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G. Molchan

Publications and source records attributed to G. Molchan.

At least 19 recordsLinked to original sources
arXiv2025

Persistence probabilities for fractionally integrated fractional Brownian noise

The main objective of this study is fractionally integrated fractional Brownian noise, I(t/a,H) where a>0 is the 'multiplicity' of integration, and H is the Hurst parameter . The subject of the analysis is the persistence exponent e(a,H) that determines the power-law asymptotic of probability that the process will not exceed a fixit level in a growing time interval (0,T). In the important cases such as fractional Brownian motion(FBM(H),a=1) and integtated Wienr process(a=2,H=1/2) these exponents are well known. To understand the problematic exponents e(2,H), we consider the (a,H) parameters from the maximum (for the task) area G= (a+H>1,0 >1, H). It is identical to the persistence exponent for a Gaussian stationary process with covariance cosh((H-1/2)t)/cosh(t/2) and generalizes the well-known case of H=1/2. Our results use well known the continuity lemma for the persistence exponents and a some generalization of Slepian's lemma for a family of Gaussian processes smoothly dependent on a parameter.

math.PR↗
arXiv2024

Number of aftershocks in epidemic-type seismicity models

Let $V_M,(m_0)$ be the number of m>M aftershocks caused by $m_0$ event. We consider the $V_M,(m_0)$ distribution within epidemic-type seismicity models, ETAS(F). These models include the Gutenberg-Richter law for magnitude and Utsu law for average productivity of m0, but differ in the type of F distribution for the number v(m0) of direct aftershocks. The class of F is quite broad and includes both the Poisson distribution, which is the basis for the regular ETAS model, and its possible alternative, the Geometric distribution. Instead of the traditional threshold $M=m_0-\Delta$, we consider $M=m_a-\Delta$, where $m_a$ is the mode in the distribution of the strongest aftershock. Under these conditions we find the limit $V_M,(m_0)$ distribution at $m_0\gg1$. In the subcritical case, the limit distribution is extremely simple and identical to the $v(m_\Delta)$ distribution with a suitable magnitude $m=m_\Delta$. Theoretical results of this kind are lacking even for the regular ETAS model. Our results provide an additional opportunity to test the type of F-distribution, Bath law, and the very concept of epidemic-type clustering.

physics.geo-ph↗
arXiv2020

Leadership exponent in the pursuit problem for 1-D random particles

For n + 1 particles moving independently on a straight line, we study the question of how long the leading position of one of them can last. Our focus is the asymptotics of the probability p(T,n) that the leader time will exceed T when n and T are large. It is assumed that the dynamics of particles are described by independent, either stationary or self-similar, Gaussian processes, not necessarily identically distributed. Roughly, the result for particles with stationary dynamics of unit variance is as follows: L= -log p(T,n) /(Tlog n)=1/d+o(1), where d/(2pi) is the power of the zero frequency in the spectrum of the leading particle, and this value is the largest in the spectrum. Previously, in some particular models, the asymptotics of L was understood as a sequential limit first over T and then over n. For processes that do not necessarily have non-negative correlations, the limit over T may not exist. To overcome this difficulty, the growing parameters T and n are considered in the domain clog T 1 . The Lamperti transform allows us to transfer the described result to self-similar processes with the normalizer of log p(T,n) becoming log T log n.

math.PR↗
arXiv2018

Integrated fractional Brownian motion: persistence probabilities and their estimates

The problem is a log-asymptotics of the probability that the Integrated fractional Brownian motion of index 0<H<1 does not exceed a fixed level during long time. For the growing time interval (0,T) the hypothetical log-asymptotics is (H(H-1)+o(1))Log T. In support of the hypothesis, we update our earlier estimates of the probability and give analytical proofs.

math.PR↗
arXiv2018

Factional Brownian motion with multivariate time in a large convex area: persistence exponents

The fractional Brownian motion of index $0 < H < 1$, H-FBM, with d-dimensional time is considered on an expanding set TG, where G is a bounded convex domain that contains 0 at its boundary. The main result: if 0 is a point of smoothness of the boundary, then the log-asymptotics of probability that H-FBM does not exceed a fixed positive level in TG is $(H - d + o(1)) \log T$, $T\to\infty$. Some generalizations of this result to isotropic but not self-similar Gaussian processes with stationary increments are also considered.

math.PR↗
arXiv2017

Comment on "Assessing CN earthquake predictions in Italy" by M. Taroni, W. Marzocchi, P. Roselli

