Arbitrary Probability Density Function Research Articles

Digital simulation of two-dimensional random fields with arbitrary power spectra and non-Gaussian probability distribution functions

Methods for simulation of two-dimensional signals with arbitrary power spectral densities and signal amplitude probability density functions are disclosed. The method relies on initially transforming a white noise sample set of random Gaussian distributed numbers into a corresponding set with the desired spectral distribution, after which this colored Gaussian probability distribution is transformed via an inverse transform into the desired probability distribution. In most cases the method provides satisfactory results and can thus be considered an engineering approach. Several illustrative examples with relevance for optics are given.

Sample Selection Algorithms for Enhanced MIMO Antenna Measurements Using Mode-Stirred Reverberation Chambers

Mode-stirred reverberation chambers (MSRCs) are a useful tool for measuring several wireless-related MIMO antenna parameters. In a conventional single-cavity MSRC, the emulated fading environment is isotropic and the amplitude of the signal is Rayleigh distributed. Previous contributions have enhanced the emulation capabilities of MSRCs so as to include the ability to emulate Rician- and non-isotropic fading environments. In this contribution, arbitrary amplitude probability density functions (PDF) emulation using a MSRC is presented by selecting parts of the sample set that forms different statistical ensembles. Several algorithms are presented and compared in terms of computation time and power accuracy using simulated as well as measured data from different MSRCs to obtain Rician, on-body and amplitude PDFs of standardized models. The technique is patent-protected by EMITE.

Generation of Non-Gaussian Wide-Sense Stationary Random Processes with Desired PSDs and PDFs

This paper describes a new method to generate discrete signals with arbitrary power spectral density (PSD) and first order probability density function (PDF) without any limitation on PDFs and PSDs. The first approximation has been achieved by using a nonlinear transform function. At the second stage the desired PDF was approximated by a number of symmetric PDFs with defined variance. Each one provides a part of energy from total signal with different ratios of remained desired PSD. These symmetric PDFs defined by sinusoidal components with random amplitude, frequency and phase variables. Both analytic results and examples are included. The proposed scheme has been proved to be useful in simulations involving non-Gaussian processes with specific PSDs and PDFs.

Continuous time anomalous diffusion in a composite medium

The one-dimensional continuous time anomalous diffusion in composite media consisting of a finite number of layers in immediate contact is investigated. The diffusion process itself is described with the help of two probability density functions (PDFs), one of which is an arbitrary jump-length PDF, and the other is a long-tailed waiting-time PDF characterized by the waiting-time index β∈(0,1). The former is assumed to be a function of the space coordinate x and the time coordinate t while the latter is a function of x and the time interval. For such an environment a very general form of the diffusion equation is derived which describes the continuous time anomalous diffusion in a composite medium. This result is then specialized to two particular forms of the jump-length PDF, namely the continuous time Lévy flight PDF and the continuous time truncated Lévy flight PDF. In both cases the PDFs are characterized by the Lévy index α∈(0,2) which is regarded to be a function of x and t. It is possible to demonstrate that for particular choices of the indices α and β other equations for anomalous diffusion, well known from the literature, follow immediately. This demonstrates the very general applicability of the derivation and of the resulting fractional differential equation discussed here.

On the Efetov–Wegner terms by diagonalizing a Hermitian supermatrix

The diagonalization of Hermitian supermatrices is studied. Such a change of coordinates is inevitable if one wishes to find certain structures in random matrix theory. However, it still poses serious problems since until now the calculation of all Rothstein contributions, known as Efetov–Wegner terms in physics, was quite cumbersome. We derive the supermatrix Bessel function with all Efetov–Wegner terms for an arbitrary rotation invariant probability density function. As applications we consider representations of generating functions for Hermitian random matrices with and without an external field as integrals over eigenvalues of Hermitian supermatrices. All results are obtained with all Efetov–Wegner terms which were previously unknown in such an explicit and compact representation.

Limiting availability of a one-unit system backed by a spare under repair or preventive maintenance

We consider a one-unit system under continuous monitoring, aided by an identical spare unit and serviced by a facility that performs repair on a failed unit or preventive maintenance on a recalled unit making it as good as new. We assume instantaneous commencement of service and installation to operation. We find the distribution of the system up time and down time when life-, recall- and service-times have arbitrary probability density functions. Hence, we obtain the limiting availability of the system. Also, we compute the servicing cost per unit time to determine whether preventive maintenance is preferable over a repair only model.

