Random Variables Research Articles

Enhanced Subspace Iteration Technique for Probabilistic Modal Analysis of Statically Indeterminate Structures

In structural stochastic dynamic analysis, the consideration of the randomness in the physical parameters of the structure necessitates the establishment of numerous stochastic finite element models and the subsequent computation of their corresponding vibration modes. When the complete analysis is applied to calculate the vibration modes for each sample of the stochastic finite element model, a substantial computational expense is incurred. To enhance computational efficiency, this work presents an extended subspace iteration method aimed at rapidly determining the vibration modal parameters of statically indeterminate structures. The essence of this proposed method revolves around efficiently constructing reduced basis vectors during the subspace iteration process, utilizing flexibility disassembly perturbation and the Krylov subspace. This extended subspace iteration method proves particularly advantageous for the modal analysis of finite element models that incorporate a multitude of random variables. The proposed modal random analysis method has been validated using both a truss structure and a beam structure. The results demonstrate that the proposed method achieves substantial savings in computational time. Specifically, for the truss structure, the calculation time of the proposed method is approximately 1.2% and 65% of that required by the comprehensive analysis method and the combined approximation method, respectively. For the beam structure, on average, the computational time of the proposed method is roughly 2.1% of a full analysis and approximately 48.2% of the Ritz vector method’s time requirement. Compared to existing stochastic modal analysis algorithms, the proposed method offers improved computational accuracy and efficiency, particularly in scenarios involving high-discreteness random analyses.

Probabilistic degenerate Dowling polynomials associated with random variables

Let Y be a random variable whose moment generating function exists in some neighborhood of the origin. The aim of this paper is to study probabilistic versions of the degenerate Whitney numbers of the second kind and those of the degenerate Dowling polynomials, namely, the probabilistic degenerate Whitney numbers of the second kind associated with and the probabilistic degenerate Dowling polynomials associated with . We derive some properties, explicit expressions, certain identities, recurrence relations and generating functions for those numbers and polynomials. In addition, we investigate their generalizations, namely, the probabilistic degenerate ‐Whitney numbers of the second kind associated with and the probabilistic degenerate ‐Dowling polynomials associated with , and get similar results to the aforementioned numbers and polynomials.

Deep Fuzzy Credibility Surfaces for Integrating External Databases in the Estimation of Operational Value at Risk

Operational risk (OR) is usually caused by losses due to human errors, inadequate or defective internal processes, system failures, or external events that affect an organization. According to the Basel II agreement, OR is defined by seven risk events: internal fraud, external fraud, labor relations, clients, damage to fixed assets, technical failures and failures in the execution and administration of processes. However, the low frequency with which a loss event occurs creates a technological challenge for insurers in estimating the operational value at risk (OpVar) for the protection derived from an organization’s business activities. Following the above, this paper develops and analyzes a Deep Fuzzy Credibility Surface model (DFCS), which allows the integration in a single structure of different loss event databases for the estimation of an operational value at risk (OpVar), overcoming the limitations imposed by the low frequency with which a risk event occurs within an organization (sparse data). For the estimation of OpVar, the DFCS model incorporates a novel activation function based on the generalized log-logistic function to model random variables of frequency and severity that define a loss event (linguistic random variables), as well as a credibility surface to integrate the magnitude and heterogeneity of losses in a single structure as a result of the integration of databases. The stability provided by the DFCS model could be evidenced through the structure exhibited by the aggregate loss distributions (ALDs), which are obtained as a result of the convolution process between frequency and severity random variables for each database and which are expected to achieve similar structures to the probability distributions suggested by Basel II agreements (lean, long tail, positive skewness) against the OR modeling. These features make the DFCS model a reference for estimating the OpVar to protect the risk arising from an organization’s business operations by integrating internal and external loss event databases.

On the degree of uniformity measure for probability distributions

Abstract A key challenge in studying probability distributions is quantifying the inherent inequality within them. Certain parts of the distribution have higher probabilities than others, and our goal is to measure this inequality using the concept of mathematical diversity, a novel approach to examining inequality. We introduce a new measure $m_D(P)$, called the {\it degree of uniformity} measure on a given discrete probability space that generalizes the idea of the slope of secant of the {\it slope of diversity} curve. This measure generalizes the idea of degree of uniformity of a contiguous part $P=\{k_1,k_2\}$ in a probability space related to a random variable $X$, to an arbitrary measurable part $P$. We also demonstrate the truly scale free and self-contained nature of the concept of {\it degree of uniformity} by relating the measure of two parts $P_1$ and $P_2$ from completely unrelated distributions with random variables $X_1$ and $X_2$ that have completely different scales of variation.

