One-Dimensional Missing-Data APES via Expectation Maximization - Spectral Analysis of Signals: The Missing Data Case - page 38

surrogate log-likelihood in (5.4), i.e.,

E ln p ( γ , µ |

θ i − 1 θ i ) γ , θ i ) θ i − 1 )

ln p ( γ | − ln p ( γ | ≥

E ln p ( γ

θ i − 1 θ i − 1 ) γ

− , µ | , . (5.5)

are overlapping, one missing sample may occur
in many snapshots (note that there is only one new sample between two adjacent data
snapshots). So two approaches are possible when we try to estimate the missing data:
estimate the missing data separately for each snapshot y l by ignoring any possible
overlapping, or jointly for all snapshots

Since the data snapshots { y l }

L − 1

0 by observing the over lappings. In
the following two sections, we make use of these ideas to develop two different
MAPES-EM algorithms, namely MAPES-EM1 and MAPES-EM2.

{ y l }

l =

5.3 MAPES-EM1
In this section we assume that the data snapshots

L − 1

1 are independent of each
other, and hence we estimate the missing data separately for different data snap-
shots. For each data snapshot y l , let γ l and µ l denote the vectors containing the
available and missing elements of y l ,respectively. In general, the indices of the
missing components could be different for different l

{ y l }

l =

. Assume that γ l has dimen-

sion g l M is the number of available elements in the snapshot

y l . (Although g l could be any integer that belongs to the interval 0

× 1, where 1 ≤ g l ≤

≤ g l ≤ M ,we

assume for now that g l = 0. Later we will explain what happens when g l = 0.)

Then γ l and µ l are related to y l by unitary transformations as follows:

S g ( l ) y l

γ l = (5.6)

S m ( l ) y l

µ l = , (5.7)

S g ( l ) and S m ( l )are M

where × g l and M × ( M − g l ) unitary selection matrices

such that

S g ( l ) S g ( l )

= I g l , (5.8)

S m ( l ) S m ( l )

= I M − g l , (5.9)