Estimation Of Probability Distribution Research Articles

Extracting Latent Variables From Forecast Ensembles and Advancements in Similarity Metric Utilizing Optimal Transport

AbstractThis study presents a novel methodology for extracting latent variables from high‐dimensional sparse data, particularly emphasizing spatial distributions such as precipitation distribution. This approach utilizes multidimensional scaling with a distance matrix derived from a new similarity metric, the Unbalanced Optimal Transport Score (UOTS). UOTS effectively captures discrepancies in spatial distributions while preserving physical units. This is similar to mean absolute error, however it considers location errors, providing a more robust measure crucial for understanding differences between observations, forecasts, and ensembles. Probability distribution estimation of these latent variables enhances the analytical utility, quantifying ensemble characteristics. The adaptability of the method to spatiotemporal data and its ability to handle errors suggest its potential as a promising tool for diverse research applications.

Distributionally robust chance-constrained optimization with Gaussian mixture ambiguity set

Conventional chance-constrained programming methods suffer from the inexactness of the estimated probability distribution of the underlying uncertainty from data. To this end, a distributionally robust approach to the problem allows for a level of ambiguity considered around a reference distribution. In this work, we propose a novel formulation for the distributionally robust chance-constrained programming problem using an ambiguity set constructed from a variant of optimal transport distance that was developed for Gaussian Mixture Models. We show that for multimodal process uncertainty, our proposed method provides an effective way to incorporate statistical moment information into the ambiguity set construction step, thus leading to improved optimal solutions. We illustrate the performance of our method on a numerical example as well as a chemical process case study. We show that our proposed methodology leverages the multimodal characteristics from the uncertainty data to give superior performance over the traditional Wasserstein distance-based method.

Elastic Shape Analysis of Tree-Like 3D Objects Using Extended SRVF Representation.

How can one analyze detailed 3D biological objects, such as neuronal and botanical trees, that exhibit complex geometrical and topological variation? In this paper, we develop a novel mathematical framework for representing, comparing, and computing geodesic deformations between the shapes of such tree-like 3D objects. A hierarchical organization of subtrees characterizes these objects - each subtree has a main branch with some side branches attached - and one needs to match these structures across objects for meaningful comparisons. We propose a novel representation that extends the Square-Root Velocity Function (SRVF), initially developed for Euclidean curves, to tree-shaped 3D objects. We then define a new metric that quantifies the bending, stretching, and branch sliding needed to deform one tree-shaped object into the other. Compared to the current metrics such as the Quotient Euclidean Distance (QED) and the Tree Edit Distance (TED), the proposed representation and metric capture the full elasticity of the branches (i.e., bending and stretching) as well as the topological variations (i.e., branch death/birth and sliding). It completely avoids the shrinkage that results from the edge collapse and node split operations of the QED and TED metrics. We demonstrate the utility of this framework in comparing, matching, and computing geodesics between biological objects such as neuronal and botanical trees. We also demonstrate its application to various shape analysis tasks such as (i) symmetry analysis and symmetrization of tree-shaped 3D objects, (ii) computing summary statistics (means and modes of variations) of populations of tree-shaped 3D objects, (iii) fitting parametric probability distributions to such populations, and (iv) finally synthesizing novel tree-shaped 3D objects through random sampling from estimated probability distributions.

Polynomial approaches in improving accuracy of probability distribution estimation using the method of moments

AbstractBACKGROUNDDetermination of a probability density function (PDF) is an area of active research in engineering sciences as it can improve process systems. A previously developed polynomial method‐of‐moments‐based PDF estimation model has been applied in the research to produce accurate approximations to both standard and more complex PDF. A model with a different polynomial basis than a monomial is still to be developed and evaluated. This is the work that is undertaken in this study.RESULTSA set of standard PDF (Normal, Weibull, Log Normal and Bimodal) and more complex distributions (solutions to the Smoluchowski coagulation equation and Population Balance equation) were approximated by the method‐of‐moments using Chebyshev, Hermite and Lagrange polynomial‐based density functions. Results show that Lagrange polynomial‐based models improve the fit compared to monomial based‐modeling in terms of RMSE and Kolmogorov–Smirnov test statistic estimates. The Kolmogorov–Smirnov test‐statistics decreased by 19% and the RMSE values were improved by around 85% compared to the standard monomial basis when using Lagrange polynomial basis.CONCLUSIONThis study indicates that the procedure using Lagrange polynomials with method‐of‐moments is a more reliable reconstruction procedure that calculates the approximate distribution using lesser number of moments, which is desirable. © 2024 The Authors. Journal of Chemical Technology and Biotechnology published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry (SCI).