On the Tail Behavior for Randomly Weighted Sums of Dependent Random Variables with its Applications to Risk Measures

On the Tail Behavior for Randomly Weighted Sums of Dependent Random Variables with its Applications to Risk Measures

Tail probability bounds for acceptable random variables of partial sums and their application in linear models

Exponential probability inequalities play an important role in many parts of probability and statistics, offering a fundamental tool for tail probability bounding and modeling stochastic processes. The present work proposes a new exponential inequality for acceptable random variables. This class covers dependencies that are not embraced by the standard independence assumptions. Our results extend the theoretical perspective by placing sharper bounds on the above-mentioned variables, which tends to increase the accuracy of probabilistic modeling. Moreover, we study complete convergence in first-order autoregressive processes with acceptable innovations. The complete convergence framework gives further insights into the long-term behavior of the wide class of processes commonly used in time series analysis, econometrics, and other areas of applied research. In turn, our results on the influence of acceptable innovations on the properties of complete convergence establish bridges between theoretical probability inequalities and statistical applications. In addition, these contributions provide new tools in investigating the stability and predictability of the autoregressive models. These considerations consequently set up significant perspectives and enrich a developing body of research related to stochastic processes by reinforcing the theoretical framework as well as the methodological apparatus of probability theory.

Violation of Leggett–Garg Inequality in Perceiving Cup-Like Objects and Cognitive Contextuality

This paper is devoted to an experimental investigation of cognitive contextuality inspired by quantum contextuality research. This contextuality is related to, but not identical to context-sensitivity which is well-studied in cognitive psychology and decision making. This paper is a part of quantum-like modeling, i.e., exploring the methodology of quantum theory outside of physics. We examined the bistable perception of cup-like objects, which strongly depends on experimental contexts. Our experimental data confirmed the existence of cognitive hysteresis, the important role of memory, and the non-commutative structure of cognitive observables. In physics, quantum contextuality is assessed using Bell-CHSH inequalities, and their violation is incorrectly believed to imply the nonlocality of Nature. The violation of Bell-type inequalities in cognitive and social science strongly indicates that the metaphysical implications of these inequalities are quite limited. In our experiments, modified Leggett–Garg inequalities were also significantly violated, but this only means that experimental data from experiments performed in different contexts cannot be modeled by a unique set of noncontextual, jointly distributed random variables. In our experiments, we know the empirical probability distributions measured in different contexts; thus, we can obtain much more detailed and reliable information about contextuality in human cognition by performing nonparametric compatibility tests.

Permutation Entropy for Signal Analysis

Shannon Entropy is the preeminent tool for measuring the level of uncertainty (and conversely, information content) in a random variable. In the field of communications, entropy can be used to express the information content of given signals (represented as time series) by considering random variables which sample from specified subsequences. In this paper, we will discuss how an entropy variant, the \textit{permutation entropy} can be used to study and classify radio frequency signals in a noisy environment. The permutation entropy is the entropy of the random variable which samples occurrences of permutation patterns from time series given a fixed window length, making it a function of the distribution of permutation patterns. Since the permutation entropy is a function of the relative order of data, it is (global) amplitude agnostic and thus allows for comparison between signals at different scales. This article is intended to describe a permutation patterns approach to a data driven problem in radio frequency communications research, and includes a primer on all non-permutation pattern specific background. An empirical analysis of the methods herein on radio frequency data is included. No prior knowledge of signals analysis is assumed, and permutation pattern specific notation will be included. This article serves as a self-contained introduction to the relationship between permutation patterns, entropy, and signals analysis for studying radio frequency signals and includes results on a classification task.

On probabilistic and causal reasoning with summation operators

Abstract Ibeling et al. (2023, Annals of Pure and Applied Logic, 103339) axiomatize increasingly expressive languages of causation and probability, and Mossé et al. (2024, The Review of Symbolic Logic, 17, 106–131) show that reasoning (specifically the satisfiability problem) in each causal language is as difficult, from a computational complexity perspective, as reasoning in its merely probabilistic or ‘correlational’ counterpart. Introducing a summation operator to capture common devices that appear in applications—such as the $do$-calculus of Pearl (Causality. Cambridge University Press, 2009) for causal inference, which makes ample use of marginalization—van der Zander et al. (The hardness of reasoning about probabilities and causality. In Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence, IJCAI-23, E. Elkind, ed., pp. 5730–5738. International Joint Conferences on Artificial Intelligence Organization. Main Track. 2023) extend these earlier complexity results to causal and probabilistic languages with marginalization. We complete this extension, fully characterizing the complexity of probabilistic and causal reasoning with summation, demonstrating that these again remain equally difficult. Surprisingly, allowing free variables for random variable values results in a system that is undecidable, so long as the ranges of these random variables are unrestricted. We finally axiomatize these languages featuring marginalization (or more generally summation), resolving open questions posed by Ibeling et al. (2023, Annals of Pure and Applied Logic, 103339).

