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reason the method has the “local” connotation. There are two steps in the LLSI. The
first step is to select
k
instances by the
L
2
-norm. The second step is regression and
estimation, regardless of how the
k
instances are selected. A heuristic
k
parameter
selection method is used by the authors.
Throughout the section, wewill use
X
n
to denote a dataset with
m
attributes
and
n
instances. Since LLSI was proposed for microarrays, it is assumed that
m
m
×
∈ R
n
.
In the data set
X
,arow
x
i
1
×
n
∈ R
represents expressions of the
i
th instance in
n
examples:
⎛
⎞
x
1
x
m
⎝
⎠
∈ R
m
×
n
X
=
AMVinthe
l
th location of the
i
th instance is denoted as
α
, i.e.
X
(
i
,
l
)
=
x
i
(
l
)
=
α
For simplicity we first assume assuming there is a MV in the first position of the
first instance, i.e.
X
(
1
,
1
)
=
x
1
(
1
)
=
α.
4.5.8.1 Selecting the Instances
m
×
n
, the KNN instance
α
in the first location
x
1
(
)
∈ R
To recover a MV
1
of
x
1
in
X
vectors for
x
1
,
x
S
i
∈ R
1
×
n
,
1
≤
i
≤
k
,
are found for LLSimpute based on the
L
2
-norm (LLSimpute). In this process of
finding the similar instances, the first component of each instance is ignored due to
the fact that
x
1
(
is missing. The LLSimpute based on the Pearson's correlation
coefficient to select the
k
instances can be consulted in [
49
].
1
)
4.5.8.2 Local Least Squares Imputation
As imputation can be performed regardless of how the
k
-instances are selected,
we present only the imputation based on
L
2
-norm for simplicity. Based on these
k
-neighboring instance vectors, the matrix
A
k
×
(
n
−
1
)
and the two vectors
b
∈ R
∈
k
×
1
and
w
∈ R
(
n
−
1
)
×
1
are formed. The
k
rows of the matrix
A
consist of the KNN
R
instances
x
S
i
1
×
n
,1
k
, with their first values deleted, the elements of the
vector b consists of the first components of the
k
vectors
x
S
i
, and the elements of the
∈ R
≤
i
≤