The paper by Taroni et al. (2016) considers results of forward prediction of Italian strong earthquakes by CN algorithm with the declared intent of providing "a careful assessment of CN prediction performances... using standard testing procedures". Given the very limited number of target events within each region, however, the considered situation is non statistical, and a priori it is clear that the standard statistical methods are not effective here. The attempt to replace the standard approaches by Pari-mutuel Gambling Score (PGS) method leads to almost complete loss of information about predicted earthquakes, even for a large sample of target events. Therefore, the conclusions based on PGS, are untenable.

physics.geo-ph↗
arXiv2016

Survival exponents for fractional Brownian motion with multivariate time

Fractional Brownian motion, H-FBM , of index with d-dimensional time is considered in a spherical domain that contains 0 at its boundary. The main result : the log-asymptotics of probability that H-FBM does not exceed a fixed positive level is (H-d)logT(1+o(1)), where T>>1 is radius of the domain.

math.PR↗
arXiv2016

The omega-square hypothesis for the seismic source

The omega-square hypothesis assumes that the ground displacement u(t) in the far-field zone decays as the inverse square of frequency in the range ~1-30 Hz. This empirical fact remains theoretically unjustified. Our analysis of the problem is based on an integral representation of u(t) in terms of the source time function f and on the spectrum analysis of local features in f. The goal is to select the local features that will enable one to generate the omega-square behavior of u(t) on a large set of receivers. We found two appropriate fragments of f : first , f exhibits a local inverse-square-root behavior near the rupture front where the frontal surface is piecewise smooth (but not smooth and not rough ); and second, f is bounded near the rupture front where the frontal surface has a slight roughness (the Hurst parameter near 1). These facts can be useful for understanding the omega-square spectral behavior of the kinematic source models.

physics.geo-ph↗
arXiv2016

A note on the ranking of earthquake forecasts

The ranking problem of earthquake forecasts is considered. We formulate simple statistical requirements to forecasting quality measure R and analyze some R-ranking methods on this basis, in particular, the pari-mutuel gambling method by Zechar&Zhuang (2014).

physics.geo-ph↗
arXiv2015

Stochastic earthquake source model: the omega-square hypothesis and the directivity effect

Recently A. Gusev suggested and numerically investigated the doubly stochastic earthquake source model. The model is supposed to demonstrate the following features in the far-field body waves: 1) the omega-square high-frequency (HF) behavior of displacement spectra; 2) lack of the directivity effect in HF radiation; and 3) a stochastic nature of the HF signal component. The model involves two stochastic elements: the local stress drop (SD) on a fault and the rupture time function (RT) with a linear dominant component. The goal of the present study is to investigate the Gusev model theoretically and to find conditions for (1, 2) to be valid and stable relative to receiver site. The models with smooth elements SD, RT are insufficient for these purposes. Therefore SD and RT are treated as realizations of stochastic fields of the fractal type. The local smoothness of such fields is characterized by the fractional (Hurst) exponent H, 0 < H < 1. This allows us to consider a wide class of stochastic functions without regard to their global spectral properties. We show that the omega-square behavior of the model is achieved approximately if the rupture time function is almost regular (H~1) while the stress drop is rough function of any index H. However, if the rupture front is linear, the local stress drop has to be function of minimal smoothness (H~0). The situation with the directivity effect is more complicated: for different RT models with the same fractal index, the effect may or may not occur. The nature of the phenomenon is purely analytical. The main controlling factor for the directivity is the degree of smoothness of the two dimensional distributions of RT random function. For this reason the directivity effect is unstable. This means that in practice the opposite conclusions relative to the statistical significance of the directivity effect are possible

physics.geo-ph↗
arXiv2013

Forecasting ability of a multi-renewal seismicity model for Italy

The inter-event time, IET, is sometimes used as a basis for prediction of large earthquakes. It is the case when theoretical analysis of prediction is possible. Quite recently a specific IET- model was suggested for dynamic probabilistic prediction of M > 5.5 events in Italy . In this study we analyze both some aspects of the statistical estimation of the model and its predictive ability. We find that more or less effective prediction is possible within 4 out of 34 seismotectonic zones where seismicity rate or clustering of events is relatively high. We show that, in the framework of the model, one can suggest a simple zone independent strategy, which practically optimizes the relative number of nonaccidental successes, or the Hanssen-Kuiper, HK, skill score. This quasi-optimal strategy declares alarm in a zone for the first 2.67 years just after the occurrence of each large event in the zone. The optimal HK skill score values are: 26% for the 3 most active zones and 2-10% for the 26 least active zones. However, the number of false alarm time intervals per one event in each of the zones is unusually high: 0.7 and 0.8-0.95 respectively. Both these theoretical estimations are important because any prospective testing of the model is unrealistic in most of the zones during a reasonable time. This particular analysis requires a discussion of the following issues of general interest: a specific approach to the analysis of predictions vs. the standard CSEP testing approach; prediction vs. forecasting; HK skill score vs. probability gain; the total forecast error diagram and connected false alarms.

