Probability Density Function Of Parameters Research Articles

Normal Faulting Movement During the 2020 Mw 6.4 Yutian Earthquake: A Shallow Rupture in NW Tibet Revealed by Geodetic Measurements

The ENE striking Longmu Co fault and the North Altyn Tagh left-lateral slip fault have led to the complex regional structure in the northwestern Tibetan Plateau, resulting in a series of normal faulting and strike slip faulting earthquakes. Using both the ascending and descending Sentinel-1A/B radar images, we depict the coseismic deformation caused by the 2020 Yutian Mw 6.4 earthquake with a peak subsidence of ~ 20 cm. We determine the seismogenic fault geometry by applying the Bayesian approach with a Markov Chain Monte Carlo sampling method, which can better characterize the posterior probability density functions of the source model parameters. The estimation results reveal that the earthquake is a normal faulting event with a moderate strike slip component. Based on the optimal fault geometry model, we extend the fault plane and invert for the distributed coseismic slip model. The optimal slip model shows that the coseismic slip is mainly concentrated at shallow depths of 3–10 km with a maximum slip of ~ 1.0 m. Our preferred geodetic coseismic model exhibits no surface rupture, which may likely be due to the shallow slip deficit in the uppermost crust. We calculate the combined loading effect of the Coulomb failure stress changes induced by the coseismic dislocations and postseismic viscoelastic relaxation of the 2008 Mw 7.1, 2012 Mw 6.4 and 2014 Mw 6.9 Yutian events. Our study demonstrates that the three preceding major Yutian shocks were insufficient to trigger the 2020 Yutian earthquake, which we consider perhaps reflects the natural release of elastic strain accumulated mainly through localized tectonic movement. We attribute the 2020 Yutian event to the release of extensional stress in a stepover zone controlled by the Longmu Co and the North Altyn Tagh sinistral strike slip fault systems. The seismic risk in the southwest end of the North Altyn Tagh fault has been elevated by the Yutian earthquake sequences, which require future attention.

Bandwidth Selection of Normal Information Spread Estimation Method for Geotechnical Parameter Probability Distribution

Normal information spread estimation method (NISEM) is an effective method to infer the geotechnical parameter probability density function. However, it is found that the optimal bandwidth calculated by the existing method is not consistent with the actual situation in engineering application. In order to determine the matching bandwidth of NISEM, the histogram grouping number and group distance according to the sample quantity and range are first determined in this paper. Then the bandwidth h of NISEM is calculated, taking h as the benchmark, 0.1h as the step to change bandwidth. And combining with judgment coefficient, the optimal matching bandwidth of NISEM is finally obtained. The validity and rationality of the proposed method are verified by geotechnical test data of two groups of large samples and four groups of small samples in practical engineering. The results show that: under the condition of limited geotechnical samples, the best matching bandwidth is related to the number of histogram groups, which is relative. And the bandwidth of each group of test samples is different. The research work in this paper perfects the application of NISEM in inferring geotechnical parameter probability density function.

A Bayesian approach to the dynamic modeling of ESP-lifted oil well systems: An experimental validation on an ESP prototype

This article presents an integrated method for estimating parameters for an electric submersible pump system with process variables data. Validation of a phenomenological model is also performed. The parameters and the associated probability density function are obtained through Bayesian inference, and the model validation is achieved in two stages. The first one is the validation of the dynamic response in which the model is compared with the experimental data. The second is achieved by comparing the regions covered by the experimental data and the model, both in steady-state. The experimental data's uncertainty is assessed using the Guide for the Expression of Uncertainty in Measurement. In turn, the uncertainty of the model's prediction is obtained by propagating the probability density function parameters. The results indicate that the method can provide a model to represent the system behavior within the existing uncertainties. Additionally, the procedure can be applied in oil production fields to provide substitute models for general purposes, such as production control, optimization, and assistance.

Molecular-level heavy petroleum hydrotreating modeling and comparison with high-resolution mass spectrometry

The establishment of a molecular-level process model is critical for heavy petroleum hydrotreating optimization. This paper focuses on constructing a heavy petroleum compositional model and hydrotreating process model with a more accurate molecular detail. Besides the bulk property data, the detail molecular characterization data from high-resolution mass spectrometry was used to guide the compositional and kinetic model development. Under the framework of structural unit and bond-electron matrix (SU-BEM), more than 12,000 representative molecules were constructed to form a heavy petroleum molecular structure library. The probability density functions (PDFs) were combined to constrain the molecular composition of the heavy petroleum. The optimal molecular composition was obtained by optimizing the PDFs parameters. 18 reaction rules were formulated to express the heavy petroleum molecular conversion pathways. The reaction network was generated automatically and transformed mathematically into a micro-kinetic model. The total numbers of involved reactions and molecules were ~ 25,000 and ~ 17,000, respectively. The model parameters were globally optimized according to experimental data. The model results show good agreement with the bulk property data. Moreover, the predicted molecular distribution and molecular conversion regulations are consistent with the high-resolution mass spectrometry data.

