Random Variables Research Articles

Discounted fully probabilistic design of decision rules

Axiomatic fully probabilistic design (FPD) of optimal decision rules strictly extends the decision making (DM) theory represented by Markov decision processes (MDP). This means that any MDP task can be approximated by an explicitly found FPD task whereas many FPD tasks have no MDP equivalent. MDP and FPD model the closed loop — the coupling of an agent and its environment — via a joint probability density (pd) relating the involved random variables, referred to as behaviour. Unlike MDP, FPD quantifies agent's aims and constraints by an ideal pd. The ideal pd is high on the desired behaviours, small on undesired behaviours and zero on forbidden ones. FPD selects the optimal decision rules as the minimiser of Kullback-Leibler's divergence of the closed-loop-modelling pd to its ideal twin. The proximity measure choice follows from the FPD axiomatics.MDP minimises the expected total loss, which is usually the sum of discounted partial losses. The discounting reflects the decreasing importance of future losses. It also diminishes the influence of errors caused by:▪ the imperfection of the employed environment model;▪ roughly-expressed aims;▪ the approximate learning and decision-rules design.The established FPD cannot currently account for these important features. The paper elaborates the missing discounted version of FPD. This non-trivial filling of the gap in FPD also employs an extension of dynamic programming, which is of an independent interest.

Reliability inference of stress-strength for a K-out-of-N: G mixed standby system with switching failure

In this paper, the statistical inference of the stress-strength reliability for a K-out-of-N: G mixed standby system with switching failure is investigated. It is assumed that the strength of the component itself and the external stress to which it is subjected are random variables that obey the same distribution. If an operating component fails, the warm standby component will be immediately switched to the operating one. During this switchover, the warm standby component is subject to switching failure with a certain probability. Based on Weibull distribution and generalized half-logistic distribution, the reliability indices of stress-strength model under two different distributions are derived, respectively. For the two types of distribution of stress and strength, the maximum likelihood estimator, maximum spacing estimator, Bayesian estimator, asymptotic confidence interval and bootstrap-p confidence interval of the reliability are obtained, respectively. The Monte Carlo method is applied in the numerical simulation to realize the simulation inference of the system reliability. The performance of the above inference methods is analyzed and compared. The influence of parameters of stress and strength on the system reliability is analyzed graphically.

A new combined analytical–numerical probabilistic method for assessing the impact of DERs on the voltage stability of bulk power systems

The integration of distributed energy resources (DERs) and unpredictable loads has increased uncertainty in power systems. Traditional methods struggle to assess performance under these uncertainties, and existing probabilistic methods face challenges with complexity and accuracy. This paper introduces a new combined analytical–numerical probabilistic method to assess the impact of DERs on voltage stability. Using Bayesian Parameter Estimation (BPE), the method derives the analytical properties of random variables (RVs) associated with DERs and loads, obtaining posterior distributions. The Metropolis–Hastings sampling technique then estimates these posteriors numerically, enabling accurate predictions of DERs and load outputs. Voltage stability analysis was performed using the continuation power flow method and validated on the IEEE 59-bus test system in MATLAB/Simulink. The results show that integrating DERs significantly improves voltage stability. The proposed method outperforms the Monte Carlo simulation (MCS)-based method in accuracy and computational speed, increasing DERs penetration and voltage stability limits by 3%. It closely matches MCS voltage estimates but requires fewer iterations (500 per loading increment) compared to MCS’s 1000, leading to faster computation times (a few hours to one day versus up to three days for MCS). This method provides an efficient solution for managing uncertainties in power systems.

