STATISTICS 512 TECHNIQUES OF MATHEMATICS FOR ...

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Application 1. Illustration of preceding theory: twopopulation
classification problem. Suppose we are
given lengths and widths of prehistoric skulls, of type
A or B (the “training sample”). We know that 1 of
these, say x 1 x 1 , are of type A, and 2 = − 1 ,
say y 1 y 2 ,areoftypeB.Nowwefind a new skull,
with length and width the components of z. We are
to classify it as A or B. (Others applications: rock
samplesingeology,riskdatainanactuarialanalysis,
etc.)
• Reduce to univariate problem: = α 0 x , =
α 0 y for some vector α. Put = α 0 z and classify
new skull as A if | − ¯| | − ¯|.
• Choose α for “maximal separation”: |¯−¯| should
be large relative to the underlying variation. Put
2 1 = 1 X
( − ¯) 2 = 1 X ³
α 0 (x − ¯x)´2
1 − 1
1 − 1
1
=
X 1 − 1 α0 (x − ¯x)(x − ¯x) 0 α = α 0 S 1 α