Ergodic transitions in continuous-time random walks

We consider continuous-time random walk models described by arbitrary sojourn time probability density functions. We find a general expression for the distribution of time-averaged observables for such systems, generalizing some recent results presented in the literature. For the case where sojourn times are identically distributed independent random variables, our results shed some light on the recently proposed transitions between ergodic and weakly nonergodic regimes. On the other hand, for the case of nonidentical trapping time densities over the lattice points, the distribution of time-averaged observables reveals that such systems are typically nonergodic, in agreement with some recent experimental evidences on the statistics of blinking quantum dots. Some explicit examples are considered in detail. Our results are independent of the lattice topology and dimensionality.

A Statistical Representation of the Matrix–Fracture Transfer Function for Porous Media

In this article, the authors present a matrix–fracture transfer function where the statistical variation in geometric properties of the matrix blocks is considered. Several particular representations with hypothetical probability density functions (PDFs) for matrix block size distributions are presented, including: (a) the single-value distribution (the limiting case); (b) the uniform distribution; (c) the Gamma distribution; and (d) an approximate representation for arbitrary PDFs. An example using experimental data from the literature, along with the single-block based transfer function developed in this study, is presented demonstrating how the statistical procedure proposed in this text can be applied in practice. It is shown with this example that significant relative errors can be introduced when the statistical variance is ignored. Furthermore, two existing dual-porosity models, the Lim and Aziz model and the Zimmerman et al. model, are also considered using the experimental data. It is shown that considerable relative errors can be introduced with these two models when the effect of statistical variance is not taken into account.

Sequential optimal experiment design for neural networks using multiple linearization

Design of an optimal input signal in system identification using a multi-layer perceptron network is treated. Neural networks of the same structure differing only in parameter values are able to approximate various nonlinear mappings. To ensure high quality of network parameter estimates, it is crucial to find a suitable input signal. It is shown that utilizing the conditional probability density function of parameters for design of the input signal provides better results than currently used procedures based on parameter point estimates only. The conditional probability density function of parameters is unknown and hence it is estimated using the Gaussian sum approach approximating arbitrary probability density function by a sum of normal distributions. This approach is less computationally demanding than the Markov Chain Monte Carlo method and achieves better results in comparison with the commonly used local prediction error methods. The properties of the proposed input signal designs are illustrated in numerical examples.

Statistics of Echoes From a Directional Sonar Beam Insonifying Finite Numbers of Single Scatterers and Patches of Scatterers

When a sonar beam sweeps across a field of scatterers and insonifies a different set of scatterers in each ping, the echoes can fluctuate significantly from ping to ping. For a homogeneous spatial distribution of scatterers in which there are a large number of scatterers in each sonar resolution cell, the echoes tend to be Rayleigh distributed. However, the echoes can be strongly nonRayleigh when there is a small number of scatterers in the sonar resolution cell-a condition that can occur with a narrow beam sonar. For scatterers randomly distributed in the sonar beam, the corresponding random weighting associated with the beampattern also significantly contributes toward a non-Rayleigh distribution when the number of scatterers is small. In this paper, a general formulation for echo statistics is developed by combining equations derived by Ehrenberg etal. [Proc. Conf. Eng. Ocean Environ., vol. 1, pp. 61-64,1972 and J. Acoust. Soc. Amer., vol. 69, pp. 955-962, 1981] and Barakat [Optica Acta, vol. 21, pp. 903-921,1974] to account for a directional sonar beam involving an arbitrary finite number of scatterers, each with an arbitrary echo probability density function (pdf), each randomly located in the beam, and each correspondingly randomly weighted by the beampattern. Theoretical predictions are made, along with numerical simulations for validation, for a range of cases including: 1) a range of number of scatterers each randomly located in the beam and 2) several different echo pdfs of the scatterers. Here, a single scatterer could be a patch of scatterers whose overall dimensions are much smaller than the resolution cell of the sonar beam and multiple scatterers could be multiple small patches. Although the application is intended for volumetric patches, the formulation could be applied to areal patches under appropriate conditions. The formulation applies directly to the geometry of the short-range direct-path sonar in which there are no reflections due to boundaries such as the seafloor and sea surface.

