Binary Random Process Research Articles

Optimizing vaccine distribution in developing countries under natural disaster risk

AbstractFor many developing countries, COVID‐19 vaccination roll‐out programs are not only slow but vaccination centers are also exposed to the risk of natural disaster, like flooding, which may slow down vaccination progress even further. Policy‐makers in developing countries therefore seek to implement strategies that hedge against distribution risk in order for vaccination campaigns to run smoothly and without delays. We propose a stochastic‐dynamic facility location model that allows policy‐makers to choose vaccination facilities while accounting for possible facility failure. The model is a multi‐stage stochastic variant of the classic facility location problem where disruption risk is modelled as a binary multivariate random process–a problem class that has not yet been studied in the extant literature. To solve the problem, we propose a novel approximate dynamic programming algorithm which trains the shadow price of opening a flood‐prone facility on historical data, thereby alleviating the need to fit a stochastic model. We trained the model using rainfall data provided by the local government of several major cities in the Philippines which are exposed to multiple flooding events per year. Numerical results demonstrate that the solution approach yields approximately 30%–40% lower cost than a baseline approach that does not consider the risk of flooding. Recommendations based on this model were implemented following a collaboration with two large cities in the Philippines which are exposed to multiple flooding events per year.

Optimal codesign of industrial networked control systems with state‐dependent correlated fading channels

SummaryThis paper examines a codesign problem in industrial networked control systems (NCS) whereby physical systems are controlled over wireless fading channels. The considered wireless channels are assumed to be stochastically dependent on the physical states of moving machineries in the industrial working space. In this paper, the moving machineries are modeled as Markov decision processes whereas the characteristics of the correlated fading channels are modeled as a binary random process whose probability measure depends on both the physical states of moving machineries and the transmission power of communication channels. Under such a state‐dependent fading channel model, sufficient conditions to ensure the stochastic safety of the NCS are first derived. Using the derived safety conditions, a codesign problem is then formulated as a constrained joint optimization problem that seeks for optimal control and transmission power policies which simultaneously minimize an infinite time cost on both communication resource and control effort. This paper shows that such optimal policies can be obtained in a computationally efficient manner using convex programming methods. Simulation results of an autonomous forklift truck and a networked DC motor system are presented to illustrate the advantage and efficacy of the proposed codesign framework for industrial NCS.

МЕТОД ГЕНЕРИРОВАНИЯ ДВУМЕРНОГО БИНАРНОГО СЛУЧАЙНОГО ПРОЦЕССА

The PURPOSE of the study is search for new and modernization of known generation methods of random processes with specified static (probability distribution law) and dynamic (correlation function) probabilistic properties. METHODS. The main research methods used in the study are the probability theory, mathematical statistics and numerical methods. RESULTS. The object of the study is a permutational technology of random process generation with a simultaneously specified probability distribution law and an autocorrelation function. This technology is simple in software implementation, has high response speed and allows to introduce the required probability properties with the accuracy of their reflection, sufficient for engineering applications (e.g. in simulation modeling). The Bernoulli (binary) law is taken as a law of probability distribution of the generated process. The conducted analysis of the probabilistic properties of the behavior of the permutation procedure components has resulted in the identification of new, previously unknown functional capabilities of this method of random process generation. Probabilistic properties (probability distribution laws and autocorrelation functions of the candidate vector components) are systematized depending on a mathematically justified permutation variant. The study is given to two variants of the permutational method that allows to generate a periodically correlated binary process. It is proved that these variants simultaneously provide the possibility to generate a two-dimensional binary random process. CONCLUSION. Conducted theoretical studies allowed to expand the functionalities of the permutational procedure of binary random process generation due to the possibility to generate a two-dimensional random binary process.

On Gated Gaussian Impulsive Noise in M-QAM with Optimum Receivers

This paper presents an approach for the analysis of gated Gaussian impulsive noise and its effects in M-ary Quadrature Amplitude Modulation (M-QAM) schemes. In the approach, both amplitude variation and noisy pulse duration can be characterized as an amplitude modulation of the impulsive component by a discrete (binary or m-ary) random process. New exact expressions are obtained for the probability density function, autocorrelation function and power spectral density of the noise as well as for the bit error probability of M-QAM considering the maximum {\em a posteriori} probability optimum receiver. An important aspect of the proposed approach is the fact that the discrete random process incorporates the main parameters of the impulsive noise, such as amplitude, duration, instants in which the noise is added and time intervals between instants in which the noise is added.

