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<!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"></HEAD><BODY bgcolor=#ffffff LANG="EN" > <A NAME="tex2html327" HREF="node26.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html325" HREF="node22.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html319" HREF="node24.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html328" HREF="node26.html">Spike trains</A><B>Up:</B> <A NAME="tex2html326" HREF="node22.html">Various Examples</A><B> Previous:</B> <A NAME="tex2html320" HREF="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 fornonlinearity in time series taken at time intervals of different length.Unevenly sampled data are quite common, examples include drill coredata, astronomical observations or stock price notations. Most observables andalgorithms cannot easily be generalised to this case which is particularly truefor nonlinear time series methods. (See [<A HREF="node36.html#XParzen83">41</A>] for material onirregularly sampled time series.) Interpolating the data to equally spacedsampling times is not recommendable for a test for nonlinearity since one couldnot <I>a posteriori</I> distinguish between genuine structure and nonlinearityintroduced spuriously by the interpolation process. Note that also zero paddingis a nonlinear operation in the sense that stretches of zeroes are unlikely tobe produced by any linear stochastic process.<P>For data that is evenly sampled except for a moderate number of gaps, surrogatesequences can be produced relatively straightforwardly by assuming the valuezero during the gaps and minimising a standard cost function likeEq.(<A HREF="node17.html#eqcost">23</A>) while excluding the gaps from the permutations tried. Theerror made in estimating correlations would then be identical for the data andsurrogates and could not affect the validity of the test. Of course, one wouldhave to modify the nonlinearity measure to avoid the gaps. For data sampled atincommensurate times, such a strategy can no longer be adopted. We then needdifferent means to specify the linear correlation structure.<P>Two different approaches are viable, one residing in the spectral domain andone 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"> thatneed not be equally spaced. The power spectrum can then be estimated by theLomb periodogram, as discussed for example in Ref. [<A HREF="node36.html#Press92">42</A>].For time series sampled at constant time intervals, the Lomb periodogram yieldsthe standard squared Fourier transformation. Except for this particular case,it does not have any inverse transformation, which makes it impossible to usethe standard surrogate data algorithms mentioned in Sec. <A HREF="node9.html#secfourier">4</A>. InRef. [<A HREF="node36.html#lomb">43</A>], we used the Lomb periodogram of the data as a constraint forthe creation of surrogates. Unfortunately, imposing a given Lomb periodogramis very time consuming because at each annealing step, the <I>O</I>(<I>N</I>) spectralestimator has to be computed at <IMG WIDTH=43 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline2314" SRC="img133.gif"> frequencies with <IMG WIDTH=56 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2316" SRC="img134.gif">.Press et al. [<A HREF="node36.html#Press92">42</A>] give an approximation algorithm that uses the fastFourier transform to compute the Lomb periodogram in <IMG WIDTH=77 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2318" SRC="img135.gif"> time ratherthan <IMG WIDTH=44 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2320" SRC="img136.gif">. The resulting code is still quite slow.<P>As a more efficient alternative to the commonly used but computationally costlyLomb periodogram, let us suggest to use binned autocorrelations. They aredefined 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">,<IMG WIDTH=54 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2326" SRC="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">. It can bebinned to a bin size <IMG WIDTH=12 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline2330" SRC="img140.gif">, giving <IMG WIDTH=207 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline2332" SRC="img141.gif">. We now have to approximate allintegrals using the available values of <IMG WIDTH=32 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2334" SRC="img142.gif">. In general, we estimate<BR><IMG WIDTH=500 HEIGHT=41 ALIGN=BOTTOM ALT="equation1080" SRC="img143.gif"><BR>Here, <IMG WIDTH=189 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2336" SRC="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"> the number of its elements. We could improve thisestimate by some interpolation of <IMG WIDTH=24 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2344" SRC="img146.gif">, as it is customary with numericalintegration 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"><BR>Here, <IMG WIDTH=243 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline2346" SRC="img148.gif">.Of course, empty bins lead to undefined autocorrelations. If we have evenlysampled data and unit bins, <IMG WIDTH=192 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline2348" SRC="img149.gif">, then thebinned autocorrelations coincide with ordinary autocorrelations at<IMG WIDTH=182 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2350" SRC="img150.gif">.