Advanced Sliding Mode Design for Optimal Automatic Generation Control in Multi-Area Multi-Source Power System Considering HVDC Link

The multi-area multi-source power system (MAMSPS), which uses a variety of power sources including gas, hydro, thermal, and renewable energy, has recently been implemented to balance the growing demand for electricity and the overall capacity for power generation. In this paper, an integral sliding mode control with a single-phase technique (ISMCSP) is applied to two areas, with each area including gas–wind–thermal power systems with HVDC system. Firstly, a two-area gas–wind–thermal power system with HVDC (TAGWTPSH) is the first model in this scheme to consider the parameter uncertainties of a MAMSPS. Secondly, sliding mode design law with a single-phase technique is introduced to alleviate chattering and oscillation problems. Then, power system stability is ensured by the Lyapunov control theory based on the new LMIs technique. Thirdly, the ISMCSP’s effectiveness in a MAMSPS is also assessed under random load patterns and parameter variations regarding settling time and over-/undershoot. The ISMCSP was created to alter the fundamental sliding mode control, and therefore the suggested approach performs better than recently published approaches. This is demonstrated by the frequency overshoot deviation value in frequency deviations: 0.7 × 10−3 to 2.8 × 10−3 for the TAGWTPSH with the suggested ISMCSP. In the last case, for random changes in load from −0.4 to +0.5 p. u, the proposed ISMCSP method still stabilizes the frequency of the areas meeting the standard requirements for AGC.

Stochastic Generation of Peak Ground Accelerations Based on Single Seismic Event Data for Safety Assessment of Structures

The Korean Peninsula, characterized by low-to-moderate seismicity, faces a shortage of strong ground motion records, posing challenges for the seismic safety assessment of critical infrastructures. Given the rarity of large-magnitude earthquakes, generating a variety of earthquakes with rational values of Peak Ground Acceleration (PGA) is essential for robust seismic fragility and risk analysis. To address this, a new stochastic approach is proposed to simulate artificial earthquakes at multiple source-to-site distances and derive the probability distribution of PGA based on recorded data from a single seismic event. Two key source parameters, seismic moment and corner frequency, are treated as random variables with a negative correlation, reflecting their uncertainties and dependence on source-to-site distance. The Monte Carlo simulation with copula sampling of the key source parameters generates Fourier spectra for artificial earthquakes, which are transformed into the time domain to yield PGA distributions at various distances. A comparison with recorded data shows that the proposed method effectively simulates ground motion intensities, with no statistically significant differences between the simulated results and recorded data (p > 0.05). The present method of determining PGA distributions provides a robust framework to enhance seismic risk analysis for the safety assessment of structures.

On the representation and computational aspects of the distribution of a linear combination of independent noncentral chi-squared random variables

Thiss article presents two new results relevant to a linear combination of χ2 random variables. The first result expresses the cumulative distribution function as a simple multiple of a certain tilted density. The second result shows that in important cases the inversion integral for the density may be expressed as a sum of relatively simple real integrals which, using suitable numerical methods, are straightforward to compute quickly and accurately.

Reliability-based topology optimization of imperfect structures considering uncertainty of load position

In this paper, a novel optimization technique is implemented to explore the effects of considering uncertain load positions. Therefore, the integration of reliability-based design into structural topology optimization, while considering imperfect geometrically nonlinear analysis, is proposed. By comparing the results obtained from perfect and imperfect geometrically and materially nonlinear analyses, this study examines the impact of nonlinearity on probabilistic and deterministic analyses. Concerning probabilistic analysis, the originality of this research lies in its incorporation of the position of the applied load as a stochastic variable. This distinctive approach complements the consideration of other relevant parameters, including volume fraction, material properties, and geometrical imperfections, with the overarching goal of capturing the variability arising from real-world conditions. For the assessment of uncertainties, normal distribution is assumed for all these parameters. Normal distributions are chosen due to their advantages in terms of simplicity, ease of implementation, and computational efficiency. These characteristics are particularly beneficial when dealing with complex optimization algorithms and extensive analyses, as is the case in our research. The proposed algorithm is validated according to the results of benchmark problems. Structural examples like cantilever beam, pinned-shell, and L-shaped beam problems are further explored within the context of imperfect geometrically nonlinear reliability-based topology optimization, with specific regard to the probabilistic aspect of the location of the externally applied loads. Moreover, the results of the suggested approach suggest that the inclusion of a probabilistic design strategy has influenced topology optimization. The reliability index acts as a controlling constraint for the resulting optimized configurations, including the mean stress values associated with the resulting topologies.