이산확률분포의 표본 분위수 계산 방법에 관한 연구

In this paper, we studied the methods of computing sample quantiles for the case of discrete probability distributions. There are two main methods which are introduced in statistics textbooks. We compared these two methods by simulation from practical and educational necessity. We considered some cases of Poisson, binomial, geometric, negative binomial, and discrete uniform distributions by setting up the parameters, and for each distribution we performed 10,000 times of simulation of drawing random samples of size 20 and 50. At each time of simulation, we computed the difference between the population quantile and the sample quantile obtained by each method. We compared the two methods by using two criteria: one is the mean square error over 10,000 times of simulation, and the other is the frequency of obtaining closer sample quantile to the population quantile than the competing method. We also obtained the estimated probabilities of exact estimation of population quantiles for the two methods, and performed the tests of homogeneity on the probability distribution of under-estimation, exact estimation, and over-estimation of the population quantiles.

Estimation of Circular Statistics in the Presence of Measurement Bias.

Circular statistics and Rayleigh tests are important tools for analyzing cyclic events. However, current methods are not robust to significant measurement bias, especially incomplete or otherwise non-uniform sampling. One example is studying 24-cyclicity but having data not recorded uniformly over the full 24-hour cycle. Our objective is to present a robust method to estimate circular statistics and their statistical significance in the presence of incomplete or otherwise non-uniform sampling. Our method is to solve the underlying Fredholm Integral Equation for the more general problem, estimating probability distributions in the context of imperfect measurements, with our circular statistics in the presence of incomplete/non-uniform sampling being one special case. The method is based on linear parameterizations of the underlying distributions. We simulated the estimation error of our approach for several toy examples as well as for a real-world example: analyzing the 24-hour cyclicity of an electrographic biomarker of epileptic tissue controlled for states of vigilance. We also evaluated the accuracy of the Rayleigh test statistic versus the direct simulation of statistical significance. Our method shows a very low estimation error. In the real-world example, the corrected moments had a root mean square error of [Formula: see text]. In contrast, the Rayleigh test statistic overestimated the statistical significance and was thus not reliable. The presented methods thus provide a robust solution to computing circular moments even with incomplete or otherwise non-uniform sampling. Since Rayleigh test statistics cannot be used in this circumstance, direct estimation of significance is the preferable option for estimating statistical significance.

Exponential Expansions for Approximation of Probability Distributions

Exponential Expansions for Approximation of Probability Distributions

Land use and land cover change simulation enhanced by asynchronous communicating cellular automata

Land use and land cover change (LUCC) modeling is crucial to urban planning and policy making. A well-used and effective paradigm for LUCC is the ANN-CA model which employs an artificial neural network (ANN) to calculate a transition probability over each land cell from various driving factors, and then uses a cellular automaton (CA) to evolve all cells consecutively to update their land usage and coverage according to the estimated probability distribution. This paper focuses on the effect of delays or perturbations possibly taking place between land cells on the LUCC modeling. To this end, a new ANN-CA model is proposed which adopts an asynchronous communicating cellular automaton (ACCA), rather than the conventional synchronous CA. Especially, the ACCA allows every cell to communicate with its neighbors independently at random times via a specific protocol, which offers a natural way to include stochastic delays in exchanging the current land usages between cells. As a result, every change of a cell's land use in the ACCA may not affect its neighbors immediately, but is subject to delays that might play an important role in modeling the practical LUCC. Numerical analysis of the new model are carried out over three regions of Chongqing city, China with different scales: Yongchuan District, Sanjiao Town, and Huangguashan Village, and experimental results demonstrate that the proposed ANN-ACCA model can achieve a higher accuracy for LUCC simulation as compared to conventional ANN-CA models.