Statistical Distribution Theory and Fractional Calculus

This is an overview paper. This paper is an attempt to show that fractional calculus can be reached through statistical distribution theory. This paper brings together results on fractional integrals and fractional derivatives of the first and second kinds in the real and complex domains in the scalar, vector, and matrix-variate cases, and shows that all these results can be reached through statistical distribution theory. It is shown that the whole area of fractional integrals can be reached through distributions of products and ratios in the scalar variable case and distributions of symmetric products and symmetric ratios in the matrix-variate cases. While summarizing the materials, the real domain results are also listed side by side with the complex domain results so that a comparative study is possible. Fractional integrals and derivatives in the real domain mean that the parameters involved could be real or complex with appropriate conditions, the arbitrary function is real-valued, and the variables involved are all real. These in the complex domain mean that the parameters could be real or complex and the arbitrary function is still real-valued but the variables involved are in the complex domain. Fully complex domain means the variables as well as the arbitrary function are in the complex domain. Most of the materials on fractional integrals and fractional derivatives involving a single matrix or a number of matrices in the real or complex domain are of this author. Slight modifications of the results, compared with the published works in various papers, are there in various sections. In the paragraph on notations, the lemmas that are taken from this author’s own book on Jacobians are common with published works and hence the similarity index with this author’s works will be high. Section Matrix-Variate Joint Distributions and Fractional Integrals in Many Matrix-Variate Cases material on a statistical approach to Kiryakova’s multi-index fractional integral and its extension to the real scalar case of second kind integrals as well as extensions of first and second kind integrals to real and complex matrix-variate cases are believed to be new. Matrix differential operators are introduced in Section Fractional Derivatives and, with the help of these operators, fractional derivatives are constructed from the corresponding fractional integrals. These operators are applicable in a large variety of functions. Applicability is shown through identities created from scale transformed gamma random variables. Some concluding remarks are given and some open problems are pointed out in Section Concluding Remarks.

Modeling Nuclear-Centric Scenarios for Ukraine’s Low-Carbon Energy Transition Using Diffusion and Regression Techniques

This study presents a mathematical model for forecasting the development of Ukraine’s Integrated Power System (IPS) until 2040, with a specific focus on the expansion of nuclear energy as a cornerstone of the nation’s low-carbon transition. The model is an extension of Frank Bass’s mixed influence diffusion model, incorporating both economic and technological factors. These factors are treated as stochastic variables essential for accurately predicting the evolution of an integrated energy system, particularly in the context of rapid renewable energy sources (RES) growth. The model employs regression techniques using generalized logistic curves, improving forecasting efficiency by aligning modeling parameters with experimental data. The study’s results indicate the potential for optimizing IPS components, including nuclear and thermal power generation, through the model’s application. The model is distinguished by its inclusion of economic and technological impacts, such as state matrices, control actions, and external influence matrices, which enhance the accuracy of simulations and predictions. The validation of the model, based on scenarios of electricity consumption and generation, shows significant alignment with observed trends, confirming the model’s reliability. The findings suggest that this model is an effective tool for developing and refining energy system scenarios, with nuclear energy playing a pivotal role in Ukraine’s sustainable energy future.

Event‐Triggered Generalized Extended State Observer‐Based Control for Nonlinear Networked Systems Under Gain Variation and Multi‐Channel Attacks

ABSTRACTThis paper investigates the event‐triggered generalized extended state observer‐based estimation issue for a class of nonlinear networked control systems with unmodeled dynamics, external disturbances and multi‐channel attacks. In order to resist different forms of attack threats on multiple communication channels from sensors to the observer, Markov chain is introduced to describe the stochastic switching or jumping behavior between different attack modes. Considering the limited network resources, the measured outputs are transmitted to the observer only when the triggering conditions are met. Moreover, the parameters modeled by the Bernoulli process are adopted to help analyze potential random gain variations of the generalized extended state observer. By employing Lyapunov stability theory and stochastic systems analysis, sufficient conditions are derived to ensure that the augmented system is exponentially bounded in mean square, and the expected observer gains are further determined through linear matrix inequalities. Finally, a numerical example and a simulation related to the RLC series circuit system are conducted, illustrating the proposed method's effectiveness.

