Probability Theory Research Articles

Order-Guided Non-Conservative Maximum Flow in Uncertain Network Interdiction Problem with Budget Constraint

Probability theory in decision science fails in the absence of sufficient data. In these situations, uncertainty theory is applied to the problem-solving process, taking into account the opinions of the domain experts. In order to minimize the maximum flow carried by the evader, we describe an interdiction problem in an uncertain network with intermediate storage and resource constraints. Such approaches would help law enforcement combat drug-related and smuggling activities. This work first converts the bi-level deterministic conservative model of the interdictor vs evader problem (IEP) into a non-conservative deterministic IEP. With intermediate storage, a binary interdiction variable, and an interdiction resource constraint, the model is further transformed into an uncertain measure-based model. Then, utilizing uncertainty theory, the uncertain model is again converted into its deterministic equivalent. We have presented two techniques to identify the arc that the interdictor needs to block and to direct the order-guided maximum flow that the evader needs to deliver to the target. The models’ and algorithms’ efficacy is confirmed by an example. Lastly, a graphic representation of the shifting trend of the value of the objective function (VOF) is shown for a range of confidence levels.

A SENSITIVITY ANALYSIS METHOD COMBINING DEMPSTER-SHAFER THEORY AND MACHINE LEARNING FOR ENERGY-SAVING EVALUATION OF BUILDING OCCUPANT BEHAVIOR

ABSTRACT For a very long time, the research of the sensitivity analysis of occupant behavior to energy assessment has been in the spotlight. The key element of the research is determining the exact probability of occupant behavior uncertainty. However, due to the specificity of occupant behavior, data on occupant behavior from different independent sources of information can differ significantly. This paper explores the use of Dempster-Shafer theory to the sensitivity analysis of energy evaluation of occupant behavior in buildings. The Dempster-Shafer theory is an imprecise probability theory that allows the system to create assumed confidence intervals based on interval values probability combined with knowledge of uncertainty factors from many different sources of information. The findings show that the data processing approach based on Dempster-Shafer theory provides effective and reliable information for evaluating energy related to human behavior in buildings. To begin with, the sensitivity analysis process might be accelerated by applying machine learning to process the data. Then, in order to ensure the accuracy of the simulation results, multiple learning methods can be used. Finally, in this paper, model parameters were chosen based on the specific circumstances as soon as the model had been built in order to effectively reduce costs related to operation and increase model accuracy. To establish the final results, the model is evaluated using global sensitivity analysis methods.

Consistency of Relations over Monoids

The interplay between local consistency and global consistency has been the object of study in several different areas, including probability theory, relational databases, and quantum information. For relational databases, Beeri, Fagin, Maier, and Yannakakis showed that a database schema is acyclic if and only if it has the local-to-global consistency property for relations, which means that every collection of pairwise consistent relations over the schema is globally consistent. More recently, the same result has been shown under bag semantics. In this paper, we carry out a systematic study of local vs. global consistency for relations over positive commutative monoids, which is a common generalization of ordinary relations and bags. Let K be an arbitrary positive commutative monoid. We begin by showing that acyclicity of the schema is a necessary condition for the local-to-global consistency property for K-relations to hold. Unlike the case of ordinary relations and bags, however, we show that acyclicity is not always sufficient. After this, we characterize the positive commutative monoids for which acyclicity is both necessary and sufficient for the local-to-global consistency property to hold; this characterization involves a combinatorial property of monoids, which we call the transportation property. We then identify several different classes of monoids that possess the transportation property. As our final contribution, we introduce a modified notion of local consistency of K-relations, which we call pairwise consistency up to the free cover. We prove that, for all positive commutative monoids K, even those without the transportation property, acyclicity is both necessary and sufficient for every family of K-relations that is pairwise consistent up to the free cover to be globally consistent.

New Combinatorial Identity for the Set of Partitions and Limit Theorems in Finite Free Probability Theory

Abstract We provide a refined combinatorial identity for the set of partitions of $\{1,\dots , n\}$, which plays an important role in investigating several limit theorems related to finite free convolutions. Firstly, we present the finite free analogue of Sakuma and Yoshida’s limit theorem. That is, we provide the limit of $\{D_{1/m}((p_{d}^{\boxtimes _{d}m})^{\boxplus _{d}m})\}_{m\in{\mathbb{N}}}$ as $m\rightarrow \infty $ in two cases: (i) $m/d\rightarrow t$ for some $t>0$ or (ii) $m/d\rightarrow 0$. The second application presents a central limit theorem for finite free multiplicative convolution. We establish a connection between this theorem and the multiplicative free semicircular distributions through combinatorial identities. Our last result gives alternative proofs for Kabluchko’s limit theorems concerning the unitary Hermite and the Laguerre polynomials.

