PREDICTION OF EARTHQUAKES, EPILEPTIC SEIZURES and STOCK MARKET CRASHES

Transcription

1 PREDICTION OF EARTHQUAKES, EPILEPTIC SEIZURES and STOCK MARKET CRASHES Péter Érdi Center for Complex Systems Studies, Kalamazoo College, Kalamazoo, MI and Department of Biophysics, KFKI Research Institute for Particle and Nuclear Physics of the Hungarian Academy of Sciences Budapest, Hungary

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3 Overview Scope and limits of predictability Phenomenology Statistical analysis of extreme events Towards prediciting seizures Dynamical models of extreme events 3

4 Scope and limits of predictability Self-organized complexity in the physical biological social sciences 4

5 Scope and limits of predictability Surprising events: no rule? Self-organized criticality (P. Bak) inherently unpredictable Intermittent criticality (D. Sornette) there are precursors 5

6 Phenomenology Gutenberg-Richter law Earthquake eruption log N(M) = bm (1) b 1 N(M): number of earthquakes of magnitude greater than M 6

7 Phenomenology Earthquake eruption Omori s law where: n(t) = K (c + t) p (2) n(t) is the number of earthquakes n measured in a certain time t, K is the decay rate, c is the time offset parameter; and the parameter p typically falls in the range

8 Phenomenology Epilepsy recent results: epileptic seizures may begin hours earlier than their clinical onset definiton of different measures relative self-excitation is a control paramater to induce/suppress epilepsy our own lab in Budapest: Zoltán Somogyvári 8

9 Phenomenology Epilepsy Post synaptic potential Time Psp(t+ t ) Psp(t+ t ) Psp(t) t=1time step complexity decrease Psp(t) t=1time step Somogyvári et al: Slow dynamics of epileptic seizure: analysis and model. Neurocomputing (2001)

10 Phenomenology Market crash 10

11 Statistical analysis of extreme events Outliers vs extreme value statistics What is the probabilty of having big earthquake in California within a year? How large might a possible stock market crash be tomorrow? lowest daily return (the minimum) highest daily return (the maximum) over a given period not normally distributed. 11

12 Statistical analysis of extreme events Generalized extreme value distribution F (x; µ, σ, ξ) = exp { [ 1 + ξ ( )] x µ 1/ξ } (3) µ is is the location parameter, σ > 0 the scale parameter and ξ is the shape parameter σ For the normal distribution, the location and scale parameters correspond to the mean and standard deviation, respectively. However, this is not necessarily true for other distributions. In fact, it is not true for most distributions. 12

13 Statistical analysis of extreme events Gumbel distribution f(x; µ, σ) = exp( (exp( x µ))/σ) (4) for < x < (5) forest fires earthquakes financial loss epileptic seizures 13

14 Statistical analysis of extreme events Power law distributions P (ξ > x) = ax k ; (6) k is the shape parameter / M>1.5 / 5km / 1 day expos= Number of bins Number of events Empirical probability density functions of the number r of earthquakes in the space time [5 5 km 2 1 day] California SCEC catalog (Saichev and Sornette) log-normal for the initial part and power law heavy tail 14

15 Statistical analysis of extreme events Power law distributions Largest value: probability that a particular sample will be larger than x: while α > 1 P (x) = (x/x min ) α+1 (7) 15

16 Statistical analysis of extreme events Power law distributions (a) 10 6 (b) (c) word frequency citations web hits books sold crater diameter in km (d) (e) 10 4 (f) telephone calls received peak intensity earthquake magnitude (g) 10 4 (h) 100 (i) intensity (j) (k) 4 (l) net worth in US dollars name frequency population of city 16

17 Towards prediciting seizures Dynamical characteristics of pre-epileptic seizures in rats with recurrence quantification analysis Xiaoli Li, Gaoxiang Ouyang, Xin Yao, Xinping Guan Physics Letters A 333(2004) measures based on recurrence points comlexity reduction seizure initation: lower signal complexity (higher synchrony) 17

18 Towards prediciting seizures Similarities in precursory features in seismic shocks and epileptic seizures P. G. Kapiris, J. Polygiannakis, X. Li, X. Yao and K. A. Eftaxias Europhys. Lett., 69( ) 2005 (mv) a Time (min) (mv) /08/99 28/08/99 31/08/99 03/09/99 06/09/99 EQ 09/09/99 a Probability ρ ρ ρ ρ b Probability ρ ρ ρ ρ b Probability r > % β 2 r > % β 3 r > % β 4 r > % β c Probability r > % β 2 r > % β 3 r > % β 4 r > % β c H H H [0,0.5) H [0,0.5) H [0.5,1] Σ A(t i ) 2 d Time (min) 5 x Seizure 1 e Time (min) pre ictal period 10 db H [0.5,1] Σ A(t i ) /08/99 28/08/99 31/08/99 03/09/99 06/09/99 09/09/99 15 x /08/99 27/08/99 30/08/99 02/09/99 05/09/99 08/09/99 10 db EQ EQ d e Period (s) Time (min) f Period (s) f 29/08 30/08 31/08 01/09 02/09 03/09 04/09 05/09 06/09 07/09 08/09 18

19 Towards prediciting seizures Transiton from pre-epileptic and pre-seizmic states toward critical states: gradually increased spatial and temporal correlations more dramatic change in energy release better supports the intermitten ciriticality than the self-organzied criticality hypothesis 19

