Joint Posterior Probability Density Function Research Articles

Adaptive tracking of non-ellipsoidal extended target with varying number of sub-objects based on variational Bayesian

High-resolution sensors make it less difficult to detect the true shape of an extended target, and the method of using a single ellipsoid to approximate the extended state of the target is no longer applicable. We can use multiple ellipsoids to approximate the shape of non-ellipsoidal extended targets, but the method of fixing number of sub-objects again fails due to changes in the target shape. In this work, we propose a novel variational Bayesian adaptive filter for non-ellipsoidal extended target tracking (NETT) with a varying number of sub-objects called VN-NETT-VBAF. The proposed filter is used to estimate the motion, extended states, and mixture probabilities of sub-objects describing a non-ellipsoidal extended target. First, the framework of VN-NETT-VBAF is established based on the spawning and combination of sub-objects. Second, the inverse Gamma (IG) distribution is used to model the unknown measurement noise covariance. Based on the variational Bayesian theory, the joint posterior probability density function (PDF) of the system parameters is iteratively updated to achieve the joint estimation of the extended target state and the measurement noise covariance. Finally, an adaptive forgetting factor is designed to update the noise covariance to further improve the robustness of the algorithm to abrupt noise. The simulation results show that the algorithm in this paper can reasonably change the number of sub-objects to adapt to the change of the target extended state, and effectively deal with the abrupt noise to obtain higher filtering accuracy.

A variational Bayesian approach for ensemble filtering of stochastically parametrized systems

AbstractModern climate models use both deterministic and stochastic parametrization schemes to represent uncertainties in their physics and inputs. This work considers the problem of estimating the involved parameters of such systems simultaneously with their state through data assimilation. Standard state‐parameter filtering schemes cannot be applied to such systems, owing to the posterior dependence between the stochastic parameters and the “dynamical” augmented state, defined as the state augmented by the deterministic parameters. We resort to the variational Bayesian (VB) approach to break this dependence, by approximating the joint posterior probability density function (pdf) of the augmented state and the stochastic parameters with a separable product of two marginal pdfs that minimizes the Kullback–Leibler divergence. The resulting marginal pdf of the augmented state is then sampled using a one‐step‐ahead smoothing‐based ensemble Kalman filter (EnKF‐OSAS), whereas a closed form is derived for the marginal pdf of the stochastic parameters. The proposed approach combines the effectiveness of the OSAS filtering approach to mitigate inconsistency issues that often arise with the joint ensemble Kalman filter (EnKF), with the advantage of obtaining a full posterior pdf for the stochastic parameters, which is not possible with the traditional maximum‐likelihood method. We demonstrate the relevance of the proposed approach through extensive numerical experiments with a one‐scale Lorenz‐96 model, which includes a stochastic parametrization representing subgrid‐scale effects.

Bayesian modal identification of non-classically damped systems using time-domain data

In this work, we present a Bayesian time–domain method for identifying the complex modal parameters of a linear dynamic system from ambient vibration data. Complex modal parameter identification is valid for both classically and non-classically damped systems, in contrast to real modal parameter identification, which is exclusively applicable to classically damped systems. A Bayesian method is adopted here which makes it possible to determine the joint posterior Probability Density Function (PDF) of the modal parameters for given measured data and modeling assumptions. For sufficient data, it is seen that the modal parameters are globally identifiable, and the posterior PDF is approximated by a Gaussian distribution with mean at the optimal parameters that maximize the posterior PDF. The uncertainty in the identified modal parameters is given by the covariance matrix of the posterior PDF. Analytical expressions are derived for computing the gradient for efficient optimization and the Hessian for determining the covariance matrix. Further, it is shown that the assumption of classical damping during analysis results in poor prediction of damping ratios in the case of non-classically damped systems. The application of the proposed approach is illustrated by means of simulated examples and an experimental study.

Wind-driven underwater ambient noise for seabed remote sensing: A review of methods and results

Estimation of seabed geoacoustic parameters in shallow water by acoustic remote sensing remains a challenging task due to constraints on hardware, data collection and analysis, and cost of maritime surveys. The work of Lisa Zurk and Martin Siderius, among many others, inspired investigation of signal processing techniques to process wind-driven underwater noise for the estimation of seabed geoacoustics. This paper presents a summary of the work on Bayesian geoacoustic inversion of ambient noise carried out at University of Victoria and Portland State University, using fixed and drifting vertical line arrays (VLA). The Bayesian inversion framework is based on Markov-chain Monte Carlo sampling. It estimates a joint posterior probability density function, from which marginal density functions, moments and covariances between geoacoustic parameters of interest can be quantified. The approach was applied to simulated and experimental ambient noise data for the estimation of seabed layer thicknesses, sound speeds, densities, and sediment attenuations. The resolution of the method was explored as a function of experiment factors such as array design and wind speed, and a method for extending the aperture of a VLA by extrapolating the noise coherence was proposed.

