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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>Mutual information</TITLE><META NAME="description" CONTENT="Mutual information"><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="tex2html154" HREF="node9.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html152" HREF="node7.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html146" HREF="node7.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html155" HREF="node9.html">False nearest neighbors</A><B>Up:</B> <A NAME="tex2html153" HREF="node7.html">Embedding parameters</A><B> Previous:</B> <A NAME="tex2html147" HREF="node7.html">Embedding parameters</A><BR> <P><H3><A NAME="SECTION00032100000000000000">Mutual information</A></H3><P>The time delayed mutual information was suggested by Fraser andSwinney [<A HREF="citation.html#fraser">25</A>] as a tool to determine a reasonable delay: Unlike theautocorrelation function, the mutual information takes into account alsononlinear correlations. One has to compute<BR><A NAME="eqdelaymutual"> </A><IMG WIDTH=500 HEIGHT=45 ALIGN=BOTTOM ALT="equation4425" SRC="img19.gif"><BR> where for some partition on the real numbers <IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6569" SRC="img20.gif"> is the probability to finda time series value in the <I>i</I>-th interval, and <IMG WIDTH=40 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline6573" SRC="img21.gif"> is the jointprobability that an observation falls into the <I>i</I>-th interval and theobservation time <IMG WIDTH=8 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6553" SRC="img16.gif"> later falls into the <I>j</I>-th. In theory this expressionhas no systematic dependence on the size of the partition elements and can bequite easily computed. There exist good arguments that if the time delayedmutual information exhibits a marked minimum at a certain value of <IMG WIDTH=8 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6553" SRC="img16.gif">,then this is a good candidate for a reasonable time delay. However, thesearguments have to be modified when the embedding dimension exceeds two.Moreover, as will become transparent in the following sections, not allapplications work optimally with the same delay. Our routine <a href="../docs_c/mutual.html">mutual</a> usesEq.(<A HREF="node8.html#eqdelaymutual"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>), where the number of boxes of identical size andthe maximal delay time has to be supplied. The adaptive algorithm usedin [<A HREF="citation.html#fraser">25</A>] is more data intensive. Since we are not really interested inabsolute values of the mutual information here but rather in the firstminimum, the minimal implementation given here seems to be sufficient. Therelated generalized mutual information of order two can be defined using thecorrelation sum concept (Sec.<A HREF="node29.html#secdimension"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>, [<A HREF="citation.html#pompe">26</A>, <A HREF="citation.html#milan">27</A>]).Estimation of the correlation entropy is explained in Sec.<A HREF="node30.html#secdimc2"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>.<P><HR><A NAME="tex2html154" HREF="node9.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html152" HREF="node7.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html146" HREF="node7.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html155" HREF="node9.html">False nearest neighbors</A><B>Up:</B> <A NAME="tex2html153" HREF="node7.html">Embedding parameters</A><B> Previous:</B> <A NAME="tex2html147" HREF="node7.html">Embedding parameters</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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