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Dionissios T. Hristopulos

Publications and source records attributed to Dionissios T. Hristopulos.

27 records · Page 2Linked to original sources
arXiv2015

Space-Time Models based on Random Fields with Local Interactions

The analysis of space-time data from complex, real-life phenomena requires the use of flexible and physically motivated covariance functions. In most cases, it is not possible to explicitly solve the equations of motion for the fields or the respective covariance functions. In the statistical literature, covariance functions are often based on mathematical constructions. We propose deriving space-time covariance functions by solving "effective equations of motion", which can be used as statistical representations of systems with diffusive behavior. In particular, we propose using the linear response theory to formulate space-time covariance functions based on an equilibrium effective Hamiltonian. The effective space-time dynamics are then generated by a stochastic perturbation around the equilibrium point of the classical field Hamiltonian leading to an associated Langevin equation. We employ a Hamiltonian which extends the classical Gaussian field theory by including a curvature term and leads to a diffusive Langevin equation. Finally, we derive new forms of space-time covariance functions.

stat.ME↗
arXiv2014

Radial Covariance Functions Motivated by Spatial Random Field Models with Local Interactions

We derive explicit expressions for a family of radially symmetric, non-differentiable, Spartan covariance functions in $\mathbb{R}^2$ that involve the modified Bessel function of the second kind. In addition to the characteristic length and the amplitude coefficient, the Spartan covariance parameters include the rigidity coefficient $η_{1}$ which determines the shape of the covariance function. If $ η_{1} >> 1$ Spartan covariance functions exhibit multiscaling. We also derive a family of radially symmetric, infinitely differentiable Bessel-Lommel covariance functions valid in $\mathbb{R}^{d}, d\ge 2$. We investigate the parametric dependence of the integral range for Spartan and Bessel-Lommel covariance functions using explicit relations and numerical simulations. Finally, we define a generalized spectrum of correlation scales $λ^{(α)}_{c}$ in terms of the fractional Laplacian of the covariance function; for $0 \le α\le1$ the $λ^{(α)}_{c}$ extend from the smoothness microscale $(α=1)$ to the integral range $(α=0)$. The smoothness scale of mean-square continuous but non-differentiable random fields vanishes; such fields, however, can be discriminated by means of $λ^{(α)}_{c}$ scales obtained for $α<1$.

math.ST↗
arXiv2014

Finite-Size Effects on Return Interval Distributions for Weakest-Link-Scaling Systems

The Weibull distribution is a commonly used model for the strength of brittle materials and earthquake return intervals. Deviations from Weibull scaling, however, have been observed in earthquake return intervals and in the fracture strength of quasi-brittle materials. We investigate weakest-link scaling in finite-size systems and deviations of empirical return interval distributions from the Weibull distribution function. We use the ansatz that the survival probability function of a system with complex interactions among its units can be expressed as the product of the survival probability functions for an ensemble of representative volume elements (RVEs). We show that if the system comprises a finite number of RVEs, it obeys the $κ$-Weibull distribution. We conduct statistical analysis of experimental data and simulations that show good agreement with the $κ$-Weibull distribution. We show the following: (1) The weakest-link theory for finite-size systems involves the $κ$-Weibull distribution. (2) The power-law decline of the $κ$-Weibull upper tail can explain deviations from the Weibull scaling observed in return interval data. (3) The hazard rate function of the $κ$-Weibull distribution decreases linearly after a waiting time $τ_c \propto n^{1/m}$, where $m$ is the Weibull modulus and $n$ is the system size in terms of representative volume elements. (4) The $κ$-Weibull provides competitive fits to the return interval distributions of seismic data and of avalanches in a fiber bundle model. In conclusion, using theoretical and statistical analysis of real and simulated data, we show that the $κ$-Weibull distribution is a useful model for extreme-event return intervals in finite-size systems.

physics.geo-ph↗
arXiv2013

A Directional Gradient-Curvature Method for Gap Filling of Gridded Environmental Spatial Data with Potentially Anisotropic Correlations