Distributed Cubature Kalman Filter Cooperative Localization Based on Parameterized-belief Propagation

<p>Location awareness has been widely used for cooperative localization of target in wireless sensor networks (WSN). But as the number of agent node increases, cooperative localization based on nonparametric belief propagation (BP) causes high communication and computation burden. In addition, high localization accuracy is also the goal of this paper. To this end, this paper based on parameterized BP strategy proposed a distributed cubature Kalman filter (DCKF) algorithm named BP-DCKF. Firstly, basing on joint posterior probability density function of all nodes, this paper constructs a factor graph (FG) model, then the edge posterior distribution of the nodes is obtained by BP strategy. Secondly, considering Gaussian parameterized BP which reconstruct the parameterized message of agents transmitting, this paper proposed the improved DCKF method, moreover, obtaining the locating model of agent node related to posterior distribution of each node in the FG. Finally, based on the localization model, the location estimation of mobile agent node can be obtained utilizing DCKF method to iteratively solve the edge posterior distribution of agent node. Simulation results show that comparing with traditional algorithm, the proposed algorithm is higher on localization accuracy, lower on communication burden.</p> <p> </p>

An Outlier-Robust Kalman Filter With Adaptive Selection of Elliptically Contoured Distributions

In this paper, elliptically contoured (EC) distributions are used to model outlier-contaminated measurement noises. Exploiting a heuristic approach to introduce an unknown parameter, we present an analytical update form of the joint posterior probability density function of the state vector and auxiliary random variable, from which a novel robust EC distributions-based Kalman filtering framework is first derived. To illustrate the effectiveness of the proposed framework, the convergence, robustness, optimality and computational complexity analyses of the proposed method are then given. In addition, to cope with complex noise environments, the interaction multiple model is employed to achieve the adaptive selection of EC distributions such that well-behaved estimation performance can be obtained for different noise cases. Simulation results demonstrate the validity and superiority of the proposed algorithm.

Realization ranking of seismic geostatistical inversion based on a Bayesian lithofacies classification - A case study from an offshore field

Seismic inversion which is defined as the process of converting seismic boundary reflections into layer elastic properties plays a vital role in different methods of seismic reservoir characterization. It can be done through either deterministic or geostatistical approaches, and geostatistical inversion is the recommended approach for heterogeneous and complicated reservoir due to its capability in capturing the details of a reservoir with the associated uncertainties. However, a common issue with the geostatistical approach is generating a large number of realizations with the same probability which needs to be ranked based on some pre-defined criteria. This study applied a pre-stack geostatistical inversion on an offshore field in the Persian Gulf using a Bayesian framework to produce the joint posterior probability density function (PDF) of P-wave seismic impedance (IP) and S-wave seismic impedance (IS). This global PDF was then, decomposed into local ones at the trace locations using a sequential Gaussian simulation technique. Furthermore, these local PDFs were sampled and 250 realizations of IP and IS generated accordingly. These elastic properties were then converted into lithofacies, and were ranked based on the oil-producing volume lithofacies. Well log data showed a useful distinction of the oil sand lithofacies in the IP vs. VP/VS crossplot, and this domain was used to compute the PDFs for classifying lithofacies derived from the seismic inversion. Moreover, the oil lithofacies volumes were calculated, and the realizations were sorted to generate the probability of P10, P50 and P90 scenarios. Also, the probability of oil presence within the reservoir was computed using all of the classified realizations.

Fixed-point iteration Gaussian sum filtering estimator with unknown time-varying non-Gaussian measurement noise

Abstract This paper addresses the state estimation problem in the presence of unknown time-varying non-Gaussian measurement noise (NGMN) modeled by Gaussian mixture distribution(GMD). Recently, the variational Bayesian (VB)-based estimators have been successfully used in the state estimation problem in the presence of unknown measurement noise. However, we reveal that when the state-space model is non-Gaussian, exploiting the traditional VB method, the joint posterior probability density function of the state and parameters of non-Gaussian measurement noise has no analytical result. To fill this gap, a fixed-point iteration Gaussian sum filtering (FPI-GSF) estimator is proposed to jointly estimate the state and the unknown time-varying NGMN online. The proposed estimator utilizes a fixed-point iteration scheme to iteratively calculate the state posterior distribution through the GSF and noise parameters posterior distribution via the VB method, thus it avoids calculating the joint posterior distribution. Simulation results demonstrate the effectiveness of the proposed estimator.

