Probability Theory Research Articles

GeosPIn: A MATLAB library for Geostatistical Petrophysical Inversion

Petrophysical inversion of seismic data is one of the key components of seismic reservoir characterization. The goal of petrophysical inversion is to estimate petrophysical properties of reservoir rocks, such as porosity, volumes of minerals or lithologies, and water and hydrocarbon saturations from seismic data. This process can be performed by combining seismic AVO modeling and rock physics relations with deterministic or probabilistic inverse theory methods, either in a two-step approach based on seismic AVO inversion and rock physics inverse mapping or in a single loop step that includes both. We present a comprehensive MATLAB library named Geostatistical Petrophysical Inversion (GeosPIn). The novelty of the library lies in a single-loop geostatistical inversion method based on the Probability Field Simulation (PFS) and the Ensemble Smoother Multiple Data Assimilation (ES-MDA), and its implementation for applications to 3D datasets. The library also includes a Bayesian petrophysical inversion based on the two-step approach for comparison. We illustrate the main algorithms, explain the structure of the source codes, and demonstrate the library applications through 1D and 3D examples.

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.

Encoding and decoding of information in general probabilistic theories

Encoding and decoding of information in general probabilistic theories

Combining Savage and Laplace: a new approach to ambiguity

AbstractThis 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.

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.

Sensitivity analysis for matched observational studies with continuous exposures and binary outcomes

Summary Matching is one of the most widely used study designs for adjusting for measured confounders in observational studies. However, unmeasured confounding may exist and cannot be removed by matching. Therefore, a sensitivity analysis is typically needed to assess a causal conclusion’s sensitivity to unmeasured confounding. Sensitivity analysis frameworks for binary exposures have been well-established for various matching designs and are commonly used in various studies. However, unlike the binary exposure case, there still lacks valid and general sensitivity analysis methods for continuous exposures, except in some special cases such as pair matching. To fill this gap in the binary outcome case, we develop a sensitivity analysis framework for general matching designs with continuous exposures and binary outcomes. First, we use probabilistic lattice theory to show our sensitivity analysis approach is finite-population- exact under Fisher’s sharp null. Second, we prove a novel design sensitivity formula as a powerful tool for asymptotically evaluating the performance of our sensitivity analysis approach. Third, to allow effect heterogeneity with binary outcomes, we introduce a framework for conducting asymptotically exact inference and sensitivity analysis on generalized attributable effects with binary outcomes via mixed- integer programming. Fourth, for the continuous outcomes case, we show that conducting an asymptotically exact sensitivity analysis in matched observational studies when both the exposures and outcomes are continuous is generally NP-hard, except in some special cases such as pair matching. As a real data application, we apply our new methods to study the effect of early-life lead exposure on juvenile delinquency. An implementation of the methods in this work is available in the R package doseSens.

Pareto and probability distributions

AbstractAlthough familiar with the developments in probability theory of his time, Vilfredo Pareto made little use of this tool in his writings, preferring theoretical constructions based on experimentation and observation. This article attempts to reconstruct Pareto’s overall approach to probability by examining his references to the distribution of income, an economic fact that lends itself to probabilistic investigation. The result of this research shows how Pareto alludes to the application of probability to income and social groups, but leaves the task to his followers.JEL classification B31, B4, C1.

Resolvent stochastic processes

A family r λ r_\lambda , λ ∈ C \lambda \in \mathbb {C} , of complex stochastic processes is introduced, which makes it possible to construct a probabilistic representation for the resolvent of the operator − 1 2 d 2 d x 2 -\frac {1}{2}\frac {d^2}{dx^2} . For λ = 0 \lambda =0 , the process r λ r_\lambda coincides with the Brownian local time process.

Data-driven modeling in digital twin for power system anomaly detection

Background: Power system anomaly detection is of great significance for realizing system situation awareness and early detection of system operating risks. In view of the complex operating conditions of the system, there are a large number of opaque links in the mechanism, and the anomaly detection approach based on physical mechanism modeling is prone to system errors due to assumptions, simplification, and transfer in the modeling process. This paper focuses on digital twin based data-driven approaches for power system anomaly detection to compensate for the limitation of physical methods in dynamical modeling. Methods: First of all, a digital twin framework for power system real-time analysis is constructed based on the concept of digital twin. Then, this paper conducts researches on the core of the designed framework, i.e., digital twin modeling. Considering the complexity of power system operating conditions, data-driven modeling is preferred and a random matrix and free probability theory based model for anomaly detection of system operating situation is constructed. Results: Simulation data with different spatiotemporal structure generated through a Monte Carlo experiment verified the sensitivity of the constructed model for data correlations. Meanwhile, the case on the system operating data generated through the IEEE 118-bus system validate the effectiveness of the proposed model for the system anomaly detection. Conclusions: The constructed data-driven model can accurately characterize the correlations among data elements, has good sensitivity to the variation of data spatial and temporal correlations, and can depict the data residuals better than the M-P law curve, which indicates the practicability and necessity of the constructed data-driven model for the digital twin modeling of power system anomaly detection.

