On infinitesimal σ-fields generated by random processes

It is proved that the infinitesimal look-ahead and look-back σ-fields of a random process disagree at atmost countably many time instants.

Approximation of Spatio-Temporal Random Processes Using Tensor Decomposition

A new representation of spatio-temporal random processes is proposed in this work. In practical applications, such processes are used to model velocity fields, temperature distributions, response of vibrating systems, to name a few. Finding an efficient representation for any random process leads to encapsulation of information which makes it more convenient for a practical implementations, for instance, in a computational mechanics problem. For a single-parameter process such as spatial or temporal process, the eigenvalue decomposition of the covariance matrix leads to the well-known Karhunen-Loeve (KL) decomposition. However, for multiparameter processes such as a spatio-temporal process, the covariance function itself can be defined in multiple ways. Here the process is assumed to be measured at a finite set of spatial locations and a finite number of time instants. Then the spatial covariance matrix at different time instants are considered to define the covariance of the process. This set of square, symmetric, positive semi-definite matrices is then represented as a third-order tensor. A suitable decomposition of this tensor can identify the dominant components of the process, and these components are then used to define a closed-form representation of the process. The procedure is analogous to the KL decomposition for a single-parameter process, however, the decompositions and interpretations vary significantly. The tensor decompositions are successfully applied on (i) a heat conduction problem, (ii) a vibration problem, and (iii) a covariance function taken from the literature that was fitted to model a measured wind velocity data. It is observed that the proposed representation provides an efficient approximation to some processes. Furthermore, a comparison with KL decomposition showed that the proposed method is computationally cheaper than the KL, both in terms of computer memory and execution time.

FIRST PASSAGE APPROXIMATION FOR NORMAL STATIONARY RANDOM PROCESSES.

A new approximate solution for the first passage probability of a stationary Gaussian random process is presented which is based on the estimation of the mean clump size. A simple expression for the mean clump size is derived in terms of the cumulative normal distribution function, which avoids the lengthy numerical integrations which are required by similar existing techniques. The method is applied to a linear oscillator and an ideal bandpass process and good agreement with published results is obtained. By making a slight modification to an existing analysis it is shown that a widely used empirical result for the asymptotic form of the first passage probability can be deduced theoretically.

Microscopic theory for random processes in weakly-coupled open systems

In order to relate macroscopic random motion (described e.g. by Langevin-type theories) to microscopic dynamics, we have undertaken the derivation of a Fokker-Planck-type equation from first microscopic principles. Both subsystems are subject to an external force field. Explicit expressions for the diffusion and drift coefficients are obtained, in terms of the field.

Refined second law of thermodynamics for fast random processes

We establish a refined version of the Second Law of Thermodynamics for Langevin stochastic processes describing mesoscopic systems driven by conservative or non-conservative forces and interacting with thermal noise. The refinement is based on the Monge-Kantorovich optimal mass transport and becomes relevant for processes far from quasi-stationary regime. General discussion is illustrated by numerical analysis of the optimal memory erasure protocol for a model for micron-size particle manipulated by optical tweezers.

Mean and variance of estimates of the bispectrum of a harmonic random process-an analysis including leakage effects

Analytical expressions are derived for the mean and variance, of estimates of the bispectrum of a real-time series assuming a cosinusoidal model. The effects of spectral leakage, inherent in discrete Fourier transform operation when the modes present in the signal have a nonintegral number of wavelengths in the record, are included in the analysis. A single phase-coupled triad of modes can cause the bispectrum to have a nonzero mean value over the entire region of computation owing to leakage. The variance of bispectral estimates in the presence of leakage has contributions from individual modes and from triads of phase-coupled modes. Time-domain windowing reduces the leakage. The theoretical expressions for the mean and variance of bispectral estimates are derived in terms of a function dependent on an arbitrary symmetric time-domain window applied to the record. the number of data, and the statistics of the phase coupling among triads of modes. The theoretical results are verified by numerical simulations for simple test cases and applied to laboratory data to examine phase coupling in a hypothesis testing framework

Online hidden Markov model parameter estimation and minimax robust quickest change detection in uncertain stochastic processes

Stochastic (or random) processes are inherent to numerous fields of human endeavour including engineering, science, and business and finance. This thesis presents multiple novel methods for quickly detecting and estimating uncertainties in several important classes of stochastic processes. The significance of these novel methods is demonstrated by employing them to detect aircraft manoeuvres in video signals in the important application of autonomous mid-air collision avoidance.

