Is there evidence of log-periodicities in the tail of the distribution of seismic moments?

Transcription

1 Is there evidence of log-periodicities in the tail of the distribution of seismic moments? Luísa Canto e Castro Sandra Dias CEAUL, Department of Statistics University of Lisbon, Portugal CEAUL, Department of Mathematics University of Trás-os-Montes e Alto Douro, Portugal Workshop: Statistical Extremes and Environmental Risk Lisbon, February - 7, 7 Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 /

2 Objective Try to fit a max-semistable distribution to the seismic moments of shallow earthquakes (h 7 km) in subduction zones (using the Harvard catalog of seismic moments from 977// to //) Motivation Works by Pisarenko and Sornette () and Pisarenko, Sornette and Rodkin () Complementary to these previous works emphasizing deviations from the Gutenberg-Richter distribution only in tail, the present paper explores the possible existence of deviations from (a) from elsewhere, that is, in the bulk of the distribution.... suggests that the deviations of the distribution of seismic moments from Gutenberg-Richter law can be modeled by log-periodic oscillations... Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 /

3 Max-stable versus Max-semistable Unifying standard expressions for max-stable d.f. { { } exp ( + γx) /γ + γx > if γ G γ (x) := exp{ e x } x R if γ = Unifying standard expressions for max-semistable d.f. (Grinevich (99,99) and Pancheva (99)) { { exp ( + γx) /γ ν(log( + γx)) } + γx > if γ G γ,ν (x) := exp{ e x ν(x)} x R if γ = where ν is a positive, bounded and periodic function with period p, where { p = γ log r if γ p = log r if γ = Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 /

4 Another characterization [Canto e Castro, de Haan and Temido ()] General to location and scale, for any d.f. G γ,ν max-semistable we have: log( log G γ,ν (s m + a m x)) = mlog r + y(x), x [,], m Z where y : [,] [,log r] is non decreasing, right continuous and continuous at x = a = r γ (a if γ and a = if γ = ) s m = m if a = s m = (a m )/(a ) if a and a > Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 /

5 Graph of the function log( log G) log r log r log r log r y log r log r a a a +a +a+a +a+a +a Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 /

6 Some graphical features of max-semistable laws Figure: D.f. s G.(x) = exp( x ) and G.,ν(x) = exp( x (8 + cos(π log x))) Figure: QQ-Plot of G.,ν against G. Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 6 /

7 Notation X i,n is the i_th ascending order statistic of a sample of size n k := k n is an intermediate sequence, that is, k is an integer sequence verifying lim k = + and lim k/n = n + n + γ + and γ are the moments estimators of γ + := max{,γ} and γ := min{,γ}, respectively γ is the maximum likelihood estimator of γ Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 7 /

8 Test Statistic E k,n [Dietrich, de Haan and Hüsler ()] E k,n = k ( ) log Xn [kt],n log X n k,n t γ+ ( γ ) t η dt γ + γ + Test Statistic PE k,n [Dietrich, de Haan and Hüsler ()] For γ PE k,n = k ( log Xn [kt],n log X n k,n γ + ) + log t t η dt Test Statistic T k,n [Drees, de Haan and Li (6)] For γ > / T k,n = k ( ( n k F x γ ) n â n/k + b n/k x) x η dx γ Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 8 /

9 Test Statistic k,n [Dias (6)] Z t (k) := X n [k/t ],n X n [k/t],n X n [k/t],n X n k,n t >. Theorem [Alves (99)] If F is in the domain of attraction of a max-stable law G γ then log t log(z t(k)) P n + γ, t >. Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 9 /

10 Test Statistic k,n (continuation) Theorem [Dias and Canto e Castro (6)] If F is in the domain of attraction of a max-semistable law G γ,ν then P Z t (k) n + rcγ if and only if t = r c, c N. Therefore, if F is in the domain of attraction of a max-semistable law G γ,ν then log t log(z P t(k)) γ if and only if t = n + rc, c N. Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 /

11 Test Statistic k,n (continuation) t =. t =. k Figure: Sample trajectory of Z t(k), t =. (F is in the domain of attraction of a max-semistable law G.,ν, r = e. ) k Figure: Sample trajectory of Z t(k), t =. (F is in the domain of attraction of a max-stable law G ) k (log Z s (ku) + γ log s) u η du Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 /

12 Test Statistic k,n (continuation) t =. t =. k Figure: Sample trajectory of Z t(k), t =. (F is in the domain of attraction of a max-semistable law G.,ν, r = e. ) k Figure: Sample trajectory of Z t(k), t =. (F is in the domain of attraction of a max-stable law G ) For γ > / k,n = k b a (log Z s (ku) + γ log s) u η du ds Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 /

13 Tests for the extreme value condition Figure: QQ-plot of G (x) = exp( x ) against G,ν(x) = exp( x ( + cos(π log x))/) H is not reject by any of the statistics Figure: QQ-plot of G.(x) = exp( x ) against G.,ν(x) = exp( x (8+cos(π log x))) H is reject by and not rejected by the others Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 /

