arXiv subjects

V. F. Pisarenko

Publications and source records attributed to V. F. Pisarenko.

11 recordsLinked to original sources

arXiv2011

Robust Statistical Tests of Dragon-Kings beyond Power Law Distributions

We ask the question whether it is possible to diagnose the existence of "Dragon-Kings" (DK), namely anomalous observations compared to a power law background distribution of event sizes. We present two new statistical tests, the U-test and the DK-test, aimed at identifying the existence of even a single anomalous event in the tail of the distribution of just a few tens of observations. The DK-test in particular is derived such that the p-value of its statistic is independent of the exponent characterizing the null hypothesis. We demonstrate how to apply these two tests on the distributions of cities and of agglomerations in a number of countries. We find the following evidence for Dragon-Kings: London in the distribution of city sizes of Great Britain; Moscow and St-Petersburg in the distribution of city sizes in the Russian Federation; and Paris in the distribution of agglomeration sizes in France. True negatives are also reported, for instance the absence of Dragon-Kings in the distribution of cities in Germany.

physics.data-an↗

arXiv2009

Distribution of Maximum Earthquake Magnitudes in Future Time Intervals, Application to the Seismicity of Japan (1923-2007)

We modify the new method for the statistical estimation of the tail distribution of earthquake seismic moments introduced by Pisarenko et al. [2009] and apply it to the earthquake catalog of Japan (1923-2007). The method is based on the two main limit theorems of the theory of extreme values and on the derived duality between the Generalized Pareto Distribution (GPD) and Generalized Extreme Value distribution (GEV). We obtain the distribution of maximum earthquake magnitudes in future time intervals of arbitrary duration tau. This distribution can be characterized by its quantile Qq(tau) at any desirable statistical level q. The quantile Qq(tau) provides a much more stable and robust characteristic than the traditional absolute maximum magnitude Mmax (Mmax can be obtained as the limit of Qq(tau) as q tends to 1, and tau tends to infinity). The best estimates of the parameters governing the distribution of Qq(tay) for Japan (1923-2007) are the following: Form parameter for GEV = -0.1901 +- 0.0717; position parameter GEV(tau=200)= 6.3387 +- 0.0380; spread parameter for GEV(tau=200)= 0.5995 +- 0.0223; Q_0.90,GEV(tau=10)= 8.34 +- 0.32. We also estimate Qq(tau) for a set of q-values and future time periods in the range for tau between 1 and 50 years from 2007. For comparison, the absolute maximum estimate Mmax from GEV, which is equal to 9.57 +- 0.86, has a scatter more than twice that of the 90 percent quantile Q_{0.90,GEV}(tau=10) of the maximum magnitude over the next 10 years counted from 2007.

physics.geo-ph↗

arXiv2008

Characterization of the tail of the distribution of earthquake magnitudes by combining the GEV and GPD descriptions of Extreme Value Theory

We present a generic and powerful approach to study the statistics of extreme phenomena (meteorology, finance, biology...) that we apply to the statistical estimation of the tail of the distribution of earthquake sizes. The chief innovation is to combine the two main limit theorems of Extreme Value Theory (EVT) that allow us to derive the distribution of T-maxima (maximum magnitude occurring in sequential time intervals of duration T) for arbitrary T. We propose a method for the estimation of the unknown parameters involved in the two limit theorems corresponding to the Generalized Extreme Value distribution (GEV) and to the Generalized Pareto Distribution (GPD). We establish the direct relations between the parameters of these distributions, which permit to evaluate the distribution of the T-maxima for arbitrary T. The duality between the GEV and GPD provides a new way to check the consistency of the estimation of the tail characteristics of the distribution of earthquake magnitudes for earthquake occurring over arbitrary time interval. We develop several procedures and check points to decrease the scatter of the estimates and to verify their consistency. We test our full procedure on the global Harvard catalog (1977-2006) and on the Fennoscandia catalog (1900-2005). For the global catalog, we obtain the following estimates: Mmax = 9.53 +- 0.52; quantile(0.97)==9.21 +- 0.20. For Fennoscandia, we obtain Mmax = 5.76 +- 0.165; quantile(0.97) =5.44 +- 0.073. The estimates of all related parameters for the GEV and GPD, including the most important form parameter, are also provided.

