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Statistical properties of surrogate time series

This page shows a study in which the statistical properties of a range of measurements are compared with those of their surrogate time series. Seven different records are studied, amongst others an in-situ measurement of cumulus and stratocumulus, historical time series of mean daily temperature, daily rain sums and runoff from two rivers. Seven different algorithms are used to generate the surrogate time series. Using this set-up, it is studied how similar the measurements and their surrogates are with respect to their power spectrum, increment distribution, structure functions, annual percentiles and return values. It is found that the surrogates that reproduce the power spectrum and the distribution of the measurements are able to match the increment distributions and the structure functions of the measurements with reasonable accuracy. In contrast surrogates that only mimic the power spectrum of the measurement can show large deviations for these statistics (see for example the second figure below). However, even the best performing surrogates do not have asymmetric increment distributions, i.e. they cannot reproduce nonlinear dynamical processes that are asymmetric in time. Furthermore, we have found systematic deviations of the structure functions on small scales.

As many figures could not be shown in the manuscript, you can find all quicklooks we have made and the data (measurements and all surrogates) on our FTP-server. At the moment the data is available in Net-CDF and in the native format of Matlab 6. If you would like us to calculate our statistics for surrogates created by your generator please contact me.

Some results

As an appetizer for the full manuscript, you can find below three figures from the study, purtaining the cumulus measurement.

This figure shows a cumulus measurement (top, red) and seven different types of surrogates. The Fourier (and FARIMA) surrogates have almost the same spectrum (autocorrelations) as the measurement. The PDF surrogates have the same distribution, but no autocorrelations. The other surrogates combine both statistics in various ways; see article for details.

The structure functions of the cumulus measurements and its PDF, Fourier and AAFT surrogates. The PDF surrogates match the structure functions well at large scales, but vary too much at small scales. The Fourier surrogates have a similar shape as the measurement, but have large offsets except for the 2nd order (q=2) structure function. The AAFT surrogates are reasonably good in both respects.

This figure "zooms in" on the previous figure and only shows the 4th order structure function for the best fitting surrogates. It shows that the (S)IAAFT surrogates fit even better to the measurement than the AAFT surrogates. The FARIMA + IAAFT surrogates fit even better. However this might be an artifact as the FARIMA part of the algorithm needs as input a time series with a distribution which is about normal, which was not achieved.
A (multi-)fractal algorithm would try to approximate this structure function by a linear fit in this log-log plot. Clearly this would have lead to much large deviations as most surrogates shown here.

Articles

The article on this topic can be found below. The poster contains the same information, but might be less comprehensible without explanation.

Statistical characteristics of surrogate data based on geophysical measurements
Nonlinear Processes in Geophysics, 13, p. 449-466, 2006
Victor Venema, Susanne Bachner, Henning Rust, and Clemens Simmer

Statistical properties of rain and river flow and their surrogates (poster | abstract)
EGU General Assembly, Vienna, Austria, 2-7 April, 2006
Victor Venema, Susanne Bachner, Henning Rust, and Clemens Simmer