Probability Density Function Of Parameters Research Articles

Probabilistic three-dimensional finite element analysis of geosynthetic-reinforced pile-supported embankments

The current study focuses on a probabilistic three-dimensional finite element analysis of geosynthetic-reinforced pile-supported embankments. The probabilistic analysis utilizes a finite element technique and was executed using Python scripting in Abaqus. A deterministic model is established for both unreinforced and geosynthetic-reinforced pile-supported embankments, employing a basic linear elastic perfectly plastic Mohr-Coulomb constitutive model. This numerical model adopted a simplified methodology that eliminates the need to simulate actual piles and subsoil. The results are validated across different embankment heights and mesh sizes. Based on results from Monte-Carlo simulations, we aimed to show how results can vary even by considering only one random variable. Probability density functions for various parameters, including pile efficacy, maximum differential settlement, and maximum strain in the reinforcement, are shown. The findings illustrate the ability of probabilistic approaches to predict the most likely occurrences for different governing parameters of geosynthetic-reinforced pile-supported embankments. They also highlight the potential of probabilistic approaches to expand further into a full-scale comprehensive study which can offer deeper insights into the stability of these embankments considering various sources of uncertainties.

Probabilistic damage identification for bridges using multiple damage-sensitive features and FE model update

A specific damage-sensitive feature (DSF) is usually sensitive to damage in a certain area, and the specific location of damage in a specific area can be detected by selecting appropriate DSFs. However, damaged components are seldom known before inspections. This makes it difficult to find an appropriate DSF, and damage identification is sometimes challenging. This paper aims to propose a natural frequency and displacement ratio-based probabilistic damage identification method for bridges using the finite element (FE) model update to solve this issue. The displacement ratio measured at two positions on the bridge is proposed as a new DSF. It is a useful DSF because it is independent of external loads. By integrating displacement ratios and natural frequencies, that is, multiple DSFs, as a decision-level data fusion approach, this paper identifies damaged components without choosing an appropriate DSF beforehand. In addition, probability density functions (PDFs) of structural parameters are estimated from PDFs of derived DSFs through the FE model update to consider errors and uncertainties in measurements. An in-house model bridge experiment is carried out to investigate the feasibility. The results demonstrated that the two kinds of damages in a bearing and girder reproduced in the experiment were successfully identified without false positives even when these damages simultaneously occurred.

Bayesian data-driven framework for structural health monitoring of composite structures under limited experimental data

Ultrasonic-guided waves can be used to monitor the health of thin-walled structures. However, the run of experimental damage tests on materials like carbon fiber-reinforced plastics can be impractical and costly. Instead, numerical models can be used to create hybrid datasets to train machine learning algorithms, integrating data from numerical and experimental tests. This paper presents a Bayesian-driven framework to compensate for limited experimental data regarding Lamb wave propagation in composite plates. Using Bayesian inference, the framework updates a numerical finite element model, considering observed uncertainties by sampling posterior probability density functions for input parameters using Markov–Chain Monte Carlo simulations with the Metropolis-Hastings algorithm. A neural network surrogate model speeds-up these simulations, leading to a model that replicates the uncertain experimental setup. This model then generates data to augment true experimental data. Finally, a one-dimensional convolutional neural network is trained on a three different datasets to analyze Lamb wave signals and assess damage. Comparing training strategies shows the hybrid dataset augmented by samples generated by the updated FE model gives the most accurate damage size predictions.

Finite element model updating and damage detection using strain-based modal data in the Bayesian framework

This paper proposes a novel methodology for Bayesian model updating and damage detection of a structural system using strain-based modal data. The strain-based modal data are assumed to consist of the posterior statistics of natural frequencies and strain mode shapes corresponding to measured strains from a classically damped system. The dynamic strain-based equation of the motion is formulated utilizing the energy functional of the structural system and the strain displacement relation. The state variables in the considered strain-based formulation consist of the strains and the elemental rigid body displacements. The methodology is developed by incorporating strain-based formulation into the Bayesian framework for Finite Element (FE) model updating. Besides, additional uncertain parameters in the form of system strain-based modal parameters are introduced in the eigenvalue equation to avoid mode-matching. The model reduction technique is utilized to eliminate the rigid body displacements and the strains that are not measured. Metropolis-Hastings (MH) within Gibbs sampling is proposed to simulate samples from the posterior Probability Density Function (PDF) of the model parameters of the FE model. The effectiveness of the proposed methodology is demonstrated using simulated examples. An experimental study is also conducted to validate the proposed methodology.

