Graphics Reference
In-Depth Information
Figure
.
.
Bivariate kernel density estimates of regional climate model output of average temperature
and the logarithm of precipitation over a grid of locations in the western USA. Black contours represent
climate in the state of Colorado while the grey contours represent climate in the (more geographically
homogenous) state of Kansas
he univariate bandwidth, h, is replaced by an invertible d
d matrix, H,whose
properties are explored in Scott (
) and Wand and Jones (
). hen the scaled
univariate kernel, K
h
(
t
)
, is generalized to be
H
−
t
K
H
(
t
)=
K
(
)
(
.
)
H
By a multivariate change of variables, K
H
(
t
)
integrates to
if K
(
t
)
does.
For multivariate data,
x
,...,
x
n
, the multivariate kernel estimator is given by
n
i
=
n
f
(
x
)=
K
H
(
x
−
x
i
)
(
.
)
he simplest product kernel is the standard multivariate normal
t
T
t
−
d
exp
K
(
t
)=(
π
)
−
(
.
)
which leads to the kernel estimate
n
i
=
n
T
H
−
T
H
−
f
(
x
)=
exp
−
(
x
−
x
i
)
(
x
−
x
i
)
(
.
)
(
π
)
d
H
While the calculations and theory of multivariate density estimation are straightfor-
ward, the challenge of identifying and understanding the structure of multivariate
