Desired Probability Distribution Research Articles

Analysis of transition path ensemble in the exactly solvable models via overdamped langevin equation

Transition of a system between two states is an important but difficult problem in natural science. In this article we study the transition problem in the framework of transition path ensemble. Using the overdamped Langevin method, we introduce the path integral formulation of the transition probability and obtain the equation for the minimum action path in the transition path space. For the effective sampling in the transition path ensemble, we derive a conditional overdamped Langevin equation. In two exactly solvable models, the free particle system and the harmonic system, we present the expression of the conditional probability density and the explicit solutions for the conditional Langevin equation and the minimum action path. The analytic results demonstrate the consistence of the conditional Langevin equation with the desired probability distribution in the transition. It is confirmed that the conditional Langevin equation is an effective tool to sample the transition path ensemble, and the stationary action principle actually leads to the most probable path.

Approximate bounding of mixing time for multiple-step Gibbs samplers

Abstract Markov chain Monte Carlo (MCMC) methods are important in a variety of statistical applications that require sampling from intractable probability distributions. Among the most common MCMC algorithms is the Gibbs sampler. When an MCMC algorithm is used, it is important to have an idea of how long it takes for the chain to become “close” to its stationary distribution. In many cases, there is high autocorrelation in the output of the chain, so the output needs to be thinned so that an approximate random sample from the desired probability distribution can be obtained by taking a state of the chain every h steps in a process called h-thinning. This manuscript extends the work of [D. A. Spade, Estimating drift and minorization coefficients for Gibbs sampling algorithms, Monte Carlo Methods Appl. 27 2021, 3, 195–209] by presenting a computational approach to obtaining an approximate upper bound on the mixing time of the h-thinned Gibbs sampler.

A Convex Duality Approach for Assigning Probability Distributions to the State of Nonlinear Stochastic Systems

In order to optimally assign a desired probability distribution to the state of a nonlinear stochastic system, a convex duality approach is proposed to arrive at the associated optimality conditions. For a general class of stochastic systems governed by controlled Itô differential equations and subject to constraints on the probability distribution of the state at a fixed terminal time, a measure theoretic formulation is presented and it is shown that the original problem is embedded in a convex linear program on the space of Radon measures and that the embedding is tight, i.e., the optimal solution of both the original and the convex relaxation problems are equal. By exploiting the duality relationship between the space of continuous functions and that of measures, the associated optimality conditions are identified in the form of Hamilton-Jacobi problems where the optimization objective, in addition to the value function evaluation at the initial conditions, includes an extra term which is the integral of the product of the value function at the terminal time and the desired probability distribution. Numerical examples are provided to illustrate the results.

Data-Driven Continuous-Time Modeling of Power System Uncertainty Using a Multi-Dimensional Stochastic Differential Equation

This paper proposes a generalized continuous random process modeling framework for imposing any desired probability distribution, autocorrelation and power spectral density of measured data of power system uncertainty by developing a multi-dimension stochastic differential equation (SDE) together with a quadratic output equation. An algorithm for determining appropriate parameters of the proposed SDE model is presented. The effectiveness of derived random modeling scheme is verified by carrying out comparative simulations using actual time series data of wind speed, solar radiation, PJM dynamic regulation, and wind power. The proposed SDE model can be readily integrated in various electrical component models for reliable operation, stability assessment and control of renewable power systems.

Chi-square of Pseudorandom Number Generator of Normal Distribution in C++17

High quality pseudorandom number generators were needed in many software solutions throughout the history of programming. Nowadays, these generators play an even more significant role in software development. Generally, these generators bring a certain level of coincidence in some algorithms which need it. This work focuses on the statistical evaluation of one of the representatives of the generators using Pearson's Chi-square goodness of fit test. The generator of pseudorandom numbers under test is the specific implementation in the modern standard of the programming language of C++ (the standard of C++17). Results presented in this paper inform whether the numbers generated by the selected generator follow the desired probability distribution (normal).

Estimating a probability distribution corresponding to the negation of a property

Suppose that we know the probability distribution corresponding to some property A. How can we then describe the distribution describing the negation of this property? In this paper, we show that, in general, we need additional information to uniquely determine the desired probability distribution. If we do not have this additional information and we still need to select one of the possible distributions for the negation, which one should we choose? We show that reasonable arguments lead to a formula proposed by Yager (IEEE Trans Fuzzy Syst 23(5):1899–1902, 2014). However, if we apply Yager’s formula to negation, we do not get the original distribution back. We show how to modify Yager’s formula so that the distribution for “not not A” will be the same as the original distribution for the property A.