20 Dynamical models of extreme events Model of self-organized criticality: Sandpile model The same effect may lead to small, but also to very large avalanches (not predictable) Toy model: Bak, Tang, and Wiesenfeld 1987 Each point z(x, y) in the grid has a number n associated with it. (say, average slope of the sand pile at that point) for a randomly selected point: z(x, y) z(x, y) + 1 if z(x, y) > threshold, than z(x, y) = z(x, y) 4 z(x ± 1, y) z(x ± 1, y) + 1 z(x, y ± 1) z(x, y ± 1) + 1 If this redistribution causes n to be too big for any nearby grid points iteration else stop and take randomly another point z(x, y) 20

21 Dynamical models of extreme events Model of self-organized criticality: Sandpile model Perturbation of different points avalanches with different size small events which don t stop unpredictability? 21

22 Dynamical models of extreme events Explosions : Finite time singularities intermittent criticality The simplest representative dynamical evolution equation leading to a finite-time singularity is: ( ) 1 de dt = tc t m 1 Em, its solution is: E(t) = E(0) t c < m m > 1 22

23 Dynamical models of extreme events General mechanism for finite time singularities Positive feedback + + E I Applications of positive feedback in: crack growth, laboratory creep rupture, simple damage model, damage model for regional seismicity, percolation model for regional seismicity, stress-shadow model for regional seismicity, Disihibition model of status epilepticus, increasing return (Brian Arthur). 23

24 Dynamical models of extreme events Medieval history of the generation of power law distribution: BB Mandelbrot, A note on a class of skew distribution function. analysis and critique of a paper by H.A. Simon, Information and Control, 2,90-99 (1959). [ABSTRACT: This note is a discussion of H.A. Simon s model (1955) concerning the class of frequency distributions generally associated with the name of G.K. Zipf. The main purpose is to show that Simon s model is analytically circular in the case of the linguistic laws of Estoup-Zipf and Willis-Yule. Insofar as the economic law of Pareto is concerned, Simon has himself noted that his model is a particular case of that of Champernowne; this is correct, with some reservation. A simplified version of Simon s model is included. ] HA Simon, Some further notes on a class of skew distribution functions, Information and Control, 3, (1960). [ABSTRACT: This note takes issue with a recent criticism by Dr. B. Mandelbrot of a certain stochastic model to explain word-frequency data. Dr. Mandelbrot s principal empirical and mathematical objections to the model are shown to be unfounded. a central question is whether the basic parameter of the distributions is larger or smaller than unity. The empirical data show it is almost always very close to unity, Sometimes slightly larger, sometimes smaller. Simple stochastic models can be constructed for either case, and give a special status, as a limiting case, to instances where the parameter is unity. More generally, the empirical data can be explained by two types of stochastic models as well as by models assuming efficient information coding. The three types of models are briefly characterized and compared. ] 24

25 Dynamical models of extreme events Medieval history of the generation of power law distribution (contd.): BB Mandelbrot, Final note on a class of skew distribution functions: analysis and critique of a model due to H.A. Simon, Information and Control, 4, (1961). [ABSTRACT: We shall restate in detail our 1959 objections to Simon s 1955 model for the Pareto-Yule-Zipf distribution. Our objections are valid quite irrespectively of the sign of p-1, so that most of Simon s (1960) reply was irrelevant. We shall also analyze the other points brought up in that reply. ] HA Simon, Reply to final note by Benoit Mandelbrot, Information and Control, 4, (1961). [ABSTRACT: Dr. Mandelbrot s original objection (1959) to using the Yule process to explain the phenomena of word frequencies were refuted in Simon (1960), and are now mostly abandoned. the present reply refutes the almost entirely new arguments introduced by Dr. Mandelbrot in his final note, and demonstrates again the adequacy of the models in (1955). ] 25

26 Dynamical models of extreme events Medieval history of the generation of power law distribution (contd.): BB Mandelbrot, Post scriptum to final note, Information and Control, 4, (1961). [ABSTRACT: My criticism has not changed since I first had the privilege of commenting upon a draft of Simon (1955). ] HA Simon, Reply to Dr. Mandelbrot s post scriptum, Information and Control, 4, (1961). [ABSTRACT: Dr. Mandelbrot has proposed a new set of objections to my 1955 models of the Yule distribution. Like his earlier objections, these are invalid.] Editorial note: Dr. Mandelbrot feels that no further comment is needed and this debate terminates herewith. 26

27 Big excitement: log-periodic corrections Log-periodic correction of power law distributions: Extension of dimensions: integer fractal complex fractal log[p(t)] = A+B(t c t) β log[p(t)] = A+B(t c t) β [1+C cos(ω log((t c t)/t ))] (8) Energy release rate of approaching rupture DM WS Examples Loma Pieta earthquake, October 1989 Kobe earthquake, 1995 Exchange Rate CHF The October Date 1987 Wall Street Crash S&P500 Wall Street crash, October

28 Take home message The theory of complex systems suggests: extreme events: may be predicted their precursors can be detected there are methodological similarities to analyze and model different critical events occurring in physical, life and social phenomena there are intial results and many open problems 28

29 ACKNOWLEDGMENTS Tamás Kiss Zoltán Somogyvári 29