Transcript Chance or Chaos? Quantifying nonlinearity and chaoticity

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Chance or Chaos?
Quantifying nonlinearity and chaoticity in
observed geophysical timeseries
Gabriele Curci
Università degli Studi dell’Aquila (ITALY)
http://www.aquila.infn.it/people/Gabriele.Curci.html/
Potsdam Institute for Climate Impact Research
13-14 January 2005
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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Summary
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• The Climate System
• Chaos useful in
practice
• Detecting nonlinearity
and chaos in
observed timeseries
• Applications: very first
results
• Conclusions and
future developments
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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Earth’s Climate System
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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Understanding the Climate System
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• Two “opposite” needs:
– Increase the number
of observations (scalar
timeseries)
– Condense the
knowledge in a theory
(e.g. to allow
predictions)

s(t )s( x (t ))


x  F (x )
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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Observation of the Climate System
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NH Temperature
Ozone Hole Area
Surface Temperature in L’Aquila
Surface Wind Speed in L’Aquila
Etc.,
etc,,
etc…
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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Chaos and Climate
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• 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 lowdimensional chaos in a
given phenomenon is
very useful for modelling
and near-term
predictability
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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DETECTING CHAOS
Practical difficulties with observed
timeseries
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• 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
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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DETECTING CHAOS
Null hypotheses and surrogate data
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• 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)
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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DETECTING CHAOS
Null hypotheses we can test against
and corresponding surrogates
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1. Independence: random draws from a fixed
probability distribution.
•
•
Random shuffling of the data
Filter with an AR linear model and shuffle the
residuals
2. 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)
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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DETECTING CHAOS
Nonlinear prediction
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A prediction on the state of the system is performed averaging on
the evolution of the neighbours of the initial state
sˆn k
1

NU n
s
s j U n
j k
Un = neighbourhoods of sn
{sj} = neighbours of sn
Un
Un+k
sn
k steps ahead
ŝn+k
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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DETECTING CHAOS
Schreiber et al. method
AR(1): x(n+1) = 0.99 x(n) + noise(n)
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AR(1) measured by y(n) = x(n)^3
obs
surr
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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DETECTING CHAOS
Schreiber et al. method
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Lorenz’ system + 10% noise
Sine wave + 50% noise
obs
surr
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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DETECTING CHAOS
Marzocchi et al. method
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Logistic map + 10% noise
1. Evaluate errors: if S/N
ratio<40-50% 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
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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DETECTING CHAOS
Basu et al.: Transportation distance
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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.
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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DETECTING CHAOS
Basu et al. method
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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 kstep ahead
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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Application: SOI and NAO
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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SOI and NAO: test against
randomness
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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SOI and NAO: test against Gaussian
linear process
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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SOI as Gaussian linear process
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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Is GW injecting randomness into the
Climate System? [Tsonis, Eos 2004]
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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Is GW injecting randomness?
Results w/ nonlinear prediction
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Degree Of
Randomness (DOR)
DOR 
err (obs)  err ( surr)
err (obs)
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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Winds over different topography
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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Winds over different topography
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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Future Developments
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• 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
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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References
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• 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
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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THE END
Thanks a lot!
Gabriele Curci, University of L’Aquila
Climate 2005, PIK, 13-14 Jan 2005 http://www.aquila.infn.it/people/Gabriele.Curci.html/
Atmospheric Physics Group: http://www.aquila.infn.it/atmosfera/
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