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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
 
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