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A simple linear
discriminant function is a projection of the
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data down to 1-D.
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So
choose the projection that gives the best
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separation
of the classes. What do we
mean by “best
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separation”?
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An obvious
direction to choose is the direction of the line
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joining the class
means.
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But
if the main direction of variance in each class is
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not
orthogonal to this line, this will not give good
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separation
(see the next figure).
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Fisher’s method
chooses the direction that maximizes
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the ratio of between class variance to within
class
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variance.
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This
is the direction in which the projected points
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contain
the most information about class membership
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(under
Gaussian assumptions)
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