Reliability Analysis of Seismic Slope Stability with Uncertain Probability Distributions

Abstract The probability density function or cumulative distribution function of shear strength parameters is commonly unknown, and the first-order reliability method, second-order reliability meth...

Joint multizonal transdimensional Bayesian inversion of surface wave dispersion and ellipticity curves for local near-surface imaging

SummaryPhysical properties of near-surface soil and rock layers play a fundamental role in the seismic site effects analysis, being an essential element of seismic hazard assessment. Site-specific mechanical properties (i.e. shear- and compressional-wave velocities and mass density) can be inferred from surface wave dispersion and horizontal-to-vertical or ellipticity data by non-linear inversion techniques. Nevertheless, results typically exhibit significant inherent non-uniqueness as different models may fit the data equally well. Standard optimization inversion techniques minimize data misfit, resulting in a single representative model, rejecting other models providing similar misfit values. An alternative inversion technique can be formulated in the Bayesian framework, where the posterior probability density on the model space is inferred. This paper introduces an inversion approach of surface wave dispersion and ellipticity data based on a novel multizonal transdimensional Bayesian formulation. In particular, we parametrize 1-D layered velocity models by the varying number of Voronoi nuclei, allowing us to treat the number of layers as an unknown parameter of the inverse problem. The chosen parametrization leads to the transdimensional formulation of the model space, sampled by a reversible jump Markov chain Monte Carlo algorithm to provide an ensemble of random samples following the posterior probability density of model parameters. The used type of the sampling algorithm controls a model complexity (i.e. the number of layers) self-adaptively based on the measured data's information content. The method novelty lies in the parsimonious selection of sampling models and in the multizonal formulation of prior assumptions on model parameters, the latter allows including additional site-specific constraints in the inversion. These assumptions may be based on, e.g. stratigraphic logs, standard penetration tests, known water table, and bedrock depth. The multizonal formulation fully preserves the validity of the transdimensional one, as demonstrated analytically. The resultant ensemble of model samples is a discrete approximation of the posterior probability density function of model parameters and associated properties (e.g. VS30, quarter-wavelength average velocity profile and theoretical SH-wave amplification function). Although the ultimate result is the posterior probability density function, some representative models are selected according to data fit and maximum of the posterior probability density function. We first validate our inversion approach based on synthetic tests and then apply it to field data acquired from the active seismic survey and single-station measurements of ambient vibrations at the SENGL seismic station site in central Switzerland.

Variational Bayesian based adaptive PDA filter in scenarios with unknown detection probability and heavy-tailed process noise

For target tracking systems, the probability of detecting a target is difficult to determine, and the process noise often has non-Gaussian heavy-tailed characteristics owing to interference from outliers. To address the issues associated with single target tracking within clutters in scenarios with an unknown detection probability and heavy-tailed process noise, this paper presents a variational Bayesian-based adaptive probabilistic data association filter (VB-APDAF). The beta distribution, Pearson type VII distribution and multinomial distribution are used to model the detection probability, the process noise, and the association events, respectively. To guarantee the conjugation, a novel parameter estimation strategy is employed. In this strategy, the previous state is introduced in the state update process to construct the joint probability density function of parameters to be estimated and data set. The VB framework is used to estimate the target state, detection probability, and associated events. An experiment was performed under simulated conditions to demonstrate the effectiveness of the proposed filter.

Formulating the history matching problem with consistent error statistics

It is common to formulate the history-matching problem using Bayes’ theorem. From Bayes’, the conditional probability density function (pdf) of the uncertain model parameters is proportional to the prior pdf of the model parameters, multiplied by the likelihood of the measurements. The static model parameters are random variables characterizing the reservoir model while the observations include, e.g., historical rates of oil, gas, and water produced from the wells. The reservoir prediction model is assumed perfect, and there are no errors besides those in the static parameters. However, this formulation is flawed. The historical rate data only approximately represent the real production of the reservoir and contain errors. History-matching methods usually take these errors into account in the conditioning but neglect them when forcing the simulation model by the observed rates during the historical integration. Thus, the model prediction depends on some of the same data used in the conditioning. The paper presents a formulation of Bayes’ theorem that considers the data dependency of the simulation model. In the new formulation, one must update both the poorly known model parameters and the rate-data errors. The result is an improved posterior ensemble of prediction models that better cover the observations with more substantial and realistic uncertainty. The implementation accounts correctly for correlated measurement errors and demonstrates the critical role of these correlations in reducing the update’s magnitude. The paper also shows the consistency of the subspace inversion scheme by Evensen (Ocean Dyn. 54, 539–560 2004) in the case with correlated measurement errors and demonstrates its accuracy when using a “larger” ensemble of perturbations to represent the measurement error covariance matrix.