Asymptotic approximations for the distribution of the product of correlated normal random variables

Asymptotic approximations for the distribution of the product of correlated normal random variables

Symbolic state-space exploration meets statistical model checking

Efficient reachability analysis, as well as statistical model checking have been proposed for the evaluation of Hybrid Petri nets with general transitions (HPnGs). Both have different (dis-)advantages. The performance of statistical simulation suffers in large models and the number of required simulation runs to achieve a relatively small confidence interval increases considerably. The approach introduced for analytical reachability analysis of HPnGs, however, becomes infeasible for a large number of random variables. To overcome these limitations, this paper applies statistical model checking (SMC) for a stochastic variant of the Signal Temporal Logic (STL) to a pre-computed symbolic state-space representation of HPnGs, i.e., the Parametric Location Tree (PLT), which has previously been used for model checking HPnGs. Furthermore, we define how to reduce the PLT for a given state-based and path-based STL property, by introducing a three-valued interpretation of a given STL property for every location of the PLT. This paper applies learning in the presence of nondeterminism and considers four different scheduler classes. The proposed improvement is especially useful if a large number of training runs is necessary to optimize the probability that a given STL property holds. A case study on a water tank model shows the feasibility of the approach, as well as improved computation times, when applying the above-mentioned reduction for varying time bounds. We validate our results with existing analytical and simulation tools, as applicable for different types of schedulers.

Convergence Rate for Randomly Weighted Sums of Random Variables and Its Application

Convergence Rate for Randomly Weighted Sums of Random Variables and Its Application

Degenerate Moments and Expectation of Monomials

The aim of this paper is twofold. Firstly, we obtain expressions of the degenerate moments of a discrete nonnegative integer valued random variable. Secondly, we get an expression for the expectation of any monomial in discrete nonnegative integer-valued random variables.

New Solutions to the Fractional Perturbed Chen Lee Liu Model with Time-Dependent Coefficients: Applications to Complex Phenomena in Optical Fibersns

In this paper, we consider the fractional perturbed Chen Lee Liu model with time-dependent coefficients (FPCLLM-TDCs). We apply the mapping method in order to get hyperbolic, elliptic, trigonometric and rational fractional solutions. These solutions are vital for understanding some fundamentally complicated phenomena. The obtained solutions will be very helpful for applications such as fiber optics and plasma physics. Finally, we show how the conformable fractional derivative order affect the exact solutions of the FPCLLM-TDCs. Furthermore, we examine the effects of time-dependent coefficients when these coefficients take on special cases such as random variables, polynomials, and hyperbolic functions.

A Markovian Gauss inequality for asymmetric deviations from the mode of symmetric unimodal distributions

For a random variable with a unimodal distribution and finite second moment Gauß (1823) proved a sharp bound on the probability of the random variable to be outside a symmetric interval around its mode. An alternative proof for it under the assumption of a finite r r -th absolute moment is given ( r ≥ 1 r\geq 1 ), based on the Khintchine representation of unimodal random variables. A special instance of the resulting Narumi–Gauß inequality is the one with finite first absolute moment, which might be called a Markovian Gauß inequality. For symmetric unimodal distributions with finite second moment Semenikhin (2019) generalized the Gauß inequality to arbitrary intervals. For the class of symmetric unimodal distributions with finite first absolute moment we construct a Markovian version of it. Related inequalities of Volkov (1969) and Sellke and Sellke (1997) will be discussed as well.

Smoothness and Lévy concentration function inequalities for distributions of random diagonal sums

We present new explicit upper bounds for the smoothness of the distribution of the random diagonal sum S n = ∑ j = 1 n X j , π ( j ) S_n=\sum _{j=1}^nX_{j,\pi (j)} of a random n × n n\times n matrix X = ( X j , r ) X=(X_{j,r}) , where X j , r X_{j,r} are independent integer valued random variables, and π \pi denotes a uniformly distributed random permutation on { 1 , … , n } \{1,\dots ,n\} independent of X X . As a measure of smoothness, we consider the total variation distance between the distributions of S n S_n and 1 + S n 1+S_n . Our approach uses new auxiliary inequalities for a generalized normalized matrix hafnian and for inverse moments of non-negative random variables, which could be of independent interest. This approach is also used to prove upper bounds of the Lévy concentration function of S n S_n in the case of independent real valued random variables X j , r X_{j,r} .