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by applying a transfer function to the input given to the matching mechanism.
Nonetheless, to keep the notation uncluttered it is assumed that the given input
x
contains all available information and both matching and the local model
selectively choose and modify the components that they require by an implicit
transfer function.
Example 3.5 (Inputs to Matching and Local Model).
Let us assume that both the
input and the output space are 1-dimensional, that is,
,and
that we perform interval matching over the interval [
l
k
,u
k
], such that
m
k
(
x
)=1
if
l
k
≤
X
=
R
and
Y
=
R
u
k
,and
m
k
(
x
) = 0 otherwise. Applying the linear model
f
(
x
;
w
k
)=
xw
k
to the input, with
w
k
being the adjustable parameter of classifier
k
,one
can only model straight lines through the origin. However, applying the transfer
function
φ
(
x
)=(1
,x
)
T
allows for the introduction of an additional bias to get
f
(
x
;
w
k
)=
w
k
φ
(
x
)=
w
k
1
+
xw
k
2
,with
w
k
=(
w
k
1
,w
k
2
)
T
x
≤
2
,whichisan
arbitrary straight line. In such a case, the input is assumed to be
x
=(1
,x
)
T
,
and the matching function to only operate on the second component of that
input. Hence, both matching and the local model can be applied to the same
input. A more detailed discussion about different transfer functions and their
resulting models is given in Sect. 5.1.1.
∈
R
3.2.4
Recovering the Global Model
To recover the global model from
K
local models, they need to be combined in
some meaningful way. For inputs that only a single classifier matches, the best
model is that of the matching classifier. However, there are no restrictions on how
many classifiers can match a single input. Therefore, in some cases, it is necessary
to mix the local models of several classifiers that match the same input.
There are several possible approaches to mixing classifier models, each corre-
sponding to different assumptions about the data-generating process. A standard
approach in introduced in Chap. 4 and alternatives are discussed in Chap. 6.
3.2.5
Finding a Good Model Structure
The model structure
is given by the number of classifiers and their localisa-
tion. As the localisation of a classifier
k
is determined by its matching function
m
k
, the model structure is completely specified by the number of classifiers
K
and their matching functions
M
=
M
.
To find a good model structure means to find a structure that allows for hypo-
theses about the data-generating process that are close to the process suggested
by the available observations. Thus, finding a good model structure implies de-
aling with over and underfitting of the training set. A detailed treatment of this
topic is postponed to Chap. 7, and for now its is assumed that a good model
structure is known.
{
m
k
}
,thatis,
M
=
{
K,
M
}
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