Chance or Chaos Quantifying noninearity and chaoticity in 機(jī)會(huì)或混沌非線性量化和混沌中

1、laquila1 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/quantifying nonlinearity and chaoticity in observed geophysical timeseries gabriele

2、curciuniversit degli studi dellaquila (italy)http:/www.aquila.infn.it/people/gabriele.curci.html/potsdam institute for climate impact research13-14 january 2005laquila2 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabrie

3、le.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/summary the climate system chaos useful in practice detecting nonlinearity and chaos in observed timeseries applications: very first results conclusions and future developmentslaquila3 of 26“chance or chaos?”climate 2005, pi

4、k, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/earths climate systemlaquila4 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www

5、.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/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)(xfx)()(txstslaquila5

6、of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/observation of the climate systemnh temperaturesurface temperature in laquilaozone hole areas

7、urface wind speed in laquilaetc., etc, etclaquila6 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/chaos and climate an “irregular” behavior

8、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 predictabilitylaquila7 of 26“chance

9、or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/detecting chaospractical difficulties with observed timeseries we observe just one or a few variables of t

10、he 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 uneffectivelaquila8 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.i

11、nfn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/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 u

12、nderlying 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)laquila9 of 26“chance or cha

13、os?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/detecting chaosnull hypotheses we can test against and corresponding surrogates1.independence: random draws from

14、 a fixed probability distribution.random shuffling of the datafilter with an ar linear model and shuffle the residuals2.gaussian linear stochastic: process completely specified by its mean, variance, and auto-correlation, or equivalently fourier amplitudes.random shuffling of fourier amplitudesgener

15、al constrained randomization (same autocorr)laquila10 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/detecting chaosnonlinear predictiona pr

16、ediction on the state of the system is performed averaging on the evolution of the neighbours of the initial statenjnuskjuknsns1snunk steps aheadn+kun+kun = neighbourhoods of snsj = neighbours of snlaquila11 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquil

17、ahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/detecting chaosschreiber et al. methodar(1): x(n+1) = 0.99 x(n) + noise(n)ar(1) measured by y(n) = x(n)3obssurrlaquila12 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele

18、 curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/detecting chaosschreiber et al. methodsine wave + 50% noiselorenz system + 10% noiseobssurrlaquila13 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005

19、gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/detecting chaosmarzocchi et al. method1.evaluate errors: if s/n ratio40-50% quit2.apply ar filter to data: a nonlinear system has correlated residual

20、s3.nonlinear prediction vs. embedding dimension4.compare with surrogateslogistic map + 10% noisehenon map + 10% noiselaquila14 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: h

21、ttp:/www.aquila.infn.it/atmosfera/detecting chaosbasu et al.: transportation distancethe 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 cos

22、t going from configuration p to q? the “transportation distance” is the combination of moves with the overall minimum costthe 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 s

23、ensitive to outliers, noise and discretization errors.laquila15 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/detecting chaosbasu et al. me

24、thod 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 aheadlorenz system + 30% noiselaquila16 of 26“chance or chaos?”climate 2005, pik, 1

25、3-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/application: soi and naolaquila17 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www

26、.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/soi and nao: test against randomnesslaquila18 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmo

27、spheric physics group: http:/www.aquila.infn.it/atmosfera/soi and nao: test against gaussian linear processlaquila19 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.a

28、quila.infn.it/atmosfera/soi as gaussian linear processlaquila20 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/is gw injecting randomness in

29、to the climate system? tsonis, eos 2004laquila21 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/is gw injecting randomness? results w/ nonli

30、near predictiondegree of randomness (dor)()()(obserrsurrerrobserrdorlaquila22 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/winds over diff

31、erent topographylaquila23 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/winds over different topographylaquila24 of 26“chance or chaos?”climate 2005, pik, 13-14 jan 2005gabriele curci, university of laquilahttp:/www.aquila.infn.it/people/gabriele.curci.html/atmospheric physics group: http:/www.aquila.infn.it/atmosfera/future developments setup a reliable procedure to determine the pres