Statistical properties of several models of fractional random point processes

3 - Time varying signals

The problem of digital modeling of random processes with given either correlation function or spectral density of the process is considered. These functions of a random process are interconnected by the Wiener–Khinchin theorem. The solution of one function can be used to solve another. The development of a mathematical representation of a stationary random process with a given correlation function based on the Karhunen-Loeve transformation, which is most often used to decorrelate the original process in order to describe it more concisely (data compression problem), has been completed. It is proposed to use the Karhunen–Loev transformation to impart the required correlation properties to the original uncorrelated random process by inverting (converting) this transformation. The form of the required transformation for a discrete (in time) representation of input and output processes of various lengths and methods for ensuring the required modeling accuracy are substantiated. A procedure for obtaining a correlation function from a given spectral density of a simulated random process is presented. An experimental study of the proposed method was carried out in the course of computer simulation in the Mathcad package which simplified the solution of the required computational problems. The initial random process was obtained as a sequence of independent (and, therefore, uncorrelated) random numbers, and the output process, as a result of the transformation, was obtained in the work. The calculated approximate correlation function is compared with the given one and the error variance is determined. The results of modeling random processes with given correlation functions and a homogeneous Markov process with a given transition probability are given as well as an example of the transition from a given spectral density of a random process to its correlation function. The results obtained confirm the effectiveness and feasibility of the developed modeling methods which will allow them to be used in computer research and design of various systems.

Cellular networks have a great transformation during the last decade. The traditional cellular network system is often overlaid with well-deployed BSs of proper power. However, more and more complex elements such as picocells, femtocells and relays have been introduced to cellular network system. BSs in these elements are unplanned, user-installed and randomly deployed. All of these things compose heterogeneous cellular networks (HCNs). In order to evaluate the performance of HCNs, people usually model the distribution of BSs in HCNs as spatial stochastic point process model. One of the most popular spatial models for these HCNs is Poisson Point Process (PPP) model. However the PPP model is not exact for some cases where BSs in PPP model may be too close to each other. In this paper, we propose a more reasonable and practical model called Matern-like point process (MLPP) model which imposes a certain minimal distance between any two BSs to overcome this problem. Simulations show that the coverage probability of the proposed model noticeably outperforms the PPP model. This improvement depends on the value of minimal distance. We also evaluate the system efficiency defined by the ratio of coverage probability and total power. The results show that the MLPP model has a significant improvement compared to PPP model.

We propose and investigate dynamic spatial-temporal point process models for independent and interacting events. The models for independent events are dynamic spatial-temporal Poisson point process (DSTPPP) model that account for temporal and spatial clustering. The models proposed for events with interaction are Markov (Gibbs) space-time point process models. We model the intensity function of a DSTPPP via conditioning arguments that allow for additional interpretations and inclusion of well-known point process models as special cases. Depending on the nature of the questions to be answered, the in- tensity function of a DSTPPP can be modeled in several ways. First, we develop models via conditioning on the time component of the events and then consider conditioning on the location component of the events. Modeling, simulation and computation are accomplished in a fully hierarchical Bayesian framework. Finally, we focus on dynamic marked Markov space-time point processes, where the events are allowed to interact with each other across time and space. Once again the hierarchical Bayesian framework is invaluable in this case, since it allows us to introduce dynamic process models via conditioning. The methodolo- gies are illustrated using simulated data, and several applications, including modeling and inference for earthquake events in California, and tornado events in Missouri.

This article proposes a technique for assessing risks related to operational reliability levels of systems of electric interlocking of track switches and signals at railroad stations. The assessment relies on the mathematical model of such random processes as failures and recoveries of the system of electric interlocking of track switches and signals and on using this system to fulfill train traffic control functions at a given station. The risk related to the operational safety of the electric interlocking system is calculated for the railroad station in general, not for a specific scenario of events. The statistics of safety violation events is based on a small sample. The risk related to the operational reliability of the electric interlocking system is assessed using risk matrices in the form of tables, where the likelihood of occurrence and the number of train delays at the station due to failures in the electric interlocking system are used together. The train delay likelihood found from the matrix is used as the frequency of events to assess operational reliability level risks. The number of trains delayed per calculated period and the average time of train delay at the considered station are used as event aftereffects. The assessment results can be used to make valid managerial decisions.

In this book, modeling and estimation problems of random processes are treated in a unified geometric framework. For this, we need some basic facts about the Hilbert space theory of stochastic vector processes that have finite second order moments and are stationary in the wide sense. Such a process \(\{y(t)\}_{t\in \mathbb{Z}}\) is a collection of real or complex-valued random variables \(y_{k}(t),k = 1,2,\ldots,m,t \in \mathbb{Z}\), which generate a Hilbert space H with inner product $$\displaystyle{\langle \xi,\eta \rangle =\mathop{ \mathrm{E}}\nolimits \{\xi \bar{\eta }\},}$$ where bar denotes conjugation. This Hilbert space is endowed with a shift, i.e., a unitary operator \(\mathcal{U}:\, \mathbf{H} \rightarrow \mathbf{H}\) with the property that $$\displaystyle{y_{k}(t + 1) = \mathcal{U}y_{k}(t),\quad k = 1,2,\ldots,t \in \mathbb{Z}.}$$ In this chapter we introduce some basic geometric facts for such Hilbert spaces. Although we shall assume that the reader has some knowledge of elementary Hilbert space theory, for the benefit of the reader, some relevant facts are collected in Appendix B.2.

