Civil Engineering Reference
In-Depth Information
3
Nonparametric Method of Selection of Decision Variables
Suppose discrete variables
X
i
1,2,...,
p
)are
q
*1 vectors,
X
j
∈
R
p
(
j
(
i
=
=
1,2,. . .
q
)
X
i
X
i
)
are continuous variables. Let
X
i
=(
,
to estimate
E
(
Y
i
|
X
i
)
but in the actually
applied settings not all of the
q
p
regressors in
X
i
are relevant with
Y
i
, We consider
cases where one or more of the regressors may be irrelevant. Without loss of
generality, we assume that only the first
p
1
(
+
components of
X
i
1
≤
p
1
≤
p
)
and the
components of
X
j
are “relevant” regressors in the sense defined
below. Note that we assume there exists at least one relevant continuous variable and
the regression model case is quite different from the conditional density estimation
case as considered in
Hall et al.
(
2007
) where one also smooths the dependent
variabl
e.
This calls for substantial changes to the proofs.
Let
X
consist of the first
p
1
relevant components of
X
c
first
q
1
(
0
≤
q
1
≤
q
)
and the first
q
1
relevant
components of
X
d
.Let
X
X
denote the remaining irrelevant components
of
X
. The way of defining
X
to be relevant and
X
to be irrelevant is to say that
(
=
X
\{
}
X
is independent of
X
clearly, which implies that
E
X
,
Y
)
(
Y
|
X
)=
E
(
Y
|
)
and so a
standard regression model, of the form
Y
=
g
(
X
)+
ε
, may equivalently be written
X
in the dimension-reduced form
Y
=
g
(
+
ε
)
.Weuse
f(x
) to denote the joint density
function of
X
i
and we use
f
to denote the marginal densities
of
X
and
X
, respe
cti
vely. We shall assume that the true regression
mo
del is
Y
i
=
and
f
(
x
)
(
x
)
g
(
X
i
)+
μ
i
where
g
0. For the discrete
regressors
X
s
, we define the kernel function for discrete variables as
(
X
i
)
is of unknown functional form and where
E
(
μ
i
|
X
i
)=
1
if X
is
=
x
s
,
,
X
i
s
d
X
s
l
(
,
,
λ
)=
if X
is
=
x
s
,
λ
s
,
For the continuous variables
x
c
x
1
,...,
=(
x
p
)
we use the product kernel given
s
=
1
h
−
1
K
x
s
−
X
is
p
K
c
x
c
X
i
)=
(
,
,
(3)
h
s
where K is a symmetric, univariate density function, and where 0
is the
smoothing parameter for
X
c
, the kernel function for the mixed regressor case
X
<
h
s
<
∞
=
X
c
X
d
is simply the product of
K
c
and
K
d
, i.e.,
(
,
)
K
c
x
c
X
i
)
K
d
x
d
X
i
)
,
K
(
x
,
X
i
)=
(
,
(
,
(4)
Thus we estimate
E
(
Y
|
X
=
x
)
by
i
=
1
Y
i
K
X
i
(
x
,
)
∑
(
)=
g
x
n
i
=
1
K
X
i
(
x
,
)
∑
X
is
−
x
s
h
s
X
is
−
x
s
h
s
.
(5)
p
1
s
q
1
s
p
s
q
s
i
X
is
,
x
s
,
λ
s
)
∏
X
is
,
x
s
,
λ
s
)
1
Y
i
∏
1
K
(
)
∏
1
K
(
)
∏
1
l
(
1
l
(
∑
=
=
=
p
1
+
=
=
q
1
+
=
X
is
−
x
s
X
is
−
x
s
p
1
s
=
1
K
q
1
s
=
1
l
p
s
=
p
1
+
1
K
q
s
=
q
1
+
1
l
i
=
1
∏
X
is
,
x
s
,
λ
s
)
∏
X
is
,
x
s
,
λ
s
)
(
)
∏
(
)
∏
(
(
∑
h
s
h
s
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