Probability Distributions Of Variables Research Articles

Predicting Electric Vehicle Charging Demand in Residential Areas Using POI Data and Decision‐Making Model

Transport electrification is a crucial element of the ongoing energy transition, essential for achieving carbon peaking and carbon neutrality goals. The proliferation of electric vehicles (EVs) introduces significant challenges to power distribution network stability due to their aggregated charging load in residential areas, particularly during peak electricity consumption periods. This paper proposes a method to predict the spatiotemporal distribution of EV charging demand in residential areas using geographic information points of interest (POI) data features and a decision‐making model. Utilizing real historical data, probability distribution models for EV users' arrival times and charging characteristics were constructed using Gaussian Mixture Models (GMM). The spatiotemporal characteristics of EV travel and charging behaviors were analyzed, and a comprehensive charging decision model incorporating both emergency and stochastic scenarios was developed. The model's efficacy in capturing the probability distributions of characteristic variables was validated through a case study. The results demonstrate the model's potential for accurately predicting EV charging demand, providing valuable insights for infrastructure planning and resource allocation. © 2024 Institute of Electrical Engineers of Japan and Wiley Periodicals LLC.

Robust optimization bidding strategy for user-side resource-side participation in the market distribution of electrical energy and peaking ancillary services considering risk expectations and opportunity constraints

Compared to traditional resources, user-side resources are of various types and have more significant uncertainty about their regulatory capacity, leading to difficulties in coordinating decisions about their simultaneous participation in the electric energy and peaking ancillary services markets. This paper proposes a joint bidding decision-making method for the day-ahead electricity energy and peak shaving auxiliary service market based on distributed robust opportunity constraints, which addresses the problem of difficulty in using an accurate probability density distribution to represent the uncertainty process of user-side resources. Firstly, a data-driven method for characterizing the uncertainty of load regulation capacity is investigated, and fuzzy sets are constructed without assuming specific probability distributions of random variables. Then, to minimize the risk expectation of the joint bidding cost on the customer side, a bidding strategy that considers the uncertainty is proposed. Finally, an example simulation verifies the reasonableness and effectiveness of the proposed joint bidding method, and the results show that the constructed model overcomes the problem of over-conservatism of the robust model, and the computational adaptability is better than that of the stochastic model, which achieves a better balance between robustness and economy.

Reformulation and Enhancement of Distributed Robust Optimization Framework Incorporating Decision-Adaptive Uncertainty Sets

Distributionally robust optimization (DRO) is an advanced framework within the realm of optimization theory that addresses scenarios where the underlying probability distribution governing the data is uncertain or ambiguous. In this paper, we introduce a novel class of DRO challenges where the probability distribution of random variables is contingent upon the decision variables, and the ambiguity set is defined through parameterization involving the mean and a covariance matrix, which also depend on the decision variables. This dependency makes DRO difficult to solve directly; therefore, first, we demonstrate that under the condition of a full-space support set, the original problem can be reduced to a second-order cone programming (SOCP) problem. Subsequently, we solve this second-order cone programming problem using a projection differential equation approach. Compared with the traditional methods, the differential equation method offers advantages in providing continuous and smooth solutions, offering inherent stability analysis, and possessing a rich mathematical toolbox, which make the differential equation a powerful and versatile tool for addressing complex optimization challenges.

Implementing Sobol's Global Sensitivity Analysis to SFRC's Flexural Strength Predictive Equation

Steel fibers are essential for SFRC since they strengthen the material’s resistance to bending and cracking stresses and guarantee its endurance. However, the uncertainty of input parameters such as concrete compositions and steel fibers causes the stochasticity of flexural strength. The study uses big data to analyze the influence of fibers and other concrete compositions on the flexural strength properties of SFRC. For this purpose, the study focuses on developing predictive models for SFRC flexural properties based on a comprehensive database comprising two hundred and seven experimental results recorded by seventeen researchers. Bayesian Model Averaging is employed to identify significant components that influence the overall flexural strength and to develop a predictive flexural strength model. Monte Carlo simulation generates big data by utilizing the probability distribution of input variables and the predictive flexural strength model. The study used Sobol’s global sensitivity analysis method to assess various input parameters’ sensitivity to SFRC flexural strength based on the generated database. The impact order of input variables on the flexural strength is identified, as determined by the Sobol’ Indice.

