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Solving a density estimation problem numerically,
the function has to be discretized.
This is done by expanding in a basis
(not necessarily orthonormal)
and,
choosing some ,
truncating the sum to terms with
,
|
(380) |
This, also called Ritz's method, corresponds
to a finite linear trial space
and is equivalent
to solving a projected stationarity equation.
Using a discretization (380)
the functional (187)
becomes
|
(381) |
Solving for the coefficients ,
to minimize the error results
according to Eq.[355) and
|
(382) |
in
|
(383) |
corresponding to the -dimensional equation
|
(384) |
with
Thus, for an orthonormal basis
Eq. (384) corresponds
to Eq. (189) projected into the trial space
by the projector
.
The so called linear models are obtained by the
(very restrictive) choice
|
(389) |
with and
= 1 and = .
Interactions, i.e., terms proportional to
products of -components like can be included.
Including all possible interaction would correspond to a
multidimensional Taylor expansion
of the function .
If the functions are also parameterized
this leads to mixture models for .
(See Section 4.4.)
Next: Mixture models
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Joerg_Lemm
2001-01-21