Qualitative Properties of Randomized Maximum Entropy Estimates of Probability Density Functions

The problem of randomized maximum entropy estimation for the probability density function of random model parameters with real data and measurement noises was formulated. This estimation procedure maximizes an information entropy functional on a set of integral equalities depending on the real data set. The technique of the Gâteaux derivatives is developed to solve this problem in analytical form. The probability density function estimates depend on Lagrange multipliers, which are obtained by balancing the model’s output with real data. A global theorem for the implicit dependence of these Lagrange multipliers on the data sample’s length is established using the rotation of homotopic vector fields. A theorem for the asymptotic efficiency of randomized maximum entropy estimate in terms of stationary Lagrange multipliers is formulated and proved. The proposed method is illustrated on the problem of forecasting of the evolution of the thermokarst lake area in Western Siberia.

Unified Reliability Measure Method Considering Uncertainties of Input Variables and Their Distribution Parameters

Aleatoric and epistemic uncertainties can be represented probabilistically in mechanical systems. However, the distribution parameters of epistemic uncertainties are also uncertain due to sparsely available or inaccurate uncertainty information. Therefore, a unified reliability measure method that considers uncertainties of input variables and their distribution parameters simultaneously is proposed. The uncertainty information for distribution parameters of epistemic uncertainties could be as a result of insufficient data or interval information, which is represented with evidence theory. The probability density function of uncertain distribution parameters is constructed through fusing insufficient data and interval information based on a Gaussian interpolation algorithm, and the epistemic uncertainties are represented using a weighted sum of probability variables based on discrete distribution parameters. The reliability index considering aleatoric and epistemic uncertainties is calculated around the most probable point. The effectiveness of the proposed algorithm is demonstrated through comparison with the Monte Carlo method in the engineering example of a crank-slider mechanism and composite laminated plate.

Large-scale Gravitational Lens Modeling with Bayesian Neural Networks for Accurate and Precise Inference of the Hubble Constant

Abstract We investigate the use of approximate Bayesian neural networks (BNNs) in modeling hundreds of time delay gravitational lenses for Hubble constant (H 0) determination. Our BNN was trained on synthetic Hubble Space Telescope quality images of strongly lensed active galactic nuclei with lens galaxy light included. The BNN can accurately characterize the posterior probability density functions (PDFs) of model parameters governing the elliptical power-law mass profile in an external shear field. We then propagate the BNN-inferred posterior PDFs into an ensemble H 0 inference, using simulated time delay measurements from a plausible dedicated monitoring campaign. Assuming well-measured time delays and a reasonable set of priors on the environment of the lens, we achieve a median precision of 9.3% per lens in the inferred H 0. A simple combination of a set of 200 test lenses results in a precision of 0.5 km s−1 Mpc−1 (0.7%), with no detectable bias in this H 0 recovery test. The computation time for the entire pipeline—including the generation of the training set, BNN training and H 0 inference—translates to 9 minutes per lens on average for 200 lenses and converges to 6 minutes per lens as the sample size is increased. Being fully automated and efficient, our pipeline is a promising tool for exploring ensemble-level systematics in lens modeling for H 0 inference.

Estimation of electricity cost of wind energy using Monte Carlo simulations based on nonparametric and parametric probability density functions

It is conventional in wind energy assessment projects to calculate the expected value of the annual energy production (AEP). However, ignoring the effect of wind speed distribution on the distribution of the annual energy production may cause wrong predictions in the cost analysis. For a better estimation of the payback period, it is essential to accurately determine the confidence levels of the electricity cost around the expected value.Wind speed distributions are commonly represented by the Weibull model. Improved functions or nonparametric functions are preferred in the case of multimodal wind speed distributions. As nonparametric probability density function, the optimum spline based functions are implemented and compared to Weibull, Weibull & Weibull for two different wind sites. The results show that the spline based probability density functions produce minimum fitting error for the analyzed cases. Once the wind speed distributions are characterized, random wind speeds are generated to calculate the AEP distributions by Monte Carlo simulations. The cost analysis is then carried out based on the AEP distributions which includes the determination of the confidence levels. It is observed that the confidence levels of the electricity costs which are not falling close to the expected cost region are much greater.

