Required Probability Distributions Research Articles

Assessing Water Management Strategies in Data-Scarce Mountain Regions under Uncertain Climate and Socio-Economic Changes

AbstractWater management in mountainous regions faces significant challenges due to deep uncertainties arising from data scarcity, knowledge gaps, and the complex interplay of climate and socio-economic changes. While existing approaches focused on uncertainty reduction and water system optimization contribute to managing uncertainties, they often require probability distributions that can be difficult to obtain in data-scarce mountain regions. To address these challenges, we demonstrate the effectiveness of Exploratory Modeling and Analysis (EMA) in assessing water management strategies and identifying operational ranges that avoid future water scarcity. Through a case study in the complex and data-scarce Peruvian Andes, we employed EMA to run 12,000 simulations by 2050, incorporating deep uncertainties from climate and socio-economic scenarios, and hydrological modeling parameters. This analysis identified specific policy combinations demonstrating greater robustness across diverse scenarios and uncertainties. EMA explicitly identifies operational ranges of policies to avoid water scarcity but also highlights the conditions that might trigger policy failure. We also delve into the roles of the different factors used in EMA and their significance in water management applications. Our research illustrates that an exploratory hydrological modeling approach based on robust decision-making can foster a more informed decision-making process for long-term water adaptation in rapidly changing mountain regions under data scarcity and deep uncertainties.

A framework for preparing a stochastic nonlinear integrate-and-fire model for integrated information theory

ABSTRACT This paper presents a framework for spiking neural networks to be prepared for the Integrated Information Theory (IIT) analysis, using a stochastic nonlinear integrate-and-fire model. The model includes the crucial dynamics of the all-or-none law and after-spike refractoriness. The noise is modelled as an additive term in the system’s equations. By preparing the model for the IIT analysis, it is meant to determine the length of the analysis time-window and the transition probability distributions required for the IIT 3.0. To this end, a system of differential equations is proposed to estimate the time evolution of the system’s mean and covariance. Assuming the binary Fired/Silent activity as the possible states of each neuron, an algorithm is proposed to calculate the required probability distributions. As long as the Fired/Silent probabilities are only concerned, the Gaussian density assumption with the estimated moments is a reasonable estimate. The synaptic inputs are treated as random variables with low variances to avoid the costs of conditioning on the system’s past activities. The Monte-Carlo simulation is used to validate the estimation methods. To increase the reliability of the inductive inference behind the Monte-Carlo method, various stimulation protocols are applied to evoke the dynamics of the equations.

Analysis of semi-Markov systems with fuzzy initial data

In real operating conditions of complex systems, random changes in their possible states occur in the course of their operation. The traditional approach to describing such systems uses Markov models. However, the real non-deterministic mechanism that controls the duration of the system's stay in each of its possible states predetermines the insufficient adequacy of the models obtained in this case. This circumstance makes it expedient to consider models that are more general than Markov ones. In addition, when choosing such models, one should take into account the fundamental often manifested feature of the statistical material actually used in the processing of an array of observations, their small sample. All this, taken together, makes it relevant to study the possibility of developing less demanding, tolerant models of the behavior of complex systems. A method for the analysis of systems described under conditions of initial data uncertainty by semi-Markov models is proposed. The main approaches to the description of this uncertainty are considered: probabilistic, fuzzy, and bi-fuzzy. A procedure has been developed for determining the membership functions of fuzzy numbers based on the results of real data processing. Next, the following tasks are solved sequentially. First, the vector of stationary state probabilities of the Markov chain embedded in the semi-Markov process is found. Then, a set of expected values for the duration of the system's stay in each state before leaving it is determined, after which the required probability distribution of the system states is calculated. The proposed method has been developed to solve the problem in the case when the parameters of the membership functions of fuzzy initial data cannot be clearly estimated under conditions of a small sample

A statistical model of secondary ion emission and attenuation clarifies disparities in quasi-simultaneous arrival coefficients measured with secondary ion mass spectrometry.