A parametric additive hazard model for time-to-event analysis

BackgroundIn recent years, the use of non- and semi-parametric models which estimate hazard ratios for analysing time-to-event outcomes is continuously criticized in terms of interpretation, technical implementation, and flexibility. Hazard ratios in particular are critically discussed for their misleading interpretation as relative risks and their non-collapsibility. Additive hazard models do not have these drawbacks but are rarely used because they assume a non- or semi-parametric additive hazard which renders computation and interpretation complicated.MethodsAs a remedy, we propose a new parametric additive hazard model that allows results to be reported on the original time rather than on the hazard scale. Being an essentially parametric model, survival, hazard and probability density functions are directly available. Parameter estimation is straightforward by maximizing the log-likelihood function.ResultsApplying the model to different parametric distributions in a simulation study and in an exemplary application using data from a study investigating medical care to lung cancer patients, we show that the approach works well in practice.ConclusionsOur proposed parametric additive hazard model can serve as a powerful tool to analyze time-to-event outcomes due to its simple interpretation, flexibility and facilitated parameter estimation.

A full‐probabilistic cloud analysis for structural seismic fragility via decoupled M‐PDEM

AbstractPerformance‐based earthquake engineering (PBEE) is essential for ensuring engineering safety. Conducting seismic fragility analysis within this framework is imperative. Existing methods for seismic fragility analysis often rely heavily on double loop reanalysis and empirical data fitting, leading to challenges in obtaining high‐precision results with a limited number of representative structural analysis instances. In this context, a new methodology for seismic fragility based on a full‐probabilistic cloud analysis is proposed via the decoupled multi‐probability density evolution method (M‐PDEM). In the proposed method, the assumption of a log‐normal distribution is not required. According to the random event description of the principle of preservation of probability, the transient probability density functions (PDFs) of intensity measure (IM) and engineering demand parameter (EDP), as key response quantities of the seismic‐structural system, are governed by one‐dimensional Li‐Chen equations, where the physics‐driven forces are determined by representative analysis data of the stochastic dynamic system. By generating ground motions based on representative points of basic random variables and performing structural dynamic analysis, the decoupled M‐PDEM is employed to solve the one‐dimensional Li‐Chen equations. This yields the joint PDF of IM and EDP, as well as the conditional PDF of EDP given IM, resulting in seismic fragility analysis outcomes. The numerical implementation procedure is elaborated in detail, and validation is performed using a six‐story nonlinear reinforced concrete (RC) frame subjected to non‐stationary stochastic ground motions. Comparative analysis against Monte Carlo simulation (MCS) and traditional cloud analysis based on least squares regression (LSR) reveals that the proposed method achieves higher computational precision at comparable structural analysis costs. By directly solving the physics‐driven Li‐Chen equations, the method provides the full‐probabilistic joint information of IM and EDP required for cloud analysis, surpassing the accuracy achieved by traditional methods based on statistical moment fitting and empirical distribution assumptions.

Statistical analysis of parameters and adsorption isotherm models.