A Model Predictive Control Approach for Stochastic Networked Control Systems

In this paper we present a stochastic model predictive control (SMPC) approach for networked control systems (NCSs) that are subject to time-varying sampling intervals and time-varying transmission delays. These network-induced uncertain parameters are assumed to be described by random processes, having a bounded support and an arbitrary continuous probability density function. Assuming that the controlled plant can be modeled as a linear system, we present a SMPC formulation based on scenario enumeration and quadratic programming that optimizes a stochastic performance index and provides closed-loop stability in the mean-square sense. Simulation results are shown to demonstrate the performance of the proposed approach.

Algorithm and software for defining the distribution of eigenvalues of random symmetric matrices via simulation

A simulation algorithm for defining the distribution of eigenvalues of a random symmetric matrix with arbitrary continuous joint probability density function of its entries is presented. The algorithm requires only a uniform random number generator. As a numerical example, the probability that eigenvalues of a certain random symmetric matrix satisfy a given condition is calculated using software implementing the algorithm.

Scalable processing of snapshot and continuous nearest-neighbor queries over one-dimensional uncertain data

In several emerging and important applications, such as location-based services, sensor monitoring and biological databases, the values of the data items are inherently imprecise. A useful query class for these data is the Probabilistic Nearest-Neighbor Query (PNN), which yields the IDs of objects for being the closest neighbor of a query point, together with the objects’ probability values. Previous studies showed that this query takes a long time to evaluate. To address this problem, we propose the Constrained Nearest-Neighbor Query (C-PNN), which returns the IDs of objects whose probabilities are higher than some threshold, with a given error bound in the answers. We show that the C-PNN can be answered efficiently with verifiers. These are methods that derive the lower and upper bounds of answer probabilities, so that an object can be quickly decided on whether it should be included in the answer. We design five verifiers, which can be used on uncertain data with arbitrary probability density functions. We further develop a partial evaluation technique, so that a user can obtain some answers quickly, without waiting for the whole query evaluation process to be completed (which may incur a high response time). In addition, we examine the maintenance of a long-standing, or continuous C-PNN query. This query requires any update to be applied to the result immediately, in order to reflect the changes to the database values (e.g., due to the change of the location of a moving object). We design an incremental update method based on previous query answers, in order to reduce the amount of I/O and CPU cost in maintaining the correctness of the answers to such a query. Performance evaluation on realistic datasets show that our methods are capable of yielding timely and accurate results.

A Growing and Pruning Method for Radial Basis Function Networks

A recently published generalized growing and pruning (GGAP) training algorithm for radial basis function (RBF) neural networks is studied and modified. GGAP is a resource-allocating network (RAN) algorithm, which means that a created network unit that consistently makes little contribution to the network's performance can be removed during the training. GGAP states a formula for computing the significance of the network units, which requires a d-fold numerical integration for arbitrary probability density function p(x) of the input data x (x in R(d)) . In this work, the GGAP formula is approximated using a Gaussian mixture model (GMM) for p(x) and an analytical solution of the approximated unit significance is derived. This makes it possible to employ the modified GGAP for input data having complex and high-dimensional p(x), which was not possible in the original GGAP. The results of an extensive experimental study show that the modified algorithm outperforms the original GGAP achieving both a lower prediction error and reduced complexity of the trained network.

Reinforcement Learning for Continuous Stochastic Actions--An Approximation of Probability Density Function by Orthogonal Wave Function Expansion--

A function approximation based on an orthonormal wave function expansion in a complex space is derived. Although a probability density function (PDF) cannot always be expanded in an orthogonal series in a real space because a PDF is a positive real function, the function approximation can approximate an arbitrary PDF with high accuracy. It is applied to an actor-critic method of reinforcement learning to derive an optimal policy expressed by an arbitrary PDF in a continuous-action continuous-state environment. A chaos control problem and a PDF approximation problem are solved using the actor-critic method with the function approximation, and it is shown that the function approximation can approximate a PDF well and that the actor-critic method with the function approximation exhibits high performance.

Blind identification of Volterra-Hammerstein systems

This paper is concerned with the blind identification of Volterra-Hammerstein systems. Two identification scenarios are covered. The first scenario assumes that, although the input is not available, the statistics of the input are a priori known. This case appears in communication applications where the input statistics of the transmitter are known to the receiver. The second scenario assumes that the input statistics are unknown. In the case of known input statistics, the input is stationary higher order white noise with arbitrary probability density function. Under the scenario of unknown input statistics, the input is restricted to Gaussian white process. New cumulant-based identification methods are described for the above scenarios. The problem is converted into a linear multivariable form and the output cumulants are calculated using Kronecker products. First, initial conditions are determined by a linear system of equations. These correspond to the boundary values of the Volterra kernels. The remaining kernel coefficients can be determined under both identification schemes from a possibly overdetermined system of linear equations.