We introduce the Directional Gradient-Curvature (DGC) method, a novel approach for filling gaps in gridded environmental data. DGC is based on an objective function that measures the distance between the directionally segregated normalized squared gradient and curvature energies of the sample and entire domain data. DGC employs data-conditioned simulations, which sample the local minima configuration space of the objective function instead of the full conditional probability density function. Anisotropy and non-stationarity can be captured by the local constraints and the direction-dependent global constraints. DGC is computationally efficient and requires minimal user input, making it suitable for automated processing of large (e.g., remotely sensed) spatial data sets. Various effects are investigated on synthetic data. The gap-filling performance of DGC is assessed in comparison with established classification and interpolation methods using synthetic and real satellite data, including a skewed distribution of daily column ozone values. It is shown that DGC is competitive in terms of cross validation performance.

stat.AP↗
arXiv2013

Classification of missing values in spatial data using spin models

A problem of current interest is the estimation of spatially distributed processes at locations where measurements are missing. Linear interpolation methods rely on the Gaussian assumption, which is often unrealistic in practice, or normalizing transformations, which are successful only for mild deviations from the Gaussian behavior. We propose to address the problem of missing values estimation on two-dimensional grids by means of spatial classification methods based on spin (Ising, Potts, clock) models. The "spin" variables provide an interval discretization of the process values, and the spatial correlations are captured in terms of interactions between the spins. The spins at the unmeasured locations are classified by means of the "energy matching" principle: the correlation energy of the entire grid (including prediction sites) is estimated from the sample-based correlations. We investigate the performance of the spin classifiers in terms of computational speed, misclassification rate, class histogram and spatial correlations reproduction using simulated realizations of spatial random fields, real rainfall data, and a digital test image. We also compare the spin-based methods with standard classifiers such as the k-nearest neighbor, the fuzzy k-nearest neighbor, and the Support Vector Machine. We find that the spin-based classifiers provide competitive choices.

physics.data-an↗
arXiv2012

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., $σ(t) = αt^β$, where $β>0$ 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) $m= βm_s$, where $m$ and $m_s$ are, respectively, the ITD and crustal shear strength Weibull moduli and (ii) the time scale $τ_s = (S_s/α)^{1/β}$ 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 $τ_s \propto e^{- ρ_{M} M_{L,c}},$ where $ρ_M \in [1.15, 3.45]$ 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.

physics.geo-ph↗
arXiv2008

Multilevel Discretized Random Field Models with "Spin" Correlations for the Simulation of Environmental Spatial Data

A problem of practical significance is the analysis of large, spatially distributed data sets. The problem is more challenging for variables that follow non-Gaussian distributions. We show that the spatial correlations between variables can be captured by interactions between "spins". The spins represent multilevel discretizations of the initial field with respect to a number of pre-defined thresholds. The spatial dependence between the "spins" is imposed by means of short-range interactions. We present two approaches, inspired by the Ising and Potts models, that generate conditional simulations from samples with missing data. The simulations of the "spin system" are forced to respect locally the sample values and the system statistics globally. We compare the two approaches in terms of their ability to reproduce the sample statistical properties, to predict data at unsampled locations, as well as in terms of their computational complexity. We discuss the impact of relevant simulation parameters, such as the domain size, the number of discretization levels, and the initial conditions.

stat.AP↗
arXiv2006

Analytic Properties and Covariance Functions of a New Class of Generalized Gibbs Random Fields

Spartan Spatial Random Fields (SSRFs) are generalized Gibbs random fields, equipped with a coarse-graining kernel that acts as a low-pass filter for the fluctuations. SSRFs are defined by means of physically motivated spatial interactions and a small set of free parameters (interaction couplings). This paper focuses on the FGC-SSRF model, which is defined on the Euclidean space $\mathbb{R}^{d}$ by means of interactions proportional to the squares of the field realizations, as well as their gradient and curvature. The permissibility criteria of FGC-SSRFs are extended by considering the impact of a finite-bandwidth kernel. It is proved that the FGC-SSRFs are almost surely differentiable in the case of finite bandwidth. Asymptotic explicit expressions for the Spartan covariance function are derived for $d=1$ and $d=3$; both known and new covariance functions are obtained depending on the value of the FGC-SSRF shape parameter. Nonlinear dependence of the covariance integral scale on the FGC-SSRF characteristic length is established, and it is shown that the relation becomes linear asymptotically. The results presented in this paper are useful in random field parameter inference, as well as in spatial interpolation of irregularly-spaced samples.

cs.IT↗