The present work intends to discuss parameter estimation and statistical analysis in adsorption. The Langmuir and Tóth isotherm models are compared for a set of carbon dioxide adsorption data on 13X zeolite from literature at different temperatures: 303, 323, 373, and 423 K. Statistical analyses were performed under frequentist and Bayesian perspectives. Under the frequentist statistical view, parameters were estimated using Maximum Likelihood estimation (MLE). Statistical analyses of parameters were performed by confidence regions in terms of elliptical approximation and likelihood region, while the evaluation of models was performed by chi-square statistics. The results showed that, for these nonlinear models, the elliptical region offers a poor approximation of the parameter estimates' confidence region, especially for the most correlated parameter pairs. Additionally, the four-parameter Tóth's equation yields less correlated parameters than the three-parameter Langmuir model. From a Bayesian perspective, the Markov chain Monte Carlo (MCMC) technique facilitated the reconstruction of the probability density functions of parameters as well as enabled the propagation of parametric uncertainties in the model responses. Finally, the accurate assessment of experimental uncertainty significantly influences the evaluation of models and their respective parameters.

Entropy-metric estimation of the small data models with stochastic parameters

The formalization of dependencies between datasets, taking into account specific hypotheses about data properties, is a constantly relevant task, which is especially acute when it comes to small data. The aim of the study is to formalize the procedure for calculating optimal estimates of probability density functions of parameters of linear and nonlinear dynamic and static small data models, created taking into account specific hypotheses regarding the properties of the studied object. The research methodology includes probability theory and mathematical statistics, information theory, evaluation theory, and stochastic mathematical programming methods. The mathematical apparatus presented in the article is based on the principle of maximization of information entropy on sets determined as a result of a small number of censored measurements of “input" and “output" entities in the presence of noise. These data structures became the basis for the formalization of linear and nonlinear dynamic and static models of small data with stochastic parameters, which include both controlled and noise-oriented input and output measurement entities. For all variants of the above-mentioned small data models, the tasks of determining the optimal estimates of the probability density functions of the parameters were carried out. Formulated optimization problems are reduced to the forms canonical for the stochastic linear programming problem with probabilistic constraints.

Statistical Treatment of Convolutional Neural Network Superresolution of Inland Surface Wind for Subgrid-Scale Variability Quantification

Abstract Machine learning models have been employed to perform either physics-free data-driven or hybrid dynamical downscaling of climate data. Most of these implementations operate over relatively small downscaling factors because of the challenge of recovering fine-scale information from coarse data. This limits their compatibility with many global climate model outputs, often available between ∼50- and 100-km resolution, to scales of interest such as cloud resolving or urban scales. This study systematically examines the capability of a type of superresolving convolutional neural network (SR-CNNs) to downscale surface wind speed data over land from different coarse resolutions (25-, 48-, and 100-km resolution) to 3 km. For each downscaling factor, we consider three convolutional neural network (CNN) configurations that generate superresolved predictions of fine-scale wind speed, which take between one and three input fields: coarse wind speed, fine-scale topography, and diurnal cycle. In addition to fine-scale wind speeds, probability density function parameters are generated through which sample wind speeds can be generated, accounting for the intrinsic stochasticity of wind speed. For assessing generalization to new data, CNN models are tested on regions with different topography and climate that are unseen during training. The evaluation of superresolved predictions focuses on subgrid-scale variability and the recovery of extremes. Models with coarse wind and fine topography as inputs exhibit the best performance when compared with other model configurations, operating across the same downscaling factor. Our diurnal cycle encoding results in lower out-of-sample generalizability when compared with other input configurations.

Response Surface Model of the Reduced Flexibility Matrix for Bayesian Damage Identification.

The present work is concerned with the use of a Response Surface Model of the reduced flexibility matrix for structural damage identification. A Response Surface Model (RSM) is fitted with the aim at providing a polynomial relationship between nodal cohesion parameters, used to describe the damage field within the structure, and elements of the reduced flexibility matrix. A design of experiment built on combinations of a relatively small number of nodal cohesion parameters is used to fit the RSM. The damage identification problem is formulated within the Bayesian framework and the Delayed Rejection Adaptive Metropolis method is used to sample the posterior probability density function of the uncertain cohesion parameters. Numerical simulations addressing damage identification in plates were carried out in order to assess the proposed approach, which succeeded in the identification of the different damage profiles considered. Besides, the use of a RSM, instead of a FEM of the structure, resulted in reductions of up almost 78% in the required computational cost.