14 Application of the tests to the data Limit distributions and simulation study, see Hüsler and Li (6) for E k,n, PE k,n and T k,n Dias (6) and Dias, Canto e Castro and Temido (6) for k,n Results for the data E k,n, PE k,n and k,n do not reject H for a significance level of % PE k,n depends of the value of k Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 /

15 QQ-plots against the max-stable law 6 x 9. seismic moment estimated G.8 Estimated G.8 quantiles Log ( F n (x)) Seismic moments quantiles x Log (x) Figure: Left: QQ-Plot of seismic moments against G.8. Right: Survival function in double-log scale Estimates for the parameters Parameters Estimates γ.8 σ. µ.8 Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 /

16 QQ-plots against the max-semistable law 7 x 6. seismic moments estimated G.,ν Estimated G.,ν quantiles Log ( F n (x)) Seismic moments quantiles x Log (x) Figure: Left: QQ-Plot of seismic moments against G.,ν. Right: Survival function in double-log scale. Estimates for the parameters Parameters Estimates r. p. γ.8 Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 /

17 Why not log-periodicities? log r.. y y y y y log r.. log r Figure: Left: Empirical function log( log F n). Right: Empirical versions of y. Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 6 /

18 Alternative ideas x 9 6. seismic moment estimated G.8 Estimated G.8 quantiles Log ( F n (x)) Seismic moments quantiles x Log (x) x 9 mixture of GEV s. estimated G.7 Estimated G.7 quantiles Log ( F n (x)) Mixture of GEV s quantiles x Log (x) Figure: Left: QQ-Plot of seismic moments against G.8 and of a simulated mixture of GEV s against G.7. Right: Survival function in double-log scale. Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 7 /

19 Alternative ideas x 9 6. seismic moment estimated G.8 Estimated G.8 quantiles Log ( F n (x)) Seismic moments quantiles x Log (x) x 9 mixture of GEV s. estimated G. Estimated G. quantiles Log ( F n (x)) Mixture of GEV s quantiles x Log (x) Figure: Left: QQ-Plot of seismic moments against G.8 and of a simulated mixture of GEV s against G.. Right: Survival function in double-log scale. Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 8 /

20 Alternative ideas x 9 6. seismic moment estimated G.8 Estimated G.8 quantiles Log ( F n (x)) Seismic moments quantiles x Log (x) x 9. truncated G.6 estimated G.6 Estimated G.6 quantiles Log ( F n (x)) Trucanted G quantiles.6 x Log (x) Figure: Left: QQ-Plot of seismic moments against G.8 and of a simulated truncated G.6 against G.6. Right: Survival function in double-log scale. Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 9 /

21 Alternative ideas x 9 6. seismic moment estimated G.8 Estimated G.8 quantiles Log ( F n (x)) Seismic moments quantiles x Log (x) 6 x 9. truncated G.6 estimated G.6 Estimated G.7 quantiles Log ( F n (x)) Trucanted G quantiles.6 x Log (x) Figure: Left: QQ-Plot of seismic moments against G.8 and of a simulated truncated G.6 against G.7. Right: Survival function in double-log scale. Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 /

22 References Canto e Castro, L., de Haan, L., and Temido, M. G. () Rarely observed sample maxima. Theory of Probab. Appl., Vol., p Dietrich, D., de Haan, L. and Hüsler, J. () Testing extreme value conditions. Extremes, vol., p Drees, H., de Haan and L., Li, D. (6) Approximations to the tail empirical distribution function with applications to testing extreme value conditions. Journal of Statistical Planning and Inference, vol. 6, p Dias, S., Canto e Castro, L., Temido, M. G. (6) Estimação de quantis elevados para distribuições no domínio de atracção das leis max-semiestáveis. In Ciência Estatística, p Edições SPE, Lisboa. Dias, S., Canto e Castro, L., Temido, M. G. (6) Contributions to the statistical inference in max-semistable models. In Notes and Communications of Center for Statistical Applications of the University of Lisbon, /6. Dias, S., Canto e Castro, L., Temido, M. G. (6) Comparação de testes para a condição de valores extremos. Submitted to Actas do XIV Congresso Anual da SPE. Dias, S. (6) Max-estável ou max-semiestável? In Notes and Communications of Center for Statistical Applications of the University of Lisbon, 6/6. Hüsler, J., Li, D. (6) On testing extreme value conditions. submitted. Pisarenko, V. F., Sornette, D. () Characterization of the frequency of extreme events by generalized Pareto distribution. Pure and Geophysics, vol. 6(), p. -6. Pisarenko, V. F., Sornette, D. and Rodkin, M. () Deviations of the distribution of seismic energies from the Gutenberg-Richer law. Computational Seismology, vol., p Canto e Castro, Dias Log-periodicities in seismic moments? SEER7 /