Statistical characteristics of surrogate data based on geophysical measurements

1. Statistical characteristics of surrogate data based on geophysical measurements Victor Venema1, Henning W. Rust 2, Susanne Bachner 1, and Clemens Simmer 1 1 Meteorological Institute University of Bonn 2 PIK, Potsdam Institute for Climate Impact Research

2. Content • Surrogate data: time series generated based on statistical properties of measurements • Distribution and/or power spectrum • 7 Geophysical time series • Generated surrogates with 7 different algorithms from their statistics • Compared the measurements to their surrogates • Increment distribution • Structure functions

3. Motivation • Need time series with a known structure • Statistical reconstruction • Bootstrap confidence intervals • Studying non-local process • … • FARIMA & Fourier methods vs. Multifractals • Multifractals vs. surrogates

4. Motivation - generator • Empirical studies • Exact measured distribution • Measured power spectrum • Scale breaks • Waves • Deviations large scales • … Satellite pictures: Eumetsat

5. 7 Generators for surrogate data • D: distribution • PDF surrogates • S: spectrum • Fourier surrogates • FARIMA surrogates • Seasonal cycle and logarithm if needed • DS: distribution + spectrum • AAFT, IAAFT, SIAAFT surrogates • FARIMA + IAAFT surrogates • seasonal cycle and log. if needed

6. Measurements

7. DS DS DS S D S DS Surrogate types

8. Increment distribution • Measurement: (t) • Increment time series for lag l: (x,l) = (t+l) - (t) • Distribution jumps sizes • Next plots: l is 1 day

9. Increment distribution temperature

10. Increment distribution Rhine

11. Structure functions • Increment time series: (x,l)=(t+l)- (t) • SF(l,q) = (1/N) Σ ||q • SF(l,2) is equivalent to auto-correlation function • Higher q focuses on larger jumps

12. Structure function Salzach

13. Structure function stratocumulus

14. RMSE 4th order structure functions • Best surrogate in bold • Multifractal means: power law fit

15. Extension IAAFT algorithm • 2D and 3D fields with PDF(z) • PDF(t), i.e. distribution varies as function of • Season, time of day • Break point • Multivariate statistics, cross correlations • Increment distribution at small scales • More accurate increment distribution • Asymmetric increment distribution (runoff) • Downscaling • Extrapolate spectrum • Iterate the original coarse mean values

16. Conclusions • DS-Surrogates of geophysical reproduce measurements accurately • spectrum • increments • structure functions • IAAFT algorithm • Flexibly • Efficiently • Many useful extensions are possible • Surrogates for empirical work • Multifractals for theoretical work (use IAAFT)

17. More information • Homepage • Papers, Matlab-programs, examples • http://www.meteo.uni-bonn.de/ venema/themes/surrogates/ • Google • surrogate clouds • multifractal surrogate time series • IAAFT in R: Tools homepage Henning Rust • http://www.pik-potsdam.de/~hrust/tools.html • IAAFT in Fortran (multivariate): search for TISEAN (Time SEries ANalysis)