A robust method for canonical correlation analysis

Canonical correlation analysis (CCA) explores the relationships between two sets of random variables. However, the conventional computation of this analysis relies on sample covariance matrices, which are highly influenced by outliers. In this study, we introduce a robust approach to perform canonical correlation analysis by employing the reweighted orthogonalized comedian estimator. To evaluate the effectiveness of the proposed method, a simulation study is conducted, comparing it with four well-established estimators across various sampling scenarios. The simulation results strongly advocate for the adoption of our method in situations where outliers are present in the data. Furthermore, we investigate the robustness of our method using breakdown plots, demonstrating its ability to withstand substantial levels of contamination. To validate the practical utility of our approach, we analyze a real-life dataset, observing consistent results with those obtained from the simulation study.

A fast and accurate numerical method for the left tail of sums of independent random variables

We present a flexible, deterministic numerical method for computing left-tail rare events of sums of non-negative, independent random variables. The method is based on iterative numerical integration of linear convolutions by means of Newtons–Cotes rules. The periodicity properties of convoluted densities combined with the Trapezoidal rule are exploited to produce a robust and efficient method, and the method is flexible in the sense that it can be applied to all kinds of non-negative continuous RVs. We present an error analysis and study the benefits of utilizing Newton–Cotes rules versus the fast Fourier transform (FFT) for numerical integration, showing that although there can be efficiency benefits to using FFT, Newton–Cotes rules tend to preserve the relative error better, and indeed do so at an acceptable computational cost. Numerical studies on problems with both known and unknown rare-event probabilities showcase the method’s performance and support our theoretical findings.

A framework for visual comparison of scalar fields with uncertainty

AbstractScientists working with uncertain data, such as climate simulations, medical images, or ensembles of physical simulations, regularly confront the problem of comparing observations, e.g., to identify similarities, differences, or patterns. Current approaches in comparative visualization of uncertain scalar fields mainly rely on juxtaposition of both data and uncertainties, where each is represented using, e.g., color mapping or volume rendering. While interpretation of uncertain scalar data from visual encodings is already cognitively challenging, comparison of uncertain fields without explicit visualization support adds a further layer of complexity. In this paper, we present a theoretical framework to devise and describe a class of techniques that directly visualize differences between two or more uncertain scalar fields in a single image. We model each such technique as a combination of one or more interpolation stages, with the application of distance measures on random variables to the resulting distributions, and an appropriate visual encoding. Our framework captures existing methods and lends itself well to formulating new comparative visualization techniques for uncertain data for different visualization scenarios. Furthermore, by modeling uncertain scalar field differences as random variables themselves, we enable additional opportunities for comparison. We demonstrate the usefulness of our framework and its properties by applying it to effective comparative visualization techniques for several synthetic and real-world data sets.

Weak laws of large numbers for weighted sums of arrays of random variables under sublinear expectation

. In this article, we introduce a new concept of integrability called extended residually h-integrability with exponent r, which deals with weighted sums of arrays under sublinear expectation. The classical copy of this concept extends the residually h-integrability given by Wang, Hu, and Volodin (2018). On this basis, we obtain three forms of weak laws of large numbers for weighted sums of arrays of rowwise negatively dependent or rowwise independent random variables, respectively, under sublinear expectation. As corollaries, we get some weak laws of large numbers for arrays of random variables with normalizing factor k n under sublinear expectation.

Regularity of laws via Dirichlet forms: application to quadratic forms in independent and identically distributed random variables

Regularity of laws via Dirichlet forms: application to quadratic forms in independent and identically distributed random variables

Efficient mass manufacturing of high-density, ultra-low-loss Si3N4 photonic integrated circuits

