Biomedical Engineering Reference
In-Depth Information
estimated recursively. By decoupling the variable selection part from the model update part in LAR,
the input variables can be selected locally with recursive estimates of correlations. The modification
to the LAR procedure using these recursions is described next.
The correlation in step a in Table
4.4
at a given step can be simply updated without comput-
ing residuals,
c
j
(k) = c
j
(k −
1
) -
γ θ
j
(4.36)
for the
k
th step of variable selection. Hence, the update procedure of step h of Table
4.4
can be
removed. Instead of computing the correlation with entire data, we can recursively estimate the cor-
relation using a forgetting factor ρ, given by
p
(n) =
ρ
p
(n − 1)+ d(n)
x
(n)
(4.37)
where
x
(
n
) is an 1 ×
M
input vector at time instance
n
.
p
(
n
) is utilized by the LAR routine such that
c
j
(0) =
p
j
(
n
). For the computation of the covariance matrix Φ in step d in Table
4.4
, we also estimate
the input covariance matrix as
R
(n) =
ρ
R
(n − 1)+
x
(n)
T
x
(n)
(4.38)
This matrix is not directly used because Φ is the covariance of only a subset of inputs. Also,
the input vectors are multiplied by the sign of correlations before computing Φ. Therefore, we need
to introduce a diagonal matrix
S
whose elements are signs of
c
j
(
k
) for
j
∈
A
. Then Φ can be com-
puted using
R
(
n
) and
S
as,
Φ
=
SR
a
S
(4.39)
where
R
a
is an
L
a
×
L
a
(
L
a
is the length of
A
) matrix representing covariance among the selected in-
put variables.
R
a
can be given by the elements of
R
(
n
), that is,
r
ij
for
i
,
j
∈
A
. To remove the compu-
tation of the equiangular vector µ that requires a batch operation, we incorporate step e in Table
4.4
into step g in Table
4.4
such that
θ
j
=
X
T
µ
=
X
T
X
a
(
αΦ
−1
1
a
) =
α
X
T
X
a
Φ
−1
1
a
(4.40)
By noting that
X
T
X
a
is the
j
th columns of
R
(
n
), for
j
∈
A
followed by multiplication with
S,
we define
R
acol
to be a submatrix of
R
(
n
) consisting of the
j
th columns for
j
∈
A.
Then step g in
Table
4.4
can be computed as
θ
j
=
α
R
acol
S
Φ
−1
1
a
(4.41 )