BAYESIAN ESTIMATE OF TELECOMMUNICATION SYSTEMS PREPAREDNESS

The paper assumes a Bayesian estimate of the telecommunication systems availability ratio. Downtime and uptime are described by gamma distributions with positive integer parameters. Estimates of the distribution parameters are obtained using the maximum likelihood method. For the set samples, the values of the desired probability distribution densities are found and an expression for estimating the availability ratio is derived. Numerical estimates for the standard and assumed estimates are given. For a system with two states, a Bayesian estimate of the availability function with consideration of downtime and serviceable condition takes into account the features of backup equipment and the effect of its failure defined by performance reliability and features that ensure the reliability of information signals. The proposed Bayesian approach has the following advantages: it is possible to conduct quantitative estimates with lack of sufficient statistics on functional use indicators; it takes into account all destabilizing factors of various nature; the presence of a lower mean square error compared to traditional methods. To implement the proposed approach that estimates the availability ratio, confidence probabilities are introduced relative to the indicator of failure flows and equipment recovery. The parameters of the a priori information can be determined by different methods or on the basis of sufficient statistical data. To illustrate the discussed calculation algorithm, a digital data transmission system of a standard satellite navigation system consisting of terminal, radio equipment, and a transponder is considered. To estimate the required values, we used data on interruptions in the operation of equipment due to its malfunction during a conditional year. The frequency of downtime caused by signal propagation conditions and equipment failures was evaluated. It was shown that the gamma distribution is suitable for describing the frequency distribution of downtime. The frequency distribution of the cyclicity coefficient with the condition of the selected time interval was also taken into account. Sample mathematical expectations and mean square deviations of the downtime coefficient were found. As a result, the numerical example shows the correctness of using the Bayesian estimate of weighted equipment preparedness.

Holographically controlled random numbers from entangled twisted photons

We present a quantum random number generator (QRNG) based on the random outcomes inherent in projective measurements on a superposition of quantum states of light. Firstly, we use multiplexed holograms encoded on a spatial light modulator to spatially map down-converted photons onto a superposition of optical paths. This gives us full digital control of the mapping process which we can tailor to achieve any desired probability distribution. More importantly, we use this method to account for any bias present within our transmission and detection system, forgoing the need for time-consuming and inefficient unbiasing algorithms. Our QRNG achieved a min-entropy of $\text{H}_{\text{min}}=0.9991\pm0.0003$ bits per photon and passed the NIST statistical test suite. Furthermore, we extend our approach to realise a QRNG based on photons entangled in their orbital angular momentum (OAM) degree of freedom. This combination of digital holograms and projective measurements on arbitrary OAM combinations allowed us to generate random numbers with arbitrary distributions, in effect tailoring the system's entropy while maintaining the inherent quantum irreproducibility. Such techniques allow access to the higher-dimensional OAM Hilbert space, opening up an avenue for generating multiple random bits per photon.

A new restricted boltzmann machine training algorithm for image restoration

A variety of approaches have been proposed for addressing different image restoration challenges. Recently, deep generative models were one of the mostly used ones. In this paper, a new Restricted Boltzmann Machines (RBM) training algorithm for addressing corrupted data has been proposed. RBMs can be trained both supervised and unsupervised, however they are very sensitive to noise and occlusion. The proposed algorithm enables the RBM to be robust against corruptions. Using the new algorithm, we have given the RBM a posterior knowledge about desired or clean data. Despite other methods, the proposed algorithm works fine without changing the architecture of the model or adding any regularization term. Concretely, the RBM can be used as a robust feature extractor, even for unclean data. By creating different corrupted versions for each image instance, and using the original version in the reconstruction phase, the RBM can learn the desired probability distribution of data. Experimental results confirm the robustness of the model against different types of corruption.