The theories of functions and polynomials of several complex and/or real variables along with their numerous applications in several areas of systems theory primarily concerned with the topics of multidimensional digital filter stability, stabilization and design, multivariate network realizability theory, digital array processing in the general framework of multidimensional signal processing techniques needed to process signals carried by propagating wave phenomena, in addition to problems occurring in control theory concerned with 2-D state-space models, notions of controllability, observability, minimality, feedback and pole-placement as well as stabilization via output feedback provided the subject matter of a Special Issue devoted to Multidimensional Systems [1]. Though the vast majority of the papers in [1] were concerned with multidimensional deterministic systems, a paper concerned with methods for inference about models of random processes on multidimensional Euclidean space from observed data was also included especially because the mathematical tools employed partially fitted those widely used in the Issue and also because it was felt at that time that the future scopes for development of multidimensional systems theory should not be restricted to deterministic systems only. In fact, broadly speaking, multidimensional systems theory spans deterministic and statistical approaches to the modeling, analysis and design of spatio-temporal continuous and discrete systems.

This article presents a continuous cascade model of volatility formulated as a stochastic differential equation. Two independent Brownian motions are introduced as random sources triggering the volatility cascade: one multiplicatively combines with volatility; the other does so additively. Assuming that the latter acts perturbatively on the system, the model parameters are estimated by the application to an actual stock price time series. Numerical calculation of the Fokker–Planck equation derived from the stochastic differential equation is conducted using the estimated values of parameters. The results reproduce the probability density function of the empirical volatility, the multifractality of the time series, and other empirical facts.

We compute the variance asymptotics for the number of real zeros of trigonometric polynomials with random dependent Gaussian coefficients and show that under mild conditions, the asymptotic behavior is the same as in the independent framework. In fact our proof goes beyond this framework and makes explicit the variance asymptotics of various models of random Gaussian processes. Our proof relies on intrinsic properties of the Kac–Rice density in order to give a short and concise proof of variance asymptotics.

First of all the paper considers the problem of verification or formalization of the online visualization and parallel computing system from the point of view of dynamic systems as a development of the theory of computational complexity for random processes. Considering problems involving truly big data inevitably leads to the use of a block approach which is also used in both information theory and stochastic differential equations. As a natural metaphor the graph signals were chosen. This is a graph in nodes, of which a spectral function is defined in the examples considered this is a function of color (RGB), height or amount of data. In parallel computing, a block can be associated with a computing unit (processor) and consider the problem of entropy (performance) maximization. In the developed on-line visualization and concurrent computing system for geometric parallelization, it is possible to implement and compare a stationary random process (equiprobable messages implemented using broadcasting and mixins) and a steady-state random process (point-to-point messages), which have different analytical solutions. Together, this allows concluding that the proposed implementation of a stationary process has a certain novelty; in addition, it was intended to be more convenient for automated parallelization. The problems of automatic load balancing (interpolation problem) and optimal scalability of parallel computing (extrapolation problem) are also considered. Not much has been done in the field of visualization verification for example a mesh visualization has been proposed to be considered as a parameterized model of a white-noise random process. Of course, this work cannot be considered complete, but the direction that the authors called stochastic semantics is obviously promising. The authors intend to take a close look at the established perturbed processes in the field of visualizations including those that take into account the human factor (the sketches of the formalization in the form of a discussion are given).

Random processes are basic to the study of environmental data, particularly data in time and space. This work presents three data analyses based on random process models: (a) a trend analysis, based on fitting a monotonic trend to river heights; (b) an analysis of point process data, with ordinal-valued marks, for damage assessment following an earthquake, and (c) an analysis of spatial-temporal meteorological data to estimate the speed of motion of a 500 mbar surface. There is discussion of stochastic processes generally.© 1997 John Wiley & Sons, Ltd.

We investigate, for several models of point processes, the (random) number N of points which have a given point as their nearest neighbor. The largedimensional limit of Poisson processes is treated by considering for n points independently and uniformly distributed in a d-dimensional cube of volume n and showing that Poisson (λ= 1). An asymptotic Poisson (λ= 1) distribution also holds for many of the other models. On the other hand, we find that . Related results concern the (random) volume, , of a Voronoi polytope (or Dirichlet cell) in the cube model; we find that while

We investigate, for several models of point processes, the (random) number N of points which have a given point as their nearest neighbor. The largedimensional limit of Poisson processes is treated by considering for n points independently and uniformly distributed in a d-dimensional cube of volume n and showing that Poisson (λ= 1). An asymptotic Poisson (λ= 1) distribution also holds for many of the other models. On the other hand, we find that . Related results concern the (random) volume, , of a Voronoi polytope (or Dirichlet cell) in the cube model; we find that while

Analysis of the feeding behavior of pigs using different models

A computer model for the simulation of the spatial and temporal distribution of acoustic signals generated by marine mammals is presented. The model incorporates information from existing references on the geographic distribution of marine mammal species. Spatial distributions of individuals of each species that may be present at a site are generated using a random point process model that incorporates a statistical representation of the spatial clustering characteristics of each species. Individual vocalizations for each species are simulated by randomly selecting sampled acoustic source signatures from either the Woods Hole Oceanographic Institution’s SOUND database or from the NRL Dual Use Acoustics Center (DUAC) database. The acoustic source signatures are generated in random time patterns that emulate known vocalization repetition characteristics for each species. Sound propagation effects are incorporated by generating acoustic transfer functions using propagation models appropriate to the frequency...