Vector Random Processes Research Articles

Dynamics of an entangled pair of spin-1/2 particles under correlated random magnetic fields.

The dynamics of a maximally entangled pair of spin-1/2 particles is obtained in the presence of random magnetic fields which are correlated. The two spin-1/2 particles are assumed to be maximally entangled initially and are then disturbed by the magnetic fields modeled as Gaussian vector random processes whose corresponding spatial components are correlated. The dynamics is derived in terms of the joint density matrix of the entangled pair using the ideas of stochastic calculus, from which the steady-state density matrix and the associated timescale for it to be reached are obtained. The asymptotic density matrix represents a state of (partial) disentanglement.

A method to characterize climate, Earth or environmental vector random processes

We propose a general methodology to characterize a non-stationary random process that can be used for simulating random realizations that keep the probabilistic behavior of the original time series. The probability distribution of the process is assumed to be a piecewise function defined by several weighted parametric probability models. The weights are obtained analytically by ensuring that the probability density function is well defined and that it is continuous at the common endpoints. Any number of subintervals and continuous probability models can be chosen. The distribution is assumed to vary periodically in time over a predefined time interval by defining the model parameters and the common endpoints as truncated generalized Fourier series. The coefficients of the expansions are obtained with the maximum likelihood method. Different sets of orthogonal basis functions are tested. The method is applied to three time series with different particularities. Firstly, it is shown its good behavior to capture the high variability of the precipitation projected at a semiarid location of Spain for the present century. Secondly, for the Wolf sunspot number time series, the Schwabe cycle and time variations close to the 7.5 and 17 years are analyzed along a 22-year cycle. Finally, the method is applied to a bivariate time series that contains (1) freshwater discharges at the last regulation point of a dam located in a semiarid zone in Andalucía (Spain) which is influenced not only by the climate variability but also by management decisions and (2) the salinity at the mouth of the river. For this case, the analysis, that was combined with a vectorial autoregressive model, focus on the assessment of the goodness of the methodology to replicate the statistical features of the original series. In particular, it is found that it reproduces the marginal and joint distributions and the duration of sojourns above/below given thresholds.

MarineTools.temporal: A Python package to simulate Earth and environmental time series

The assessment of the uncertainty about the evolution of complex processes usually requires different realizations consisting of multivariate temporal signals of environmental data. However, it is common to have only one observational set. MarineTools.temporal is an open-source Python package for the non-stationary parametric statistical analysis of vector random processes suitable for environmental and Earth modelling. It takes a single timeseries of observations and allows the simulation of many time series with the same probabilistic behavior. The software generalizes the use of piecewise and compound distributions with any number of arbitrary continuous distributions. The code contains, among others, multi-model negative log-likely functions, wrapped-normal distributions, and generalized Fourier timeseries expansion. Its programming philosophy significantly improves the computing time and makes it compatible with future extensions of scipy.stats. We apply it to the analysis of freshwater river discharge, water currents, and the simulation of ensemble projections of sea waves, to show its capabilities.

Features of construction of evaluation algorithms multidimensional random processes

The problem of evaluating the information which is present in random signals from various sources-meters is considered. It is assumed that the random process which is evaluated and the meter output according mathematical description of the problem are vector random processes. In this case, the dimension of the vector of the measurement can be larger than the dimension of the vector of the process being evaluated. The analysis of analytical methods and algorithms of the estimation that based on the determination of the main probabilistic characteristics of a random process by both the a priori and the a posteriori methods with various optimality criteria is carried out. Based on the analysis, the problem of complexing of the meters of the random process is considered according the proposed criterion for the maximum of posterior verisimilitude, combining the criterion of maximum verisimilitude and the criterion of maximum the a posteriori probability, general methodology complexing is developed. Proposed example of the complexing shows the efficiency of the proposed method. This approach to the construction of the algorithms of the evaluation for multidimensional random processes allows to increase the accuracy of estimation, since it takes into account additional information and its complex processing.

Probabilistic analysis of two-dimensional vector random processes on the phase plane

Introduction: The models of vector random processes are characterized by a variety of practical applications and, at the same time, by the complexity of studying the detailed probabilistic structure. Such features often lead to considering only separate components of vector functions, forcing the shift to scalar process analysis. All this significantly reduces the overall information content of the research, increasing the relevance of the search for promising approaches to probabilistic analysis of vector processes. Purpose: To present vector random processes in a phase space of states, and to apply the general theory of level crossings of random functions to the study of probabilistic structure of phase trajectories. Results: We have identified the characteristic features of displaying random processes on a phase plane, and introduced numerical characteristics for the description and analysis of probabilistic behavior of sample functions. A probabilistic analysis has been performed for the characteristics of the «level crossing» type for vector processes with various definitions of the areas of acceptable values. We used the typical models of two-dimensional Gaussian processes and «signal plus noise» models to demonstrate how the probabilistic structure of the phase trajectories depends on the given threshold levels, major distribution parameters and spectral-correlation properties of the studied processes. Practical relevance : The article contributes to the research information content and its visualization in the analysis of probabilistic behavior of vector random processes. It combines general methods of phase space of vector processes, methods of phase plane in the analysis of scalar processes and the theory of level crossings of random functions. The classical approaches to the visual description of phase trajectories are supplemented by the methods of quantitative analysis of detailed probabilistic structure of random functions.