Two-stage multi-objective optimization based on knowledge-driven approach: A case study on production and transportation integration

The multi-objective evolutionary algorithm (MOEA) has been widely applied to solve various optimization problems. Existing search models based on dominance and decomposition are extensively used in MOEAs to balance convergence and diversity during the search process. In this paper, we propose for the first time a two-stage MOEA based on a knowledge-driven approach (TMOK). The first stage aims to find a rough Pareto front through an improved nondominated sorting algorithm, whereas the second stage incorporates a dynamic learning mechanism into a decomposition-based search model to reasonably allocate computational resources. To further speed up the convergence of TMOK, we present a Markov chain-based TMOK (MTMOK), which can potentially capture variable dependencies. In particular, MTMOK employs a marginal probability distribution of single variables and an N-state Markov chain of two adjacent variables to extract valuable knowledge about the problem solved. Moreover, a simple yet effective local search is embedded into MTMOK to improve solutions through variable neighborhood search procedures. To illustrate the potential of the proposed algorithms, we apply them to solve a distributed production and transportation-integrated problem encountered in many industries. Numerical results and comparisons on 54 test instances with different sizes verify the effectiveness of TMOK and MTMOK. We have made the 54 instances and the source code of our algorithms publicly available to support future research and real-life applications.

HYACINTH: HYdrogen And Carbon chemistry in the INTerstellar medium in Hydro simulations

Aims. We present a new sub-grid model, HYACINTH – HYdrogen And Carbon chemistry in the INTerstellar medium in Hydro simulations – for computing the non-equilibrium abundances of H2 and its carbon-based tracers, namely CO, C, and C+, in cosmological simulations of galaxy formation. Methods. The model accounts for the unresolved density structure in simulations using a variable probability distribution function of sub-grid densities and a temperature-density relation. Included is a simplified chemical network that has been tailored for hydrogen and carbon chemistry within molecular clouds and easily integrated into large-scale simulations with minimal computational overhead. As an example, we applied HYACINTH to a simulated galaxy at redshift z ~ 2.5 in post-processing and compared the resulting abundances with observations. Results. The chemical predictions from HYACINTH are in reasonable agreement with high-resolution molecular-cloud simulations at different metallicities. By post-processing a galaxy simulation with HYACINTH, we reproduced the H I − H2 transition as a function of the hydrogen column density NH for both Milky-Way-like and Large-Magellanic-Cloud-like conditions. We also matched the NCO versus NH2 relation inferred from absorption measurements towards Milky-Way molecular clouds, although most of our post-processed regions occupy the same region as (optically) dark molecular clouds in the NCO – NH2 plane. Column density maps reveal that CO is concentrated in the peaks of the H2 distribution, while atomic carbon more broadly traces the bulk of H2 in our post-processed galaxy. Based on both the column density maps and the surface density profiles oŕ the different gas species in the post-processed galaxy, we find that C+ maintains a substantially high surŕace density out to ~10 kpc as opposed to other components that exhibit a higher central concentration. This is similar to the extended [C II] emission ŕound in some recent observations at high redshifts.

Probabilistic model for fatigue damage estimation of wind turbines with hidden markov model and neural network

The growing demand for sustainable and affordable energy sources has led to rapid development in wind energy systems. Ensuring the long-term performance and reliability of wind turbines requires accurate estimation of fatigue damage under diverse environmental conditions. By incorporating probabilistic methods, this paper presents a comprehensive framework for long-term fatigue damage evaluation for wind turbines with high accuracy and efficiency. The proposed framework adopts the multivariate kernel density estimation to construct the high-dimensional joint probability distribution of environmental variables. A hidden Markov model serves as the driver to build the connection between historical records and future events. The short-term fatigue damage is estimated by the deep neural network based on future environmental parameters. The long-term fatigue damage is then achieved by accumulating the short-term fatigue damage evaluations with sufficient event instances. To demonstrate the applicability and effectiveness of the proposed framework, a case study is conducted with a wind turbine in the North Sea region. The fatigue damage analysis results highlight the necessity of regular inspections and maintenance for wind turbines to ensure their long-term service life and optimal performance. The framework is capable of considering a wide range of environmental factors and accounting for the inherent uncertainties and stochastic nature, providing a robust and comprehensive approach for estimating fatigue damage of wind turbines as well as other structures under long-term environmental conditions.