Secondary ion mass spectrometry data collected using electron multiplier detectors are subject to a correction for the quasi-simultaneous arrival (QSA) effect. Published Poisson statistical models indicate that the QSA coefficients, β, should have an invariant value of 0.5, whereas, with one exception, published experimental determinations vary between 0.6 and 1.0, with a mean value of 0.75. We developed a more complex model, combining both ion emission and attenuation, that predicts the observed range in measured β and elucidates the mechanism of secondary ion formation. For a given aperture setting, any secondary ion has an equal probability of successful transit to the electron multiplier. Binomial statistics can model pass-fail aperture attenuation but require probability distributions of the quasi-simultaneously emitted (QSE) ion tally, per primary ion, as input. Assuming (a) that each primary ion impact results in 0, 1, 2,… secondary ion emissions, randomly, with an average Ks and (b) that there is finite probability (P2) of a further emission process dependent on Ks , the required QSE probability distributions were generated via a combined Poisson-binomial statistical model. The value of β was output as a function of Ks and P2. For values of P2 > 0 and any value of Ks , β always exceeds 0.5. As P2 → 0, β → 0.5; for values of increasing P2 > 0.5 and decreasing Ks < 0.5, β → >1. Were the emission of one ion not to influence the probability of the formation of a second (i.e. model output for P2 = 0), β should always be 0.5. Yet measurements have never reported this value. Consequently, assuming that published β values are correct, emissions of QSE secondary ions do not occur independently, and it may be inferred that there are linked mechanisms of secondary ion formation as shown here.

Decision support for offshore asset construction using expert judgments for supply disruptions risk

Offshore asset construction is a complex and costly process that is subject to various uncertainties within the entire supply chain. Hence, both the construction management optimization and the reduction of deployment expenditures should be supported by automated decision support models which include proper representations of predominant uncertainties. One of these is the supply disruption risk that is often ignored in existing models. Therefore, this article proposes a methodology to properly take this construction risk into account. An algorithm to model this risk was developed and a study was conducted to obtain the required probability distributions of disruption delays using real data and expert judgments for an offshore wind farm construction application. The simulation of a realistic test case with an appropriately modified stochastic simulation tool showed that it is important to consider this risk in order to make optimal decisions for different offshore wind farm construction strategies. © 2019 Elsevier B.V.

Reliability analysis of structures based on a probability‐uncertainty hybrid model

AbstractThe traditional reliability analysis method based on probabilistic method requires probability distributions of all the uncertain parameters. However, in practical applications, the distributions of some parameters may not be precisely known due to the lack of sufficient sample data. The probabilistic theory cannot directly measure the reliability of structures with epistemic uncertainty, ie, subjective randomness and fuzziness. Hence, a hybrid reliability analysis (HRA) problem will be caused when the aleatory and epistemic uncertainties coexist in a structure. In this paper, by combining the probability theory and the uncertainty theory into a chance theory, a probability‐uncertainty hybrid model is established, and a new quantification method based on the uncertain random variables for the structural reliability is presented in order to simultaneously satisfy the duality of random variables and the subadditivity of uncertain variables; then, a reliability index is explored based on the chance expected value and variance. Besides, the formulas of the chance theory‐based reliability and reliability index are derived to uniformly assess the reliability of structures under the hybrid aleatory and epistemic uncertainties. The numerical experiments illustrate the validity of the proposed method, and the results of the proposed method can provide a more accurate assessment of the structural system under the mixed uncertainties than the ones obtained separately from the probability theory and the uncertainty theory.

Stochastic modelling for capturing the behaviour of irregular-shaped non-spherical particles in confined turbulent flows

For calculating dispersed particle-laden flows in confined systems, the well-known Euler/Lagrange approach is most suitable. Lagrangian tracking of non-spherical particles with certain shapes is mostly performed by additionally solving for the orientation of particles in the flow and using resistance coefficients (i.e. drag, lift and torque) which depend on this orientation. For that in many cases theoretical results for Stokes flow around such particles are used. In practical situations where very often irregular shaped non-spherical particles are transported in a flow, such an approach cannot be adopted since the particles have mostly a statistical distribution of shape and hence it is difficult to define a major and minor axis of the particles. The novel approach developed here is based on a statistical treatment of the fluid forces and moments acting on irregular-shaped particles as well as the wall collision process in order to mimic their stochastic behaviour. The required probability distribution functions (PDF's) for the resistance coefficients were derived by applying direct numerical simulations (DNS) based on the Lattice-Boltzmann method (LBM). The PDF's for the wall normal and parallel restitution ratios were developed based on an experimental analysis of the wall collision of irregular-shaped particles using stereoscopic high-speed imaging. Preliminary Euler/Lagrange calculations applying these statistical models were conducted for a horizontal channel flow laden with irregular-shaped particles and compared to measurements. The results revealed that the calculation of the particle phase assuming the standard models for spherical particles yields completely wrong cross-stream profiles of particle mass flux, an under-prediction of the stream-wise particle mean velocity and an over-prediction of the associated fluctuating component. The stochastic models for the flow resistance coefficients and the wall collision process on the other hand provided much better agreement with the measurements.