physics.geo-ph↗

arXiv2007

Properties of a simple bilinear stochastic model: estimation and predictability

We analyze the properties of arguably the simplest bilinear stochastic multiplicative process, proposed as a model of financial returns and of other complex systems combining both nonlinearity and multiplicative noise. By construction, it has no linear predictability (zero two-point correlation) but a certain nonlinear predictability (non-zero three-point correlation). It can thus be considered as a paradigm for testing the existence of a possible nonlinear predictbility in a given time series. We present a rather exhaustive study of the process, including its ability to produce fat-tailed distribution from Gaussian innovations, the unstable characteristics of the inversion of the key nonlinear parameters and of the two initial conditions necessary for the implementation of a prediction scheme and an analysis of the associated super-exponential sensitivity of the inversion of the innovations in the presence of a large impluse. Our study emphasizes the conditions under which a degree of predictability can be achieved and describes a number of different attempts, which overall illuminates the properties of the process. In conclusion, notwithstanding its remarkable simplicity, the bilinear stochastic process exhibits remarkably rich and complex behavior, which makes it a serious candidate for the modeling of financial times series and of other complex systems.

physics.data-an↗

arXiv2007

New Approach to the Characterization of Mmax and of the Tail of the Distribution of Earthquake Magnitudes

We develop a new method for the statistical esitmation of the tail of the distribution of earthquake sizes recorded in the Worldwide Harvard catalog of seismic moments converted to mW-magnitudes (1977-2004 and 1977-2006). We show that using the set of maximum magnitudes (the set of T-maxima) in windows of duration T days provides a significant improvement over existing methods, in particular (i) by minimizing the negative impact of time-clustering of foreshock / main shock /aftershock sequences in the estimation of the tail of the magnitude distribution, and (ii) by providing via a simulation method reliable estimates of the biases in the Moment estimation procedure (which turns out to be more efficient than the Maximum Likelihood estimation). Using a simulation method, we have determined the optimal window size of the T-maxima to be T=500 days. We have estimated the following quantiles of the distribution of T-maxima of earthquake magnitudes for the whole period 1977-2006: Q_{0.16}(Mmax)=9.3, Q_{0.5}(Mmax)=9.7 and Q_{0.84}(Mmax)=10.3. Finally, we suggest two more stable statistical characterristics of the tail of the distribution of earthquake magnitudes: the quantile QT(q) of a high probability level q for the T-maxima, and the probability of exceedence for a high threshold magnitude. We obtained the following sample estimates for the global Harvard catalog: QT(q=0.98)=8.6 +- 0.2 and a probability for the T-maxima to exceed magnitude 8 equal to 0.13-0.20. The comparison between our estimates for the two periods 1977-2004 and 1977-2006, where the later period includes the great Sumatra earthquake, 24.12.2004, mW=9.0, confirms the instability of the estimation of the parameter Mmax and the stability of the two other estimates.

physics.geo-ph↗

arXiv2004

New statistic for financial return distributions: power-law or exponential?

We introduce a new statistical tool (the TP-statistic and TE-statistic) designed specifically to compare the behavior of the sample tail of distributions with power-law and exponential tails as a function of the lower threshold u. One important property of these statistics is that they converge to zero for power laws or for exponentials correspondingly, regardless of the value of the exponent or of the form parameter. This is particularly useful for testing the structure of a distribution (power law or not, exponential or not) independently of the possibility of quantifying the values of the parameters. We apply these statistics to the distribution of returns of one century of daily data for the Dow Jones Industrial Average and over one year of 5-minutes data of the Nasdaq Composite index. Our analysis confirms previous works showing the tendency for the tails to resemble more and more a power law for the highest quantiles but we can detect clear deviations that suggest that the structure of the tails of the distributions of returns is more complex than usually assumed; it is clearly more complex that just a power law.

physics.comp-ph↗

arXiv2003

On Statistical Methods of Parameter Estimation for Deterministically Chaotic Time-Series

We discuss the possibility of applying some standard statistical methods (the least square method, the maximum likelihood method, the method of statistical moments for estimation of parameters) to deterministically chaotic low-dimensional dynamic system (the logistic map) containing an observational noise. A ``pure'' Maximum Likelihood (ML) method is suggested to estimate the structural parameter of the logistic map along with the initial value $x_1$ considered as an additional unknown parameter. Comparisons with previously proposed techniques on simulated numerical examples give favorable results (at least, for the investigated combinations of sample size $N$ and noise level). Besides, unlike some suggested techniques, our method does not require the a priori knowledge of the noise variance. We also clarify the nature of the inherent difficulties in the statistical analysis of deterministically chaotic time series and the status of previously proposed Bayesian approaches. We note the trade-off between the need of using a large number of data points in the ML analysis to decrease the bias (to guarantee consistency of the estimation) and the unstable nature of dynamical trajectories with exponentially fast loss of memory of the initial condition. The method of statistical moments for the estimation of the parameter of the logistic map is discussed. This method seems to be the unique method whose consistency for deterministically chaotic time series is proved so far theoretically (not only numerically).

physics.data-an↗

arXiv2003

Empirical Distributions of Log-Returns: between the Stretched Exponential and the Power Law?