Bayesian Mechanistic Inference, Statistical Mechanics, and a New Era for Monte Carlo.

On the one hand, much of computational chemistry is concerned with "bottom-up" calculations which elucidate observable behavior starting from exact or approximated physical laws, a paradigm exemplified by typical quantum mechanical calculations and molecular dynamics simulations. On the other hand, "top down" computations aiming to formulate mathematical models consistent with observed data, e.g., parametrizing force fields, binding or kinetic models, have been of interest for decades but recently have grown in sophistication with the use of Bayesian inference (BI). Standard BI provides an estimation of parameter values, uncertainties, and correlations among parameters. Used for "model selection," BI can also distinguish between model structures such as the presence or absence of individual states and transitions. Fortunately for physical scientists, BI can be formulated within a statistical mechanics framework, and indeed, BI has led to a resurgence of interest in Monte Carlo (MC) algorithms, many of which have been directly adapted from or inspired by physical strategies. Certain MC algorithms─notably procedures using an "infinite temperature" reference state─can be successful in a 5-20 parameter BI context which would be unworkable in molecular spaces of 103 coordinates and more. This Review provides a pedagogical introduction to BI and reviews key aspects of BI through a physical lens, setting the computations in terms of energy landscapes and free energy calculations and describing promising sampling algorithms. Statistical mechanics and basic probability theory also provide a reference for understanding intrinsic limitations of Bayesian inference with regard to model selection and the choice of priors.

Semantic analysis of two-syllabic terms of metrology. Part 2: Risk in measurements and calculations

Prior to the requirement for testing and calibration laboratories to take into account the risk of statistical assumptions, false positive and false negative decisions in international documents, it was found that the methodology of the “Guide to Expressing Measurement Uncertainty” based on the Bayes approach and the Monte Carlo method for calculating probabilistic risk characteristics is not applicable. A later draft revision of the “Manual on the Expression of Measurement Uncertainty” attempted to shift the interpretation of measurement uncertainty from the scattering parameter to the probability distribution. An attempt to contribute to solving the problem of definitional uncertainty in the International Dictionary of Basic and Basic Terms of Metrology was also unsuccessful. In the new version of the dictionary of general statistical terms and probability theory terms, the term measurement uncertainty is excluded, and one of the notes states that “the probability distribution fully describes the probabilistic properties of the uncertainty of the result”. However, due to the new requirements for risk calculations, international documents were urgently put into effect without radical assessments of the inapplicability of the Bayes approach and the Monte Carlo method, the disadvantages were renamed limitations, but there are no specific instructions for calculating risks. Based on the experience of the compositional approach to estimating accuracy, a procedure based on the convolution of probability distributions in the form of a modified reversal formula is recommended, which allows taking into account the definitive uncertainty in the moment approach. It is established that the method of accounting for the definitive uncertainty by convolution of uniform distributions is practically suggested in the text of the “Manual on the expression of measurement uncertainty”, but not used.

Theories of Uncertainty Communication: An Interdisciplinary Literature Review

This systematic literature review presents an interdisciplinary overview of theories tested in experiments on the effects of communicating uncertainty. Using a machine learning-aided pipeline, we selected and manually coded 413 experimental studies. We discuss core assumptions and predictions of the main theories of uncertainty communication. Most normative theorizing (e.g., Bayesianism, Expected Utility Theory) is rooted in Probability Theory, which is only suitable for addressing shallow and medium uncertainty. This explains the underrepresentation of experimental research into deep uncertainty communication. To foster a more comprehensive understanding of uncertainty communication effects, we identify research questions and theories deserving greater attention.

