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Chance or Chaos?Quantifying nonlinearity and
chaoticity in observed geophysical timeseries

• Gabriele Curci
• Università degli Studi dellAquila (ITALY)
• http//www.aquila.infn.it/people/Gabriele.Curci.ht
  ml/
• Potsdam Institute for Climate Impact Research
• 13-14 January 2005

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Summary

• The Climate System
• Chaos useful in practice
• Detecting nonlinearity and chaos in observed
  timeseries
• Applications very first results
• Conclusions and future developments

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Earths Climate System
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Understanding the Climate System

• Two opposite needs
• Increase the number of observations (scalar
  timeseries)
• Condense the knowledge in a theory (e.g. to allow
  predictions)

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Observation of the Climate System
Ozone Hole Area
NH Temperature
Surface Wind Speed in LAquila
Surface Temperature in LAquila
Etc., etc,, etc
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Chaos and Climate

• An irregular behavior is natural in system with
  a large number of degrees of freedom
  (stochasticity)
• Deterministic chaos could explain irregular
  dynamics also with a few degrees of freedom
• Detecting low-dimensional chaos in a given
  phenomenon is very useful for modelling and
  near-term predictability

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DETECTING CHAOSPractical difficulties with
observed timeseries

• We observe just one or a few variables of the
  system
• Noise if very high, it masks the chaotic signal
• Finite length and missing data
• The common tools for detecting chaos (Lyapunov
  exp, correlation dimension) are uneffective

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DETECTING CHAOSNull hypotheses and surrogate data

• Before attempting to use complicated timeseries
  analysis tools one should try to establish the
  presence of nonlinearity
• First, a null hypothesis for the underlying
  process is formulated (e.g. Gaussian linear)
• Second, we build surrogate data that accurately
  represent the null hypothesis
• Third, we try to find a system parameter that is
  capable to detect a meaningful deviation of the
  data from the null hypothesis (surrogates)

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DETECTING CHAOSNull hypotheses we can test
against and corresponding surrogates

• Independence random draws from a fixed
  probability distribution.
• Random shuffling of the data
• Filter with an AR linear model and shuffle the
  residuals
• Gaussian linear stochastic process completely
  specified by its mean, variance, and
  auto-correlation, or equivalently Fourier
  amplitudes.
• Random shuffling of Fourier amplitudes
• General constrained randomization (same autocorr)

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DETECTING CHAOSNonlinear prediction

• A prediction on the state of the system is
  performed averaging on the evolution of the
  neighbours of the initial state

Un neighbourhoods of sn
sj neighbours of sn
Un
Unk
sn
snk
k steps ahead
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DETECTING CHAOSSchreiber et al. method
AR(1) x(n1) 0.99 x(n) noise(n)
AR(1) measured by y(n) x(n)3
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DETECTING CHAOSSchreiber et al. method
Sine wave 50 noise
Lorenz system 10 noise
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DETECTING CHAOSMarzocchi et al. method
Logistic map 10 noise

1. Evaluate errors if S/N ratiolt40-50 quit
2. Apply AR filter to data a nonlinear system has
   correlated residuals
3. Nonlinear prediction vs. embedding dimension
4. Compare with surrogates

Henon map 10 noise
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DETECTING CHAOSBasu et al. Transportation
distance

• The difference between two timeseries is usually
  measured in a geometrical sense. We can include
  information about the similarity of the
  attractors introducing the transportation
  distance

Problem how does it cost going from
configuration P to Q? The transportation
distance is the combination of moves with the
overall minimum cost
The transportation distance is efficiently solved
by a transshipment problem algorithm Moeckel and
Murray, 1997. It is based on both geometrical
and probabilistic and it is less sensitive to
outliers, noise and discretization errors.
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DETECTING CHAOSBasu et al. method
Lorenz system 30 noise

• Compare the distribution of the transportation
  distance between original data and surrogates
  (OS) and among surrogates (MS)
• Transportation distance between original
  timeseries and its nonlinear prediction k-step
  ahead

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Application SOI and NAO
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SOI and NAO test against randomness
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SOI and NAO test against Gaussian linear process
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SOI as Gaussian linear process
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Is GW injecting randomness into the Climate
System? Tsonis, Eos 2004
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Is GW injecting randomness? Results w/ nonlinear
prediction
Degree Of Randomness (DOR)
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Winds over different topography
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Winds over different topography
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Future Developments

• Setup a reliable procedure to determine the
  presence and the degree of nonlinearity of a
  timeseries using the mentioned ideas
• Model-observation comparison (degree of
  nonlinearity, variability)
• Model parameters tuning

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References

• Schreiber, T. (1999), Interdisciplinary
  application of nonlinear time series methods,
  Physics Reports, 308, 1-64
• Marzocchi, W., F. Mulargia and G. Gonzato (1997),
  Detecting low-dimensional chaos in geophysical
  time series, J. Geophys. Res., 102(B2), 3195-3209
• Basu S. and E. Foufoula-Georgiou (2002),
  Detection of nonlinearity and chaoticity in time
  series using the transportation distance
  function, Phys. Let. A, 301, 413-423

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THE END

• Thanks a lot!