Generation of a service loading with the desired probability distribution and autocorrelation for fatigue tests using MATLAB

Generation of a service loading with the desired probability distribution and autocorrelation for fatigue tests using MATLAB

Exploring matter wave scattering by means of the phase diagram

For matter wave scattering from passive quantum obstacles, based on probability conservation we propose a phase diagram in terms of phase and modulus of scattering coefficients to explore all possible directional scattering probabilities. In the phase diagram, we can not only have the physical bounds on scattering coefficients for all channels, but also indicate the probability competitions among absorption, extinction, and scattering cross-sections. With the help of this phase diagram, we discuss different scenarios to steer scattering probability distribution, through the interference between s- and p-channels. In particular, we reveal the required conditions to implement a quantum scatterer, i.e., a quantum dot in semiconductor matrix, with a minimum (or zero) value in the scattering probability toward any direction. Two sets of realizable semiconductor materials and geometry are proposed for the desired probability distribution. Our results provide a guideline in designing quantum scatterers with controlling and sensing matter waves.

Algorithms for improving efficiency of discrete Markov chains

Sampling from an intractable probability distribution is a common and important problem in scientific computing. A popular approach to solve this problem is to construct a Markov chain which converges to the desired probability distribution, and run this Markov chain to obtain an approximate sample. In this paper, we provide two methods to improve the performance of a given discrete reversible Markov chain. These methods require the knowledge of the stationary distribution only up to a normalizing constant. Each of these methods produces a reversible Markov chain which has the same stationary distribution as the original chain, and dominates the original chain in the ordering introduced by Peskun [11]. We illustrate these methods on two Markov chains, one connected to hidden Markov models and one connected to card shuffling. We also prove a result which shows that the Metropolis-Hastings algorithm preserves the Peskun ordering for Markov transition matrices.

TreeSimGM: Simulating phylogenetic trees under general Bellman-Harris models with lineage-specific shifts of speciation and extinction in R.

Understanding macroevolutionary processes using phylogenetic trees is a challenging and complex process that draws on mathematics, computer science and biology. Given the development of complex mathematical models and the growing computational processing power, simulation tools are becoming increasingly popular.In order to simulate phylogenetic trees, most evolutionary biologists are forced to build their own algorithms or use existing tools built on different platforms and/or as standalone programmes. The absence of a simulation tool accommodating for user‐chosen model specifications limits, amongst others, model testing and pipelining with approximate Bayesian computation methods or other subsequent statistical analysis.We introduce “TreeSimGM,” an r‐package simulation tool for phylogenetic trees under a general Bellman and Harris model. This package allows the user to specify any desired probability distribution for the waiting times until speciation and extinction (e.g. age‐dependent speciation/extinction). Upon speciation, the user can specify whether one descendant species corresponds to the ancestor species inheriting its age or whether both descendant species are new species of age 0. Moreover, it is possible to scale the waiting time to speciation/extinction for newly formed species. Thus, “TreeSimGM” not only allows the user to simulate stochastic phylogenetic trees assuming several popular existing models, such as the Yule model, the constant‐rate birth–death model, and proportional to distinguishable arrangement models, but it also allows the user to formulate new models for exploration. A short explanation of the supported models and a few examples of how to use our package are presented here.As an r‐package, “TreeSimGM” allows flexible and powerful stochastic phylogenetic tree simulations. Moreover, it facilitates the pipelining of outputs or inputs with other functions in r. “TreeSimGM” contributes to the tools available to the r community in the fields of ecology and evolution, is freely available under the GPL‐2 licence and can be downloaded at https://cran.r-project.org/web/packages/TreeSimGM.

Derivation and analysis of simplified filters

Filtering is concerned with the sequential estimation of the state, and uncertainties, of a Markovian system, given noisy observations. It is particularly difficult to achieve accurate filtering in complex dynamical systems, such as those arising in turbulence, in which effective low-dimensional representation of the desired probability distribution is challenging. Nonetheless recent advances have shown considerable success in filtering based on certain carefully chosen simplifications of the underlying system, which allow closed form filters. This leads to filtering algorithms with significant, but judiciously chosen, model error. The purpose of this article is to analyze the effectiveness of these simplified filters, and to suggest modifications of them which lead to improved filtering in certain time-scale regimes. We employ a Markov switching process for the true signal underlying the data, rather than working with a fully resolved DNS PDE model. Such Markov switching models haven been demonstrated to provide an excellent surrogate test-bed for the turbulent bursting phenomena which make filtering of complex physical models, such as those arising in atmospheric sciences, so challenging.

