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T
y
. Nowwe assume the existence ofMVs. In PC regression,
the missing part
y
miss
in the expression vector
y
is estimated from the observed part
y
obs
by using the PCA result. Let
w
l
obs
and
w
l
miss
be parts of each principal axis
w
l
,
corresponding to the observed and missing parts, respectively, in
y
. Similarly, let
W
y
is given by
x
l
=
(
w
l
/λ
l
)
=
(
W
obs
,
W
miss
)
where
W
obs
or
W
miss
denotes a matrix whose column vectors are
w
obs
,...,
w
obs
or
w
miss
,...,
w
miss
, respectively.
Factor scores
x
=
(
x
1
,...,
x
K
)
for the example vector
y
are obtained by mini-
mization of the residual error
2
err
=
y
obs
−
W
obs
x
.
This is a well-known regression problem, and the least square solution is given by
W
obsT
W
obs
)
−
1
W
obsT
y
obs
.
x
=
(
Using
x
, the missing part is estimated as
y
miss
=
W
miss
x
(4.23)
In the PC regression above,
W
should be known beforehand. Later, we will discuss
the way to determine the parameter.
4.4.3.2 Bayesian Estimation
A parametric probabilistic model, which is called probabilistic PCA (PPCA), has
been proposed recently. The probabilistic model is based on the assumption that the
residual error
and the factor scores
x
l
(
1
≤
l
≤
K
)
in Equation (reflinearcomb)
obey normal distributions:
p
(
x
)
=
N
K
(
x
|
0
,
I
K
),
p
()
=
N
D
(
|
0
,(
1
/τ )
I
D
),
where
N
K
(
x
|
μ, Σ)
denotes a
K
-dimensional normal distribution for
x
, whose mean
and covariance are
μ
and
Σ
, respectively.
I
K
is a
(
K
×
K
)
identity matrix and
τ
is a
scalar inverse variance of
. In this PPCA model, a complete log-likelihood function
is written as:
ln p
(
y
,
x
|
θ)
≡
ln p
(
y
,
x
|
W
,μ,τ)
=−
2
+
1
2
D
2
ln
K
D
2
2
−
−
τ
−
+
τ
−
,
y
Wx
x
ln
2
2
