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<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.2//EN"><!--Converted with LaTeX2HTML 96.1-h (September 30, 1996) by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds --><HTML><HEAD><TITLE>Principal components</TITLE><META NAME="description" CONTENT="Principal components"><META NAME="keywords" CONTENT="TiseanHTML"><META NAME="resource-type" CONTENT="document"><META NAME="distribution" CONTENT="global"><LINK REL=STYLESHEET HREF="TiseanHTML.css"></HEAD><BODY bgcolor=ffffff LANG="EN" > <A NAME="tex2html172" HREF="node11.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html170" HREF="node5.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html164" HREF="node9.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html173" HREF="node11.html">Poincaré sections</A><B>Up:</B> <A NAME="tex2html171" HREF="node5.html">Phase space representation</A><B> Previous:</B> <A NAME="tex2html165" HREF="node9.html">False nearest neighbors</A><BR> <P><H2><A NAME="SECTION00033000000000000000">Principal components</A></H2><A NAME="secsvd"> </A>It has been shown in Ref. [<A HREF="citation.html#embed">22</A>] that the embedding technique can begeneralized to a wide class of smooth transformations applied to a time delayembedding. In particular, if we introduce time delay coordinates <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6675" SRC="img35.gif">,then almost every linear transformation of sufficient rank again leads to anembedding. A specific choice of linear transformation is known as <EM>principal component analysis, singular value decomposition, empiricalorthogonal functions, Karhunen-Loéve decomposition</EM>, and probably a few othernames. The technique is fairly widely used, for example to reduce multivariatedata to a few major modes. There is a large literature, including textbookslike that by Jolliffe [<A HREF="citation.html#PC">31</A>]. In the context of nonlinear signal processing,the technique has been advocated among others by Broomhead and King [<A HREF="citation.html#svd">32</A>].<P>The idea is to introduce a new set of orthonormal basis vectors in embeddingspace such that projections onto a given number of these directions preservethe maximal fraction of the variance of the original vectors. In other words,the error in making the projection is minimized for a given number ofdirections. Solving this minimization problem [<A HREF="citation.html#PC">31</A>] leads to an eigenvalueproblem. The desired <EM>principal directions</EM> can be obtained as theeigenvectors of the symmetric autocovariance matrix that correspond to thelargest eigenvalues. The alternative and formally equivalent approach via thetrajectory matrix is used in Ref. [<A HREF="citation.html#svd">32</A>]. The latter is numerically morestable but involves the singular value decomposition of an <IMG WIDTH=47 HEIGHT=20 ALIGN=MIDDLE ALT="tex2html_wrap_inline6677" SRC="img36.gif"> matrixfor <I>N</I> data points embedded in <I>m</I> dimensions, which can easily exceedcomputational resources for time series of even moderate length [<A HREF="citation.html#numrec">33</A>].<P>In almost all the algorithms described below, simple time delay embeddings canbe substituted by principal components. In the TISEAN project (routines<a href="../docs_c/pca.html">pca</a>, <a href="../docs_f/pc.html">pc</a>), principal components are only provided as a stand-alonevisualization tool and for linear filtering [<A HREF="citation.html#Vautard">34</A>], seeSec. <A HREF="node12.html#secsvdfilter"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> below. In any case, one first has to choose aninitial time delay embedding and then a number of principal components to bekept. For the purpose of visualization, the latter is immediately restrictedto two or at most three. In order to take advantage of the noise averagingeffect of the principal component scheme, it is advisable to choose a muchshorter delay than one would for an ordinary time delay embedding, while atthe same time increasing the embedding dimension. Experimentation isrecommended. Figure <A HREF="node10.html#figmcg_pc"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> shows the contributions of the first twoprincipal components to the magneto-cardiogram shown in Fig. <A HREF="node6.html#figmcgd"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>.<P><P><blockquote><A NAME="4545"> </A><IMG WIDTH=225 HEIGHT=233 ALIGN=BOTTOM ALT="figure417" SRC="img34.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figmcg_pc"> </A> Phase space representation of a human magneto-cardiogram using the two largest principal components. An initial embedding was chosen in <I>m</I>=20 dimensions with a delay of <IMG WIDTH=38 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline6649" SRC="img33.gif"> ms. The two components cover 70% of the variance of the initial embedding vectors.<BR></blockquote><P><HR><A NAME="tex2html172" HREF="node11.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html170" HREF="node5.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html164" HREF="node9.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html173" HREF="node11.html">Poincaré sections</A><B>Up:</B> <A NAME="tex2html171" HREF="node5.html">Phase space representation</A><B> Previous:</B> <A NAME="tex2html165" HREF="node9.html">False nearest neighbors</A><P><ADDRESS><I>Thomas Schreiber <BR>Wed Jan 6 15:38:27 CET 1999</I></ADDRESS></BODY></HTML>⌨️ 快捷键说明
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