Analysis of the structural reliability of communication networks supporting protective switching mechanisms for one protected section and one backup section

Objectives. The service level agreement is an important tool used in building reasonable relations between subscribers and operators of telecommunication networks. This includes the quality of services provided. One key component is reliability as assessed by the availability factor. The most suitable model for assessing the reliability of the service provided is a random graph model based on the service contour. This is the set of technical resources involved in the provision of this service. In this formulation, the assessment of the reliability of the service is based on the reliability of elements which constitute the telecommunications network (graph), nodes (vertices) and communication lines (edges). At the same time, the availability factors of nodes and lines are determined by the design features of the distribution environment, as well as the technical means used to organize them. The purpose of this work is to develop an approach to analyzing the reliability of telecommunication networks which support protective switching mechanisms for one protected and one backup sections.Methods. The following methods are used: theory of random graphs, matrices, probabilities and computer modeling.Results. The elements of the route, both basic and reserving, are divided into three groups. The first indicates permanent unchangeable parts of the paths, the second group identifies the reserved sections, and the third group indicates the reserving sections. At the same time, each of the reserved and reserving sections is formed on the basis of specified preferences. They are usually aimed at increasing the resulting reliability, although other rules may be used. In the case of protective switching schemes for one protected section and one backup sections, a variant of forming routes used for further calculations of the reliability indicator is shown.Conclusions. Using the example of a backbone network, the study shows that the use of protective switching mechanisms for the case of one required transmission route demonstrates a significant increase in reliability, with the exception of the use of protective switching in sections. This is primarily due to the topology features of the network under consideration.

Diophantine conditions in the law of the iterated logarithm for lacunary systems

AbstractIt is a classical observation that lacunary function systems exhibit many properties which are typical for systems of independent random variables. However, it had already been observed by Erdős and Fortet in the 1950s that probability theory’s limit theorems may fail for lacunary sums $$\sum f(n_k x)$$ ∑ f ( n k x ) if the sequence $$(n_k)_{k \ge 1}$$ ( n k ) k ≥ 1 has a strong arithmetic “structure”. The presence of such structure can be assessed in terms of the number of solutions $$k,\ell $$ k , ℓ of two-term linear Diophantine equations $$a n_k - b n_\ell = c$$ a n k - b n ℓ = c . As the first author proved with Berkes in 2010, saving an (arbitrarily small) unbounded factor for the number of solutions of such equations compared to the trivial upper bound, rules out pathological situations as in the Erdős–Fortet example, and guarantees that $$\sum f(n_k x)$$ ∑ f ( n k x ) satisfies the central limit theorem (CLT) in a form which is in accordance with true independence. In contrast, as shown by the first author, for the law of the iterated logarithm (LIL) the Diophantine condition which suffices to ensure “truly independent” behavior requires saving this factor of logarithmic order. In the present paper we show that, rather surprisingly, saving such a logarithmic factor is actually the optimal condition in the LIL case. This result reveals the remarkable fact that the arithmetic condition required of $$(n_k)_{k \ge 1}$$ ( n k ) k ≥ 1 to ensure that $$\sum f(n_k x)$$ ∑ f ( n k x ) shows “truly random” behavior is a different one at the level of the CLT than it is at the level of the LIL: the LIL requires a stronger arithmetic condition than the CLT does.

Plasma profile reconstruction supported by kinetic modeling

Combining the analysis of multiple diagnostics and well-chosen prior information in the framework of Bayesian probability theory, the Integrated Data Analysis code (IDA Fischer et al 2010 Fusion Sci. Technol. 58 675–84) can provide density and temperature radial profiles of fusion plasmas. These IDA-fitted measurements are then used for further analysis, such as discharge simulations and other experimental data analysis. Since IDA considers measurement data, which is frequently fragmentary, with statistical and systematic uncertainties, which are often difficult to quantify, from a heterogeneous set of diagnostics, the fitted profiles and their gradients may be in contradiction to well-established expectations from transport theory. Using the modeling suite ASTRA coupled with the quasi-linear transport solver TGLF, we have created a loop in which simulated profiles and their uncertainties are fed back into IDA as an additional prior, thus providing constraints about the physically reasonable parameter space. We apply this physics-motivated prior to several different plasma scenarios and find improved heat flux match, while still matching the experimental data. This work feeds into a broader effort to make IDA more robust against measurement uncertainties or lack of measurements by combining multiple transport solvers with different levels of complexity and computing costs in a multi-fidelity approach.

Uncertain stochastic hybrid zero-sum games based on forward uncertain difference equations and backward stochastic difference equations

We investigate the interplay between forward uncertain difference equations and backward stochastic difference equations, which belong to distinct mathematical frameworks: uncertainty theory and probability theory, respectively. By establishing the connections between uncertain expectation, probabilistic expectation, and chance expectation, we develop the chance theory framework for analyzing uncertain stochastic hybrid zero-sum games constrained by both forward uncertain difference equations and backward stochastic difference equations. Through an equivalent transformation, we convert these games into deterministic difference equation solving problems. By backward-solving the difference equations and applying mathematical induction, analytical expressions for the equilibrium solutions of two types of uncertain stochastic hybrid zero-sum games are obtained. Furthermore, we explore solution methods for zero-sum games with different operational orders of uncertain and probabilistic expectations and highlight the fundamental differences through numerical examples.