A Comparison of Different Statistics for Detecting Multiplicative Faults in Multivariate Statistics-Based Fault Detection Approaches

The explosion of different fault detection (FD) statistics in multivariate statistics-based FD approaches has meant that the practitioner is faced with the unenviable job of determining which to use in a given circumstance. Moreover, compared to extensive investigations on additive faults, the performance of commonly used FD statistics for detecting multiplicative faults has not been holistically evaluated. Therefore, this paper seeks to investigate the different statistics that can be applied to detect multiplicative faults in order to provide users and practitioners in the FD field with guidance to select an appropriate method. The considered statistics are broadly classified into three groups: traditional methods ( e.g. $T^{2}$ -statistic) and their extensions; the Wishart distribution-based methods; and those methods created in the information and communication fields to describe the measurement variance and covariance ( e.g. Kullback-Leibler divergence). These three groups are compared by considering the required probability distributions, interconnections, and detection performance for multiplicative faults. Using simulated data from numerical examples and the Tennessee Eastman benchmark process, the theoretical results are validated, and the applicability of multivariate statistics-based FD methods incorporating all considered statistics for detecting multiplicative faults is examined at the end of this paper.

On The Nearby-Tip Strain Investigation and Failure-Propability Evaluation for Impacted Thin Plates Using the 2-Random-Variables Multi-Canonical-Based Joint Propability Distributions

The study of the validity and probability of failure in solids and structures is highly considered as one of the most incredibly-highlighted study fields in many science and engineering applications, the design analysts must therefore seek to investigate the points where the failing strains may be occurred, the probabilities of which these strains can cause the existing cracks to propagate through the fractured medium considered, and thereafter the solutions by which the analysts can adopt the approachable techniques to reduce/arrest these propagating cracks.In the present study a theoretical investigation upon simply-supported thin plates having surface cracks within their structure is to be accomplished, and the applied impact load to these thin plates tends to induce almost infinite strains nearby the crack tip of the existing cracks. The distribution of these strains and the probability distribution of failure due to these strains are to be of a particular importance within the current research.Within the current study a modified theoretical technique, which is derived from the classical plate theory, whose concepts are illustrating the required plane-stress conditions for fractured thin plates, taking into consideration the impact-load effects in conjunction with the fracture-mechanics concepts, is to be followed and obeyed so as to arrive at the required equations representing the nearby-tip strains within the thin plates made from the pure aluminum 1100 type alloys. A further statistically-based analysis must lead into the utilization of the joint probability distributions having two random variables in order to construct the required probability distributions of the failure which may be occurred due to the highly-localized nearby-tip strains.

On The Nearby-Tip Strain Investigation and Failure-Propability Evaluation for Impacted Thin Plates Using the 2-Random-Variables Multi-Canonical-Based Joint Propability Distributions

The study of the validity and probability of failure in solids and structures is highly considered as one of the most incredibly-highlighted study fields in many science and engineering applications, the design analysts must therefore seek to investigate the points where the failing strains may be occurred, the probabilities of which these strains can cause the existing cracks to propagate through the fractured medium considered, and thereafter the solutions by which the analysts can adopt the approachable techniques to reduce/arrest these propagating cracks.In the present study a theoretical investigation upon simply-supported thin plates having surface cracks within their structure is to be accomplished, and the applied impact load to these thin plates tends to induce almost infinite strains nearby the crack tip of the existing cracks. The distribution of these strains and the probability distribution of failure due to these strains are to be of a particular importance within the current research.Within the current study a modified theoretical technique, which is derived from the classical plate theory, whose concepts are illustrating the required plane-stress conditions for fractured thin plates, taking into consideration the impact-load effects in conjunction with the fracture-mechanics concepts, is to be followed and obeyed so as to arrive at the required equations representing the nearby-tip strains within the thin plates made from the pure aluminum 1100 type alloys. A further statistically-based analysis must lead into the utilization of the joint probability distributions having two random variables in order to construct the required probability distributions of the failure which may be occurred due to the highly-localized nearby-tip strains.

Response-surface-based structural reliability analysis with random and interval mixed uncertainties

Traditional reliability analysis requires probability distributions of all the uncertain parameters. However, in many practical applications, the variation bounds can be only determined for the parameters with limited information. A complex hybrid reliability problem then will be caused when the random and interval variables coexist in a same structure. In this paper, by introducing the response surface technique, we develop a new hybrid reliability method to efficiently compute the interval of the failure probability of the structure due to the probability-interval hybrid uncertainty. The present method consists of a sequence of iterations. At each step, a response surface model is constructed for the limit-state function by using a quadratic polynomial and a modified axial experimental design method. An approximate hybrid reliability problem is created based on the response surface model, which is subsequently solved by an efficient decoupling approach. An updating strategy is suggested to improve the quality of the response surface and whereby ensure the reliability analysis precision. A computational procedure is then summarized for the whole iterations. Four numerical examples and also a practical application are provided to demonstrate the effectiveness of the present method.