<P>Once we are able to specify the linear properties of a time series, we can alsodefine a cost function as usual and generate surrogates that realise the binnedautocorrelations of the data. A delicate point however is the choice of binsize. If we take it too small, we get bins that are almost empty. Within thespace of permutations, there may be only a few ways then to generate preciselythat value of <IMG WIDTH=43 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline2352" SRC="img151.gif">, in other words, we over-specifythe problem. If we take the bin size too large, we might not capture importantstructure in the autocorrelation function.<P>As an application, let us construct randomised versions of part of an ice coredata set, taken from the Greenland Ice Sheet Project Two (GISP2) [<A HREF="node36.html#gisp2">44</A>].An extensive data base resulting from the analysis of physical and chemicalproperties of Greenland ice up to a depth of 3028.8 m has been published by theNational Snow and Ice Data Center together with the World Data Center-A forPalaeoclimatology, National Geophysical Data Center, Boulder,Colorado [<A HREF="node36.html#CDROM">45</A>]. A long ice core is usually cut into equidistant slicesand initially, all measurements are made versus depth. Considerable expertisethen goes into the dating of each slice [<A HREF="node36.html#dating">46</A>]. Since the density of theice, as well as the annual total deposition, changes with time, the final timeseries data are necessarily unevenly sampled. Furthermore, often a few valuesare missing from the record. We will study a subset of the data ranging back10000 years in time, corresponding to a depth of 1564 m, and continuing until2000 years before present. Figure <A HREF="node25.html#figicetime">14</A> shows the sampling rateversus time for the particular ice core considered.<P><blockquote><A NAME="968"> </A><IMG WIDTH=364 HEIGHT=195 ALIGN=BOTTOM ALT="figure1084" SRC="img152.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figicetime"> </A> Sampling rate versus time for an ice core time series.<BR></blockquote><P>We use the <IMG WIDTH=20 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap_inline2354" SRC="img153.gif">O time series which indicates the deviation of the<IMG WIDTH=26 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline2356" SRC="img154.gif"> <IMG WIDTH=58 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2358" SRC="img155.gif"> ratio from its standard value <IMG WIDTH=16 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline2360" SRC="img156.gif">:<IMG WIDTH=174 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2362" SRC="img157.gif">. Since the ratio of thecondensation rates of the two isotopes depends on temperature, the isotoperatio can be used to derive a temperature time series. The upper trace inFig. <A HREF="node25.html#figice">15</A> shows the recording from 10000 years to 2000 years beforepresent, comprising 538 data points.<P>In order to generate surrogates with the same linear properties, we estimateautocorrelations up to a lag of <IMG WIDTH=62 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline2364" SRC="img158.gif"> years by binning to a resolution of5 y. A typical surrogate is shown as the lower trace in Fig. <A HREF="node25.html#figice">15</A>. Wehave not been able to detect any nonlinear structure by comparing thisrecording with 19 surrogates, neither using time asymmetry nor predictionerrors. It should be admitted, however, that we haven't attempted to providenonlinearity measures optimised for the unevenly sampled case. For thatpurpose, also some interpolation is permissible since it is then part of thenonlinear statistic. Of course, in terms of geophysics, we are asking a verysimplistic question here. We wouldn't really expect strong nonlinear signaturesor even chaotic dynamics in such a single probe of the global climate. All theinteresting information -- and expected nonlinearity -- lies in theinterrelation between various measurements and the assessment of long termtrends we have deliberately excluded by selecting a subset of the data.<P><blockquote><A NAME="970"> </A><IMG WIDTH=365 HEIGHT=253 ALIGN=BOTTOM ALT="figure1085" SRC="img159.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figice"> </A> Oxygen isotope ratio time series derived from an ice core (upper trace) and a corresponding surrogate (lower trace) that has the same binned autocorrelations up to a lag of 1000 years at a resolution of 5 years.</blockquote><BR><P><HR><A NAME="tex2html327" HREF="node26.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html325" HREF="node22.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html319" HREF="node24.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html328" HREF="node26.html">Spike trains</A><B>Up:</B> <A NAME="tex2html326" HREF="node22.html">Various Examples</A><B> Previous:</B> <A NAME="tex2html320" HREF="node24.html">Multivariate data</A><P><ADDRESS><I>Thomas Schreiber <BR>Mon Aug 30 17:31:48 CEST 1999</I></ADDRESS></BODY></HTML>⌨️ 快捷键说明
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