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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>Locally projective nonlinear noise reduction</TITLE><META NAME="description" CONTENT="Locally projective 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="tex2html316" HREF="node25.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html314" HREF="node22.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html308" HREF="node23.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html317" HREF="node25.html">Nonlinear noise reduction in </A><B>Up:</B> <A NAME="tex2html315" HREF="node22.html">Nonlinear noise reduction</A><B> Previous:</B> <A NAME="tex2html309" HREF="node23.html">Simple nonlinear noise reduction</A><BR> <P><H2><A NAME="SECTION00062000000000000000">Locally projective nonlinear noise reduction</A></H2><P>A more sophisticated method makes use of the hypotheses that the measured datais composed of the output of a low-dimensional dynamical system and of randomor high-dimensional noise. This means that in an arbitrarily high-dimensionalembedding space the deterministic part of the data would lie on alow-dimensional manifold, while the effect of the noise is to spread the dataoff this manifold. If we suppose that the amplitude of the noise issufficiently small, we can expect to find the data distributed closely aroundthis manifold. The idea of the projective nonlinear noise reduction scheme isto identify the manifold and to project the data onto it. The strategiesdescribed here go back to Ref. [<A HREF="citation.html#on">61</A>]. A realistic case study is detailedin Ref. [<A HREF="citation.html#buzug">62</A>].<P>Suppose the dynamical system, Eq. (<A HREF="node5.html#eqode"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) or Eq. (<A HREF="node5.html#eqmap"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>), form a<I>q</I>-dimensional manifold <IMG WIDTH=18 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline7171" SRC="img78.gif"> containing the trajectory. According to theembedding theorems, there exists a one-to-one image of the attractor in the embedding space, if the embedding dimension is sufficientlyhigh. Thus, if the measured time series were not corrupted with noise, all theembedding vectors <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline7173" SRC="img79.gif"> would lie inside another manifold<IMG WIDTH=18 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline7175" SRC="img80.gif"> in the embedding space. Due to the noisethis condition is no longer fulfilled. The idea of the locally projective noisereduction scheme is that for each <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline7173" SRC="img79.gif"> there exists a correction<IMG WIDTH=21 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7179" SRC="img81.gif">, with <IMG WIDTH=36 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7181" SRC="img82.gif"> small, in such a way that <IMG WIDTH=95 HEIGHT=30 ALIGN=MIDDLE ALT="tex2html_wrap_inline7183" SRC="img83.gif"> and that <IMG WIDTH=21 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7179" SRC="img81.gif"> is orthogonalon <IMG WIDTH=18 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline7175" SRC="img80.gif">. Of course a projection to the manifold can only be areasonable concept if the vectors are embedded in spaces which are higherdimensional than the manifold <IMG WIDTH=18 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline7175" SRC="img80.gif">. Thus we have to over-embed in<I>m</I>-dimensional spaces with <I>m</I>><I>q</I>.<P>The notion of orthogonality depends on the metric used. Intuitively one wouldthink of using the Euclidean metric. But this is not necessarily the bestchoice. The reason is that we are working with delay vectors which containtemporal information. Thus even if the middle parts of two delayvectors are close, the late parts could be far away from each other due to theinfluence of the positive Lyapunov exponents, while the first parts coulddiverge due the negative ones. Hence it is usually desirable to correct onlythe center part of delay vectors and leave the outer parts mostly unchanged,since their divergence is not only a consequence of the noise, but also of thedynamics itself. It turns out that for most applications it is sufficient tofix just the first and the last component of the delay vectors and correct therest. This can be expressed in terms of a metric tensor <IMG WIDTH=12 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline7195" SRC="img84.gif"> which wedefine to be [<A HREF="citation.html#on">61</A>]<BR><IMG WIDTH=500 HEIGHT=39 ALIGN=BOTTOM ALT="equation5225" SRC="img85.gif"><BR>where <I>m</I> is the dimension of the ``over-embedded'' delay vectors.<P>Thus we have to solve the minimization problem<BR><IMG WIDTH=500 HEIGHT=36 ALIGN=BOTTOM ALT="equation5227" SRC="img86.gif"><BR>with the constraints<BR><IMG WIDTH=500 HEIGHT=19 ALIGN=BOTTOM ALT="equation5229" SRC="img87.gif"><BR>and<BR><IMG WIDTH=500 HEIGHT=20 ALIGN=BOTTOM ALT="equation5231" SRC="img88.gif"><BR>where the <IMG WIDTH=17 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline7199" SRC="img89.gif"> are the normal vectors of <IMG WIDTH=18 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline7175" SRC="img80.gif"> at the point<IMG WIDTH=57 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7203" SRC="img90.gif">.<P>This ideas are realized in the programs <ahref="../docs_c/ghkss.html">ghkss</a>, and <ahref="../docs_f/project.html">project</a> inTISEAN. While the first two work as <EM>a posteriori</EM> filters on completedata sets, the last one can be used in a data stream. This means that it ispossible to do the corrections online, while the data is coming in (for moredetails see section <A HREF="node25.html#subsecnoise_stream"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>). All three algorithms mentionedabove correct for curvature effects. This is done by either post-processing thecorrections for the delay vectors (<a href="../docs_c/ghkss.html">ghkss</a>) or by preprocessing the centres ofmass of the local neighborhoods (<a href="../docs_f/project.html">project</a>).<P>The idea used in the <a href="../docs_c/ghkss.html">ghkss</a> program is the following. Suppose the manifoldwere strictly linear. Then, provided the noise is white, the corrections in thevicinity of a point on the manifold would point in all directions with the sameprobability. Thus, if we added all the corrections <IMG WIDTH=12 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline7205" SRC="img91.gif"> we expectthem to sum to zero (or <IMG WIDTH=60 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7207" SRC="img92.gif">). On the otherhand, if the manifold is curved, we expect that there is a trend towards thecentre of curvature (<IMG WIDTH=73 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7209" SRC="img93.gif">). Thus, to correct for this trend eachcorrection <IMG WIDTH=12 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline7205" SRC="img91.gif"> is replaced by<IMG WIDTH=59 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7213" SRC="img94.gif">.<P>A different strategy is used in the program <a href="../docs_f/project.html">project</a>. The projections aredone in a local coordinate system which is defined by the condition that theaverage of the vectors in the neighborhood is zero. Or, in other words, theorigin of the coordinate systems is the centre of mass <IMG WIDTH=36 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7215" SRC="img95.gif"> of the neighborhood <IMG WIDTH=11 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline6891" SRC="img49.gif">. This centre of mass has abias towards the centre of the curvature [<A HREF="citation.html#KantzSchreiber">2</A>]. Hence, aprojection would not lie on the tangent at the manifold, but on a secant. Now⌨️ 快捷键说明
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