FLEXURAL VIBRATIONS OF THE VERTICAL ROD VARIABLE SECTION WITH A CONCENTRATED MASS

Consider bending vibrations of the vertical rod of variable section with a concentrated mass. We study the oscillations excited vector harmonic and random processes with the components of a kinematic and dynamic disturbances. Determine the amplitudes and standard deviations.

Asymptotically Equivalent Sequences of Matrices and Multivariate ARMA Processes

The present paper considers a special class of vector random processes that we call multivariate asymptotically wide sense stationary (WSS) processes. A multivariate random process is said to be asymptotically WSS if it has constant mean and the sequence of its autocorrelation matrices is asymptotically equivalent (a.e.) to the sequence of autocorrelation matrices of some multivariate WSS process. It is shown that this class of processes contains meaningful processes other than multivariate WSS processes. In particular, we give sufficient conditions for multivariate moving average (MA) processes, multivariate autoregressive (AR) processes and multivariate autoregressive moving average (ARMA) processes to be asymptotically WSS. Furthermore, in order to solve multiple-input-multiple-output (MIMO) problems in communications and signal processing involving this kind of processes, we extend the Gray definition of a.e. sequences of matrices and his main results on these sequences to non-square matrices. As an example, the derived results on a.e. sequences of non-square matrices are applied to compute the differential entropy rate and the minimum mean square error (MMSE) for a linear predictor of a multivariate asymptotically WSS process.

Vector Time-Frequency AR Models for Nonstationary Multivariate Random Processes

We introduce the vector time-frequency autoregressive (VTFAR) model for a parsimonious parametric description of nonstationary vector random processes. The VTFAR model generalizes the recently proposed scalar TFAR model to the multivariate case. It is physically meaningful because nonstationarity and spectral correlation are represented in terms of frequency shifts, and it is parsimonious for the practically relevant class of underspread vector processes (i.e., nonstationary vector processes with rapidly decaying correlation in time and frequency). For vector processes with decaying correlation across the signals, we introduce a variant of the VTFAR model with banded parameter matrices. Furthermore, we present a VTFAR parameter estimator that is based on a system of linear equations with two-level block-Toeplitz structure, and we develop an efficient order-recursive algorithm for solving these equations. We also present information criteria for estimating the VTFAR model order and the matrix bandwidth of the banded VTFAR model. The performance of the proposed VTFAR parameter and order estimators is assessed through numerical simulations. Finally, an application to nonstationary multivariate spectral analysis is presented.

Stochastic Spectral and Fourier-Wavelet Methods for Vector Gaussian Random Fields

Randomized Spectral Models (RSM) and Randomized Fourier-Wavelet Models (FWM) for simulation of homogeneous Gaussian random fields based on spectral representations and plane wave decomposition of random fields are developed. Extensions of FWM to vector random processes are constructed. Convergence of the constructed Fourier-Wavelet models (in the sense of finite-dimensional distributions) under some general conditions on the spectral tensor is given. A comparative analysis of RSM and FWM is made by calculating Eulerian statistical characteristics of a 3D isotropic incompressible random field through an ensemble and space averaging.

Stochastic Spectral and Fourier-Wavelet Methods for Vector Gaussian Random Fields

Randomized Spectral Models (RSM) and Randomized Fourier-Wavelet Models (FWM) for simulation of homogeneous Gaussian random fields based on spectral representations and plane wave decomposition of random fields are developed. Extensions of FWM to vector random processes are constructed. Convergence of the constructed Fourier-Wavelet models (in the sense of finite-dimensional distributions) under some general conditions on the spectral tensor is given. A comparative analysis of RSM and FWM is made by calculating Eulerian statistical characteristics of a 3D isotropic incompressible random field through an ensemble and space averaging.

Description and Simulation of Nonstationary Processes Based on Hilbert Spectra

A new method is proposed for the description and simulation of nonstationary random processes based on Hilbert spectra of their sample observations. A sample of a random process is first decomposed into intrinsic mode functions (IMFs) by the method of empirical mode decomposition. The Hilbert transforms of the IMFs yield the instantaneous amplitude and frequency, from which the Hilbert spectrum can be obtained as a function of time and frequency. The average of the Hilbert spectra over the samples is then defined as the Hilbert spectrum of the process and used as the target in the simulation of the process. The method is also extended to vector random processes. Unlike current procedures such as those based on the evolutionary process, no assumptions of functional forms for the spectra are necessary which are unknown a priori; and no assumptions of piecewise stationarity and egodicity of the process are required in parameter estimation. Applications to spectral characterization and simulation of multivariate earthquake ground motions show that the Hilbert spectra give a clear description of the time-varying spectral content of the motions and the simulated samples represent an accurate statistical image of the records. The response spectra compare well with those of the records and retain the jagged look. The method has great potential for engineering applications when dealing with nonstationary, nonlinear random processes.