Silicon nitride (Si3N4) photonic integrated circuits (PICs) offer significant advantages over traditional silicon photonics, including low loss and superior power handling at optical communication wavelength bands. To facilitate high-density integration and effective nonlinearity, the use of thick, stoichiometric Si3N4 films is crucial. However, when using low-pressure chemical vapor deposition (LPCVD) to achieve high optical material transparency, Si3N4 films exhibit large tensile stress on the order of GPa, leading to wafer cracking that challenges mass production. Methods for crack prevention are therefore essential. The photonic Damascene process has addressed this issue, attaining record low-loss Si3N4 PICs, but it lacks control of the waveguide height, leading to large random variations of waveguide dispersion and unpredictable spectrum responses of critical functional devices such as optical couplers. Conversely, subtractive processes achieve better dimension control but rely on techniques unsuitable for large-scale production. To date, an outstanding challenge is to attain both lithographic precision and ultra-low loss in high-confinement Si3N4 PICs that are compatible with large-scale foundry manufacturing. Here, we present a single-step deposited, DUV-based subtractive method for producing wafer-scale ultra-low-loss Si3N4 PICs that harmonize these necessities. By employing deep etching of densely distributed, interconnected trenches into the substrate, we effectively mitigate the tensile stress in the Si3N4 layer, enabling direct deposition of thick films without cracking and substantially prolonged storage duration. A secondary ion mass spectrometry (SIMS) analysis reveals that these deep trenches simultaneously serve as gettering centers for metal impurities, in particular copper, thereby reducing the absorption loss in Si3N4 waveguides. Lastly, we identify ultraviolet (UV)-radiation-induced damage that can be remedied through a rapid thermal annealing. Collectively, we develop ultra-low-loss Si3N4 microresonators and 0.5-m-long spiral waveguides with losses down to 1.4 dB/m at 1550 nm with high production yield. This work addresses the long-standing challenges toward scalable and cost-effective production of tightly confined, low-loss Si3N4 PICs as used for quantum photonics, large-scale linear and nonlinear photonics, photonic computing, and narrow-linewidth lasers.

Tractable optimal experimental design using transport maps*

Abstract We present a flexible method for computing Bayesian optimal experimental designs (BOEDs) for inverse problems with intractable posteriors. The approach is applicable to a wide range of BOED problems and can accommodate various optimality criteria, prior distributions and noise models. The key to our approach is the construction of a transport-map-based surrogate to the joint probability law of the design, observational and inference random variables. This order-preserving transport map is constructed using tensor trains and can be used to efficiently sample from (and evaluate approximate densities of) conditional distributions that are required in the evaluation of many commonly-used optimality criteria. The algorithm is also extended to sequential data acquisition problems, where experiments can be performed in sequence to update the state of knowledge about the unknown parameters. The sequential BOED problem is made computationally feasible by preconditioning the approximation of the joint density at the current stage using transport maps constructed at previous stages. The flexibility of our approach in finding optimal designs is illustrated with some numerical examples inspired by disease modeling and the reconstruction of subsurface structures in aquifers.

Remarks on Limit Theorems for the Free Quadratic Forms.

In 2021, Ejsmont and Biernacki showed that the free tangent distribution can be used to measure household satisfaction with durable consumer goods. This distribution arises as the limit of free random variables. This, new article serves as the theoretical introduction to the continuation of the research presented in the paper from 2021. We continue the study of the limit of specific quadratic forms in free probability, which is the first step towards constructing a new distribution for the evaluation of satisfaction with material affluence among household. We formulate a non-central limit theorem for weighted sums of commutators and square of the sums for free random variable. In addition we give the random matrix models for these limits.

Constrained Bayesian Method for Testing Equi-Correlation Coefficient

The problem of testing the equi-correlation coefficient of a standard symmetric multivariate normal distribution is considered. Constrained Bayesian and classical Bayes methods, using the maximum likelihood estimation and Stein’s approach, are examined. For the investigation of the obtained theoretical results and choosing the best among them, different practical examples are analyzed. The simulation results showed that the constrained Bayesian method (CBM) using Stein’s approach has the advantage of making decisions with higher reliability for testing hypotheses concerning the equi-correlation coefficient than the Bayes method. Also, the use of this approach with the probability distribution of linear combinations of chi-square random variables gives better results compared to that of using the integrated probability distributions in terms of providing both the necessary precisions as well as convenience of implementation in practice. Recommendations towards the use of the proposed methods for solving practical problems are given.