A process‐based water stable isotope mixing model for plant water sourcing

AbstractStable isotopes of hydrogen and oxygen in water are common tools for investigating water uptake apportionment, but many of the existing methods rely on simple linear mixing approaches that do not mechanistically incorporate additional information about site physical properties and conditions. Here, we develop a ‘physically based root water uptake isotope mixing estimation’ model (PRIME) that combines a continuous and parametric probability density function for root water uptake with site physical data in a process‐based linear mixing framework. To demonstrate the application of PRIME, water uptake patterns of boreal forest Pinus banksiana trees were estimated on four dates in 2019. To aid in validation, estimates were compared with that of the Bayesian linear mixing model framework, MixSIAR. The two approaches provided similar results, but due to its continuous and parametric nature, PRIME provided estimates of superior resolution, certainty, and model parsimony. Although both models incorporate additional physical information into their mixing frameworks, PRIME does so in a mechanistic manner, thereby reflecting the relevant hydrological processes more effectively than the purely empirical approach taken by MixSIAR. Furthermore, because PRIME uses a continuous function to describe the predicted uptake pattern, it allows users to quantify water uptake with essentially infinite resolution, through integration over the desired depth ranges. These findings demonstrate the advantages of utilizing a continuous, parametric, and process‐based mixing model to estimate root water uptake apportionment, thus providing a relatively simple yet powerful tool with which to approach plant water sourcing.

Stochastic energy management of a multi-microgrid system with battery/ supercapacitor energy storages considering demand response and transactive energy

This paper proposes stochastic energy management for an electrical system consisting of a multi-microgrid and a distribution company (DISCO). The objective function aims at finding an optimal day-ahead energy schedule of dispatchable resources, energy storages, and demand response in the presence of intermittent renewable energy sources. Two different energy storage devices, i.e., battery and supercapacitor (SC), are modeled in the system to provide both high power and energy density. Wind speed, solar irradiation, and electrical demand uncertainties have been modeled using probability density functions of uncertain parameters, and the number of scenarios has been decreased using mixed-integer linear programming (MILP) based scenario reduction with a novel characteristic. In the proposed model, the DISCO, as a leader, attempts to maximize its profit while the microgrids (MGs), as followers, tend to minimize their operational costs. Karush-Kuhn-Tucker, strong duality, and Fortuny-Amat transformation are used to convert the non-linear bi-level problem into a linear single-level problem. The proposed model was tested on a hypothetical distribution system and the results showed the efficiency of the model in reducing operational costs of MGs by 7.5%, increasing the DISCO profit by 6.9%, and improving the reliability of the system by reducing the energy not supplied.

Gaussian process metamodel and Markov chain Monte Carlo-based Bayesian inference framework for stochastic nonlinear model updating with uncertainties

The estimation of the posterior probability density function (PDF) of unknown parameters remains a challenge in stochastic nonlinear model updating with uncertainties; thus, a novel Bayesian inference framework based on the Gaussian process metamodel (GPM) and the advanced Markov chain Monte Carlo (MCMC) method is proposed in this paper. The instantaneous characteristics (ICs) of the decomposed measurement response, calculated using the Hilbert transform and the discrete analytical mode decomposition methodology, are extracted as nonlinear indices and further used to construct the likelihood function. Then, the posterior PDFs of structural nonlinear model parameters are derived based on the Bayesian theorem. To precisely calculate the posterior PDF, an advanced MCMC approach, i.e., delayed rejection adaptive Metropolis-Hastings (DRAM) algorithm, is adopted with the advantages of a high acceptance ratio and strong robustness. However, as a common shortage in most MCMC methods, the resampling technology is still applied, and numerous iterations of nonlinear simulations are conducted to ensure accuracy, thus directly reducing the computational efficiency of the DRAM. To address the abovementioned issue, a mathematical regression metamodel of the GPM with a polynomial kernel function is adopted in this paper instead of the traditional finite element model (FEM) to simulate a nonlinear response for the reduction of computational cost, and the hyperparameters are further estimated using the conjugate gradient optimization methodology. Finally, numerical simulations concerning a Giuffré–Menegotto–Pinto modeled steel-frame structure and a seven-storey base-isolated structure are conducted. Furthermore, a shake-table experimental test of a nonlinear steel framework is investigated to validate the accuracy of the Bayesian inference method. Both simulations and experiment demonstrate that the proposed GPM and DRAM-based Bayesian method effectively estimates the posterior PDF of unknown parameters and is appropriate for stochastic nonlinear model updating even with multisource uncertainties.