Structural damage identification via time domain response and Markov Chain Monte Carlo method

The present work addresses the problem of structural damage identification built on the statistical inversion approach. Here, the damage state of the structure is continuously described by a cohesion parameter, which is spatially discretized by the finite element method. The inverse problem of damage identification is then posed as the determination of the posterior probability densities of the nodal cohesion parameters. The Markov Chain Monte Carlo method, implemented with the Metropolis–Hastings algorithm, is considered in order to approximate the posterior probabilities by drawing samples from the desired joint posterior probability density function. With this approach, prior information on the sought parameters can be used and the uncertainty concerning the known values of the material properties can be quantified in the estimation of the cohesion parameters. The assessment of the proposed approach has been performed by means of numerical simulations on a simply supported Euler–Bernoulli beam. The damage identification and assessment are performed considering time domain response data. Different damage scenarios and noise levels were addressed, demonstrating the feasibility of the proposed approach.

A comparison of two Bayesian approaches for uncertainty quantification

Statistical calibration of model parameters conditioned on observations is performed in a Bayesian framework by evaluating the joint posterior probability density function (pdf) of the parameters. The posterior pdf is very often inferred by sampling the parameters with Markov Chain Monte Carlo (MCMC) algorithms. Recently, an alternative technique to calculate the so-called Maximal Conditional Posterior Distribution (MCPD) appeared. This technique infers the individual probability distribution of a given parameter under the condition that the other parameters of the model are optimal. Whereas the MCMC approach samples probable draws of the parameters, the MCPD samples the most probable draws when one of the parameters is set at various prescribed values. In this study, the results of a user-friendly MCMC sampler called DREAM(ZS) and those of the MCPD sampler are compared. The differences between the two approaches are highlighted before running a comparison inferring two analytical distributions with collinearity and multimodality. Then, the performances of both samplers are compared on an artificial multistep outflow experiment from which the soil hydraulic parameters are inferred. The results show that parameter and predictive uncertainties can be accurately assessed with both the MCMC and MCPD approaches.

Novel Gauss–Hermite integration based Bayesian inference on optimal wavelet parameters for bearing fault diagnosis

Rolling element bearings are commonly used in machines to provide support for rotating shafts. Bearing failures may cause unexpected machine breakdowns and increase economic cost. To prevent machine breakdowns and reduce unnecessary economic loss, bearing faults should be detected as early as possible. Because wavelet transform can be used to highlight impulses caused by localized bearing faults, wavelet transform has been widely investigated and proven to be one of the most effective and efficient methods for bearing fault diagnosis. In this paper, a new Gauss–Hermite integration based Bayesian inference method is proposed to estimate the posterior distribution of wavelet parameters. The innovations of this paper are illustrated as follows. Firstly, a non-linear state space model of wavelet parameters is constructed to describe the relationship between wavelet parameters and hypothetical measurements. Secondly, the joint posterior probability density function of wavelet parameters and hypothetical measurements is assumed to follow a joint Gaussian distribution so as to generate Gaussian perturbations for the state space model. Thirdly, Gauss–Hermite integration is introduced to analytically predict and update moments of the joint Gaussian distribution, from which optimal wavelet parameters are derived. At last, an optimal wavelet filtering is conducted to extract bearing fault features and thus identify localized bearing faults. Two instances are investigated to illustrate how the proposed method works. Two comparisons with the fast kurtogram are used to demonstrate that the proposed method can achieve better visual inspection performances than the fast kurtogram.

Enhancing the abilities in assessing slurry pumps' performance degradation and estimating their remaining useful lives by using captured vibration signals

