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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>Simple nonlinear noise reduction</TITLE><META NAME="description" CONTENT="Simple nonlinear noise reduction"><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="tex2html306" HREF="node24.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html304" HREF="node22.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html298" HREF="node22.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html307" HREF="node24.html">Locally projective nonlinear noise </A><B>Up:</B> <A NAME="tex2html305" HREF="node22.html">Nonlinear noise reduction</A><B> Previous:</B> <A NAME="tex2html299" HREF="node22.html">Nonlinear noise reduction</A><BR> <P><H2><A NAME="SECTION00061000000000000000">Simple nonlinear noise reduction</A></H2><P>The simplest nonlinear noise reduction algorithm we know of replacesthe central coordinate of each embedding vector by the local average of thiscoordinate. This amounts to a locally constant approximation of the dynamicsand is based on the assumption that the dynamics is continuous. The algorithmis described in [<A HREF="citation.html#lazy">59</A>], a similar approach is proposed in [<A HREF="citation.html#Arkady">43</A>].In an unstable, for example chaotic, system, it is essential not to replace the first and last coordinates of the embedding vectors by local averages.Due to the instability, initial errors in these coordinates are magnifiedinstead of being averaged out.<P>This noise reduction scheme is implemented quite easily. First an embeddinghas to be chosen. Except for extremely oversampled data, it is advantageous tochoose a short time delay. The program <a href="../docs_f/lazy.html">lazy</a> always uses unit delay. Theembedding dimension <I>m</I> should be chosen somewhat higher than that required bythe embedding theorems. Then for each embedding vector <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6675" SRC="img35.gif">, aneighborhood <IMG WIDTH=28 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline7109" SRC="img70.gif"> is formed in phase space containingall points <IMG WIDTH=34 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7111" SRC="img71.gif"> such that <IMG WIDTH=97 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7113" SRC="img72.gif">. Theradius of the neighborhoods <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif"> should be taken large enough in orderto cover the noise extent, but still smaller than a typical curvature radius.These conditions cannot always be fulfilled simultaneously, in which case onehas to repeat the process with several choices and carefully evaluate theresults. If the noise level is substantially smaller than the typical radiusof curvature, neighborhoods of radius about 2-3 times the noise level gave thebest results with artificial data. For each embedding vector<IMG WIDTH=163 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline7117" SRC="img73.gif"> (the delay time has been set to unity), acorrected middle coordinate <IMG WIDTH=48 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline7119" SRC="img74.gif"> is computed by averaging overthe neighborhood <IMG WIDTH=28 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline7109" SRC="img70.gif">:<BR><A NAME="eqlazy"> </A><IMG WIDTH=500 HEIGHT=48 ALIGN=BOTTOM ALT="equation5091" SRC="img75.gif"><BR>After one complete sweep through the time series, all measurements <IMG WIDTH=14 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6689" SRC="img37.gif"> arereplaced by the corrected values <IMG WIDTH=14 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7125" SRC="img76.gif">. Of course, for the first and last(<I>m</I>-1)/2 points (if <I>m</I> is odd), no correction is available. The averagecorrection can be taken as a new neighborhood radius for the next iteration.Note that the neighborhood of each point at least contains the point itself.If that is the only member, the average Eq.(<A HREF="node23.html#eqlazy"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) is simply theuncorrected measurement and no change is made. Thus one can safely performmultiple iterations with decreasing values of <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif"> until no furtherchange is made.<P>Let us illustrate the use of this scheme with an example, a recording of theair flow through the nose of a human as an indicator of breathactivity. (Thedata is part of data set B of the Santa Fe time series contest held in1991/92 [<A HREF="citation.html#SFI">17</A>], see Rigney et al. [<A HREF="citation.html#b.dat">60</A>] for a description.)The result of simple nonlinear noise reduction is shown in Fig. <A HREF="node23.html#figb"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>.<P><P><blockquote><A NAME="5218"> </A><IMG WIDTH=214 HEIGHT=489 ALIGN=BOTTOM ALT="figure1024" SRC="img77.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figb"> </A> Simple nonlinear noise reduction of human breath rate data. Three iterations have been carried out, staring with neighborhoods of size 0.4 units. Embeddings in 7 dimensions at unit delay have been used. Arguably, the resulting series (lower panel) is less noisy. However, in Sec. <A HREF="node35.html#secsurro"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> we will show evidence that the noise is not just additive and independent of the signal. <BR></blockquote><P><HR><A NAME="tex2html306" HREF="node24.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html304" HREF="node22.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html298" HREF="node22.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html307" HREF="node24.html">Locally projective nonlinear noise </A><B>Up:</B> <A NAME="tex2html305" HREF="node22.html">Nonlinear noise reduction</A><B> Previous:</B> <A NAME="tex2html299" HREF="node22.html">Nonlinear noise reduction</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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