Estimation of site-specific multivariate probability distribution of soil properties using a mixed sampling technique

Estimation of multivariate probability density function (PDF) of soil properties based on limited site investigation data plays an important role in data driven-based geotechnical engineering analysis and design. However, this estimation is often intractable because the site-specific investigation data generally are multivariate, uncertain and unique, sparse, incomplete, sometimes corrupt and spatially variable. The estimation is more challenging when the uncertainties in model parameters of the distribution (i.e., mean, standard deviations and scale of fluctuation of soil properties) are taken into consideration. To address these difficulties, this study proposed a mixed sampling technique by combing the Gibbs sampler and transitional Markov Chain Monte Carlo sampler to effectively estimate the multivariate probability density function of soil properties. The proposed method is demonstrated through a virtual site example while the effect of statistical uncertainty in the scale of fluctuation on reliability analyses is illustrated in an infinite slope example. It is shown that the proposed method can not only handle the complex features of site investigation data but also effectively quantify the statistical uncertainties of the mean values, standard deviations and scale of fluctuation of the investigated soil properties. Ignoring the statistical uncertainties of scale of fluctuation can affect the estimation of mean and standard deviations of the soil parameters. When considering the statistical uncertainty of scale of fluctuation, the 95% confidence intervals for the profiles of the soil properties can rationally cover the actual soil data at the unobserved points. Ignoring the statistical uncertainty of scale of fluctuation can lead to the underestimation of the probability of failure of the infinite slope.

Semi-supervised diagnosis method for coupling faults of key rotating components based on EEMD-KPCA under cross working conditions

To address the challenges of extracting coupled fault features from key rotating components and classifying them under changeable operating conditions, a semi-supervised fault diagnosis method is proposed. First, Ensemble Empirical Mode Decomposition and Kernel Principal Component Analysis are employed to decompose the original coupled fault signals and reduce feature dimensionality. Experiments are conducted on labeled datasets, yielding an average classification accuracy of 92.43%. To further classify unlabeled datasets under various working conditions, a probability distribution estimation function is incorporated and a confidence threshold is set. For unlabeled data with probabilities greater than the confidence threshold, a pseudo-label is added to increase the labeled data quantity. Thus, it makes learning from these unlabeled data possible. A comparison with the other three methods under cross working conditions showcases the superiority of the proposed approach.

Approximating Probability Distributions by Using Wasserstein Generative Adversarial Networks

Approximating Probability Distributions by Using Wasserstein Generative Adversarial Networks

Estimating Probability Distributions of Travel Times by Fitting a Markovian Velocity Model

Estimating Probability Distributions of Travel Times by Fitting a Markovian Velocity Model

APPLICATION OF LANCHESTER'S MATHEMATICAL LAWS IN MILITARY STRATEGY

Mathematics, as the science of numbers, structures, and models, plays an important role in many aspects of military operations and strategies. From calculating the probability of success or failure of military operations to determining the best ways to deploy troops and resources, mathematics provides military strategists with powerful tools to make informed and effective decisions. One of the most important aspects of using mathematics in the military is developing strategies for combat operations. From determining the location of troops and the size of forces to the conditions of operations, mathematical methods and models help to optimize decisions. To determine the probability of success of military operations, mathematics uses probability theory and statistics. This allows us to estimate the probability of achieving the goal, taking into account various factors such as military equipment, enemy location, and other external conditions. The analysis of previous military conflicts and data allows us to statistically estimate probability distributions and predict military events. Mathematics also plays an important role in solving logistics and resource allocation problems. Determining the optimal route for the movement of troops and resources can be formulated as a path optimization problem. Mathematical methods are also used in the field of intelligence and the development of new technologies. Cryptography, which protects important information from unauthorized access, is based on complex mathematical algorithms. Mathematical models can also be used to simulate military operations, study the impact of new weapons systems, or analyze missile trajectories. It is worth adding that mathematical laws help to analyze and predict the outcomes of military conflicts, in particular, to determine the impact of the size and effectiveness of the enemy forces on the probability of success. In this article, the main focus is on the laws of Lanchester's mathematical models. The article presents the derivation of Lanchester's linear law and an example of its application. The mathematical principles of the quadratic Lanchester's law are considered on an example. It is indicated and highlighted how these mathematical laws can be applied in the context of the Russian-Ukrainian conflict.