Parametric Information Geometry with the Package Geomstats

We introduce the information geometry module of the Python package Geomstats. The module first implements Fisher–Rao Riemannian manifolds of widely used parametric families of probability distributions, such as normal, gamma, beta, Dirichlet distributions, and more. The module further gives the Fisher–Rao Riemannian geometry of any parametric family of distributions of interest, given a parameterized probability density function as input. The implemented Riemannian geometry tools allow users to compare, average, interpolate between distributions inside a given family. Importantly, such capabilities open the door to statistics and machine learning on probability distributions. We present the object-oriented implementation of the module along with illustrative examples and show how it can be used to perform learning on manifolds of parametric probability distributions.

Bayesian calibration for Lamb wave propagation on a composite plate using a machine learning surrogate model

This paper presents a new framework for stochastic updating of a finite element model for a composite plate, considering the influence of temperature on Lamb wave propagation. The framework involves deterministic updating to optimize mechanical properties and stochastic updating to derive probability density functions for key parameters. It utilizes sensitivity analysis and Bayesian inference with Markov-Chain Monte Carlo simulations and the Metropolis–Hastings sampling algorithm. This paper proposes a machine learning surrogate model based on artificial neural networks to improve computational efficiency. This surrogate modeling approach allows parallelized Monte Carlo simulations, reducing updating time significantly without compromising the accuracy of the resulting probability density functions for model parameters. These advancements show a promising way to enhance composite plate modeling and Lamb wave propagation studies, providing a more efficient and accurate approach to verify and validate finite element models with potential applications in engineering simulations.

Stochastic characterisation of unstable lean hydrogen–air annular premixed flames

This study is devoted to the characterisation of lean premixed hydrogen–air laminar flames stabilised on an annular burner. The flames studied are chosen to lie close to the lean flammability limit, spanning a range of Damköhler and Reynolds numbers, and correspond to situations where instabilities arise. The experimental characterisation is achieved by examining the statistical moments of the flame hydroxyl radical (OH*) chemiluminescence and schlieren. The standard deviation and the skewness, that are needed to fully characterise the measured probability density functions, are shown to exhibit a non-monotonic behaviour with the distance from the burner surface, first increasing, then decreasing downstream the flame tip. An exponentially modified Gaussian distribution model is proposed to describe the skewed schlieren fluctuations probability density function. This model is closed by developing relations for the probability density function parameters based on the Damköhler and Reynolds numbers, and then used to describe the OH* stochastic behaviour. The relation between the distribution parameters and the transport equation of a reactive scalar gradient is addressed also.

Non-probabilistic credible reliability analysis of the composite laminate

For the reliability analysis of composite laminate structures, there is a problem of difficult data acquisition. Traditional reliability analysis methods require a large amount of data to maintain calculation accuracy. Therefore, the traditional reliability analysis method is not applicable to composite laminates. For the above-mentioned problems, this paper presents a method to calculate the credible reliability of composite laminate structures using the Bayesian theorem. The method is distinguished from the traditional uncertainty analysis methods. The likelihood function and the newly introduced samples are first used to correct the prior distribution of the uncertain parameters. Furthermore, the posterior distribution, containing information from both the sample and prior distributions, can be obtained. Ultimately, based on the acquired posterior distribution, the probability density function can be updated with newly introduced samples. Using the Hashin failure criterion, the failure probabilities of the fibers and matrix of the composite laminate structure are quantified, respectively. Then, the credible reliability of the composite laminate is obtained. Numerical example results show the proposed method can maintain high accuracy with fewer samples and demonstrate the validity of the model. For the numerical calculations, the deterministic parameters of the distribution function and the functional distribution parameters were verified, composite laminates and aircraft wing structures were analyzed. The experimental results show that the error between the probability density function of the parameters and the probability density function predicted by the sample is less than 0.045. Compared with the traditional reliability analysis methods for composite structures, this method can effectively measure the credible reliability of composite laminate structures with fewer samples and similar accuracy. The method enables real-time updating of the parameter probability density function by newly introduced samples and improves the confidence level.