One and Done? Optimal Decisions From Very Few Samples

In many learning or inference tasks human behavior approximates that of a Bayesian ideal observer, suggesting that, at some level, cognition can be described as Bayesian inference. However, a number of findings have highlighted an intriguing mismatch between human behavior and standard assumptions about optimality: People often appear to make decisions based on just one or a few samples from the appropriate posterior probability distribution, rather than using the full distribution. Although sampling-based approximations are a common way to implement Bayesian inference, the very limited numbers of samples often used by humans seem insufficient to approximate the required probability distributions very accurately. Here, we consider this discrepancy in the broader framework of statistical decision theory, and ask: If people are making decisions based on samples--but as samples are costly--how many samples should people use to optimize their total expected or worst-case reward over a large number of decisions? We find that under reasonable assumptions about the time costs of sampling, making many quick but locally suboptimal decisions based on very few samples may be the globally optimal strategy over long periods. These results help to reconcile a large body of work showing sampling-based or probability matching behavior with the hypothesis that human cognition can be understood in Bayesian terms, and they suggest promising future directions for studies of resource-constrained cognition.

Robust Scheduling of Task Graphs under Execution Time Uncertainty

Effective multicore computing requires to make efficient usage of the computational resources on a chip. Offline mapping and scheduling can be applied to improve the performance, but classical approaches require considerable a priori knowledge of the target application. In a practical setting, precise information is often unavailable; one can then resort to approximate time and resource usage figures, but this usually requires to make conservative assumptions. The issue is further stressed if real-time guarantees must be provided. We tackle predictable and efficient nonpreemptive scheduling of multitask applications in the presence of duration uncertainty. Hard real-time guarantees are provided with limited idle time insertion, by exploiting a hybrid offline/online technique known as Precedence Constraint Posting (PCP). Our approach does not require probability distributions to be specified, relying instead on simple and cheaper-to-obtain information (bounds, average values). The method has been tested on synthetic applications/platforms and compared with an offline optimized Fixed Priority Scheduling (FPS) approach and a pure online FIFO scheduler; the results are very promising, as the PCP schedules exhibit good stability and improved average execution time (14 percent on average, up to 30 percent versus FPS and up to 40 percent versus the FIFO scheduler).

Structural analysis with probability-boxes

Probability-box (p-box) is a rigorous and practical way to represent epistemic sources of uncertainty where the available knowledge is insufficient to construct the required probability distributions. In this paper, interval finite element (FE) methods are combined with the concept of p-box to analyse structures subjected to uncertain loads modelled by p-boxes. Two methods, namely the discrete p-box convolution and interval Monte Carlo methods, are presented along with example problems. The computational efficiency of the p-box FE method is also presented.

Methods of plant breeding in the genome era

Methods of genomic value prediction are reviewed. The majority of the methods are related to mixed model methodology, either explicitly or implicitly, by treating systematic environmental effects as fixed and quantitative trait locus (QTL) effects as random. Six different methods are reviewed, including least squares (LS), ridge regression, Bayesian shrinkage, least absolute shrinkage and selection operator (Lasso), empirical Bayes and partial least squares (PLS). The LS and PLS methods are non-Bayesian because they do not require probability distributions for the data. The PLS method is introduced as a special dimension reduction scheme to handle high-density marker information. Theory and methods of cross-validation are described. The leave-one-out cross-validation approach is recommended for model validation. A working example is used to demonstrate the utility of genome selection (GS) in barley. The data set contained 150 double haploid lines and 495 DNA markers covering the entire barley genome, with an average marker interval of 2·23 cM. Eight quantitative traits were included in the analysis. GS using the empirical Bayesian method showed high predictability of the markers for all eight traits with a mean accuracy of prediction of 0·70. With traditional marker-assisted selection (MAS), the average accuracy of prediction was 0·59, giving an average gain of GS over MAS of 0·11. This study provided strong evidence that GS using marker information alone can be an efficient tool for plant breeding.