Quantum model regression for generating fuzzy numbers in adiabatic quantum computing

This paper addresses the problem of inherent fuzziness in real-world data, arising from uncertainties, complexities, and limitations of traditional statistical methods. We introduce a pioneering method leveraging Adiabatic Quantum Computing (AQC), based on an adiabatic quantum regression model, to generate fuzzy numbers—ideal tools owing to their extension of real numbers and robust arithmetic properties. This innovative approach overcomes challenges in Quantum Machine Learning (QML) and limitations of Noisy Intermediate-Scale Quantum (NISQ) computers, providing a solution superior to conventional statistical methods and addressing the crucial issue of exponential power increase. Unique to this work, we offer rare closed-form expressions to produce fuzzy numbers through AQC and emphasise the integration of random variables and fuzzy numbers to encapsulate uncertainty fully. We propose a novel transformation of the Cumulative Distribution Function (CDF) into triangular and trapezoidal fuzzy numbers using AQC, enabling a comprehensive description of probability distributions of random variables. Experimental results are detailed, demonstrating the significant applicability and breakthroughs of this research in addressing data fuzziness.

Causal Structure Learning with Conditional and Unique Information Groups-Decomposition Inequalities.

The causal structure of a system imposes constraints on the joint probability distribution of variables that can be generated by the system. Archetypal constraints consist of conditional independencies between variables. However, particularly in the presence of hidden variables, many causal structures are compatible with the same set of independencies inferred from the marginal distributions of observed variables. Additional constraints allow further testing for the compatibility of data with specific causal structures. An existing family of causally informative inequalities compares the information about a set of target variables contained in a collection of variables, with a sum of the information contained in different groups defined as subsets of that collection. While procedures to identify the form of these groups-decomposition inequalities have been previously derived, we substantially enlarge the applicability of the framework. We derive groups-decomposition inequalities subject to weaker independence conditions, with weaker requirements in the configuration of the groups, and additionally allowing for conditioning sets. Furthermore, we show how constraints with higher inferential power may be derived with collections that include hidden variables, and then converted into testable constraints using data processing inequalities. For this purpose, we apply the standard data processing inequality of conditional mutual information and derive an analogous property for a measure of conditional unique information recently introduced to separate redundant, synergistic, and unique contributions to the information that a set of variables has about a target.

An adaptive data-driven subspace polynomial dimensional decomposition for high-dimensional uncertainty quantification based on maximum entropy method and sparse Bayesian learning

Polynomial dimensional decomposition (PDD) is a surrogate method originated from the ANOVA (analysis of variance) decomposition, and has shown powerful performance in uncertainty quantification (UQ) accuracy and convergence recently. However, complex high-dimensional problems result in a large number of polynomial basis functions, leading to heavy computational burden, and the probability distributions of the input random variables are indispensable for PDD modeling and UQ, which may be unavailable in practical engineering. This study establishes an adaptive data-driven subspace PDD (ADDSPDD) for high-dimensional UQ, which employs two types of data for modeling the PDD basis function and the low-dimensional subspace directly, namely, the data of input random variables and the input-response samples. Firstly, we propose a data-driven zero-entropy criterion-based maximum entropy method for reconstructing the probability density functions (PDF) of input variables. Then, with the aid of the established PDFs, a data-driven subspace PDD (DDSPDD) is proposed based on the whitening transformation. To recover the subspace of the function of interest accurately and efficiently, we put forward an approximate active subspace method (AAS) based on the Taylor expansion under some mild premises. Finally, we integrate an adaptive learning algorithm into the DDSPDD framework based on the sparse Bayesian learning theory, obtaining our ADDSPDD; thus, the real subspace and the significant PDD basis functions can be identified with limited computational budget. We validate the proposed method by using four examples, and systematically compare four existing dimension-reduction methods with the AAS. Results show that the proposed framework is effective and the AAS is a good choice when the corresponding assumptions are satisfied.

