node25.html

an analysis software with souce code for the time series with methods based on the theory of nonline

binned autocorrelations coincide with ordinary autocorrelations at
<IMG WIDTH=182 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2350" SRC="img150.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img150.gif">.
<P>
Once we are able to specify the linear properties of a time series, we can also
define a cost function as usual and generate surrogates that realise the binned
autocorrelations of the data. A delicate point however is the choice of bin
size. If we take it too small, we get bins that are almost empty. Within the
space of permutations, there may be only a few ways then to generate precisely
that value of <IMG WIDTH=43 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline2352" SRC="img151.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img151.gif">, in other words, we over-specify
the problem. If we take the bin size too large, we might not capture important
structure in the autocorrelation function.
<P>
As an application, let us construct randomised versions of part of an ice core
data set, taken from the Greenland Ice Sheet Project Two (GISP2)&nbsp;[<A HREF="node36.html#gisp2" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#gisp2">44</A>].
An extensive data base resulting from the analysis of physical and chemical
properties of Greenland ice up to a depth of 3028.8&nbsp;m has been published by the
National Snow and Ice Data Center together with the World Data Center-A for
Palaeoclimatology, National Geophysical Data Center, Boulder,
Colorado&nbsp;[<A HREF="node36.html#CDROM" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#CDROM">45</A>]. A long ice core is usually cut into equidistant slices
and initially, all measurements are made versus depth. Considerable expertise
then goes into the dating of each slice&nbsp;[<A HREF="node36.html#dating" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node36.html#dating">46</A>]. Since the density of the
ice, as well as the annual total deposition, changes with time, the final time
series data are necessarily unevenly sampled. Furthermore, often a few values
are missing from the record.  We will study a subset of the data ranging back
10000&nbsp;years in time, corresponding to a depth of 1564&nbsp;m, and continuing until
2000&nbsp;years before present. Figure&nbsp;<A HREF="node25.html#figicetime" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node25.html#figicetime">14</A> shows the sampling rate
versus time for the particular ice core considered.
<P>
<blockquote><A NAME="968">&#160;</A><IMG WIDTH=364 HEIGHT=195 ALIGN=BOTTOM ALT="figure1084" SRC="img152.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img152.gif"><BR>
<STRONG>Figure:</STRONG> <A NAME="figicetime">&#160;</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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img154.gif"> <IMG WIDTH=58 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2358" SRC="img155.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img155.gif"> ratio from its standard value <IMG WIDTH=16 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline2360" SRC="img156.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img156.gif">:
<IMG WIDTH=174 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline2362" SRC="img157.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img157.gif">. Since the ratio of the
condensation rates of the two isotopes depends on temperature, the isotope
ratio can be used to derive a temperature time series. The upper trace in
Fig.&nbsp;<A HREF="node25.html#figice" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node25.html#figice">15</A> shows the recording from 10000&nbsp;years to 2000&nbsp;years before
present, comprising 538 data points.
<P>
In order to generate surrogates with the same linear properties, we estimate
autocorrelations up to a lag of <IMG WIDTH=62 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline2364" SRC="img158.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img158.gif">&nbsp;years by binning to a resolution of
5&nbsp;y. A typical surrogate is shown as the lower trace in Fig.&nbsp;<A HREF="node25.html#figice" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/node25.html#figice">15</A>.  We
have not been able to detect any nonlinear structure by comparing this
recording with 19 surrogates, neither using time asymmetry nor prediction
errors. It should be admitted, however, that we haven't attempted to provide
nonlinearity measures optimised for the unevenly sampled case. For that
purpose, also some interpolation is permissible since it is then part of the
nonlinear statistic. Of course, in terms of geophysics, we are asking a very
simplistic question here. We wouldn't really expect strong nonlinear signatures
or even chaotic dynamics in such a single probe of the global climate.  All the
interesting information -- and expected nonlinearity -- lies in the
interrelation between various measurements and the assessment of long term
trends we have deliberately excluded by selecting a subset of the data.
<P>
<blockquote><A NAME="970">&#160;</A><IMG WIDTH=365 HEIGHT=253 ALIGN=BOTTOM ALT="figure1085" SRC="img159.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/surropaper/img159.gif"><BR>
<STRONG>Figure:</STRONG> <A NAME="figice">&#160;</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&nbsp;years at a resolution of
   5&nbsp;years.</blockquote>
<BR>
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<P><ADDRESS>
<I>Thomas Schreiber <BR>
Mon Aug 30 17:31:48 CEST 1999</I>
</ADDRESS>
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