Neutrosophic Topp-Leone Distribution for Interval-Valued Data Analysis

Many issues in real life are riddled with confusion, vagueness, and ambiguity. The Topp-Leone distribution is a significant one-parameter probability distribution in the context of classical probability theory. There is a gap in the literature when it comes to dealing with circumstances involving interval-valued data with a classical Topp-Leone distribution. So, in this connection, neutrosophic Topp-Leone distribution is presented in this paper as an extension of the traditional Topp-Leone distribution. The new neutrosophic distribution takes into account the indeterminacy and crisp form of interval-valued distributions. The suggested distribution's mathematical features were derived, including moments and related measures, quantile function, survival, hazard, inverted hazard functions, and mills ratio. Maximum likelihood is used to estimate model parameters. Finally, a complex dataset is utilized to show the usefulness of the proposed distributions.

Switching particle systems for foraging ants showing phase transitions in path selections

Switching interacting particle systems studied in probability theory are the stochastic processes of hopping particles on a lattice made up of slow and fast particles, where the switching between these types of particles occurs randomly at a given transition rate. This paper explores how such stochastic processes involving multiple particles can model group behaviors of ants. Recent experimental research by the last author’s group has investigated how ants switch between two types of primarily relied cues to select foraging paths based on the current situation. Here, we propose a discrete-time interacting random walk model on a square lattice, incorporating two types of hopping rules. Numerical simulation results demonstrate global changes in selected homing paths, transitioning from trailing paths of the ‘pheromone road’ to nearly optimal paths depending on the switching parameters. By introducing two types of order parameters characterizing the dependence of homing duration distributions on switching parameters, we discuss these global changes as phase transitions in ant path selections. We also study critical phenomena associated with continuous phase transitions.

The generalised distribution semantics and projective families of distributions

We generalise the distribution semantics underpinning probabilistic logic programming by distilling its essential concept, the separation of a free random component and a deterministic part. This abstracts the core ideas beyond logic programming as such to encompass frameworks from probabilistic databases, probabilistic finite model theory and discrete lifted Bayesian networks. To demonstrate the usefulness of such a general approach, we completely characterise the projective families of distributions representable in the generalised distribution semantics and we demonstrate both that large classes of interesting projective families cannot be represented in a generalised distribution semantics and that already a very limited fragment of logic programming (acyclic determinate logic programs) in the deterministic part suffices to represent all those projective families that are representable in the generalised distribution semantics at all.

Forecasting technique of transport need for population of cities with high tourist potential during peak seasons

Introduction. One of the key problems in ensuring the quality of recreation for the population is the provision of transport services that meet the transport needs of vacationers. The study of the transport needs of the population of resort cities during the holiday season is caused by the need to assess changing passenger flows as a result of a multiple increase due to vacationers in accordance with their place of attraction and possible adjustments to the routes of regular passenger transportation by road. This determines the relevance of the topic of this article. The purpose of the article is to develop a methodology for determining the needs for transport services for the population and vacationers in cities with high resort potential during the busiest periods, which makes it possible to design transportation processes along regular routes.Materials and methods. General scientific methods of analysis and synthesis, the provisions of probability theory and mathematical statistics, mathematical modeling, as well as the provisions of the technology of transport processes were used as the main research methods. The work carried out the zoning of the city into transport areas with a center of gravity at stopping points of urban passenger transport. Calculations of potential transport needs were carried out on the basis of a survey of vacationers and field observations, which were processed using standard Microsoft Excel and Statistika software.Results. The main result of the work is a methodology for determining the potential transport needs of the population and vacationers in cities with high tourism potential for the development and adjustment of routes for regular passenger transportation by road during the holiday season, which represents the scientific novelty of the research. The technique contains a mathematical model and algorithm.Discussion and conclusion. The use of this methodology will enable the organizer of transportation in resort cities to optimize the organization of regular passenger transportation by road during peak seasons.

The expanding universe of the geometric mean

AbstractIn this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital $$C^*$$ C ∗ -algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the $$C^*$$ C ∗ -algebra of operators on a Hilbert space, even to general unital $$C^*$$ C ∗ -algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.