Aftershock probabilistic seismic hazard analysis based on enhanced Bayesian network considering frequency information

AbstractBayesian network (BN) is an important tool in probabilistic seismic risk analysis (PSRA) due to its holistic nature and powerful probabilistic inference capabilities. However, while the information that can be stored by BN includes variable values, probability distributions, and relationships between variables, it does not involve event quantities. The results obtained from PSRA and probabilistic seismic hazard analysis (PSHA) are commonly frequency‐related (the number of occurrences per unit time), fundamentally describing event numbers. BN assumes a fixed event number relationship between nodes, but the aftershock counts vary for different mainshock magnitudes. This limitation further restricts its effectiveness in lifespan PSRA for structures. Therefore, this study aims to propose an enhanced BN (EBN) for modeling aftershock PSHA (APSHA) considering frequency information. This paper first introduces the modeling method using BN, including PSHA and a simplified model of APSHA using Båth's law. Next, an EBN that can effectively address frequency issues between the mainshock and aftershocks is presented for modeling APSHA considering the modified Omori's law, while the mathematical principles of both forward and backward inferences for the EBN are elaborated in detail. Then, the article carefully analyzes the utilization of the EBN for APSHA modeling under different scenarios where the information regarding the mainshock is known to varying extents, followed by the extension of the EBN in a time domain to obtain time‐dependent aftershock hazard. Finally, an example of APSHA in a specific location in China is illustrated.

Multi-period descriptive sampling for scenario generation applied to the stochastic capacitated lot-sizing problem

Using scenarios to model a stochastic system’s behavior poses a dilemma. While a large(r) set of scenarios usually improves the model’s accuracy, it also causes drastic increases in the model’s size and the computational effort required. Multi-period descriptive sampling (MPDS) is a new way to generate a small(er) set of scenarios that yield a good fit both to the periods’ probability distributions and to the convoluted probability distributions of stochastic variables (e.g., period demands) over time. MPDS uses descriptive sampling to draw a sample of S representative random numbers from a period’s known (demand) distribution. Now, to create a set of S representative scenarios, MPDS heuristically combines these random numbers (period demands) period by period so that a good fit is achieved to the convoluted (demand) distributions up to any period in the planning interval. A further contribution of this paper is an (accuracy) improvement heuristic, called fine-tuning, executed once the fix-and-optimize (FO) heuristic to solve a scenario-based mixed integer programming model has been completed. Fine-tuning uses linear programming (LP) with fixed binary variables (e.g., setup decisions) generated by FO and iteratively adapts production quantities so that compliance with given expected service level constraints is reached. The LP is solved with relatively little computational effort, even for large(r) sets of scenarios. We show the advancements possible with MPDS and fine-tuning by solving numerous test instances of the stochastic capacitated lot-sizing problem under a static uncertainty approach.

Analysis on Probability Mass Function and Probability Density Function

Probability Mass Function (PMF) and Probability Density Function (PDF) are fundamental concepts in probability theory and statistics that play a crucial role in describing the probability distribution of random variables. This abstract provides a comprehensive overview of these concepts, highlighting their definitions, characteristics, and applications. The Probability Mass Function is a concept primarily associated with discrete random variables. It defines the probability of a specific outcome occurring. The PMF assigns probabilities to individual values in the sample space, providing a clear picture of the likelihood of each possible outcome. Commonly denoted as P(X=x), where X is the random variable and x is a specific value, the PMF must satisfy two essential properties: non-negativity and the sum of probabilities over all possible outcomes equals one. On the other hand, the Probability Density Function is a concept applied to continuous random variables. Unlike the PMF, which deals with specific values, the PDF deals with ranges of values. The PDF represents the probability that a continuous random variable falls within a given interval. Denoted as f(x), it is essential to note that the probability of any specific point is zero, and instead, probabilities are defined for intervals. The area under the PDF curve over a given interval corresponds to the probability of the random variable falling within that interval. Understanding the differences and similarities between PMF and PDF is crucial for statistical analysis. While PMF is discrete and deals with specific values, PDF is continuous and provides probabilities for intervals. Both functions are integral to the calculation of various statistical measures, including expected values, variance, and standard deviation. This abstract concludes with a discussion of practical applications in diverse fields, such as finance, engineering, and natural sciences, where a deep understanding of PMF and PDF is essential for making informed decisions and drawing meaningful conclusions from data. The integration of these concepts into statistical models and analyses enhances the accuracy and reliability of predictions, making PMF and PDF indispensable tools in the field of probability and statistics

