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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"><!--Converted with LaTeX2HTML 2002-2 (1.70)original version by: Nikos Drakos, CBLU, University of Leeds* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan* with significant contributions from: Jens Lippmann, Marek Rouchal, Martin Wilck and others --><HTML><HEAD><TITLE>Cross Validation</TITLE><META NAME="description" CONTENT="Cross Validation"><META NAME="keywords" CONTENT="web1"><META NAME="resource-type" CONTENT="document"><META NAME="distribution" CONTENT="global"><META NAME="Generator" CONTENT="LaTeX2HTML v2002-2"><META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css"><LINK REL="STYLESHEET" HREF="web1.css"><LINK REL="next" HREF="node18.html"><LINK REL="previous" HREF="node16.html"><LINK REL="up" HREF="node6.html"><LINK REL="next" HREF="node18.html"></HEAD><BODY ><!--Navigation Panel--><A NAME="tex2html386" HREF="node18.html"><IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="file:/home/depot/swtree/depot/latex2html-2002-2/latex2html-2002-2/icons/next.png"></A> <A NAME="tex2html384" HREF="node6.html"><IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="file:/home/depot/swtree/depot/latex2html-2002-2/latex2html-2002-2/icons/up.png"></A> <A NAME="tex2html378" HREF="node16.html"><IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="file:/home/depot/swtree/depot/latex2html-2002-2/latex2html-2002-2/icons/prev.png"></A> <BR><B> Next:</B> <A NAME="tex2html387" HREF="node18.html">Bootstrapping a Principal Component</A><B> Up:</B> <A NAME="tex2html385" HREF="node6.html">Lectures</A><B> Previous:</B> <A NAME="tex2html379" HREF="node16.html">The jackknife</A><BR><BR><!--End of Navigation Panel--><!--Table of Child-Links--><A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A><UL><LI><A NAME="tex2html388" HREF="node17.html#SECTION002111000000000000000">Cross-Validation when there is a response variable</A></UL><!--End of Table of Child-Links--><HR><H1><A NAME="SECTION002110000000000000000">Cross Validation</A></H1><BODY bgcolor="#FFFFFF"><B>Separate Diagnostic Data Sets</B>When having iterated through several exploratory methods, varied the projections and looked for the best `fit', there seems onlyone honest method of verifying whether one is overfittingnoise or whether there really is a latent variable ora good prediction available and that is to have another set of data of exactly the same type.<P>The best thing to do is at the beginning of the study to take a random sub-sample, without any particular stratificationand to put it aside for the confirmatory stage.Many scientists are mean with their data, and only have just enoughto model, but nowadays the expense of an extra 25 % or so,should be made - especially when the consequencesof the study are medical, this is what tukey and mallows calla careful serarate diagnostic.<H2><A NAME="SECTION002111000000000000000">Cross-Validation when there is a response variable</A></H2>When the above prescription is not followed and one of the variableshas the status of variable to be explained, it is possible - at computationalexpense, but who cares ? - to redo the analysis leaving out part ofthe data and comparing with the reference set.<P>For instance in Discriminant Analysis<BR>For each observation, do the analysis without that one, andlook whether or not it is well classified, this will give an unbiasedestimate of the percentage of badly classified.Cross Validation can thus be used when one variablehas the particular status of being explained.<P>And in regresssion<BR>We want to estimate the prediction error:<BR><P></P><DIV ALIGN="CENTER"><!-- MATH \begin{displaymath}PE=E_F(y-\hat{y})^2\end{displaymath} --><IMG WIDTH="143" HEIGHT="33" BORDER="0" SRC="img236.png" ALT="\begin{displaymath}PE=E_F(y-\hat{y})^2\end{displaymath}"></DIV><BR CLEAR="ALL"><P></P>This can be done by cross validation,writing:<BR><P></P><DIV ALIGN="CENTER"><!-- MATH \begin{displaymath}PRESS=\frac{1}{n}\sum_{i=1}^n(\hat{y}_{(i)}-y)^2\end{displaymath} --><IMG WIDTH="213" HEIGHT="55" BORDER="0" SRC="img237.png" ALT="\begin{displaymath}PRESS=\frac{1}{n}\sum_{i=1}^n(\hat{y}_{(i)}-y)^2\end{displaymath}"></DIV><BR CLEAR="ALL"><P></P><P>However it has also been used at the diagnostic stage inprincipal components, and in classification and regression trees where it helps choose the size of an `optimal tree'.<P><HR><!--Navigation Panel--><A NAME="tex2html386" HREF="node18.html"><IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="file:/home/depot/swtree/depot/latex2html-2002-2/latex2html-2002-2/icons/next.png"></A> <A NAME="tex2html384" HREF="node6.html"><IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="file:/home/depot/swtree/depot/latex2html-2002-2/latex2html-2002-2/icons/up.png"></A> <A NAME="tex2html378" HREF="node16.html"><IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="file:/home/depot/swtree/depot/latex2html-2002-2/latex2html-2002-2/icons/prev.png"></A> <BR><B> Next:</B> <A NAME="tex2html387" HREF="node18.html">Bootstrapping a Principal Component</A><B> Up:</B> <A NAME="tex2html385" HREF="node6.html">Lectures</A><B> Previous:</B> <A NAME="tex2html379" HREF="node16.html">The jackknife</A><!--End of Navigation Panel--><ADDRESS>Susan Holmes2004-05-19</ADDRESS></BODY></HTML>⌨️ 快捷键说明
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