Graphics Reference
In-Depth Information
Regression is commonly used to describe and analyze the relation between explana-
tory input variables X and one or multiple responses Y. In many applications such
relations aretoocomplicated tomodelwith aparametric regression function. Classi-
cal nonparametric regression (see e.g., Fan and Gijbels,
; Wand and Jones,
;
Loader,
; Simonoff,
) and varying coe
cient models (see e.g., Hastie and
Tibshirani,
; Fan and Zhang,
; Carroll et al.,
; Cai et al.,
), allow for
a more flexible form. In this article we describe an approach that allows us to e
-
ciently handle discontinuities and spatial inhomogeneities of the regression function
in such models.
Nonparametric Regression
8.1
Let us assume that we have a random sample Z
,...,Z
n
of the form Z
i
.
Every X
i
is a vector of explanatory variables which determines the distribution of an
observedresponse Y
i
.LettheX
i
'sbevaluedinthefinitedimensionalEuclideanspace
X=
=(
X
i
, Y
i
)
R
q
. he explanatory variables X
i
may quantify
some experimental conditions, coordinates within an image, or a time. he response
Y
i
in these cases identifies the observed outcome of the experiment: the gray value
or color at the given location and the value of a time series, respectively.
We assume that the distribution of each Y
i
is determined by a finite dimensional
parameter θ
R
d
and the Y
i
's belong to
Y
θ
X
i
whichmaydependonthevalue X
i
of the explanatory variable.
=
(
)
Examples
8.1.1
We use the following examples to illustrate the situation.
Example
1:
homoscedastic nonparametric regression model his model is specified
by the regression equation Y
i
1
=
θ
(
X
i
)+
ε
i
with a regression function θ and additive
,σ
i.i.d. Gaussian errors ε
i
. We will use this model to illustrate the main
properties of our algorithms in a univariate (d
N(
)
) setting. he model also serves as
a reasonable approximation to many imaging problems. Here the explanatory vari-
ables X
i
define a two-dimensional (d
=
=
) or three-dimensional (d
=
) grid with
observed gray values Y
i
at each grid point.
Example
2:
inhomogeneous binary response model Here Y
i
is a Bernoulli random
variable with parameter θ
2
(
X
i
)
;thatis,
P
(
Y
i
=
X
i
)=
θ
(
X
i
)
and
P
(
Y
i
=
X
i
)=
−
θ
(
X
i
)
. his model occurs in classification. It is also adequate for binary images.
Example 3:
inhomogeneous Poisson model Every Y
i
follows a Poisson distribution
with parameter θ
3
=
θ
(
X
i
)
, i.e., Y
i
attains nonnegative integer values and
