Spartan Random Processes in Time Series Modeling

A Spartan random process (SRP) is used to estimate the correlation structure of time series and to predict (extrapolate) the data values. SRP's are motivated from statistical physics, and they can be viewed as Ginzburg-Landau models. The temporal correlations of the SRP are modeled in terms of `interactions' between the field values. Model parameter inference employs the computationally fast modified method of moments, which is based on matching sample energy moments with the respective stochastic constraints. The parameters thus inferred are then compared with those obtained by means of the maximum likelihood method. The performance of the Spartan predictor (SP) is investigated using real time series of the quarterly S&P 500 index. SP prediction errors are compared with those of the Kolmogorov-Wiener predictor. Two predictors, one of which explicit, are derived and used for extrapolation. The performance of the predictors is similarly evaluated.Comment: 10 pages, 3 figures, Proceedings of APFA

Spatial Random Field Models Inspired from Statistical Physics with Applications in the Geosciences

The spatial structure of fluctuations in spatially inhomogeneous processes can be modeled in terms of Gibbs random fields. A local low energy estimator (LLEE) is proposed for the interpolation (prediction) of such processes at points where observations are not available. The LLEE approximates the spatial dependence of the data and the unknown values at the estimation points by low-lying excitations of a suitable energy functional. It is shown that the LLEE is a linear, unbiased, non-exact estimator. In addition, an expression for the uncertainty (standard deviation) of the estimate is derived.Comment: 10 pages, to appear in Physica A v4: Some typos corrected and grammatical change

Stochastic Stick - Slip Model Linking Crustal Shear Strength and Earthquake Interevent Times

The current understanding of the earthquake interevent times distribution (ITD) is incomplete. The Weibull distribution is often used to model the earthquake ITD. We link the earthquake ITD on single faults with the Earth's crustal shear strength distribution by means of a phenomenological stick - slip model. We obtain Weibull ITD for power-law stress accumulation, i.e., $\sigma(t) = \alpha t^{\beta}$, where $\beta >0$ for single faults or systems with homogeneous strength statistics. We show that logarithmic stress accumulation leads to the log-Weibull ITD. For the Weibull ITD, we prove that (i) $m= \beta m_s$, where $m$ and $m_s$ are, respectively, the ITD and crustal shear strength Weibull moduli and (ii) the time scale $\tau_s = (S_s/\alpha)^{1/\beta}$ where $S_s$ is the scale of crustal shear strength. We generalize the ITD model for fault systems. We investigate deviations of the ITD tails from the Weibull due to sampling bias, magnitude selection, and non-homogeneous strength parameters. Assuming the Gutenberg - Richter law and independence of $m$ on the magnitude threshold, $M_{L,c},$ we deduce that $\tau_s \propto e^{- \rho_{M} M_{L,c}},$ where $\rho_M \in [1.15, 3.45]$ for seismically active regions. We demonstrate that a microearthquake sequence conforms reasonably well to the Weibull model. The stochastic stick - slip model justifies the Weibull ITD for single faults and homogeneous fault systems, while it suggests mixtures of Weibull distributions for heterogeneous fault systems. Non-universal deviations from Weibull statistics are possible, even for single faults, due to magnitude thresholds and non-uniform parameter values.Comment: 32 pages, 11 figures Version 2; minor correction