Slurry pumps are widely used to transport abrasive slurry that contains oil and sands. Because of the abrasive nature, the impellers inside the pumps wear easily. Severe impeller wear may cause unexpected pump failure that leads to substantial oil production loss. To assess the impeller performance degradation and then estimate its remaining useful life (RUL), an efficient prognostic method has been designed. For assessing the impeller performance degradation, statistical features were extracted from vibration signals collected from on-site operating slurry pumps. Their corresponding frequency spectra were generated after the vibration signals were processed by a low-pass filter. Here, the low-pass filter aims to retain impeller-related vibration components, such as the pump vane-passing frequency and its harmonics. Principal component analysis was then applied to reduce the dimensionality of the extracted statistical features to one dimensionality, which was used to construct a health indicator to reflect the health evolution of the impeller over time. For estimating the impeller's RUL, a nonlinear state space model was designed to track its temporal health indicator. An efficient unscented transform method was employed to iteratively estimate the joint posterior probability density function of the parameters of the nonlinear state space model. After the proper nonlinear state space model had been determined, extrapolations of the nonlinear state space model to a specified alert threshold were used to estimate the impeller's RUL. Vibration signals captured from on-site operating slurry pumps were used to verify the effectiveness of the proposed prognostic method. The results show that prediction accuracy of the estimated RULs have been improved as compared to those generated by other recently developed slurry pump prognostic methods. Moreover, the more the temporal vibration data is available, the better the performance of the state space model; hence, the higher the accuracy in predicting the impeller's RUL.

Probabilistic calibration of a Greenland Ice Sheet model using spatially resolved synthetic observations: toward projections of ice mass loss with uncertainties

Abstract. Computer models of ice sheet behavior are important tools for projecting future sea level rise. The simulated modern ice sheets generated by these models differ markedly as input parameters are varied. To ensure accurate ice sheet mass loss projections, these parameters must be constrained using observational data. Which model parameter combinations make sense, given observations? Our method assigns probabilities to parameter combinations based on how well the model reproduces the Greenland Ice Sheet profile. We improve on the previous state of the art by accounting for spatial information and by carefully sampling the full range of realistic parameter combinations, using statistically rigorous methods. Specifically, we estimate the joint posterior probability density function of model parameters using Gaussian process-based emulation and calibration. This method is an important step toward calibrated probabilistic projections of ice sheet contributions to sea level rise, in that it uses data–model fusion to learn about parameter values. This information can, in turn, be used to make projections while taking into account various sources of uncertainty, including parametric uncertainty, data–model discrepancy, and spatial correlation in the error structure. We demonstrate the utility of our method using a perfect model experiment, which shows that many different parameter combinations can generate similar modern ice sheet profiles. This result suggests that the large divergence of projections from different ice sheet models is partly due to parametric uncertainty. Moreover, our method enables insight into ice sheet processes represented by parameter interactions in the model.

A general sequential Monte Carlo method based optimal wavelet filter: A Bayesian approach for extracting bearing fault features

A general sequential Monte Carlo method, particularly a general particle filter, attracts much attention in prognostics recently because it is able to on-line estimate posterior probability density functions of the state functions used in a state space model without making restrictive assumptions. In this paper, the general particle filter is introduced to optimize a wavelet filter for extracting bearing fault features. The major innovation of this paper is that a joint posterior probability density function of wavelet parameters is represented by a set of random particles with their associated weights, which is seldom reported. Once the joint posterior probability density function of wavelet parameters is derived, the approximately optimal center frequency and bandwidth can be determined and be used to perform an optimal wavelet filtering for extracting bearing fault features. Two case studies are investigated to illustrate the effectiveness of the proposed method. The results show that the proposed method provides a Bayesian approach to extract bearing fault features. Additionally, the proposed method can be generalized by using different wavelet functions and metrics and be applied more widely to any other situation in which the optimal wavelet filtering is required.

Fast probabilistic petrophysical mapping of reservoirs from 3D seismic data

A fast probabilistic inversion method for 3D petrophysical property prediction from inverted prestack seismic data has been developed and tested on a real data set. The inversion objective is to estimate the joint probability density function (PDF) of model vectors consisting of porosity, clay content, and water saturation components at each point in the reservoir, from data vectors with compressional- and shear-wave-impedance components that are obtained from the inversion of seismic data. The proposed inversion method is based on mixture density network (MDN), which is trained by a given set of training samples, and provides an estimate of the joint posterior PDF’s of the model parameters for any given data point. This method is much more time and memory efficient than conventional nonlinear inversion methods. The training data set is constructed using nonlinear petrophysical forward relations and includes different sources of uncertainty in the inverse problem such as variations in effective pressure, bulk modulus and density of hydrocarbon, and random noise in recorded data. Results showed that the standard deviations of all model parameters were reduced after inversion, which shows that the inversion process provides information about all parameters. The reduction of uncertainty in water saturation was smaller than that for porosity or clay content; nevertheless the maximum of the a posteriori (MAP) of model PDF clearly showed the boundary between brine saturated and oil saturated rocks at wellbores. The MAP estimates of different model parameters show the lateral and vertical continuity of these boundaries. Errors in the MAP estimate of different model parameters can be reduced using more accurate petrophysical forward relations. This fast, probabilistic, nonlinear inversion method can be applied to invert large seismic cubes for petrophysical parameters on a standard desktop computer.

