📄 node6.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>Delay coordinates</TITLE><META NAME="description" CONTENT="Delay coordinates"><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="tex2html132" HREF="node7.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html130" HREF="node5.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html124" HREF="node5.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html133" HREF="node7.html">Embedding parameters</A><B>Up:</B> <A NAME="tex2html131" HREF="node5.html">Phase space representation</A><B> Previous:</B> <A NAME="tex2html125" HREF="node5.html">Phase space representation</A><BR> <P><H2><A NAME="SECTION00031000000000000000">Delay coordinates</A></H2><P>The most important phase space reconstruction technique is the <EM>method ofdelays</EM>. Vectors in a new space, the embedding space, are formed from timedelayed values of the scalar measurements:<BR><A NAME="eqdelay"> </A><IMG WIDTH=500 HEIGHT=18 ALIGN=BOTTOM ALT="equation4416" SRC="img15.gif"><BR>The number <I>m</I> of elements is called the <EM>embedding dimension</EM>, the time<IMG WIDTH=8 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6553" SRC="img16.gif"> is generally referred to as the <EM>delay</EM> or <EM>lag</EM>. Celebratedembedding theorems by Takens [<A HREF="citation.html#takens">21</A>] and by Sauer et al. [<A HREF="citation.html#embed">22</A>]state that if the sequence <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6549" SRC="img14.gif"> does indeed consist of scalar measurementsof the state of a dynamical system, then under certain genericity assumptions,the time delay embedding provides a one-to-one image of the original set<IMG WIDTH=23 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6557" SRC="img17.gif">, provided <I>m</I> is large enough.<P>Time delay embeddings are used in almost all methods described in thispaper. The implementation is straightforward and does not require furtherexplanation. If <I>N</I> scalar measurements are available, the number of embeddingvectors is only <IMG WIDTH=96 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6563" SRC="img18.gif">. This has to be kept in mind for the correctnormalization of averaged quantities. There is a large literature on the``optimal'' choice of the embedding parameters <I>m</I> and <IMG WIDTH=8 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6553" SRC="img16.gif">. It turns out,however, that what constitutes the optimal choice largely depends on theapplication. We will therefore discuss the choice of embedding parametersoccasionally together with other algorithms below.<P><P><blockquote><A NAME="4404"> </A><IMG WIDTH=222 HEIGHT=481 ALIGN=BOTTOM ALT="figure190" SRC="img7.gif"><br><STRONG>Figure:</STRONG> <A NAME="figmcgd"> </A> Time delay representation of a human magneto-cardiogram. In the upper panel, a short delay time of 10 ms is used to resolve the fast waveform corresponding to the contraction of the ventricle. In the lower panel, the slower recovery phase of the ventricle (small loop) is better resolved due to the use of a slightly longer delay of 40 ms. Such a plot can be conveniently be produced by a graphic tool such as <a href="http://www.cs.dartmouth.edu/gnuplot_info.html">gnuplot</a> without generating extra data files.<BR></blockquote><P>A stand-alone version of the delay procedure (<ahref="../docs_c/delay.html">delay</a>) is animportant tool for the visual inspection of data, even though visualization isrestricted to two dimensions, or at most two-dimensional projections ofthree-dimensional renderings. A good unfolding already in two dimensions maygive some guidance about a good choice of the delay time for higherdimensional embeddings. As an example let us show two differenttwo-dimensional delay coordinate representations of a human magneto-cardiogram(Fig. <A HREF="node6.html#figmcgd"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>). Note that we do neither assume nor claim that themagneto- (or electro-) cardiogram is deterministic or even chaotic. Althoughin the particular case of cardiac recordings the use of time delay embeddingscan be motivated theoretically [<A HREF="citation.html#marcus">23</A>], we here only want to use theembedding technique as a visualization tool.<P><HR><A NAME="tex2html132" HREF="node7.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html130" HREF="node5.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html124" HREF="node5.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html133" HREF="node7.html">Embedding parameters</A><B>Up:</B> <A NAME="tex2html131" HREF="node5.html">Phase space representation</A><B> Previous:</B> <A NAME="tex2html125" HREF="node5.html">Phase space representation</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 + -