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<BR> <P>
<H2><A NAME="SECTION00061000000000000000">Simple nonlinear noise reduction</A></H2>
<P>
The simplest nonlinear noise reduction algorithm we know of replaces
the central coordinate of each embedding vector by the local average of this
coordinate. This amounts to a locally constant approximation of the dynamics
and is based on the assumption that the dynamics is continuous. The algorithm
is described in [<A HREF="citation.html#lazy" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#lazy">59</A>], a similar approach is proposed in [<A HREF="citation.html#Arkady" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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 magnified
instead of being averaged out.
<P>
This noise reduction scheme is implemented quite easily. First an embedding
has to be chosen. Except for extremely oversampled data, it is advantageous to
choose a short time delay. The program <a href="../wuppertal/lazy.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/lazy.html">lazy</a> always uses unit delay. The
embedding dimension <I>m</I> should be chosen somewhat higher than that required by
the embedding theorems. Then for each embedding vector <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6675" SRC="img35.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img35.gif">, a
neighborhood <IMG WIDTH=28 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline7109" SRC="img70.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img70.gif"> is formed in phase space containing
all points <IMG WIDTH=34 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7111" SRC="img71.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img71.gif"> such that <IMG WIDTH=97 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7113" SRC="img72.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img72.gif">. The
radius of the neighborhoods <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img3.gif"> should be taken large enough in order
to cover the noise extent, but still smaller than a typical curvature radius.
These conditions cannot always be fulfilled simultaneously, in which case one
has to repeat the process with several choices and carefully evaluate the
results. If the noise level is substantially smaller than the typical radius
of curvature, neighborhoods of radius about 2-3 times the noise level gave the
best results with artificial data. For each embedding vector
<IMG WIDTH=163 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline7117" SRC="img73.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img73.gif"> (the delay time has been set to unity), a
corrected middle coordinate <IMG WIDTH=48 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline7119" SRC="img74.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img74.gif"> is computed by averaging over
the neighborhood <IMG WIDTH=28 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline7109" SRC="img70.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img70.gif">:
<BR><A NAME="eqlazy"> </A><IMG WIDTH=500 HEIGHT=48 ALIGN=BOTTOM ALT="equation5091" SRC="img75.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img37.gif"> are
replaced by the corrected values <IMG WIDTH=14 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7125" SRC="img76.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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 average
correction 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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node23.html#eqlazy"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/cross_ref_motif.gif"></A>) is simply the
uncorrected measurement and no change is made. Thus one can safely perform
multiple iterations with decreasing values of <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img3.gif"> until no further
change is made.
<P>
Let us illustrate the use of this scheme with an example, a recording of the
air flow through the nose of a human as an indicator of breath
activity. (The
data is part of data set B of the Santa Fe time series contest held in
1991/92 [<A HREF="citation.html#SFI" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#SFI">17</A>], see Rigney et al. [<A HREF="citation.html#b.dat" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node23.html#figb"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node35.html#secsurro"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/cross_ref_motif.gif"></A> we will show evidence that the noise is not just
additive and independent of the signal. <BR>
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<P><ADDRESS>
<I>Thomas Schreiber <BR>
Wed Jan 6 15:38:27 CET 1999</I>
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