On Gated Gaussian Impulsive Noise in M-QAM with Optimum Receivers

This paper presents an approach for the analysis of gated Gaussian impulsive noise and its effects in M-ary Quadrature Amplitude Modulation (M-QAM) schemes. In the approach, both amplitude variation and noisy pulse duration can be characterized as an amplitude modulation of the impulsive component by a discrete (binary or m-ary) random process. New exact expressions are obtained for the probability density function, autocorrelation function and power spectral density of the noise as well as for the bit error probability of M-QAM considering the maximum {\em a posteriori} probability optimum receiver. An important aspect of the proposed approach is the fact that the discrete random process incorporates the main parameters of the impulsive noise, such as amplitude, duration, instants in which the noise is added and time intervals between instants in which the noise is added.

Tracking Multiple Visual Targets via Particle-Based Belief Propagation

Multiple-target tracking in video (MTTV) presents a technical challenge in video surveillance applications. In this paper, we formulate the MTTV problem using dynamic Markov network (DMN) techniques. Our model consists of three coupled Markov random fields: 1) a field for the joint state of the multitarget; 2) a binary random process for the existence of each individual target; and 3) a binary random process for the occlusion of each dual adjacent target. To make the inference tractable, we introduce two robust functions that eliminate the two binary processes. We then propose a novel belief propagation (BP) algorithm called particle-based BP and embed it into a Markov chain Monte Carlo approach to obtain the maximum a posteriori estimation in the DMN. With a stratified sampler, we incorporate the information obtained from a learned bottom-up detector (e.g., support-vector-machine-based classifier) and the motion model of the target into the message propagation. Other low-level visual cues such as motion and shape can be easily incorporated into our framework to obtain better tracking results. We have performed extensive experimental verification, and the results suggest that our method is comparable to the state-of-art multitarget tracking methods in all the cases we tested.

Bayesian Estimation of Stochastic Parameterizations in a Numerical Weather Forecasting Model

Abstract Parameterizations in numerical models account for unresolved processes. These parameterizations are inherently difficult to construct and as such typically have notable imperfections. One approach to account for this uncertainty is through stochastic parameterizations. This paper describes a methodological approach whereby existing parameterizations provide the basis for a simple stochastic approach. More importantly, this paper describes systematically how one can “train” such parameterizations with observations. In particular, a stochastic trigger function has been implemented for convective initiation in the Kain–Fritsch (KF) convective parameterization scheme within the fifth-generation Pennsylvania State University–National Center for Atmospheric Research (Penn State–NCAR) Mesoscale Model (MM5). In this approach, convective initiation within MM5 is modeled by a binary random process. The probability of initiation is then modeled through a transformation in terms of the standard KF trigger variables, but with random parameters. The distribution of these random parameters is obtained through a Bayesian Monte Carlo procedure informed by radar reflectivities. Estimates of these distributions are then incorporated into the KF trigger function, giving a meaningful stochastic (distributional) parameterization. The approach is applied to cases from the International H2O project (IHOP). The results suggest the stochastic parameterization/Bayesian learning approach has potential to improve forecasts of convective precipitation in mesoscale models.

Diffraction efficiency of random binary-amplitude diffracting screens

The diffraction characteristics of random binary-amplitude diffracting screens are investigated. It is shown that, for such a random diffracting screen with complex amplitude transmittance t equal to either 0 or 1, the diffraction efficiency of the screen is maximized when 50% of the screen has transmittance t = 1, independent of other characteristics of the screen. The maximum diffraction efficiency equals ¼. For this result to hold the screen must be adequately modeled by a stationary binary random process.

Second order effect of binary sources on characteristics of queue and loss rate

A wideband source in high speed networks is typically represented by a binary random process. In this paper we characterize the second-order properties of each binary source by a multi-state Markov modulated Poisson process (MMPP). A comprehensive numerical study is carried out to identify the individual effect of the source second-order dynamics on the queue length and loss rate. The results can be used to verify the validity of the two-state Markov chain binary source assumption which is commonly made within the framework of input rate control and bandwidth allocation in high speed networks. The concept of input power spectrum is then developed as a unified source characterization for multimedia traffic queueing analyses.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On filtered binary processes

The problem of calculating the probability density function of the output of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</tex> filter driven by a binary random process with intervals generated by an equilibrium renewal process is studied. New integral equations, closely related to McFadden's original integral equations, are derived and solved by a matrix approximation method and by iteration. Transformations of the integral equations into differential equations are investigated and a new closed-form solution is obtained in one special case. Some numerical results that compare the matrix and iteration solutions with both exact solutions and approximate solutions based upon the Fokker-Planck equation are presented.

