Estimation Of Probability Distribution Research Articles

A Multi-Medium Analysis of Human Health Risk of Toxic Elements in Rice-Crayfish System: A Case Study from Middle Reach of Yangtze River, China.

Rice-crayfish system has been extensively promoted in China in recent years. However, the presence of toxic elements in soil may threaten the quality of agricultural products. In this study, eight toxic elements were determined in multi-medium including soil, rice, and crayfish from the rice-crayfish system (RCS) and conventional rice culture (CRC) area. Crayfish obtained a low level of toxic element content, and mercury (Hg) in rice from RCS showed the highest bioavailability and mobility. Health risk assessment, coupled with Monte Carlo simulation, revealed that the dietary exposure to arsenic (As) and Hg from rice and crayfish consumption was the primary factor for non-carcinogenic risk, while Cd and As were the dominant contributors to the high carcinogenic risk of rice intake for adults and children, respectively. Based on the estimated probability distribution, the probabilities of the total cancer risk (TCR) of rice intake for children from RCS were lower than that from CRC.

The selection of an optimal segmentation region in physiological signals

AbstractPhysiological signals are often corrupted by noisy sources. Usually, artificial intelligence algorithms analyze the whole signal, regardless of its varying quality. Instead, experienced cardiologists search for a high‐quality signal segment, where more accurate conclusions can be draw. We propose a methodology that simultaneously selects the optimal processing region of a physiological signal and determines its decoding into a state sequence of physiologically meaningful events. Our approach comprises two phases. First, the training of a neural network that then enables the estimation of the state probability distribution of a signal sample. Second, the use of the neural network output within an integer program. The latter models the problem of finding a time window by maximizing a likelihood function defined by the user. Our method was tested and validated in two types of signals, the phonocardiogram and the electrocardiogram. In phonocardiogram and electrocardiogram segmentation tasks, the system's sensitivity increased on average from 95.1% to 97.5% and from 78.9% to 83.8%, respectively, when compared to standard approaches found in the literature.

Probability distribution estimation for harmonisable loads and responses of linear elastic structures

In probabilistic distribution-based structural dynamics representing joint probability density functions (JPDFs) of loads and those of structural responses at any collection of time instants is a challenging and important issue. In this study, based on the mean square (m.s.) stochastic calculus, a novel method for estimating the JPDFs of external loads and those of structural responses is proposed. In particular, an external load is modelled as a harmonisable process, and its two approximate representations are proposed by the discrete Fourier transform. The JPDFs of the Fourier coefficients are directly estimated from multiple load samples. Using the load JPDFs in the frequency domain can estimate the JPDFs of the load at any finite time instants. By applying the load representations to the dynamic governing equation of linear elastic structures, the JPDFs of the structural responses at any finite time instants can be estimated. The two load representations are proved to converge in m.s. to the exact load process under certain conditions. Correspondingly, the estimated JPDFs of the load and those of structural responses in the time domain converge to their corresponding exact ones. The proposed method is numerically verified using a linear elastic structure under a nonstationary earthquake ground motion.

Methods for probability distributions estimation of indoor environmental parameters and long-term IEQ assessment

ABSTRACT To overcome the difficulties in comparing enormous time series for indoor environmental parameters and make use of technological developments in previous research, this study considers environmental parameters from a new perspective, their probability density functions (PDFs). PDFs are combined with existing indoor environmental quality (IEQ) mathematical models to assess the long-term IEQ. Two methodologies were developed: probability distribution estimation of indoor environmental parameters and long-term IEQ distribution assessment based on the Monte Carlo method. The effectiveness of the developed methodologies was illustrated in a three-month IEQ assessment of an office. PDFs were obtained with specific mathematical expressions for the three-month air temperature, sound level and illuminance. The three-month distributions for thermal, visual, acoustic and overall environmental quality were presented using eight previous IEQ mathematical models. PDFs have the advantage of using only a few main parameters instead of an enormous time series to explain the behaviour and characteristics of environmental parameters. PDFs can also potentially determine the commonalities of environmental distributions for different buildings. The obtained IEQ distributions present a straightforward and comprehensive impression of the long-term IEQ, rather than a simple index.