Interpreting gas sorption isotherms in glassy polymers using a Bayesian framework: A view on parameter uncertainty propagation into mixture sorption predictions

Investigation of mixed-gas sorption is necessary for robust design and optimization of membrane-based processes. While sorption models for glassy polymers are well-established, they often deviate from observed mixture data. In doing so, however, parameters of these models are typically estimated via traditional least-squares optimization methods, and so parametric uncertainty is often ignored in mixture sorption predictions. As an alternative, we use Bayesian Inference (BI) to estimate probability distributions for sorption models' parameters and thus provide statistically meaningful mixture sorption forecasts that reflect parameter uncertainty rigorously. To this end, we exploit molecular sorption simulations, combining Grand Canonical Monte Carlo (GCMC) and Equilibrium Molecular Dynamics (EMD), to focus on two different glassy polymer systems (i.e., single- and mixed-gas sorption of CO2/CH4 in a fluorinated polyimide and CH4/H2 in a polymer of intrinsic microporosity) and three popular mixture sorption models (the Dual-Mode Sorption model, Non-Equilibrium Thermodynamics for Glassy Polymers model, and the Ideal Adsorbed Solution Theory) to demonstrate the benefits of this technique for uncertainty quantification and propagation in membrane applications. We show that observed sorption data at typical working pressures (e.g., 0−25atm) are often insufficient to accurately estimate model parameters, and consequently, these models often fail to represent observed mixture data adequately. Furthermore, we show that sorption data (e.g., high-pressure data), able to capture intrinsic isotherm nonlinearities of sorption models, are critical to considerably improve parameter inference from collective model-data fits and thus accurately predict mixture sorption.

Recursive Probability Mass Function Method to Calculate Probability Distributions of Pulse-Shaped Signals

This paper proposes an accurate and efficient method to calculate probability distributions of pulse-shaped complex signals. We show that the distribution over the in-phase and quadrature-phase (I/Q) complex plane is obtained by a recursive probability mass function of the accumulator for a pulse-shaping filter. In contrast to existing analytical methods, the proposed method provides complex-plane distributions in addition to instantaneous power distributions. Since digital signal processing generally deals with complex amplitude rather than power, the complex-plane distributions are more useful when considering digital signal processing. In addition, our approach is free from the derivation of signal-dependent functions. This fact results in its easy application to arbitrary constellations and pulse-shaping filters like Monte Carlo simulations. Since the proposed method works without numerical integrals and calculations of transcendental functions, the accuracy degradation caused by floating-point arithmetic is inherently reduced. Even though our method is faster than Monte Carlo simulations, the obtained distributions are more accurate. These features of the proposed method realize a novel framework for evaluating the characteristics of pulse-shaped signals, leading to new modulation, predistortion and peak-to-average power ratio (PAPR) reduction schemes.

Reliability assessment of civil structures with incomplete probability distribution information

The probability distribution of random variables is usually incomplete owing to a limited amount of data, making it a challenging problem for structural reliability assessments. This paper presents a novel reliability analysis method for civil structures with incomplete data. The probability distributions of the random variables were estimated using an improved Bayesian inference method. In the proposed inference method, the frequencies and mode shapes were selected as the observation data. The likelihood function is replaced by homogeneous chaos, and the computational efficiency of Bayesian inference is improved. The affine invariant ensemble algorithm was used to generate samplers for the posterior distribution. After estimating the probability distribution of the incomplete random variables, a reliability analysis was performed, and the failure probability of the structure was estimated. Numerical studies on a two-span, two-story reinforced concrete building and a five-span bridge are presented. Experimental studies were also performed on an eight-story steel frame structure. The results show that the probability distributions of random variables can be significantly updated using the proposed technique and modal properties. The proposed technique was more efficient than the Gibbs sampling method. The failure probabilities of the structures were estimated using the estimated probability distributions.

