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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>SVD filters</TITLE><META NAME="description" CONTENT="SVD filters"><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="tex2html190" HREF="node13.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html188" HREF="node5.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html184" HREF="node11.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html191" HREF="node13.html">Visualizationnon-stationarity</A><B>Up:</B> <A NAME="tex2html189" HREF="node5.html">Phase space representation</A><B> Previous:</B> <A NAME="tex2html185" HREF="node11.html">Poincaré sections</A><BR> <P><H2><A NAME="SECTION00035000000000000000">SVD filters</A></H2><A NAME="secsvdfilter"> </A>There are at least two reasons to apply an SVD filter to time series data:Either, if one is working with flow data, one can implicitly determine theoptimal time delay, or, when deriving a stroboscopic map from synchronouslysampled data of a periodically driven system, one might use the redundancy tooptimize the signal to noise ratio.<P>In both applications the mathematics is the same: One constructs thecovariance matrix of all data vectors (e.g. in an <I>m</I>-dimensionaltime delay embedding space), <BR><IMG WIDTH=500 HEIGHT=17 ALIGN=BOTTOM ALT="equation4638" SRC="img41.gif"><BR>and computes its singular vectors. Then one projects onto the<I>m</I>-dimensional vectorscorresponding to the <I>q</I> largest singular values. To work with flow data, <I>q</I>should be at least the correct embedding dimension, and <I>m</I> considerably larger (e.g. <I>m</I>=2<I>q</I> or larger). The result is a vector valued time series, andin [<A HREF="citation.html#embed">22</A>] the relation of these components to temporal derivatives onthe one hand and to Fourier components on the other hand were discussed. If,in the non-autonomous case, one wants to compress flow data to map data,<I>q</I>=1. In this case, the redundancy of the flow is implicitly used for noise reduction of the map data. The routine <a href="../docs_c/pca.html">pca</a> can be used for both purposes.<P><HR><A NAME="tex2html190" HREF="node13.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html188" HREF="node5.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html184" HREF="node11.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html191" HREF="node13.html">Visualizationnon-stationarity</A><B>Up:</B> <A NAME="tex2html189" HREF="node5.html">Phase space representation</A><B> Previous:</B> <A NAME="tex2html185" HREF="node11.html">Poincaré sections</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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