The strong law of large numbers for dependent vector processeswith decreasing correlation: “Double averaging concept”

A new form of the strong law of large numbers for dependent vector sequences using the “double averaged” correlation function is presented. The suggested theorem generalizes the well‐known Cramer–Lidbetter′s theorem and states more general conditions for fulfilling the strong law of large numbers within the class of vector random processes generated by a non stationary stable forming filters with an absolutely integrable impulse function.

Coefficient of coherence in the case of two vector random processes

A general method for estimating coherence and phase cross-spectra of vector random processes is described, and illustrated by examples. It is shown that the coherence matrix for two vector random processes with arbitrary dimension is the Hermitian matrix, which can be transformed to a diagonal form with real eigenvalues. For two-dimensional processes, the greater eigenvalue characterises the coherence between components of analysed vectors with respect to the direction of minimum noise variance, and the lesser value corresponds to the direction of maximum noise variance.

Results on lattice vector quantization with dithering

The statistical properties of the error in uniform scalar quantization have been analyzed by a number of authors in the past, and is a well-understood topic today. The analysis has also been extended to the case of dithered quantizers, and the advantages and limitations of dithering have been studied and well documented in the literature. Lattice vector quantization is a natural extension into multiple dimensions of the uniform scalar quantization. Accordingly, there is a natural extension of the analysis of the quantization error. It is the purpose of this paper to present this extension and to elaborate on some of the new aspects that come with multiple dimensions. We show that, analogous to the one-dimensional case, the quantization error vector can be rendered independent of the input in subtractive vector-dithering. In this case, the total mean square error is a function of only the underlying lattice and there are lattices that minimize this error. We give a necessary condition on such lattices. In nonsubtractive vector dithering, we show how to render moments of the error vector independent of the input by using appropriate dither random vectors. These results can readily be applied for the case of wide sense stationary (WSS) vector random processes, by use of iid dither sequences. We consider the problem of pre- and post-filtering around a dithered lattice quantifier, and show how these filters should be designed in order to minimize the overall quantization error in the mean square sense. For the special case where the WSS vector process is obtained by blocking a WSS scalar process, the optimum prefilter matrix reduces to the blocked version of the well-known scalar half-whitening filter.

Maximum likelihood estimation for multivariate observations of Markov sources

Parameter estimation for multivariate functions of Markov chains, a class of versatile statistical models for vector random processes, is discussed. The model regards an ordered sequence of vectors as noisy multivariate observations of a Markov chain. Mixture distributions are a special case. The foundations of the theory presented here were established by Baum, Petrie, Soules, and Weiss. A powerful representation theorem by Fan is employed to generalize the analysis of Baum, {\em et al.} to a larger class of distributions.

Envelopes of Vector Random Processes and Their Crossing Rates

Vector-valued random processes, $\mathbf{X}(t)$, can be "enveloped" by set-valued random processes, $\mathscr{S}(t)$, to which they belong with probability 1 during any finite length of time. When applied to scalar processes, the set-definition of envelope includes and is richer than the familiar point-definitions. Several random set-envelope processes in $n$-dimensional space, $R_n$, are defined and the mean rates at which they "cross" given regions of $R_n$ are calculated. Comparison is made with the mean crossing rates of associated enveloped Gaussian processes, $\mathbf{X}(t)$.

On the spectral factorization of nonstationary vector random processes

Conditions which depend on the covariance of a vector random process, sufficient to ensure the process can be generated by a linear, invertible system of finite order driven by white noise are derived, and equations which determine the parameters of the system are found. Some structural properties of lumped covariances are given; these stress the close relation between the structure of linear systems and that of lumped covariances and provide a means of establishing the minimal order of generating systems.

Simulation of vector random processes in problems of reliability prediction

Simulation of vector random processes in problems of reliability prediction

Generalizations and extensions of the Fokker- Planck-Kolmogorov equations

In this paper, the classical Fokker-Planck-Kolmogorov equations are generalized to hold for conditional probability density functions of arbitrary random processes. Conditions are derived under which the generalized equations are of finite order both for one-dimensional and for vector random processes. An extension of the generalized equations which overcomes degeneracy occurring in the steady-state case is also presented.

Symbolic Calculus of the Wiener Process and Wiener-Hermite Functionals

A new definition is given for the ``ideal random function'' (derivative of the Wiener function), which separates out infinite factors by fullest exploitation of the possibilities of the Dirac delta function. By allowing all integrals to be written formally as sums, this facilitates the definition and manipulation of the Wiener-Hermite functionals, especially for vector random processes of multiple argument. Expansion of a random function in Wiener-Hermite functionals is discussed. An expression is derived for the expectation value of the product of any number of Wiener-Hermite functionals; this is all that is needed in principle to obtain full statistical information from the Wiener-Hermite functional expansion of a random function. The method is illustrated by the calculation of the first correction to the flatness factor (measure of Gaussianity) of a nearly-Gaussian random function.