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Multicategory Classification: The Multicategory Support Vector Machine
and the Multichotomous Penalized Likelohood Estimate
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Grace Wahba
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Abstract
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We describe two modern methods for statistical
model building and classification,
penalized likelihood methods and
and support vector machines (SVM's). Both are obtained
as solutions to optimization problems in
reproducing kernel Hilbert spaces (RKHS).
A training set is given, and an algorithm
for classifiying future observations is built from it.
The ($k$-category) multichotomous penalized likelihood method returns
a vector of probabilities $(p_1(t), \cdots p_k(t))$
where $t$ is the attribute vector
of the object to be classified. The multicategory
support vector vachine returns a classifier
vector $(f_1(t), \cdots f_k(t))$ satisfying
$\sum_\ell f_{\ell}(t) = 0$, where $max_{\ell}f_{\ell}(t)$
identifies the category. The two category SVM's
are very well known,
while the multi-category SVM (MSVM) described here,
which includes modifications for unequal misclassification costs and
unrepresentative training sets, is new.
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We describe applications of each method: For penalized likelihood,
estimating the 10-year probability of death due to
several causes, as a function of several risk
factors observed in a demographic study,
and for MSVM's, classifying radiance profiles from
the MODIS instrument according to clear, water cloud or
ice cloud.
See also G. Wahba, PNAS,2002 -go back to my home
page and the TRLIST for the link.
Some computational and tuning issues are noted.
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