physics.geo-ph↗
arXiv2010

Hot-Cold Spots in Italian Macroseismic Data

The site effect is usually associated with local geological conditions, which increase or decrease the level of shaking compared with standard attenuation relations. We made an attempt to see in the macroseismic data of Italy some other effects, namely, hot/cold spots in the terminology of Olsen (2000), which are related to local fault geometry rather than to soil conditions. We give a list of towns and villages liable to amplify (+) or to reduce (-) the level of shaking in comparison with the nearby settlements. Relief and soil conditions cannot always account for the anomalous sites. Further, there are sites where both (+) and (-) effects are observed depending on the earthquake. The opposite effects can be generated by events from the same seismotectonic zone and along the same direction to the site. Anomalous sites may group themselves into clusters of different scales. All isolated anomalous patterns presented in this paper can be used in hazard analysis, in particular, for the modeling and testing of seismic effects.

physics.geo-ph↗
arXiv2010

Gambling scores in earthquake prediction analysis

The number of successes 'n' and the normalized measure of space-time alarm 'tau' are commonly used to characterize the strength of an earthquake prediction method and the significance of prediction results. To evaluate better the forecaster's skill, it has been recently suggested to use a new characteristic, the gambling score R, which incorporates the difficulty of guessing each target event by using different weights for different alarms. We expand the class of R-characteristics and apply these to the analysis of results of the M8 prediction algorithm. We show that the level of significance 'alfa' strongly depends (1) on the choice of weighting alarm parameters, (2) on the partitioning of the entire alarm volume into component parts, and (3) on the accuracy of the spatial rate of target events, m(dg). These tools are at the disposal of the researcher and can affect the significance estimate in either direction. All the R-statistics discussed here corroborate that the prediction of 8.0<=M<8.5 events by the M8 method is nontrivial. However, conclusions based on traditional characteristics (n,tau) are more reliable owing to two circumstances: 'tau' is stable since it is based on relative values of m(.), and the 'n' statistic enables constructing an upper estimate of 'alfa' taking into account the uncertainty of m(.).

physics.geo-ph↗
arXiv2010

Earthquake prediction analysis: The M8 algorithm

The quality of space-time earthquake prediction is usually characterized by a two-dimensional error diagram (n,tau), where 'n' is the rate of failures-to-predict and 'tau' is the normalized measure of space-time alarm. The most interesting space measure for analysis of a prediction strategy is the rate of target events m(dg) in a sub-area 'dg'. In this case the quantity H=1-(n+tau) determines the prediction capability of the strategy. The uncertainty of m(dg) causes difficulties in estimating 'H' and the statistical significance, 'alfa', of prediction results. We investigate this problem theoretically and show how the uncertainty of the measure can be taken into account in two situations, viz., the estimation of 'alfa' and the construction of a confidence zone for (n,tau)-parameters of the random strategies. We use our results to analyse the M8 earthquake prediction algorithm.

physics.geo-ph↗
arXiv2008

Space-Time Earthquake Prediction: the Error Diagrams

The quality of earthquake prediction is usually characterized by a two-dimensional diagram 'n' vs. 'tau', where 'n' is the rate of failures-to-predict and 'tau' is a characteristic of space- time alarm. Unlike the time prediction case, the quantity 'tau' is not defined uniquely, so that the properties of the (n,tau) diagram require a theoretical analysis, which is the main goal of the present study. This note is based on a recent paper by Molchan and Keilis-Borok in GJI, 173 (2008), 1012-1017.

physics.geo-ph↗
arXiv2008

Earthquake Prediction: Probabilistic Aspect

A theoretical analysis of the earthquake prediction problem in space-time is presented. We find an explicit structure of the optimal strategy and its relation to the generalized error diagram. This study is a generalization of the theoretical results for time prediction. The possibility and simplicity of this extension is due to the choice of the class of goal functions. We also discuss issues in forecasting versus prediction, scaling laws versus predictability, and measure of prediction efficiency at the research stage.

physics.geo-ph↗
arXiv2006

The Problem of Small Unilateral Deviations: the Existence of Decay Exponents

Let x(s), s in R^d be a Gaussian self-similar random process of index H. We consider the problem of log-asymptotics for the probability p(T) that x(s), x(0)=0 does not exceed a fixed level in a star-shaped expanding domain TxG as T>>1. We solve the problem of the existence of the limit, theta:=lim (-log p(T))/(log T)^D, T>>1, for the fractional Brownian sheet x(s)on [0,T]^2 then D=2 and we estimate the theta for the integrated fractional Brownian motion then D=1.

math.PR↗