L’Aquila 1 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: Chance or Chaos? Chance or Chaos? Quantifying nonlinearity and chaoticity in observed geophysical timeseries Gabriele Curci Università degli Studi dell’Aquila (ITALY) Potsdam Institute for Climate Impact Research January 2005

L’Aquila 2 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: Summary The Climate System Chaos useful in practice Detecting nonlinearity and chaos in observed timeseries Applications: very first results Conclusions and future developments

L’Aquila 3 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: Earth’s Climate System

L’Aquila 4 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: 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)

L’Aquila 5 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: Observation of the Climate System NH Temperature Surface Temperature in L’Aquila Ozone Hole Area Surface Wind Speed in L’Aquila Etc., etc,, etc…

L’Aquila 6 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: 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

L’Aquila 7 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: DETECTING CHAOS Practical 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

L’Aquila 8 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: DETECTING CHAOS Null 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)

L’Aquila 9 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: DETECTING CHAOS Null hypotheses we can test against and corresponding surrogates 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)

L’Aquila 10 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: DETECTING CHAOS Nonlinear prediction A prediction on the state of the system is performed averaging on the evolution of the neighbours of the initial state snsn UnUn k steps ahead ŝ n+k U n+k U n = neighbourhoods of s n {s j } = neighbours of s n

L’Aquila 11 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: DETECTING CHAOS Schreiber et al. method AR(1): x(n+1) = 0.99 x(n) + noise(n)AR(1) measured by y(n) = x(n)^3 obs surr

L’Aquila 12 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: DETECTING CHAOS Schreiber et al. method Sine wave + 50% noiseLorenz’ system + 10% noise obs surr

L’Aquila 13 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: DETECTING CHAOS Marzocchi et al. method 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 Logistic map + 10% noise Henon map + 10% noise

L’Aquila 14 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: DETECTING CHAOS Basu 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.

L’Aquila 15 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: DETECTING CHAOS Basu et al. method 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 Lorenz’ system + 30% noise

L’Aquila 16 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: Application: SOI and NAO

L’Aquila 17 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: SOI and NAO: test against randomness

L’Aquila 18 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: SOI and NAO: test against Gaussian linear process

L’Aquila 19 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: SOI as Gaussian linear process

L’Aquila 20 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: Is GW injecting randomness into the Climate System? [Tsonis, Eos 2004]

L’Aquila 21 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: Is GW injecting randomness? Results w/ nonlinear prediction Degree Of Randomness (DOR)

L’Aquila 22 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: Winds over different topography

L’Aquila 23 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: Winds over different topography

L’Aquila 24 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: 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

L’Aquila 25 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: 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), Basu S. and E. Foufoula-Georgiou (2002), Detection of nonlinearity and chaoticity in time series using the transportation distance function, Phys. Let. A, 301,

L’Aquila 26 of 26 “Chance or Chaos?” Climate 2005, PIK, Jan 2005 Gabriele Curci, University of L’Aquila Atmospheric Physics Group: THE END Thanks a lot!