Inference in hybrid Bayesian networks with mixtures of truncated exponentials

Mixtures of truncated exponentials (MTE) potentials are an alternative to discretization for solving hybrid Bayesian networks. Any probability density function (PDF) can be approximated with an MTE potential, which can always be marginalized in closed form. This allows propagation to be done exactly using the Shenoy–Shafer architecture for computing marginals, with no restrictions on the construction of a join tree. This paper presents MTE potentials that approximate an arbitrary normal PDF with any mean and a positive variance. The properties of these MTE potentials are presented, along with examples that demonstrate their use in solving hybrid Bayesian networks. Assuming that the joint density exists, MTE potentials can be used for inference in hybrid Bayesian networks that do not fit the restrictive assumptions of the conditional linear Gaussian (CLG) model, such as networks containing discrete nodes with continuous parents.

Symmetric PH and EP distributions and their applications to the probability Hough transform

In this paper, we first introduce a symmetric phase type (PH) distribution, and then propose a symmetric exponential-polynomial type (EP) distribution based on the symmetric PH distribution. The class of symmetric EP distributions is illustrated to be so large that an arbitrary symmetric probability density function in L 2(−∞, +∞), which is the space of square integrable functions on the real line, can be approximated in (−∞, +∞) by a sequence of symmetric EP probability density functions. We prove this result by means of a constructive approach based on two orthogonal decompositions. Laguerre spectrum decomposition and Hermite spectrum decomposition. We also provide a moment-based approach for simply determining a symmetric EP probability density function to approximatively express a symmetric random variable under which some moments of the symmetric random variable are given. We further propose multidimensional symmetric PH and EP distributions and provide their useful structures and properties. Finally, we apply the symmetric PH and EP distributions to study the probability Hough transform.

One- and two-dimensional shadowing functions for any height and slope stationary uncorrelated surface in the monostatic and bistatic configurations

The approaches developed by Wagner (1966) and Smith (1967) for computing the shadowing properties from a one-dimensional randomly stationary surface are investigated for an arbitrary surface uncorrelated height and slope probability density function (PDF) and extended to a two-dimensional surface in the monostatic and bistatic configurations. Bourlier et al. (see Progress in Electromagnetic Research, J. A. Kong, Ed., vol.27, p.226-87, 2000) have expressed, from Brown's (1980) work , the Smith and Wagner average shadowing functions, for a one-dimensional surface, whatever the assumed uncorrelated slope and height PDF. They are then completely defined from both integrations over the surface slope PDF. The shadowing function is performed for Gaussian, Laplacian, and exponential slope probability density functions. With the method presented by Bourlier, Saillard, and Berginc, vol.48, p.437-46, Mar. 2000, the one-dimensional monostatic shadowing function is also compared with the exact solution. It is obtained by generating the slope-height surfaces. The Gaussian and Laplacian slope PDFs are treated with a Gaussian surface height. The analytical results are extended to a one-dimensional bistatic configuration, and the case of a two-dimensional surface is investigated with a Gaussian and Laplacian surface slope PDFs. The last point is very relevant, because the classical shadowing functions of Smith and Wagner are assumed to be one-dimensional or isotropic.

Comparative Performance of Cellular Communication Systems with Adaptive Modulation and Coding and Systems with Power Control

We carry out a comparative analysis of performances of a cellular communication system with adaptive modulation and coding (AMC) and a system with power control (PC). It is shown that the performance of an AMC system is higher than the performance of a PC system for an arbitrary probability density function of the signal fading. To achieve identical throughputs of the two systems, the mean transmitted power for PC must be greater than that for AMC. For identical mean transmitted powers, the throughput of an AMC system is always greater than that of a PC system. We consider code division multiple-access (CDMA) systems with arbitrary numbers of fading time-taps (paths), having a statistically independent Rayleigh fading of signals from path to path. The influence of self-interference on AMC and PC performance is studied. This self-interference results from the nonideal path separation in the receiver.