A deep clustering framework integrating pairwise constraints and a VMF mixture model

<abstract><p>We presented a novel deep generative clustering model called Variational Deep Embedding based on Pairwise constraints and the Von Mises-Fisher mixture model (VDEPV). VDEPV consists of fully connected neural networks capable of learning latent representations from raw data and accurately predicting cluster assignments. Under the assumption of a genuinely non-informative prior, VDEPV adopted a von Mises-Fisher mixture model to depict the hyperspherical interpretation of the data. We defined and established pairwise constraints by employing a random sample mining strategy and applying data augmentation techniques. These constraints enhanced the compactness of intra-cluster samples in the spherical embedding space while improving inter-cluster samples' separability. By minimizing Kullback-Leibler divergence, we formulated a clustering loss function based on pairwise constraints, which regularized the joint probability distribution of latent variables and cluster labels. Comparative experiments with other deep clustering methods demonstrated the excellent performance of VDEPV.</p></abstract>

Law of Probability Distribution Function of The Module of Difference of Discrete Random Variables

The use of the absolute value symbol in theoretical studies often leads to difficulties in analytical operations. However, from a practical point of view, transformations based on absolute differences are the most visual and effective. The article discusses functional transformations, which are modules of the differences of discrete random variables. The concept of the first difference modulus is given. The laws of the probability distribution of random variables obtained as the modulus of the difference of two independent random variables that have the same discrete distribution (Bernoulli or geometric), as well as the distribution laws of new random variables that are the sums of the moduli of the first difference, have been found. The main numerical characteristics of new random variables are calculated.

Systematic framework for handling uncertainty in probabilistic failure analysis of corroded concretes

Due to ambiguous correlations between input random variables and multi-source uncertainties from the electrochemical model, significant differences between deterministic corrosion prediction models and actual measurements are often observed. A systematic framework of quantification of uncertainties is developed for structures with correlated random variables originated from multiple sources, which allows efficiently estimating the failure probability distribution of the steel corrosion over time considering the randomness of the cover depth, the surface chloride concentration, and the chloride diffusion coefficient. After partitioning correlated random variables into different groups based on their uncertain sources, the Morris one-step-at-a-time and Sobol model is established to rank with respect to the importance of each correlated random variable. Based on polynomial chaos expansions and genetic programming methods, a more condensed set of random variables is created to propagate parametric problems. The unknown probability distribution of the input random variables is formulated by the Markov chain Monte Carlo to realize rigorous uncertainty quantification of the structural reliability. The application of the systematic framework to a set of numerical examples of steel corrosion includes experimental validation and uncertainty quantification and propagation of environmental, material and geometric properties. The results show that the framework can be integrated with parametric electrochemical models to allow robustness and reliability of corrosion prediction.

Analysis of statistical properties of variables in log data for advanced anomaly detection in cyber security

Log lines consist of static parts that characterize their structure and enable assignment of event types, and event parameters, i.e., variable parts that provide specific information on system processes, such as host and user names, IP addresses, and file operations. Many detection approaches only focus on anomalous event type occurrences, i.e., they parse log lines to derive unique event identifiers and subsequently detect anomalies in event sequences or event count vectors, but neglect variable parts of log lines entirely during analysis. This is especially problematic, when monitoring strongly structured log data that contains only a small number of distinct event types, for example, logs that consist of strict key value pairs, i.e., parameters that occur consistently throughout all log lines, such as it is case in access and audit logs. Thus, novel approaches are required, which focus on analysis of log lines' variable parts. In this paper, we propose the variable type detector (VTD), a novel unsupervised approach that autonomously analyzes variable log line parts to enable anomaly detection. It assigns data types to each variable, which also include probability distributions for discrete and continuous variables. The VTD raises an alarm if a variable's data type changes. Furthermore, it implements a robust indicator function that reduces false positives by tracking the data type history of each variable and reports only significant data type changes. Additionally, an event indicator enables event-based anomaly detection by taking into account the data types of all variables of a single event type. The evaluation conducted on open-source log data, demonstrates the effectiveness of the VTD compared to conventional anomaly detection approaches, such as time series analysis and PCA. Consequently, the VTD acts as a solution that extends the intrusion detection capabilities of security information and event management (SIEM) and integrates with modern concepts of endpoint detection and response (EDR) and extended detection and responses (XDR), while simultaneously serving as an asset for process monitoring that supports user and entity behavior analytics (UEBA).