A large consensus now seems to take for granted that the distributions of empirical returns of financial time series are regularly varying, with a tail exponent close to 3. We revisit this results and use standard tests as well as develop a battery of new non-parametric and parametric tests (in particular with stretched exponential (SE) distributions) to characterize the distributions of empirical returns of financial time series, with application to the 100 years of daily return of the Dow Jones Industrial Average and over 1 years of 5-minutes returns of the Nasdaq Composite index. Based on the discovery that the SE distribution tends to the Pareto distribution in a certain limit such that the Pareto (or power law) distribution can be approximated with any desired accuracy on an arbitrary interval by a suitable adjustment of the parameters of the SE distribution, we demonstrate that Wilks' test of nested hypothesis still works for the non-exactly nested comparison between the SE and Pareto distributions. The SE distribution is found significantly better over the whole quantile range but becomes unnecessary beyond the 95% quantiles compared with the Pareto law. Similar conclusions hold for the log-Weibull model with respect to the Pareto distribution. Our main result is that the tails ultimately decay slower than any stretched exponential distribution but probably faster than power laws with reasonable exponents. Implications of our results on the ``moment condition failure'' and for risk estimation and management are presented.

physics.soc-ph↗

arXiv2002

Fractal Plate Tectonics

We analyze in details the statistical significance of the claim by Bird [2002] of a power law distribution of plate areas covering the Earth and confirm that the power law with exponent 0.25 +- 0.05 is the most robust and parsimonious model for all plates, including the very largest plates, when taking into account the constraint that the plates areas must sum up to 4 pi steradians. We propose a general class of fragmentation models that rationalize this observation and discuss the implications for the earth dynamics and the general self-organization of tectonic deformations at multiple scales.

cond-mat.stat-mech↗

arXiv2002

Rigorous statistical detection and characterization of a deviation from the Gutenberg-Richter distribution above magnitude 8 in subduction zones

We present a quantitative statistical test for the presence of a crossover c0 in the Gutenberg-Richter distribution of earthquake seismic moments, separating the usual power law regime for seismic moments less than c0 from another faster decaying regime beyond c0. Our method is based on the transformation of the ordered sample of seismic moments into a series with uniform distribution under condition of no crossover. The bootstrap method allows us to estimate the statistical significance of the null hypothesis H0 of an absence of crossover (c0=infinity). When H0 is rejected, we estimate the crossover c0 using two different competing models for the second regime beyond c0 and the bootstrap method. For the catalog obtained by aggregating 14 subduction zones of the Circum Pacific Seismic Belt, our estimate of the crossover point is log(c0) =28.14 +- 0.40 (c0 in dyne-cm), corresponding to a crossover magnitude mW=8.1 +- 0.3. For separate subduction zones, the corresponding estimates are much more uncertain, so that the null hypothesis of an identical crossover for all subduction zones cannot be rejected. Such a large value of the crossover magnitude makes it difficult to associate it directly with a seismogenic thickness as proposed by many different authors in the past. Our measure of c0 may substantiate the concept that the localization of strong shear deformation could propagate significantly in the lower crust and upper mantle, thus increasing the effective size beyond which one should expect a change of regime.

cond-mat.stat-mech↗

arXiv2000

Characterization of the frequency of extreme events by the Generalized Pareto Distribution

Based on recent results in extreme value theory, we use a new technique for the statistical estimation of distribution tails. Specifically, we use the Gnedenko-Pickands-Balkema-de Haan theorem, which gives a natural limit law for peak-over-threshold values in the form of the Generalized Pareto Distribution (GPD). Useful in finance, insurance, hydrology, we investigate here the earthquake energy distribution described by the Gutenberg-Richter seismic moment-frequency law and analyze shallow earthquakes (depth h < 70 km) in the Harvard catalog over the period 1977-2000 in 18 seismic zones. The whole GPD is found to approximate the tails of the seismic moment distributions quite well above moment-magnitudes larger than mW=5.3 and no statistically significant regional difference is found for subduction and transform seismic zones. We confirm that the b-value is very different in mid-ocean ridges compared to other zones (b=1.50=B10.09 versus b=1.00=B10.05 corresponding to a power law exponent close to 1 versus 2/3) with a very high statistical confidence. We propose a physical mechanism for this, contrasting slow healing ruptures in mid-ocean ridges with fast healing ruptures in other zones. Deviations from the GPD at the very end of the tail are detected in the sample containing earthquakes from all major subduction zones (sample size of 4985 events). We propose a new statistical test of significance of such deviations based on the bootstrap method. The number of events deviating from the tails of GPD in the studied data sets (15-20 at most) is not sufficient for determining the functional form of those deviations. Thus, it is practically impossible to give preference to one of the previously suggested parametric families describing the ends of tails of seismic moment distributions.

cond-mat.stat-mech↗