Modeling and Performance Evaluation of Dual Polarized MIMO Land Mobile Satellite Channels

paper is concerned with generating of narrow-band dual polarized Multi Input Multi Output (MIMO) over land Mobile Satellite (LMS) fading channels. In the absence of accurate experimental results of MIMO-LMS channels, a statistical model for the characterization of MIMO-LMS is proposed based on available experimental results for Single Input Single output (SISO) LMS and MIMO wireless channels as well as on their extrapolation to the MIMO-LMS case of interest. In this paper a step-by-step methodology for transforming SISO model (Loo model) to MIMO model with detailed description and block diagrams for the simulation and time series signal generation for MIMO-LMS, with the desired power, probability distribution, covariance relations and spectrum. Moreover, based on the proposed channel model, the channel capacity statistics as well as the error performance of MIMO- LMS are evaluated through computer simulation. Useful numerical results on the capacity are also provided taking into account several operational system parameters. The simulation model and results are useful for LMS-MIMO physical layer researches and system designer, who need an easy to implement and realizable model, representative of typical MIMO-LMS communication systems.

Evaluating Data Assimilation Algorithms

Abstract Data assimilation leads naturally to a Bayesian formulation in which the posterior probability distribution of the system state, given all the observations on a time window of interest, plays a central conceptual role. The aim of this paper is to use this Bayesian posterior probability distribution as a gold standard against which to evaluate various commonly used data assimilation algorithms. A key aspect of geophysical data assimilation is the high dimensionality and limited predictability of the computational model. This paper examines the two-dimensional Navier–Stokes equations in a periodic geometry, which has these features and yet is tractable for explicit and accurate computation of the posterior distribution by state-of-the-art statistical sampling techniques. The commonly used algorithms that are evaluated, as quantified by the relative error in reproducing moments of the posterior, are four-dimensional variational data assimilation (4DVAR) and a variety of sequential filtering approximations based on three-dimensional variational data assimilation (3DVAR) and on extended and ensemble Kalman filters. The primary conclusions are that, under the assumption of a well-defined posterior probability distribution, (i) with appropriate parameter choices, approximate filters can perform well in reproducing the mean of the desired probability distribution, (ii) they do not perform as well in reproducing the covariance, and (iii) the error is compounded by the need to modify the covariance, in order to induce stability. Thus, filters can be a useful tool in predicting mean behavior but should be viewed with caution as predictors of uncertainty. These conclusions are intrinsic to the algorithms when assumptions underlying them are not valid and will not change if the model complexity is increased.

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.

Mean level signal crossing rate for an arbitrary stochastic process: comment

In J. Opt. Soc. Am. A27, 797 (2010), Yura and Hanson derived what they claim to be "a general expression for the mean level crossing rate for an arbitrary" random process following any desired probability distribution function. On the other hand, the authors themselves assert that in some cases their results "differ somewhat from the result reported in the literature." This discrepancy "remains unexplained" by the authors "and is laid open for future discussion." In this note, we explain the reason for such discrepancy and show that the Yura-Hanson formula is indeed a special-case solution. A more general solution is then given that is applicable to arbitrary random processes and that is fully consistent with the Rice mean level crossing rate formula.

마코비안 도착과정을 이용한 축구경기 득점결과의 예측

We develop a stochastic model to predict the score of a soccer match. We describe the scoring process of the soccer match as a markovian arrival process (MAP). To do this, we define a two-state underlying Markov chain, in which the two states represent the offense and defense states of the two teams to play. Then, we derive the probability vector generating function of the final scores. Numerically inverting this generating function, we obtain the desired probability distribution of the scores. Sample numerical examples are given at the end to demonstrate how to utilize this result to predict the final score of the match.

Digital simulation of an arbitrary stationary stochastic process by spectral representation

In this paper we present a straightforward, efficient, and computationally fast method for creating a large number of discrete samples with an arbitrary given probability density function and a specified spectral content. 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 contrast to previous work, where the analyses were limited to auto regressive and or iterative techniques to obtain satisfactory results, we find that a single application of the inverse transform method yields satisfactory results for a wide class of arbitrary probability distributions. Although a single application of the inverse transform technique does not conserve the power spectra exactly, it yields highly accurate numerical results for a wide range of probability distributions and target power spectra that are sufficient for system simulation purposes and can thus be regarded as an accurate engineering approximation, which can be used for wide range of practical applications. A sufficiency condition is presented regarding the range of parameter values where a single application of the inverse transform method yields satisfactory agreement between the simulated and target power spectra, and a series of examples relevant for the optics community are presented and discussed. Outside this parameter range the agreement gracefully degrades but does not distort in shape. Although we demonstrate the method here focusing on stationary random processes, we see no reason why the method could not be extended to simulate non-stationary random processes.