The Vasicek distribution autoregressive time-series model for aggregated data of default and delinquency rates

Abstract Is it possible to analyse time series of aggregated data of default and delinquency rates with a simple model that includes both cross-sectional and serial dependence? To address this question, this paper proposes an autoregressive time-series model for Vasicek-distributed random variables. A direct result from the proposed model is that it leads to a novel out-of-time validation test for aggregated default and delinquency rates. The motivation for the model comes from an analysis of a time series of aggregated US credit card delinquencies, for which the Poisson or binomial distribution cannot be used. The proposed model can also be employed to analyse a time series of the average default probability of a portfolio.

Majorization and randomness measures

Abstract A series of papers by Hickey (1982, 1983, 1984) presents a stochastic ordering based on randomness. This paper extends the results by introducing a novel methodology to derive models that preserve stochastic ordering based on randomness. We achieve this by presenting a new family of pseudometric spaces based on a majorization property. This class of pseudometrics provides a new methodology for deriving the randomness measure of a random variable. Using this, the paper introduces the Gini randomness measure and states its essential properties. We demonstrate that the proposed measure has certain advantages over entropy measures. The measure satisfies the value validity property, provides an adequate extension to continuous random variables, and is often more appropriate (based on sensitivity) than entropy in various scenarios.

Modelling of Steady-State Seepage of an Embankment Dam Using Teaching-Learning Based Optimization Algorithm

The goal of the this study is to investigate the applicability of the teaching-learning based optimization (TLBO) algorithm for modeling seepage in embankment dams. The input parameters selected for the models to be built are the values of permeability (ks), van Genuchten's suitability parameters α and n, whose effect on seepage has been investigated over the years due to their uncertainties. The validity of the TLBO was compared with that of conventional regression analysis (CRA) methods. Both methods were utilized with different regression forms. The parameters chosen as input are modeled as random variables with a log-normal distribution, and total discharge (Q) was obtained. Four statistical indices, that is, root mean square error, mean absolute error, average relative error and coefficient of determination, were used to evaluate the performance of the models. The equations obtained using TLBO algorithms can predict the total discharge in embankment dams better than CRA. In addition, the reliability of TLBO has been demonstrated by conducting analyses using the outputs of CRA as a benchmark.

Expectation analysis for bounding solutions of the 0–1 knapsack problem

In this paper, an entirely novel discrete probabilistic model is presented to generate 0–1 Knapsack Problem instances. We analyze the expected behavior of the greedy algorithm, the eligible-first algorithm and the linear relaxation algorithm for these instances; all used to bound the solution of the 0–1 Knapsack Problem (0–1 KP) and/or its approximation. The probabilistic setting is given and the main random variables are identified. The expected performance for each of the aforementioned algorithms is analytically established in closed forms in an unprecedented way.

Multi‐objective and multi‐stage low‐carbon planning of park integrated energy system considering random outages from superior power grid

AbstractThis article proposes a multi‐objective and multi‐stage low‐carbon planning approach for park integrated energy systems (PIES) considering the impacts of random outages from the superior electrical grid. This approach incorporates optimal multi‐stage construction sequencing and stepped carbon emission trading to leverage the economic and low‐carbon benefits of long‐term planning. First, the islanding modes of PIES are described using four random variables: island type, duration, start time, and typical day of occurrence, from which islanding scenarios are generated based on scenario tree. Next, a multi‐objective planning model that considers both economics and reliability is constructed, with the objectives of minimizing the total lifecycle planning cost and the expected economic loss during islanding. The improved Normalized Normal Constraint (NNC) method is proposed to solve the multi‐objective planning problem. Then, the fuzzy membership function is used to determine the optimal compromise solution, resulting in a planning scheme that balances economic efficiency and supply reliability. Finally, simulations indicate that, at the cost of a slight increase in planning expenses, the proposed model significantly reduces the loss costs under islanding modes compared with single‐objective economic‐focused planning. Additionally, the improved NNC method can achieve a more uniform Pareto frontier compared with the conventional NNC method.