The fuzzy Kullback–Leibler divergence for estimating parameters of the probability distribution in fuzzy data: an application to classifying Vietnamese Herb Leaves

In this paper, we address the challenge of estimating probability distributions which are typically represented by parameter-based values. However, this estimation is prone to errors and does not comprehensively capture the nature of real-world data. Additionally, real-world data often follows a mixed form of probability distributions, where sub-datasets may contain incomplete information. To enhance flexibility, especially in classification problems, we propose a new method for describing parameters estimated through Bayesian statistics. Our method introduces fuzzy parameters and assesses the similarity between probability distributions using the fuzzy extended Kullback–Leibler divergence. We demonstrate the practical application of our approach in Vietnamese Herb Leaves classification. By incorporating fuzzy parameters and leveraging Bayesian statistics, our method provides more robust estimations of probability distributions and enables improved flexibility in classification tasks.

RETRACTED ARTICLE: Intelligent monitoring methodology for large-scale logistics transport vehicles based on parallel Internet of Vehicles

The essence of the Internet of Vehicles is a social and physical information system, including the psychological and organizational factors of human beings. The complexity of the Internet will lead to certain deviations when monitoring vehicles. Therefore, the Parallel Internet of Vehicles is employed to monitor the information on large-scale logistics transport vehicles. This platform is built based on the ACP intelligent approach, which consists of three parts: An artificial system (A), a computational experiment (C), and parallel execution (P). The Adaboost algorithm is used to extract information on large-scale logistics transport vehicles from the ACP parallel Internet of Vehicles, and the Tabu search strategy is applied to optimize the Monte Carlo positioning algorithm. The approximate optimal estimation is obtained by optimizing the filtering to eliminate vehicle positions with fewer possibilities. The weight of important sampling values of the independent vehicle node positions is integrated to complete the posterior probability distribution estimation of the possible positions of vehicles, in order to realize vehicle position monitoring. It is verified that the root-mean square error of the algorithm when positioning a vehicle is less than 0.18, and the monitoring deviation is quite small.

Adaptive Bayesian learning for making risk-aware decisions: A case of trauma survival prediction

Decision tree (DT) models provide a transparent approach to prediction of patient’s outcomes within a probabilistic framework. Averaging over DT models under certain conditions can deliver reliable estimates of predictive posterior probability distributions, which is of critical importance in the case of predicting an individual patient’s outcome. Reliable estimations of the distribution can be achieved within the Bayesian framework using Markov chain Monte Carlo (MCMC) and its Reversible Jump extension enabling DT models to grow to a reasonable size. Existing MCMC strategies however have limited ability to control DT structures and tend to sample overgrown DT models, making unreasonably small partitions, thus deteriorating the uncertainty calibration. This happens because the MCMC explores a DT model parameter space within a limited knowledge of the distribution of data partitions. We propose a new adaptive strategy which overcomes this limitation, and show that in the case of predicting trauma outcomes the number of data partitions can be significantly reduced, so that the unnecessary uncertainty of estimating the predictive posterior density is avoided. The proposed and existing strategies are compared in terms of entropy which, being calculated for predicted posterior distributions, represents the uncertainty in decisions. In this framework, the proposed method has outperformed the existing sampling strategies, so that the unnecessary uncertainty in decisions is efficiently avoided.

Two-layer multiple scenario optimization framework for integrated energy system based on optimal energy contribution ratio strategy

Rational design and advanced energy management considering multiple uncertainties are imperative for the superior integrated energy system (IES). This work proposed a novel two-layer stochastic multiple scenario optimization framework for the collaborative optimization of capacity and operation of IES. To improve the accuracy of probability density estimation, the improved kernel density estimation (KDE) was employed to obtain the probability density distributions of wind speed, sunlight and multi-demands. Then the scenario sets were generated by Latin hypercube sampling (LHS) simulation and self-organization map (SOM) clustering. To decouple the output, efficiency and part load factor of devices during operation, the following optimal contribution rate (FOCR) strategy was proposed, which could actively adjust the output ratio of energy conversion devices to realize the flexible energy supply. The developed optimization methodology was used for a case study in an office building. The results indicate that the improved KDE for probability distribution estimation of uncertainties achieves the accuracy percentage enhancement, for the average value, of 37.1% and 49.6% for root-mean-square error (RMSE), respectively, compared with the conventional KDE and parametric model. Considering energy saving, economy and environmental indicators, the performance of the FOCR strategy is superior to those of the three traditional strategies.