Worst-case effect in bearing capacity of spread foundations considering safety factors and anisotropy in soil spatial variability

ABSTRACT Random field theory is currently the most important tool used when modelling spatial variabilities in soil properties. However, there are practical difficulties in indicating spatial correlation parameters, such as horizontal and vertical fluctuation scales. Therefore, the performed analyses touch upon the occurrence of the worst-case effect of the horizontal fluctuation scale. The random finite element (RFEM) approach is utilised to estimate probability distributions of the bearing capacity of a spread foundation. The analyses show that the observed location of the worst-case phenomenon is closely related to the adopted safety factor. Therefore, it is difficult to directly indicate the horizontal fluctuation scale for which the failure probability has the greatest value. Nevertheless, some general guidelines are formulated and proposed. These general rules can be applied to the two-dimensional bearing capacity problem.

Risk mitigation in model-based experiment design: A continuous-effort approach to optimal campaigns

A key challenge in maximizing the effectiveness of model-based design of experiments for calibrating nonlinear process models is the inaccurate prediction of information that is afforded by each new experiment. We present a novel methodology to exploit prior probability distributions of model parameter estimates in a bi-objective optimization formulation, where a conditional-value-at-risk criterion is considered alongside an average information criterion. We implement a tractable numerical approach that discretizes the experimental design space and leverages the concept of continuous-effort experimental designs in a convex optimization formulation. We demonstrate effectiveness and tractability through three case studies, including the design of dynamic experiments. In one case, the Pareto frontier comprises experimental campaigns that significantly increase the information content in the worst-case scenarios. In another case, the same campaign is proven to be optimal irrespective of the risk attitude. An open-source implementation of the methodology is made available in the Python software Pydex.

Risk analysis in social housing delivery: A public–private partnerships approach

AbstractFaced with fiscal pressure and concerns over bureaucratic inefficiency, policymakers at Australia's state government level are evaluating instruments to improve social housing services. A public–private partnerships (PPPs) procurement approach is one such instrument for increasing housing supply to address housing affordability in Australia. To determine the value of pursuing a project through a PPP delivery versus a traditional procurement approach, public administrators need to determine discount rates for comparing different cash flows streams on a consistent basis. Additionally, they must estimate public sector comparator (PSC) to determine public sector's risk exposure in social housing delivery and allocating constrained housing budgets. This research assesses the discount rates and estimates the risk profile of social housing delivery faced by public agencies. Three case studies were utilised to present project estimate probability distributions for gaining insight into risk exposure in social housing delivery and helping housing budget allocation at the state government level. The findings recommend the use of an estimated 30‐year treasury bond rate (3.14%) as the PSC discount rate for social infrastructure. Additionally, the results suggest the use of a PPP discount rate for social housing of 6.01%, given a risk‐free rate of 3.14% and a systematic risk premium of 2.87%.

A data-enhanced distributionally robust optimization method for economic dispatch of integrated electricity and natural gas systems with wind uncertainty

With growing penetrations of wind power in electricity systems, the coordinated dispatch of integrated electricity and natural gas systems is becoming a popular research topic. Distributionally robust optimization can cope with the wind uncertainty of integrated electricity and natural gas systems by providing optimal solutions for the worst-case probability distribution. However, limited historical wind data hinder the estimation of worst-case probability distribution. As a breakthrough in artificial intelligence, generative adversarial networks can be established to approximate a complex uncertain probability distribution from raw data and generate realistic data subject to the identical distribution. This paper proposes a data-driven optimization method for economic dispatch of integrated electricity and natural gas systems with wind uncertainty, whose probability distribution is free. Based on limited historical data, the data-driven generative adversarial network generates artificial wind power data, which helps to improve the estimation of worst-case probability distribution in distributionally robust optimization. Moreover, the robustness of optimization solutions can be adjusted cost-effectively by controlling the auxiliary data number. In a case study, optimization solutions of the proposed method are shown to achieve a lower probability of chance constraint violation at a nearly negligible cost increase compared with those from four typical optimization methods.