Beyond green borders: an innovative model for sustainable transportation in supply chains

Modern requirements necessitate the establishment of sustainable transportation systems, considering the substantial growth in transportation activities over recent years, which is expected to continue. Companies are facing the challenge of modeling their system transport to align with green principles. Sustainable transport relied on involving diverse stakeholders, particularly scientific research, in the development of this field. In light of this, maintaining sustainable transport quality involves conducting thorough investigations into an innovative study focusing on an uncertain interval programming model for a multi-stage, multi-objective, multi-product transportation challenge within budget constraints and safety measures in a green supply chain. Human languages often contain imperfect or unknown information, inherently lacking certainty; achieving precision in describing existing states or future outcomes is frequently unattainable. In probability theory, sufficient historical information is crucial for estimating probability distributions; while in fuzzy theory, determining a reliable membership function proves challenging; hence, there is often a hesitant estimation of the degree of belief in the occurrence of each condition. Addressing such uncertainties, the theory of uncertain intervals proves highly valuable. Given these considerations, the elements of the specified problem are recognized as uncertain intervals. To manage this lack of assurance, a fusion of interval theory and methods from uncertain programming is used to formulate two distinct models: an expected value model and a chance-constrained model. The equivalent deterministic models are then formulated and solved utilizing Weighted Sum Method, fuzzy programming, and goal programming. Following this, a numerical example is utilized to assess the model’s performance, and the results obtained are compared. Finally, the document concludes with a sensitivity analysis and outlines future directions.

Derivation of standard quantum theory via state discrimination

It is a key issue to characterize the model of standard quantum theory out of general models by an operational condition. The framework of general probabilistic theories (GPTs) is a new information theoretical approach to single out standard quantum theory. It is known that traditional properties, for example, Bell-CHSH inequality, are not sufficient to single out standard quantum theory among possible models in GPTs. As a more precise property, we focus on the bound of the performance for an information task called state discrimination in general models. We give an equivalent condition for outperforming the minimum discrimination error probability under the standard quantum theory given by the trace norm. Besides, by applying the equivalent condition, we characterize standard quantum theory out of general models in GPTs by the bound of the performance for state discrimination.

Estimating the Duration of Construction Works Using Fuzzy Modeling to Assess the Impact of Risk Factors

One of the most pressing issues in the implementation of construction projects is the extension of planned deadlines, significantly impacting project costs. This situation often arises due to inaccurate estimation of construction durations, which rely on normative values without accounting for factors hindering construction progress. Consequently, this article aims to develop an innovative approach for assessing construction durations, considering specific risk factors and their influence on construction activities. Given the difficulty of determining risk factors and their effects during the design phase using classical probability theory, characterized by unknown probability distributions, it is highlighted that this scenario represents planning and implementation under conditions of non-statistical uncertainty. Therefore, the article proposes an approach utilizing elements of fuzzy set theory, particularly fuzzy rules and linguistic variables, to determine delays in individual construction tasks. The proposed approach involves estimating extensions of construction timelines based on a specified probability level of occurrence for risk events and their impact. Additionally, the article provides a theoretical description of the proposed approach and practical calculation examples, demonstrating that the authors’ approach significantly enhances the accuracy of construction timeline forecasts, providing more reliable data for project planning and management.

Enhancing UCB-tuned and Asymptotically Optimal UCB Algorithms through Weighted Average Techniques in Multi-Armed Bandit Scenarios

This paper delves into the complexities of the Multi-Armed Bandit (MAB) problem, a fundamental concept in reinforcement learning and probability theory, with a focus on its application in recommendation systems and dynamic fields such as dynamic pricing and investment. It begins by shedding light on the essential paradox at the heart of the MAB problem – the balance between exploration and exploitation within limited parameters. The study primarily centers on Upper Confidence Bound (UCB) policies, especially UCB-tuned and Asymptotically Optimal UCB, noted for their adept balance between exploration and utilization. The novel contribution of this research is the enhancement of these UCB policies via an innovative weighted average method, leading to the development of WA-UCB-tuned and WA Asymptotically Optimal UCB algorithms. The research rigorously compares these optimized iterations with traditional UCB1, UCB-tuned, and Asymptotically Optimal UCB across varied MAB models featuring different numbers of arms. This study provides an exhaustive introduction to the MAB problem and pertinent UCB policies, the methodology behind the weighted average optimization, extensive experimental analysis, and comprehensive evaluations of the findings. The results showcase marked improvements in algorithmic performance, suggesting significant advancements in the domain of recommendation systems and other applications of the MAB problem.

Combining Savage and Laplace: a new approach to ambiguity

This paper presents a new representation of preference orderings for the study of ambiguity-related decision-making. The central feature is a preference-based decomposition of subjective probabilities that provides information about inherent ambiguity. The probability decomposition is combined with a utility function reflecting the decision-maker’s attitude toward ambiguity. The proposed theory generalizes Savage’s SEU and allows for a straightforward measurement of ambiguity and ambiguity aversion while keeping concepts for measuring risk and risk attitudes unaffected. For the measurement of ambiguity, concepts of probability theory can be used since decision acts can be interpreted as two-dimensional probability distributions. The proposed measure of ambiguity aversion exploits the properties of the utility function in the same way as the Arrow/Pratt measure of risk aversion.