Quantification and Analysis of Uncertainties in Parameters of Rotors Supported by Bearings

Rotating unbalance is one of the most important critical parameters that causes operational failures of rotating machinery. The uneven distribution of mass on the structure of the rotors creates heavy spots, which must be eliminated to avoid generating excessive stress on the rotor bearings. The main objective of this work is to perform the uncertainty analysis on rotating machine systems supported with rolling bearings. A computational procedure is implemented to achieve this objective that can qualitatively represent the main behaviors and parameters of rotating machines. Further, methods of uncertainty quantification are applied to verify the behavior of the system given the probability density functions of the input parameters. One of the most commonly used methods is the Monte Carlo method, which requires thousands of simulations to produce accurate results. This method is used to obtain means and standard deviations of the system responses, given the means and standard deviations of the inputs. In our work, Monte Carlo simulation has been successfully used as a reference stochastic solver to evaluating the variability of the dynamic responses.

Quantification and Analysis of Uncertainties in Parameters of Rotors Supported by Bearings

Rotating unbalance is one of the most important critical parameters that causes operational failures of rotating machinery. The uneven distribution of mass on the structure of the rotors creates heavy spots, which must be eliminated to avoid generating excessive stress on the rotor bearings. The main objective of this work is to perform the uncertainty analysis on rotating machine systems supported with rolling bearings. A computational procedure is implemented to achieve this objective that can qualitatively represent the main behaviors and parameters of rotating machines. Further, methods of uncertainty quantification are applied to verify the behavior of the system given the probability density functions of the input parameters. One of the most commonly used methods is the Monte Carlo method, which requires thousands of simulations to produce accurate results. This method is used to obtain means and standard deviations of the system responses, given the means and standard deviations of the inputs. In our work, Monte Carlo simulation has been successfully used as a reference stochastic solver to evaluating the variability of the dynamic responses.

Uso de parâmetros estatísticos para a classificação de regiões homogêneas de temperatura do ar.

Time series of hourly temperature from 146 weather stations located in Santa Catarina State – South Brazil were used to show that a compact data representation, using probability density functions (pdf) parameters, could be useful to classify homogeneous air temperature areas. The normal distribution fitted well the 146 weather stations temperature time series, presenting a median value of 0.9721 for the Pearson correlation coefficient. The means and standard deviations obtained by adjusting the Gaussian functions for the 146 stations were used as input parameters for two different classifiers: hierarchical and k-means. Both classifiers separated Santa Catarina's weather stations into four distinct groups. These groups had direct relationship with altitude ranges and with the influence of sea. The classification of weather stations in different homogeneous groups was useful to identify climatic behaviors of hourly temperatures. In addition to the characterization of the climate itself, this classification can be useful as a support for the validation of numerical weather forecast models, and for the identification of abnormal temperature time series in a regional spatial context.

Fully Bayesian Human–Machine Data Fusion for Robust Online Dynamic Target Characterization

This work examines the problem of fusing human operator observations with probabilistic information extracted by an automated data fusion system, in the context of dynamic multitarget track characterization for large-scale surveillance. This soft data fusion problem is challenging because human operator observation errors are difficult to calibrate a priori and in general exhibit subtle conditional dependencies. Furthermore, the efficacy of combining inputs from operators depends heavily on operator error characteristics as well as track characterization uncertainty. A hierarchical fully Bayesian probabilistic model is developed to explicitly account for human sensor quality and Markovian dependencies in human reports. This model is used to perform online Bayesian inference via Gibbs sampling to simultaneously update the data fusion system’s knowledge of human sensor characteristics and target type probabilities. A strategy is also developed for value of information assessments to automatically query human operators for inputs as needed. Efficient approximations of high-dimensional human sensor parameter posterior distributions via Dirichlet probability density function (pdf) moment matching and parameter tying are also developed to provide significant computational speedups with little information loss. Results for different simulated operator profiles and automated target characterization baselines show that fully Bayesian fusion improves track characterization performance, while intelligently accounting for uncertain operator characteristics.