LEARNING HIGH-DIMENSIONAL PROBABILITY DISTRIBUTIONS USING TREE TENSOR NETWORKS

We consider the problem of the estimation of a high-dimensional probability distribution from i.i.d. samples of the distribution using model classes of functions in tree-based tensor formats, a particular case of tensor networks associated with a dimension partition tree. The distribution is assumed to admit a density with respect to a product measure, possibly discrete for handling the case of discrete random variables. After discussing the representation of classical model classes in tree-based tensor formats, we present learning algorithms based on empirical risk minimization using a $L^2$ contrast. These algorithms exploit the multilinear parametrization of the formats to recast the nonlinear minimization problem into a sequence of empirical risk minimization problems with linear models. A suitable parametrization of the tensor in tree-based tensor format allows to obtain a linear model with orthogonal bases, so that each problem admits an explicit expression of the solution and cross-validation risk estimates. These estimations of the risk enable the model selection, for instance when exploiting sparsity in the coefficients of the representation. A strategy for the adaptation of the tensor format (dimension tree and tree-based ranks) is provided, which allows to discover and exploit some specific structures of high-dimensional probability distributions such as independence or conditional independence. We illustrate the performances of the proposed algorithms for the approximation of classical probabilistic models (such as Gaussian distribution, graphical models, Markov chain).

Quantile estimation of probability distributions for maximum daily precipitation and short time series of observational data for engineering design

Quantile estimation of probability distributions for maximum daily precipitation and short time series of observational data for engineering design

Deep learning-assisted analysis of automobiles handling performances

Abstract The luxury car market has demanding product development standards aimed at providing state-of-the-art features in the automotive domain. Handling performance is amongst the most important properties that must be assessed when developing a new car model. In this work, we analyse the problem of predicting subjective evaluations of automobiles handling performances from objective records of driving sessions. A record is a multi-dimensional time series describing the temporal evolution of the mechanical state of an automobile. A categorical variable quantifies the evaluations of handling properties. We describe an original deep learning system, featuring a denoising autoencoder and hierarchical attention mechanisms, that we designed to solve this task. Attention mechanisms intrinsically compute probability distributions over their inputs’ components. Combining this feature with the saliency maps technique, our system can compute heatmaps that provide a visual aid to identify the physical events conditioning its predictions.

An equivalent expectation evaluation method for approximating the probability distribution of performance functions

This paper proposes an equivalent expectation evaluation method for approximating the probability distribution of performance functions. The innovation of the proposed method is to formulate the probability distribution of the performance function as an expectation regardless of the dimensionality and complexity of the problem. When a monotonic inverse function (MIF) of the performance function with respect to a certain random variable can be obtained, the probability distribution can be exactly formulated as an expectation related to the distribution of this variable and the MIF. While if such a MIF does not exist or cannot be easily obtained, by constructing two simple auxiliary functions whose MIF can be obtained, a generalized method is proposed to formulate the probability distribution of the performance function as an expectation related to the distribution of the auxiliary functions. Then, the Sobol sequence and point estimate method are employed to determine the expectation. The efficiency and accuracy of the proposed method are demonstrated through four numerical examples. The results show that the proposed method is applicable to approximating probability distributions of both static and dynamic reliability problems involving high-dimensional, nonlinear, or implicit performance functions.

Gaussian mixture model deep neural network and its application in porosity prediction of deep carbonate reservoir

To estimate the spatial distribution of porosity, model-driven or data-driven methods are usually used to establish the relationship between porosity and seismic elastic parameters. However, due to the strong heterogeneity and complex pore structures of carbonate reservoirs, porosity estimation of carbonates still represents a great challenge. The existing conventional model-driven and data-driven-based porosity estimation methods have high uncertainty. To characterize the complex statistical distribution of porosity, the nonlinear relationship between porosity and seismic elastic parameters, and the uncertainty of porosity estimation, we have used a Gaussian mixture model deep neural network (GMM-DNN) to invert porosity from seismic elastic parameters. We use a Gaussian mixture model to describe the complex distribution of porosity, and we apply a deep neural network to establish the nonlinear relationship among seismic compressional-wave (P-wave) velocity, density, and porosity. The outputs of the GMM-DNN provide an estimated probability distribution of porosity conditioned on the input seismic elastic parameters. The synthetic data example verifies the feasibility of this method. We further apply the GMM-DNN-based porosity inversion method to a deep complex carbonate reservoir in the Tarim Basin, Northwest China. The well-logging data are used to train the GMM-DNN; then, the P-wave velocity and density obtained by prestack amplitude-variation-with-offset inversion are fed into the trained network to reasonably estimate the porosity distribution of the whole target reservoir and evaluate its uncertainties.

