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<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 data
is composed of the output of a low-dimensional dynamical system and of random
or high-dimensional noise. This means that in an arbitrarily high-dimensional
embedding space the deterministic part of the data would lie on a
low-dimensional manifold, while the effect of the noise is to spread the data
off this manifold. If we suppose that the amplitude of the noise is
sufficiently small, we can expect to find the data distributed closely around
this manifold. The idea of the projective nonlinear noise reduction scheme is
to identify the manifold and to project the data onto it. The strategies
described here go back to Ref. [<A HREF="citation.html#on" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#on">61</A>]. A realistic case study is detailed
in Ref. [<A HREF="citation.html#buzug" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#buzug">62</A>].
<P>
Suppose the dynamical system, Eq. (<A HREF="node5.html#eqode" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node5.html#eqode"><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>) or Eq. (<A HREF="node5.html#eqmap" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node5.html#eqmap"><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>), form a
<I>q</I>-dimensional manifold <IMG WIDTH=18 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline7171" SRC="img78.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img78.gif"> containing the trajectory. According to the
embedding theorems, there exists a one-to-one image of the attractor
in the embedding space, if the embedding dimension is sufficiently
high. Thus, if the measured time series were not corrupted with noise, all the
embedding vectors <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline7173" SRC="img79.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img79.gif"> would lie inside another manifold
<IMG WIDTH=18 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline7175" SRC="img80.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img80.gif"> in the embedding space. Due to the noise
this condition is no longer fulfilled. The idea of the locally projective noise
reduction scheme is that for each <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline7173" SRC="img79.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img79.gif"> there exists a correction
<IMG WIDTH=21 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7179" SRC="img81.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img81.gif">, with <IMG WIDTH=36 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7181" SRC="img82.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img82.gif"> small, in such a way that <IMG WIDTH=95 HEIGHT=30 ALIGN=MIDDLE ALT="tex2html_wrap_inline7183" SRC="img83.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img83.gif"> and that <IMG WIDTH=21 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7179" SRC="img81.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img81.gif"> is orthogonal
on <IMG WIDTH=18 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline7175" SRC="img80.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img80.gif">. Of course a projection to the manifold can only be a
reasonable concept if the vectors are embedded in spaces which are higher
dimensional than the manifold <IMG WIDTH=18 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline7175" SRC="img80.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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 would
think of using the Euclidean metric. But this is not necessarily the best
choice. The reason is that we are working with delay vectors which contain
temporal information. Thus even if the middle parts of two delay
vectors are close, the late parts could be far away from each other due to the
influence of the positive Lyapunov exponents, while the first parts could
diverge due the negative ones. Hence it is usually desirable to correct only
the 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 the
dynamics itself. It turns out that for most applications it is sufficient to
fix just the first and the last component of the delay vectors and correct the
rest. This can be expressed in terms of a metric tensor <IMG WIDTH=12 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline7195" SRC="img84.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img84.gif"> which we
define to be [<A HREF="citation.html#on" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#on">61</A>]
<BR><IMG WIDTH=500 HEIGHT=39 ALIGN=BOTTOM ALT="equation5225" SRC="img85.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img86.gif"><BR>
with the constraints
<BR><IMG WIDTH=500 HEIGHT=19 ALIGN=BOTTOM ALT="equation5229" SRC="img87.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img87.gif"><BR>
and
<BR><IMG WIDTH=500 HEIGHT=20 ALIGN=BOTTOM ALT="equation5231" SRC="img88.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img88.gif"><BR>
where the <IMG WIDTH=17 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline7199" SRC="img89.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img89.gif"> are the normal vectors of <IMG WIDTH=18 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline7175" SRC="img80.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img80.gif"> at the point
<IMG WIDTH=57 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7203" SRC="img90.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img90.gif">.
<P>
This ideas are realized in the programs <a
href="../dresden/ghkss.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/dresden/ghkss.html">ghkss</a>, and <a
href="../wuppertal/project.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/project.html">project</a> in
TISEAN. While the first two work as <EM>a posteriori</EM> filters on complete
data sets, the last one can be used in a data stream. This means that it is
possible to do the corrections online, while the data is coming in (for more
details see section <A HREF="node25.html#subsecnoise_stream" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node25.html#subsecnoise_stream"><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>). All three algorithms mentioned
above correct for curvature effects. This is done by either post-processing the
corrections for the delay vectors (<a href="../dresden/ghkss.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/dresden/ghkss.html">ghkss</a>) or by preprocessing the centres of
mass of the local neighborhoods (<a href="../wuppertal/project.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/project.html">project</a>).
<P>
The idea used in the <a href="../dresden/ghkss.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/dresden/ghkss.html">ghkss</a> program is the following. Suppose the manifold
were strictly linear. Then, provided the noise is white, the corrections in the
vicinity of a point on the manifold would point in all directions with the same
probability. Thus, if we added all the corrections <IMG WIDTH=12 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline7205" SRC="img91.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img91.gif"> we expect
them to sum to zero (or <IMG WIDTH=60 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7207" SRC="img92.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img92.gif">). On the other
hand, if the manifold is curved, we expect that there is a trend towards the
centre of curvature (<IMG WIDTH=73 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7209" SRC="img93.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img93.gif">). Thus, to correct for this trend each
correction <IMG WIDTH=12 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline7205" SRC="img91.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img91.gif"> is replaced by
<IMG WIDTH=59 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7213" SRC="img94.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img94.gif">.
<P>
A different strategy is used in the program <a href="../wuppertal/project.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/project.html">project</a>. The projections are
done in a local coordinate system which is defined by the condition that the
average of the vectors in the neighborhood is zero. Or, in other words, the
origin of the coordinate systems is the centre of mass <IMG WIDTH=36 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7215" SRC="img95.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img95.gif"> of the neighborhood <IMG WIDTH=11 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline6891" SRC="img49.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img49.gif">. This centre of mass has a
bias towards the centre of the curvature [<A HREF="citation.html#KantzSchreiber" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#KantzSchreiber">2</A>]. Hence, a
projection would not lie on the tangent at the manifold, but on a secant. Now
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