Recruitment as an evolving random process of aggregation and mortality

The dynamics of the survival of recruiting fish are analyzed as evolving random processes of aggregation and mortality. The analyses draw on recent advances in the physics of complex networks and, in particular, the scale-free degree distribution arising from growing random networks with preferential attachment of links to nodes. In this study simulations were conducted in which recruiting fish 1) were subjected to mortality by using alternative mortality encounter models and 2) aggregated according to random encounters (two schools randomly encountering one another join into a single school) or preferential attachment (the probability of a successful aggregation of two schools is proportional to the school sizes). The simulations started from either a “disaggregated” (all schools comprised a single fish) or an aggregated initial condition. Results showed the transition of the school-size distribution with preferential attachment evolving toward a scale-free school size distribution, whereas random attachment evolved toward an exponential distribution. Preferential attachment strategies performed better than random attachment strategies in terms of recruitment survival at time when mortality encounters were weighted toward schools rather than to individual fish. Mathematical models were developed whose solutions (either analytic or numerical) mimicked the simulation results. The resulting models included both Beverton-Holt and Ricker-like recruitment, which predict recruitment as a function of initial mean school size as well as initial stock size. Results suggest that school-size distributions during recruitment may provide information on recruitment processes. The models also provide a template for expanding both theoretical and empirical recruitment research.

A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star's mass. As a consequence, the pdf's of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars. © 2009 American Institute of Physics.

Appraisal of random and systematic land cover transitions for regional water balance and revegetation strategies

This study describes the use of landscape transition analysis as a means for effective basin management. Land cover transitions from 1995 to 2002 were analyzed using a cross-tabulation matrix for an important economic zone in south-west Victoria, Australia. Specifically, the matrix was used to determine whether the transitions were random or systematic. Random landscape transitions occur when a land cover replaces other land covers in proportion to their availability. Systematic landscape transitions occur when there are deviations from random patterns, and land use types ‘target’ other land use types for replacement. The analysis was conducted with 11 land cover categories and showed that dryland pastures have been systematically losing area to dryland crops and blue gum (Eucalyptus globulus) plantations. Dryland crops have systematically expanded in the north-east of the catchment, an area where increasing in-stream salinization has occurred concurrently with this transition. The systematic expansion of the blue gum plantations has been predominantly in the south-west of the catchment and has the potential to reduce stream flows and groundwater recharge in an already water-stressed region, as blue gums use more water than the dryland pastures they are replacing. All other transitions were largely random. These findings have implications for land use planning in the study area for regional water balance and revegetation strategies. Landscape transition analysis is a cost-effective means of contributing to the management of water resources at a regional scale, and is highly recommended for future basin planning.

A Markov random field model for combining optimum-path forest classifiers using decision graphs and game strategy approach

The research on multiple classifiers systems includes the creation of an ensemble of classifiers and the proper combination of the decisions. In order to combine the decisions given by classifiers, methods related to fixed rules and decision templates are often used. Therefore, the influence and relationship between classifier decisions are often not considered in the combination schemes. In this paper we propose a framework to combine classifiers using a decision graph under a random field model and a game strategy approach to obtain the final decision. The results of combining Optimum-Path Forest (OPF) classifiers using the proposed model are reported, obtaining good performance in experiments using simulated and real data sets. The results encourage the combination of OPF ensembles and the framework to design multiple classifier systems. © 2011 Springer-Verlag.

Probabilistic approaches in civil engineering: generation of random fields and structural identification with genetic algorithms

The inherent stochastic character of most of the physical quantities involved in engineering models has led to an always increasing interest for probabilistic analysis. Many approaches to stochastic analysis have been proposed. However, it is widely acknowledged that the only universal method available to solve accurately any kind of stochastic mechanics problem is Monte Carlo Simulation. One of the key parts in the implementation of this technique is the accurate and efficient generation of samples of the random processes and fields involved in the problem at hand. In the present thesis an original method for the simulation of homogeneous, multi-dimensional, multi-variate, non-Gaussian random fields is proposed. The algorithm has proved to be very accurate in matching both the target spectrum and the marginal probability. The computational efficiency and robustness are very good too, even when dealing with strongly non-Gaussian distributions. What is more, the resulting samples posses all the relevant, welldefined and desired properties of “translation fields”, including crossing rates and distributions of extremes. The topic of the second part of the thesis lies in the field of non-destructive parametric structural identification. Its objective is to evaluate the mechanical characteristics of constituent bars in existing truss structures, using static loads and strain measurements. In the cases of missing data and of damages that interest only a small portion of the bar, Genetic Algorithm have proved to be an effective tool to solve the problem.

DUI client characteristics - an interim analysis of the random assignment process. Interim report.

National Highway Traffic Safety Administration, Office of Driver and Pedestrian Programs, Washington, D.C.