Modeling conditional dependencies for bus travel time estimation

In the age of Intelligent Transportation Systems it is essential to provide operators and passengers with reliable information. The estimation of probability distributions of public transport travel times is crucial as it directly informs about the reliability of travel times. Thus, the probability distributions of travel times are useful for timetabling and route choice. This work estimates probability distributions of segment (multi-section) bus running and dwell times. We propose a hidden Markov chain framework, which captures the dependency structure of consecutive section running times and includes conditional correlations. The dependency structure of consecutive segment dwell times is modeled as a combination of correlation and operation-specific dependencies. Such a model allows describing the relationship between section-level running/station-level dwell time distributions and segment level distributions. The model is interpretable, as the dependency structure is explicitly modeled. Finally, the proposed model is evaluated on the operation of the trolley bus network of Zurich, Switzerland, and shows an average increase in fitting quality (measured by Wasserstein distance) of 26% for running times and 29% for dwell times compared to an approach not including conditional dependencies, i.e., convolution of link running and dwell times.

Ultralarge-scale approximations and galaxy clustering: Debiasing constraints on cosmological parameters

ABSTRACT Upcoming galaxy surveys will allow us to probe the growth of the cosmic large-scale structure with improved sensitivity compared to current missions, and will also map larger areas of the sky. This means that in addition to the increased precision in observations, future surveys will also access the ultralarge-scale regime, where commonly neglected effects such as lensing, redshift-space distortions, and relativistic corrections become important for calculating correlation functions of galaxy positions. At the same time, several approximations usually made in these calculations such as the Limber approximation break down at those scales. The need to abandon these approximations and simplifying assumptions at large scales creates severe issues for parameter estimation methods. On the one hand, exact calculations of theoretical angular power spectra become computationally expensive, and the need to perform them thousands of times to reconstruct posterior probability distributions for cosmological parameters makes the approach unfeasible. On the other hand, neglecting relativistic effects and relying on approximations may significantly bias the estimates of cosmological parameters. In this work, we quantify this bias and investigate how an incomplete modelling of various effects on ultralarge scales could lead to false detections of new physics beyond the standard ΛCDM model. Furthermore, we propose a simple debiasing method that allows us to recover true cosmologies without running the full parameter estimation pipeline with exact theoretical calculations. This method can therefore provide a fast way of obtaining accurate values of cosmological parameters and estimates of exact posterior probability distributions from ultralarge-scale observations.

Bimonotone subdivisions of point configurations in the plane

Bimonotone subdivisions in two dimensions are subdivisions all of whose sides are either vertical or have nonnegative slope. They correspond to statistical estimates of probability distributions of strongly positively dependent random variables. The number of bimonotone subdivisions compared to the total number of subdivisions of a point configuration provides insight into how often the random variables are positively dependent. We give recursions as well as formulas for the numbers of bimonotone and total subdivisions of $2\times n$ grid configurations in the plane. Furthermore, we connect the former to the large Schr\"oder numbers. We also show that the numbers of bimonotone and total subdivisions of a $2\times n$ grid are asymptotically equal. We then provide algorithms for counting bimonotone subdivisions for any $m \times n$ grid. Finally, we prove that all bimonotone triangulations of an $m \times n$ grid are connected by flips. This gives rise to an algorithm for counting the number of bimonotone (and total) triangulations of an $m\times n$ grid.

