performance, as might be expected (see the examples in Section 5.7).

{ }

5.5.3 Missing-Sample Estimation
For many applications, such as data restoration, estimating the missing samples
is needed and can be done via the MAPES-EM algorithms. For MAPES-EM2,
at each frequency of interest ,wetake the conditional mean b as an estimate of

the missing sample vector. The final estimate of the missing sample vector is the
average of all b obtained from all frequencies of interest. For MAPES-EM1, at

ω