Forecasting Long-Lead Rainfall Probability with Application to Australia’s Northeastern Coast

Abstract The authors develop a method for the long-lead forecasting of El Niño–influenced rainfall probability and illustrate it using the economically important prediction, from the beginning of the year, of September–November (SON) rainfall in the coastal sugarcane producing region of Australia’s northeastern coast. The method is based on two probability distributions. One is the Gaussian error distribution of the long-lead prediction of the El Niño index Niño-3.4 by the Clarke and Van Gorder forecast method. The other is the relationship of the rainfall distribution to the Niño-3.4 index. The rainfall distribution can be approximated by a gamma distribution whose two parameters depend on Niño-3.4. To predict the rainfall at, say, the Tully Sugar, Ltd., mill on the north Queensland coast in SON 2009, the June–August (JJA) value of Niño-3.4 is predicted and then 1000 possible “observed” JJA Niño-3.4 values calculated from the error distribution. Each one of these observed Niño-3.4 values is then used, with the Niño-3.4-dependent gamma distribution for that location, to calculate 1000 possible SON rainfall totals. The result is one million possible SON rainfalls. A histogram of these rainfalls is the required probability distribution for the rainfall at that location predicted from the beginning of the year. Cross-validated predictions suggest that the method is successful.

Combining content-based and collaborative recommendations: A hybrid approach based on Bayesian networks

Recommender systems enable users to access products or articles that they would otherwise not be aware of due to the wealth of information to be found on the Internet. The two traditional recommendation techniques are content-based and collaborative filtering. While both methods have their advantages, they also have certain disadvantages, some of which can be solved by combining both techniques to improve the quality of the recommendation. The resulting system is known as a hybrid recommender system. In the context of artificial intelligence, Bayesian networks have been widely and successfully applied to problems with a high level of uncertainty. The field of recommendation represents a very interesting testing ground to put these probabilistic tools into practice. This paper therefore presents a new Bayesian network model to deal with the problem of hybrid recommendation by combining content-based and collaborative features. It has been tailored to the problem in hand and is equipped with a flexible topology and efficient mechanisms to estimate the required probability distributions so that probabilistic inference may be performed. The effectiveness of the model is demonstrated using the MovieLens and IMDB data sets.

Opening Two Envelopes

In the two-envelope problem, one is offered a choice between two envelopes, one containing twice as much money as the other. After seeing the contents of the chosen envelope, the chooser is offered the opportunity to make an exchange for the other envelope. However, it appears to be advantageous to switch, regardless of what is observed in the chosen envelope. This problem has an extensive literature with connections to probability and decision theory. The literature is roughly divided between those that attempt to explain what is flawed in arguments for the advantage of switching and those that attempt to explain when such arguments can be correct if counterintuitive. We observe that arguments in the literature of the two-envelope problem that the problem is paradoxical are not supported by the probability distributions meant to illustrate the paradoxical nature. To correct this, we present a distribution that does support the usual arguments. Aside from questions about the interpretation of variables, algebraic ambiguity, modal confusions and the like, most of the interesting aspects of the two-envelope problem are assumed to require probability distributions on an infinite space. Our next main contribution is to show that the same counterintuitive arguments can be reflected in finite versions of the problem; thus they do not inherently require reasoning about infinite values. A topological representation of the problem is presented that captures both finite and infinite cases, explicating intuitions underlying the arguments both that there is an advantage to switching and that there is not.

Commitment of Electric Power Generators Under Stochastic Market Prices

A formulation for the commitment of electric power generators under a deregulated electricity market is proposed. The problem is expressed as a stochastic optimization problem in which expected profits are maximized while meeting demand and standard operating constraints. Under an assumption of perfect competition, when an electric power producer has the option of trading electricity at market prices, a unit commitment schedule can be obtained by optimizing the self-commitment of each unit separately subject to stochastic prices. Three certainty-equivalent formulations of the stochastic self-commitment problem are provided. The procedures involve application of dynamic programming, statistical analysis, and asymptotic probability computations. The price of electricity is represented by a stochastic model depending on demand, generating unit reliabilities, and temperature fluctuations. We use several approximation methods (normal, Edgeworth series expansion, and Monte Carlo simulation) for computing the required probability distributions. Numerical examples are provided for a market consisting of 150 generating units.

DISTRIBUTIONALLY ROBUST MONTE CARLO SIMULATION: A TUTORIAL SURVEY

Whereas the use of traditional Monte Carlo simulation requires probability distributions for the uncertain parameters entering the system, distributionally robust Monte Carlo simulation does not. According to this new theory, instead of carrying out simulations using some rather arbitrary probability distribution such as Gaussian for the uncertain parameters, we provide a rather different prescription based on distributional robustness considerations. Motivated by manufacturing considerations, a class of distributions ℱ is specified and the results of the simulation hold for all f ∈ ℱ. This new method of Monte Carlo simulation was developed with the robustician in mind in that we begin only with bounds on the uncertain parameters and no a priori probability distribution is assumed.