📄 node34.html
字号:
<!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>Entropy estimates</TITLE><META NAME="description" CONTENT="Entropy estimates"><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="tex2html416" HREF="node35.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html414" HREF="node33.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html410" HREF="node33.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html417" HREF="node35.html">Testing for nonlinearity</A><B>Up:</B> <A NAME="tex2html415" HREF="node33.html">Information dimension</A><B> Previous:</B> <A NAME="tex2html411" HREF="node33.html">Information dimension</A><BR> <P><H3><A NAME="SECTION00082100000000000000">Entropy estimates</A></H3><P>The correlation dimension characterizes the <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif"> dependence of thecorrelation sum inside the scaling range. It is natural to ask what we canlearn form its <I>m</I>-dependence, once <I>m</I> is larger than <IMG WIDTH=19 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7885" SRC="img166.gif">. The number of<IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif">-neighbors of a delay vector is an estimate of the local probabilitydensity, and in fact it is a kind of joint probability: All <I>m</I>-components ofthe neighbor have to be similar to those of the actual vector simultaneously.Thus when increasing <I>m</I>, joint probabilities covering larger time spans getinvolved. The scaling of these joint probabilities is related to thecorrelation entropy <IMG WIDTH=14 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7893" SRC="img167.gif">, such that<BR><A NAME="eqdimfullC2"> </A><IMG WIDTH=500 HEIGHT=19 ALIGN=BOTTOM ALT="equation5956" SRC="img168.gif"><BR>As for the scaling in <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif">, also the dependence on <I>m</I> is validonly asymptotically for large <I>m</I>, which one will not reach due to the lackof data points. So one will study <IMG WIDTH=40 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7901" SRC="img169.gif"> versus <I>m</I> and try to extrapolate tolarge <I>m</I>. The correlation entropy is a lower bound of the Kolmogorov Sinaientropy, which in turn can be estimated by the sum of the positive Lyapunovexponents. The program <a href="../docs_c/d2.html">d2</a> produces as output the estimates of <IMG WIDTH=14 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7893" SRC="img167.gif">directly, from the other correlation sum programs it has to be extracted bypost-processing the output.<P>The entropies of first and second order can be derived from the output of <a href="../docs_f/c1.html">c1</a> and <a href="../docs_c/d2.html">d2</a> respectively. An alternate means of obtaining these and theother generalized entropies is by a box counting approach. Let <IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6569" SRC="img20.gif"> be theprobability to find the system state in box <I>i</I>, then the order <I>q</I> entropy isdefined by the limit of small box size and large <I>m</I> of<BR><A NAME="eqhq"> </A><IMG WIDTH=500 HEIGHT=34 ALIGN=BOTTOM ALT="equation5961" SRC="img170.gif"><BR> To evaluate <IMG WIDTH=38 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline7917" SRC="img171.gif"> over a fine mesh of boxes in <IMG WIDTH=46 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline7919" SRC="img172.gif"> dimensions,economical use of memory is necessary: A simple histogram would take <IMG WIDTH=45 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7921"SRC="img173.gif"> storage. Therefore the program <a href="../docs_c/boxcount.html">boxcount</a> implementsthe mesh of boxes as a tree with <IMG WIDTH=33 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7923" SRC="img174.gif">-fold branching points. Thetree is worked through recursively so that at each instance at most onecomplete branch exists in storage. The current version does not implementfinite sample corrections to Eq.(<A HREF="node34.html#eqhq"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>).<P><HR><A NAME="tex2html416" HREF="node35.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html414" HREF="node33.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html410" HREF="node33.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html417" HREF="node35.html">Testing for nonlinearity</A><B>Up:</B> <A NAME="tex2html415" HREF="node33.html">Information dimension</A><B> Previous:</B> <A NAME="tex2html411" HREF="node33.html">Information dimension</A><P><ADDRESS><I>Thomas Schreiber <BR>Wed Jan 6 15:38:27 CET 1999</I></ADDRESS></BODY></HTML>⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -