📄 fisher's linear discriminant.mht
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Subject: Fisher's linear discriminant
Date: Sun, 20 Aug 2000 20:36:23 +0800
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originally by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of =
Leeds=0A=
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan=0A=
* with significant contributions from:=0A=
Jens Lippmann, Marek Rouchal, Martin Wilck and others =
--><HTML><HEAD><TITLE>Fisher's linear discriminant</TITLE>
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<H2><A name=3DSECTION00057000000000000000>Fisher's linear =
discriminant</A> </H2>
<P>Fisher's linear discriminant is a classification method that projects =
high-dimensional data onto a line and performs classification in this=20
one-dimensional space. The projection maximizes the distance between the =
means=20
of the two classes while minimizing the variance within each class. This =
defines=20
the Fisher criterion, which is maximized over all linear projections, =
<I>w</I>:=20
<BR>
<P></P>
<DIV align=3Dcenter><!-- MATH: \begin{equation}=0A=
J(w) =3D \frac{ |m_1 - m_2 |^2}{s_1^2 +s_2^2}=0A=
\end{equation} -->
<TABLE align=3Dcenter width=3D"100%">
<TBODY>
<TR vAlign=3Dcenter>
<TD align=3Dmiddle noWrap><IMG=20
alt=3D"\begin{displaymath}J(w) =3D \frac{ \vert m_1 - m_2 =
\vert^2}{s_1^2 +s_2^2} \end{displaymath}"=20
height=3D52=20
=
src=3D"http://www.cse.ucsc.edu/research/compbio/genex/genexTR2html/img35.=
gif"=20
width=3D157></TD>
<TD align=3Dright width=3D10>(9)</TD></TR></TBODY></TABLE></DIV><BR =
clear=3Dall>
<P></P>where <I>m</I> represents a mean, <I>s</I><SUP>2</SUP> represents =
a=20
variance, and the subscripts denote the two classes. In signal theory, =
this=20
criterion is also known as the signal-to-interference ratio. Maximizing =
this=20
criterion yields a closed form solution that involves the inverse of a=20
covariance-like matrix. This method has strong parallels to linear =
perceptrons.=20
We learn the threshold by optimizing a cost function on the training =
set.=20
<P><BR>
<HR>
<ADDRESS><I>Michael Brown</I> <BR><I>1999-11-05</I> =
</ADDRESS></BODY></HTML>
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