On Optimum Receivers for Analog Communication in Impulsive Noise

In this paper the problem of impulsive noise suppression in analog communication systems is considered. The problem of demodulation of amplitude and phase modulated signals in the presence of impulsive noise is formulated by introducing a binary random process to indicate the presence of impulsive noise. It is shown that sub-optimal receiver structures can be derived using this model and these are similar to some experimental receivers reported earlier in the literature. The receivers derived are non-linear in nature and their working is demonstrated through digital computer simulations.

Statistical Properties of Ramplike Random Processes

The work of Lampard and Redman [5] on the statistical properties of the integral of a binary random process is generalized and extended to include more general ramplike random processes z(t) . These ramplike processes are defined to be ramps of random slope that switch slope at random times. Both the intervals between switching times and the slopes of the ramp segments are assumed to be independent random variables. The general properties of the one-dimensional and multidimensional probability density functions of z(t) are discussed. A Laplace transform technique is employed to obtain the Laplace transform of the characteristic function of these probability density functions in terms of the Laplace transform of the characteristic function of the ramp slopes multipled by the probability functions of the switching times. For Poisson switching times, these expressions are relatively compact. However, the inversion problem is formidable, and closed form results are presently available only for a binary slope probability density function (a previously known result). Due to the properties of characteristic functions, the moments of z(t) are obtained by a single Laplace inversion of the derivative of the Laplace transform expressions. These moments may be used to characterize the probability density functions of z(t) with a series expansion involving the Gaussian density function and its derivatives. Examples are given for the cases of Gaussian and rectangular slope probability density functions with Poisson switching times.

A Note on the Integral of a Binary Random Process

A Note on the Integral of a Binary Random Process

Stochastic Sampling of a Binary Random Process--II

A sampling scheme is considered in which a binary process with statistically independent axis-crossing intervals is sampled according to a given logic by a Poisson pulse process, thereby producing a new binary process. The results derived supplement the results of a previous paper on the sampling of a binary process by a random pulse process. The probability density of the time interval between successive zeros of the resulting binary process is derived, and it is shown how higher-order time interval statistics may be obtained. As an example, a periodic binary process is sampled by a Poisson pulse process, and it is shown that the first-order time interval density function of the resultant process is multimodal, which under certain limiting conditions becomes a symmetric density function.

Spectral Properties of a Binary Random Process

Spectral Properties of a Binary Random Process

Stochastic Sampling of a Binary Random Process

When a binary random process is sampled by an independent random pulse process a new binary process is generated. Statistical properties of this new binary process are investigated in this paper. Two methods of sampling are considered, according as an even or odd number of zeros of the binary process being sampled occur between successive sampling pulses. When the binary process is one in which the number of zeros in a given time interval obeys the Poisson distribution, and the time intervals between successive sampling pulses are independent, then the time intervals between successive zeros of the output binary process are also statistically independent, in which case all statistical properties of the output process are obtained. By iteration of this sampling procedure it is shown that a whole class of binary random processes, all having statistically independent intervals, is made accessible. The autocorrelation function of the output binary process is obtained in the case where an arbitrary binary process is sampled by a pulse process having independent intervals. The mean rate of the output process is discussed when the sampling intervals are not independent. Four special cases are considered as examples. Finally a brief description of an experimental sampling device and some results obtained with it, is given. This paper is envisaged as the first of a series of three papers dealing with binary random processes and their application in the analysis of simple filters containing a randomly switched parameter.

Statistical Properties of the Integral of a Binary Random Process

Statistical properties of the output z(t) of a finite time integrator are discussed. The input considered is a binary random y(t) having successive axis-crossing intervals which are statistically independent. Transform expressions are derived for the first- and second-order transition probability densities of the integrated process, and it is shown how these results may be extended to three or more dimensions. Four processes are considered as examples. The integrated z(t) is shown to be a projection of a Markov in three dimensions. The other two components are the original binary y(t) , and an associated ramp process x(t) . Various statistical properties of this ramp are considered and it is shown that x(t) is Markovian in one dimension. The first-order probability density and the transition probability density are discussed. Also, the transition probability density for the joint [x(t), y(t), z(t)] is given. Finally, in Appendix II, results are given for the first passage and recurrence time probability densities of x(t) , together with a relation between these two density functions.

The probability density of the output of an RC filter when the input is a binary random process

A new method is given for obtaining the probability density of the output of an RC filter when the input is a stationary binary random process. The axis-crossing intervals of the input are assumed to be statistically independent and identically distributed, but with an arbitrary density function. The method involves a linear integral equation which can be reduced by Laplace transforms. A new family of solutions is given which includes two previously known cases: the random square wave of Poisson type, and the periodic square wave with random time origin. The general result of this family is given in terms of tabulated functions. For other solutions, a recursive technique may be necessary.