Reconciliation of Measured and Forecast Data for Topology Identification in Distribution Systems

The status of switches in distribution systems might be changed for protection purposes or to achieve the operation objectives. However, the system topology should be known for the efficacious management and control of these systems. In most practical systems, due to the lack of enough measurements, the topology cannot be identified accurately and speedily. The forecast output can be transformed into the joint probability density function (<small>pdf</small>) of uncertain parameters, e.g., power demands. The measured data and this <small>pdf</small> are first conflated to update the joint <small>pdf</small> to best comply with the measurements. The topology identification problem, i.e., finding the probability of each topology given a set of measured values, is converted to a simpler form, i.e., finding the probability of observing these measured values in this topology. These probabilities are then calculated by tracing how the measurements affect the shape of the new consolidated <small>pdf</small> in each topology compared to the original <small>pdf</small>. The proposed technique employs as much data as available, is able to find the accurate probabilities, and is yet quite fast. The mathematical treatment of extracting such probabilities is presented and the proposed technique is validated in numerical studies in terms of accuracy, speed, and versatility.

On Bayesian Estimation of Loss and Risk Functions

Loss functions and Risk functions play very important role in Bayesian estimation. This paper aims at the Bayesian estimation for the loss and risk functions of the unknown parameter of the H(r, theta), (theta being the unknown parameter) distribution The estimation has been performed under Rukhin’s loss function. The importance of this distribution is that it contains some important distributions such as the Half Normal distribution, Rayleigh distribution and Maxwell’s distribution as particular cases. The inverse Gamma distribution has assumed as the prior distribution for the unknown parameter theta. This prior distribution is a Natural Conjugate prior distribution for the unknown parameter because the posterior probability density function of the unknown parameter is also inverse gamma distribution The Rukhin’s loss function involves another loss function denoted by w(theta, delta) he form of w(theta, delta) is important as it changes the estimate. In this paper, three forms of w(theta, delta) have been taken and corresponding estimates have been derived. The three, forms are, the Squared Error Loss Function (SELF) and two different forms of Weighted Squared Error Loss Function (WSELF) namely, the Minimum Expected Loss (MELO) Function and the Exponentially Weighted Minimum Expected Loss (EWMELO) Function have been considered. A criterion of performance of various form of w(theta, delta) has ben defined. It has been proved that among three forms of w(theta, delta), considered here, the form corresponding to EWMELO is most dominant.

Minimum Variance Pole Placement in Uncertain Linear Control Systems

The pole-placement issue for linear multi-input multi-output (MIMO) dynamic systems with uncertain parameters has been addressed in this article. A static feedback matrix has been designed for minimizing variances of closed-loop poles (CLPs) and for assigning poles to the nominal system at the desired places. It is assumed that the joint probability density function (PDF) of uncertain parameters is known and the system has more than one input. A new unknown vector is used like an eigenvector for a stochastic closed-loop system matrix to state the problem. The variances of poles are considered as cost functions, and the means of poles are termed constraints. This form of the problem statement has helped us to simply find a solution. In the first step, the optimization problem with constraints was handled by solving the equality constraint, and then, the problem was converted to a classic extended eigenvalue optimization problem. Later, the eigenvalue optimization problem was solved by the Rayleigh quotient and the feedback matrix was accomplished. Finally, this approach was simulated and validated using the MATLAB simulations, and the results were compared with a robust pole-placement method, which MATLAB control toolbox uses. The Monte Carlo simulations showed lower covariance for CLPs around the mean poles as compared to the robust pole-placement method.

Bayesian Method for Parameter Estimation in Transient Heat Transfer Problem

Like many other scientific fields, an inverse technique is used to find unknown material properties (such as thermal conductivity, specific heat) or boundary conditions (e.g. heat flux, heat transfer coefficient) in the heat transfer problem. The role of the inverse technique is even more important if the properties or boundary conditions change with location and/or time. In this study, a Bayesian inference based inverse technique is used to find local heat transfer coefficient as well as local steam temperature at the inner surface of a steam header where the header was first heated by supplying steam followed by the cooling of the header due to the release of steam from the chamber. A finite volume based numerical model is developed to solve the forward unsteady heat conduction problem in the steam header. For inverse problem, a Markov chain Monte Carlo method is used to evaluate the posterior probability density functions (PPDFs) of unknown parameters in Bayesian framework, while a Metropolis-Hastings algorithm is used for random sample generation. Here PPDFs are used to extract the point estimates of the parameters and their credible intervals from their possible distributions. With the estimated parameters, the calculated outer surface temperatures agree well with experimental measurements in both heating and cooling process. At the end of the discharge process, the temperature distribution in the radial direction approaches flat profiles, but a steady state is only achieved at the bottom of the header. Our stochastic based Bayesian technique can be used to estimate unknown boundary conditions efficiently at an inaccessible location.