📄 node25.html
字号:
<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.0//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>Uneven sampling</TITLE>
<META NAME="description" CONTENT="Uneven sampling">
<META NAME="keywords" CONTENT="Surrogates">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<LINK REL=STYLESHEET HREF="Surrogates.css" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/Surrogates.css">
</HEAD>
<BODY bgcolor=#ffffff LANG="EN" >
<A NAME="tex2html327" HREF="node26.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node26.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/next_motif.gif"></A> <A NAME="tex2html325" HREF="node22.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node22.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/up_motif.gif"></A> <A NAME="tex2html319" HREF="node24.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node24.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/previous_motif.gif"></A> <BR>
<B> Next:</B> <A NAME="tex2html328" HREF="node26.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node26.html">Spike trains</A>
<B>Up:</B> <A NAME="tex2html326" HREF="node22.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node22.html">Various Examples</A>
<B> Previous:</B> <A NAME="tex2html320" HREF="node24.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node24.html">Multivariate data</A>
<BR> <P>
<H2><A NAME="SECTION00063000000000000000">Uneven sampling</A></H2>
<A NAME="secuneven"> </A>
Let us show how the constrained randomisation method can be used to test for
nonlinearity in time series taken at time intervals of different length.
Unevenly sampled data are quite common, examples include drill core
data, astronomical observations or stock price notations. Most observables and
algorithms cannot easily be generalised to this case which is particularly true
for nonlinear time series methods. (See [<A HREF="node36.html#XParzen83" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#XParzen83">41</A>] for material on
irregularly sampled time series.) Interpolating the data to equally spaced
sampling times is not recommendable for a test for nonlinearity since one could
not <I>a posteriori</I> distinguish between genuine structure and nonlinearity
introduced spuriously by the interpolation process. Note that also zero padding
is a nonlinear operation in the sense that stretches of zeroes are unlikely to
be produced by any linear stochastic process.
<P>
For data that is evenly sampled except for a moderate number of gaps, surrogate
sequences can be produced relatively straightforwardly by assuming the value
zero during the gaps and minimising a standard cost function like
Eq.(<A HREF="node17.html#eqcost" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node17.html#eqcost">23</A>) while excluding the gaps from the permutations tried. The
error made in estimating correlations would then be identical for the data and
surrogates and could not affect the validity of the test. Of course, one would
have to modify the nonlinearity measure to avoid the gaps. For data sampled at
incommensurate times, such a strategy can no longer be adopted. We then need
different means to specify the linear correlation structure.
<P>
Two different approaches are viable, one residing in the spectral domain and
one in the time domain. Consider a time series sampled at times <IMG WIDTH=28 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2310" SRC="img132.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img132.gif"> that
need not be equally spaced. The power spectrum can then be estimated by the
Lomb periodogram, as discussed for example in Ref. [<A HREF="node36.html#Press92" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#Press92">42</A>].
For time series sampled at constant time intervals, the Lomb periodogram yields
the standard squared Fourier transformation. Except for this particular case,
it does not have any inverse transformation, which makes it impossible to use
the standard surrogate data algorithms mentioned in Sec. <A HREF="node9.html#secfourier" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node9.html#secfourier">4</A>. In
Ref. [<A HREF="node36.html#lomb" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#lomb">43</A>], we used the Lomb periodogram of the data as a constraint for
the creation of surrogates. Unfortunately, imposing a given Lomb periodogram
is very time consuming because at each annealing step, the <I>O</I>(<I>N</I>) spectral
estimator has to be computed at <IMG WIDTH=43 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline2314" SRC="img133.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img133.gif"> frequencies with <IMG WIDTH=56 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2316" SRC="img134.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img134.gif">.
Press et al. [<A HREF="node36.html#Press92" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#Press92">42</A>] give an approximation algorithm that uses the fast
Fourier transform to compute the Lomb periodogram in <IMG WIDTH=77 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2318" SRC="img135.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img135.gif"> time rather
than <IMG WIDTH=44 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2320" SRC="img136.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img136.gif">. The resulting code is still quite slow.
<P>
As a more efficient alternative to the commonly used but computationally costly
Lomb periodogram, let us suggest to use binned autocorrelations. They are
defined as follows. For a continuous signal <I>s</I>(<I>t</I>) (take <IMG WIDTH=48 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2324" SRC="img137.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img137.gif">,
<IMG WIDTH=54 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2326" SRC="img138.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img138.gif"> for simplicity of notation here), the autocorrelation function is
<IMG WIDTH=338 HEIGHT=33 ALIGN=MIDDLE ALT="tex2html_wrap_inline2328" SRC="img139.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img139.gif">. It can be
binned to a bin size <IMG WIDTH=12 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline2330" SRC="img140.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img140.gif">, giving <IMG WIDTH=207 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline2332" SRC="img141.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img141.gif">. We now have to approximate all
integrals using the available values of <IMG WIDTH=32 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2334" SRC="img142.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img142.gif">. In general, we estimate
<BR><IMG WIDTH=500 HEIGHT=41 ALIGN=BOTTOM ALT="equation1080" SRC="img143.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img143.gif"><BR>
Here, <IMG WIDTH=189 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2336" SRC="img144.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img144.gif"> denotes the bin ranging from <I>a</I>
to <I>b</I> and <IMG WIDTH=52 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2342" SRC="img145.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img145.gif"> the number of its elements. We could improve this
estimate by some interpolation of <IMG WIDTH=24 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2344" SRC="img146.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img146.gif">, as it is customary with numerical
integration but the accuracy of the estimate is not the central issue here.
For the binned autocorrelation, this approximation simply gives
<BR><A NAME="eqcdelta"> </A><IMG WIDTH=500 HEIGHT=43 ALIGN=BOTTOM ALT="equation1082" SRC="img147.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img147.gif"><BR>
Here, <IMG WIDTH=243 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline2346" SRC="img148.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img148.gif">.
Of course, empty bins lead to undefined autocorrelations. If we have evenly
sampled data and unit bins, <IMG WIDTH=192 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2348" SRC="img149.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img149.gif">, then the
⌨️ 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 + -