Cosmic birefringence from the Axiverse

We revisit the evidence for CMB birefringence in the context of a rich Axiverse. Using probability density functions (PDFs) for various axion parameters,such as the mass and axion decay constant, we construct the PDF for thecosmic birefringence angle and investigate its properties.By relating the observed value of the birefringence angle to the mean or standard deviationof the constructed PDF, we constrain the shape of the input PDFs, providing insightsinto the statistical distribution of the Axiverse. We focus on three different types of axionpotentials: cosine, quadratic, and asymptotically linear axion monodromy.Ouranalysis showcases the potential of cosmic birefringence in constraining the distributionof axion parameters and uncovering possible correlations among them.We additionally offerpredictions for “birefringence tomography”, anticipating future measurements of birefringence fromlower multipoles, and show how it can be used to rule out simpler versions of the Axiverse. Our findings contribute to theongoing exploration of the Axiverse and its implications for cosmic birefringence.

Probability distributions for stochastic comfort in railway vehicles

Comfort has traditionally been assessed as a single deterministic index that can be assigned to each railway vehicle type (at a given speed on a given track quality level). However, it is now well established that passengers influence vibration and, as a consequence, occupancy level and seating arrangement influence comfort, thus turning it into a stochastic variable. The concept of Compound Comfort, presented in Palomares et al. [Is the standard ride comfort index an actual estimation of railway passenger comfort? Veh Syst Dyn. 2022;1–14. doi: 10.1080/00423114.2022.2148543], is used in this paper to evaluate several train types. Simplified models are used as a first approximation for vibration analysis. Despite model simplicity, since the number of possible seating arrangements is intractable, a Monte Carlo procedure is used to obtain compound comfort probability density distributions from a randomised subset of cases. The results from the work presented here will show that the information provided by stochastic distributions is much richer than a single deterministic index. Nevertheless, obtaining distributions from Monte Carlo tests (numerical or experimental) is impractical for any commercial assessment of comfort. Therefore, an inference procedure is called for and has been devised in order to obtain probability density function parameters from just one tare test, along with ten runs of a half laden test. This recipe has been adapted from one presented previously to be able to accommodate the skewed distributions that emerge when a wide range of train configurations is considered.

Bayesian damage identification of an unsymmetrical frame structure with an improved PSO algorithm

With the development of artificial intelligence technology, intelligent algorithms are widely utilized in the field of probabilistic model updating to solve the optimization problems. In this study, the opposition-based learning based on population centroid and boundary rebound strategy was incorporated into PSO to improve the optimization performance on structural damage detection, and the damage identification of an unsymmetrical frame based on Bayesian model updating with the improved PSO algorithm is proposed for the first time. Six benchmark functions were firstly employed to test the optimization performance of the improved PSO algorithm, which overcomes the premature convergence of the original PSO algorithm in which trapped particles often fall into local optimum. Then three damage scenarios of a laboratory unsymmetrical frame were utilized to validate the effectiveness of the proposed method in structural damage identification. The encouraging experimental results exhibit that the proposed method not only can successfully identify the damage locations and extents, but the associated uncertainties of the identified results can be quantified by calculating the posterior probability density functions of the uncertain model parameters. In addition, the effects of the quantities of measured information on the posterior uncertainties of the model parameters were investigated, and the uncertainty analysis results reveal that the posterior uncertainties associated with the identified model parameters are sensitive to the amount of measurement used in model updating. The proposed method is expected to help in damage identification and condition assessment of the practical engineering structures.