Statistical characteristics of surrogate data based on geophysical measurements

1
Statistical characteristics of surrogate data
based on geophysical measurements

• Victor Venema 1, 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
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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

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Measurements
7
Surrogate types
DS DS DS S D S DS
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Increment distribution

• Measurement ?(t)
• Increment time series for lag l
• ?(x,l) ?(tl) - ?(t)
• Distribution jumps sizes
• Next plots l is 1 day

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Increment distribution temperature
10
Increment distribution Rhine
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Structure functions

• Increment time series ?(x,l)?(tl)- ?(t)
• SF(l,q) (1/N) S ?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

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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/surrog
  ates/
• 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)