Early field planning using optimisation and considering uncertainties

During early phases of oil field development, field planners must decide upon the optimal number of wells and optimal field plateau rate, usually by performing sensitivity studies. These design choices are then “frozen” in subsequent development stages. However, they often end up being suboptimal when the field is built and produced and the uncertainty is reduced. In this work, we employ non-linear numerical optimisation, latin hypercube sampling and the Schwartz & Smith oil price model to compute probability distributions of the optimal number of wells, plateau rate and project value. We also employ an analytical model to compute production profiles and project value and consider uncertainties in in-place oil volume, well productivity and oil price. Then, we study how do these distributions change from early field planning until when the field is abandoned, when uncertainties are reduced to a minimum. The variation in time of the in-place oil volume uncertainty is modelled with a random walk. The well productivity is a step function altered randomly after production startup. The actual oil price trajectory is picked randomly from possible trajectories computed with the Schwartz and Smith model. The results show that the distributions of the optimal number of wells, plateau rate and project value depend greatly on the uncertainties in the input data. Field designs based on the average of the distributions during the early phase are profitable, but suboptimal. A potential upside of such designs is that they entail less capital investment and therefore less financial risk when compared against the optimal field design. • Robust approach to consider uncertainties when computing optimal field design. • The optimal design strongly depends on the uncertainties in input parameters. • Uncertainties in input parameters are considered to vary during the field’s life. • The value of perfect information is modest for the cases studied in this paper.

Non-Linear Probabilistic Modification of Miner's Rule for Damage Accumulation.

A non-linear modification to Miner’s rule for damage accumulation is proposed to reduce the scatter between experimental fatigue life and fatigue life predicted according to the original Miner’s sum. Based on P-s-n probability distribution and design s-n curves, the modification satisfies the assumption of equality between the mean damage degree (at the critical level) and fatigue life random variables, which is not covered in the original formulation. The adopted formulation shows the discrepancies between the fatigue lives predicted according to the design s-n curves and the estimated probability distribution. It also proves that it is inappropriate to apply a normal distribution to fatigue life analysis and that the model becomes non-linear only for non-normal distributions. The predictions according to the established model were compared to the predictions obtained with Miner’s rule.

Prediction for Various Drought Classes Using Spatiotemporal Categorical Sequences

Drought frequently spreads across large spatial and time scales and is more complicated than other natural disasters that can damage economic and other natural resources worldwide. However, improved drought monitoring and forecasting techniques can help to minimize the vulnerability of society to drought and its consequent influences. This emphasizes the need for improved drought monitoring tools and assessment techniques that provide information more precisely about drought occurrences. Therefore, this study developed a new method, Model-Based Clustering for Spatio-Temporal Categorical Sequences (MBCSTCS), that uses state selection procedures through finite mixture modeling and model-based clustering. The MBCSTCS uses the functional structure of first-order Markov model components for modeling each data group. In MBCSTCS, the suitable order K of the components is selected by Bayesian information criterion (BIC). In MBCSTCS, the estimated mixing proportions and the posterior probabilities are used to compute probability distribution associated with the future steps of transitions. Furthermore, MBCSTCS predicts drought occurrences in future time using spatiotemporal categorical sequences of various drought classes. The MBCSTCS is applied to the six meteorological stations in the northern area of Pakistan. Moreover, it is found that MBCSTCS provides expeditious information for the long-term spatiotemporal categorical sequences. These findings may be helpful to make plans for early warning systems, water resource management, and drought mitigation policies to decrease the severe effects of drought.

Probabilistic characterization of soil properties based on the maximum entropy method from fractional moments: Model development, case study, and application

In probabilistic reliability analysis for geotechnical engineering applications, determining the underlying probability distributions of soil properties and corresponding parameters from observed data is a critical initial step because subsequent risk and reliability analyses depend upon these evaluations. Conventionally, the choice of probability distribution is dictated by subjective familiarity with a classical (e.g., normal or lognormal) distribution. This paper proposes an objective and unbiased method to estimate probability distributions of a soil property using the maximum entropy method from fractional moments of observed data. The probability distribution is based on the concept of maximum entropy and is free from the assumptions of classical distributions. A case study is presented for the undrained shear strength of soil in the Nipigon River landslide area, Ontario, Canada. The maximum entropy distributions of the soil property from fractional moments are compared to the frequency histogram, normal and lognormal distributions, and maximum entropy distributions from integral moments. The maximum entropy distribution with two-order fractional moments, almost equivalent to that with four-order integral moments, is verified with the chi-square goodness-of-fit test. Issues related to overfitting/underfitting, minimum sample size, fractional orders, negative moments, and limitations are discussed. Application of the method to geotechnical reliability analysis is illustrated.