CSC 411 / CSC D11
Nonlinear Regression
3
Nonlinear Regression
Sometimes linear models are not sufficient to capture the realworld phenomena, and thus nonlinear
models are necessary. In regression, all such models will have the same basic form, i.e.,
y
=
f
(
x
)
(1)
In linear regression, we have
f
(
x
) =
Wx
+
b
; the parameters
W
and
b
must be fit to data.
What nonlinear function do we choose? In principle,
f
(
x
)
could be anything: it could involve
linear functions, sines and cosines, summations, and so on. However, the form we choose will
make a big difference on the effectiveness of the regression: a more general model will require
more data to fit, and different models are more appropriate for different problems. Ideally, the
form of the model would be matched exactly to the underlying phenomenon. If we’re modeling a
linear process, we’d use a linear regression; if we were modeling a physical process, we could, in
principle, model
f
(
x
)
by the equations of physics.
In many situations, we do not know much about the underlying nature of the process being
modeled, or else modeling it precisely is too difficult. In these cases, we typically turn to a few
models in machine learning that are widelyused and quite effective for many problems. These
methods include basis function regression (including Radial Basis Functions), Artificial Neural
Networks, and
k
Nearest Neighbors.
There is one other important choice to be made, namely, the choice of objective function for
learning, or, equivalently, the underlying noise model. In this section we extend the LS estimators
introduced in the previous chapter to include one or more terms to encourage smoothness in the
estimated models. It is hoped that smoother models will tend to overfit the training data less and
therefore generalize somewhat better.
3.1
Basis function regression
A common choice for the function
f
(
x
)
is a basis function representation
1
:
y
=
f
(
x
) =
s
k
w
k
b
k
(
x
)
(2)
for the 1D case. The functions
b
k
(
x
)
are called basis functions. Often it will be convenient to
express this model in vector form, for which we define
b
(
x
) = [
b
1
(
x
)
, . . . , b
M
(
x
)]
T
and
w
=
[
w
1
, . . . , w
M
]
T
where
M
is the number of basis functions. We can then rewrite the model as
y
=
f
(
x
) =
b
(
x
)
T
w
(3)
Two common choices of basis functions are
polynomials
and
Radial Basis Functions (RBF)
.
A simple, common basis for polynomials are the
monomials
, i.e.,
b
0
(
x
) = 1
,
b
1
(
x
) =
x,
b
2
(
x
) =
x
2
,
b
3
(
x
) =
x
3
,
...
(4)
1
In the machine learning and statistics literature, these representations are often referred to as linear regression,
since they are linear functions of the “features”
b
k
(
x
)
Copyright c
c
2009 Aaron Hertzmann and David Fleet
9
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Nonlinear Regression
2
1.5
1
0.5
0
0.5
1
1.5
2
8
6
4
2
0
2
4
6
8
x
Polynomial basis functions
x
0
x
1
x
2
x
3
2
1.5
1
0.5
0
0.5
1
1.5
2
1
0.5
0
0.5
1
1.5
2
x
Radial Basis Functions
Figure 1: The first three basis functions of a polynomial basis, and Radial Basis Functions
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 Spring '10
 DavidFleet
 Linear Regression, Regression Analysis, Machine Learning, basis functions, Fleet Aircraft, Fleet Model 1

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