METHODOLOGICAL FEATURES OF TEACHING RANDOM VARIABLES AT THE COMPREHENSIVE SECONDARY SCHOOL

In the evolving landscape of educational methodologies, the integration of complex mathematical concepts such as random variables into secondary school curriculums poses significant challenges and opportunities. As the global educational framework increasingly emphasizes the importance of a robust mathematical foundation, understanding and teaching stochastic elements within the school system has become crucial. This article delves into the nuanced methodological features essential for imparting knowledge of random variables effectively to secondary school students. We explore the historical context of mathematical education reforms in Armenia, highlighting the recent inclusion of stochastic topics in the school programs since 2004. The reformed educational standards introduced in 2020 and implemented in 2023 have further emphasized the need for a structured approach to teaching random variables, alongside other advanced mathematical topics. Through this lens, we examine the preparedness of educators, the adequacy of existing teaching methods, and the potential cognitive hurdles faced by students. Our analysis is rooted in both qualitative assessments and a theoretical framework that aims to bridge the gap between abstract mathematical theories and practical understanding, ensuring that students not only learn but also appreciate the significance of probability theory in everyday contexts .

On the properties of the anisotropic multivariate Hermite-Gauss functions

The Hermite-Gauss basis functions have been extensively employed in classical and quantum optics due to their convenient analytic properties. A class of multivariate Hermite-Gauss functions, the anisotropic Hermite-Gauss functions, arise by endowing the standard univariate Hermite-Gauss functions with a positive definite quadratic form. These multivariate functions admit useful applications in optics, signal analysis and probability theory, however they have received little attention in literature. In this paper, we examine the properties of these functions, with an emphasis on applications in computational optics.

Application of active and passive filter-compensating devices to reduce the negative effects of harmonic distortions

Subject of research: processes occurring in electrical networks with induction heating installations. Purpose of research: filter compensators. Object of research: study of design features of active and passive harmonic filters. Methods of research: in the process of performing research, scientific and technical generalization of literary sources according to the initial prerequisites for research, methods of probability theory and mathematical statistics were used. Experimental studies were carried out by a complex method using the Resurs-PKE device. Main results of research: on the basis of conducted researches it is shown that filter-compensating devices can effectively reduce the negative impact of harmonic distortions.

Algebraic groups in non-commutative probability theory revisited

Algebraic groups in non-commutative probability theory revisited

A new approach for the assessment of coastal flooding risk. Application in Rhodes island, Greece

A directional, copula-based, Monte Carlo approach is proposed for the quantitative assessment of a new definition of coastal flooding risk, namely a probability-based index due to extreme combinations of wave conditions and sea levels, which could be a key element for coastal zone planning and management. This approach differs from the standard practice, which focuses on the extreme events and studies either their specific return values or their worst-case scenarios. The adopted approach incorporates mean wave direction as a covariate into the probabilistic representation of sea states, and via the use of a variety of probabilistic and statistical tools, represents a total of five stochastic variables, namely the significant wave height, the peak wave period, the mean wave direction, the storm surge, and tide. Two directional, alternative methods have been developed; a trivariate copula- and a bivariate copula-based method, that seemed to produce quite similar results. Four types of copulas have been examined for their goodness-of-fit, i.e., three Archimedean copulas, namely, the Clayton, the Gumbel, and the Frank copulas, as well as the elliptical Gaussian copula. A comparative analysis of the two methods is performed aiming to point out their pros, as well as their accuracy and complexity issues. The application of the proposed method to the natural beaches of the Rhodes Island in Greece, in combination with a recently developed formula for wave run-up, highlighted the most crucial parameters affecting the coastal flooding risk in the study area.

Probabilistic Analysis of Random Check Intrusion Detection System

The ubiquitous adoption of network-based technologies has left organizations vulnerable to malicious attacks. It has become vital to have effective intrusion detection systems (IDS) that protect the network from attacks. In this paper, we study the intrusion detection problem through the lens of probability theory. We consider a situation where a network receives random malicious signals at discrete time instances, and an IDS attempts to capture these signals via a random check process. We aim to develop a probabilistic framework for intrusion detection under the given scenario. Concretely, we calculate the detection rate of a network attack by an IDS and determine the expected number of detections. We perform extensive theoretical and experimental analyses of the problem. The results presented in this paper would be helpful tools for designing and analyzing intrusion detection systems. We propose a probabilistic framework that could be useful for IDS experts; for a network-based IDS that monitors in real-time, analyzing the entire traffic flow can be computationally expensive. By probabilistically sampling only a fraction of the network traffic, the IDS can still perform its task effectively while reducing the computational cost. However, checking only a fraction of the traffic increases the possibility of missing an attack. This research can help IDS designers achieve appropriate detection rates while maintaining a low false alarm rate. The groundwork laid out in this paper could be used for future research on understanding the probabilities related to intrusion detection.