Modelo probabilístico para análises de ruído do tráfego urbano usando sinais sonoros reais

Abstract Vehicular traffic is pointed out as a major source of urban noise pollution today. In this paper, we evaluated the precision of a new probabilistic model for urban traffic noise analyses. The proposed model adopts real sound signals and the Monte Carlo method in simulations. Probability distributions of traffic variables were obtained in-situ on two urban roads. The acoustic signals and corresponding energies of single pass-by of vehicles were obtained usingsound signal recordings on test tracks under free-field condition. The model simulates vehicular traffic noise on urban roads in free or in trafficlight controlled flow and considers the influence of bus stops.The proposed model calculates different acoustic descriptors, such as Statistical sound levels (LA10 and LA90), Equivalent continuous sound level (LAeq), Trafficnoise index (TNI) and Noise pollution level (LNP). Furthermore, it allowsthe listening of simulated noise. The experimental results indicate that theproposed model is reliable and accurate for vehicular traffic noise prediction.

Frequency-Control-Aware Probabilistic Load Flow: An Analytical Method

Probabilistic load flow (PLF) calculation, as a fundamental tool to analyze transmission system behavior, has been studied for decades. Despite a variety of available methods, existing PLF approaches rarely take system control into account. However, system control, as an automatic buffer between the fluctuations in random variables and the variations in system variables (e.g., nodal voltages), has a significant impact on the final PLF result. To consider control actions' influence, this paper proposes the first analytical PLF method for the transmission grid that takes into account primary and secondary frequency controls. This method is based on a high-precision linear power flow model, whose precision is even further improved in this paper by an original correction approach. This paper also proves that if the joint probability distribution (JPD) of random variables is expressed by a Gaussian mixture model (GMM), then the JPD of system variables is an infinite GMM. By leveraging this proposition, the proposed method generates the joint PLF of the whole system at the quasi-steady and steady states. The high accuracy and satisfactory efficiency of this method are verified on multiple test systems with different percentages of renewables.

Application of the Monte-Carlo Method to Assess the Operational Reliability of a Household-Constructed Wetland with Vertical Flow: A Case Study in Poland

The objective of this study was to model the operation of a vertical-flow constructed wetland (VF-CW) for domestic wastewater, using Monte-Carlo simulations and selected probability distributions of various random variables. The analysis was based on collected wastewater quality data, including the values of the pollutant indicators BOD5 (biochemical oxygen demand), CODCr (chemical oxygen demand), and TSS (total suspended solids), in the 2017–2020 period. Anderson–Darling (A–D) statistics were applied to assess the fit of the theoretical distributions to the empirical distributions of the random variables under study. The selection of the best-fitting statistical distributions was determined using the percentage deviation (PBIAS) criterion. Based on the analyses that were performed, the best-fitting statistical distributions for the pollution indicators of the raw wastewater were the generalised extreme value distribution for BOD5, the Gaussian distribution for CODCr, and the log-normal distribution for TSS. For treated effluent, the log-normal distribution was the best fit for BOD5 and CODCr; the semi-normal distribution, for TSS. The new data generated using the Monte-Carlo method allowed the reliability of the VF-CW operation to be assessed by determining the reliability indices, i.e., the average efficiency of the removal of pollutants (η), the technological efficiency index (R), the reliability index (CR), and the risk index of the negative control of the sewage treatment plant operation (Re). The obtained results indicate that only in the case of CODCr, the analysed treatment facility may fail to meet the requirements related to the reduction of organic pollutants to the required level, which is evidenced by the values of the indicators CR = 1.10, R = 0.49, and η = 0.82. In addition, the risk index of the negative operation of the facility (Re) assumes a value of 1, which indicates that during the period of its operation, the VF-CW system will not operate with the required efficiency in relation to this indicator. The novelty of this work is the implementation of the indicated mathematical simulation methods for analysing the reliability of the operation of the domestic wastewater treatment facility.