Source reconstruction: a statistical mechanics perspective

A Bayesian inference scheme is used to address the problem of general source reconstruction. To this purpose, starting from a simple, physically motivated structure-based representation for a general source distribution, Bayesian probability theory is used to formulate the full joint posterior probability density function for the parameters in this representation. The exploration of the high-dimensional parameter space using simulated annealing and a Markov chain Monte Carlo sampler is described. The explicit linkage between the computational tools used for posterior sampling and concepts in statistical mechanics is emphasised. The proposed methodology is validated using a real dispersion experiment involving a multiple source release.

Probability Theory as Logic: Data Assimilation for Multiple Source Reconstruction

Probability theory as logic (or Bayesian probability theory) is a rational inferential methodology that provides a natural and logically consistent framework for source reconstruction. This methodology fully utilizes the information provided by a limited number of noisy concentration data obtained from a network of sensors and combines it in a consistent manner with the available prior knowledge (mathematical representation of relevant physical laws), hence providing a rigorous basis for the assimilation of this data into models of atmospheric dispersion for the purpose of contaminant source reconstruction. This paper addresses the application of this framework to the reconstruction of contaminant source distributions consisting of an unknown number of localized sources, using concentration measurements obtained from a sensor array. To this purpose, Bayesian probability theory is used to formulate the full joint posterior probability density function for the parameters of the unknown source distribution. A simulated annealing algorithm, applied in conjunction with a reversible-jump Markov chain Monte Carlo technique, is used to draw random samples of source distribution models from the posterior probability density function. The methodology is validated against a real (full-scale) atmospheric dispersion experiment involving a multiple point source release.

Bayesian ambient noise inversion for geoacoustic uncertainty estimation.

The noise produced by wind-driven breaking waves in shallow water provides a method for probing the seabed, and drifting vertical arrays have been deployed for remote sensing of geoacoustic parameters by estimating the frequency- and angle-dependent reflection coefficients. In addition, techniques such as spectral factorization allow obtaining the impulse response of the multilayered seabed environment. This impulse response carries information of the sediment acoustic properties that can be extracted and passed as prior information to a Bayesian framework for the estimation of geoacoustic parameters and its corresponding uncertainties, which ultimately determine the resolution of the method. The Bayesian formulation estimates a joint posterior probability density function, from which marginal density functions, moments, and covariances between geoacoustic parameters of interest can be quantified. In this work, Bayesian inversion based on Markov-chain Monte Carlo sampling is applied to simulated ambient noise data for the estimation of layer thicknesses, compressional sound speed, density, and sediment attenuation, and the resolution of the method is explored as a function of qualities of the data such as array design and wind speed. The approach is applied to experimental data collected near Sicily.

Bayesian Model Updating Using Hybrid Monte Carlo Simulation with Application to Structural Dynamic Models with Many Uncertain Parameters

In recent years, Bayesian model updating techniques based on measured data have been applied to system identification of structures and to structural health monitoring. A fully probabilistic Bayesian model updating approach provides a robust and rigorous framework for these applications due to its ability to characterize modeling uncertainties associated with the underlying structural system and to its exclusive foundation on the probability axioms. The plausibility of each structural model within a set of possible models, given the measured data, is quantified by the joint posterior probability density function of the model parameters. This Bayesian approach requires the evaluation of multidimensional integrals, and this usually cannot be done analytically. Recently, some Markov chain Monte Carlo simulation methods have been developed to solve the Bayesian model updating problem. However, in general, the efficiency of these proposed approaches is adversely affected by the dimension of the model parameter space. In this paper, the Hybrid Monte Carlo method is investigated (also known as Hamiltonian Markov chain method), and we show how it can be used to solve higher-dimensional Bayesian model updating problems. Practical issues for the feasibility of the Hybrid Monte Carlo method to such problems are addressed, and improvements are proposed to make it more effective and efficient for solving such model updating problems. New formulae for Markov chain convergence assessment are derived. The effectiveness of the proposed approach for Bayesian model updating of structural dynamic models with many uncertain parameters is illustrated with a simulated data example involving a ten-story building that has 31 model parameters to be updated.

A Fully Bayesian Inversion for Spatial Distribution of Fault Slip with Objective Smoothing

Abstract In inversion of geodetic data for spatial distribution of fault slip under smoothing and positivity constraints, subjective or incorrect techniques are often employed to select a smoothing parameter that determines the relative weight placed on fitting the data versus smoothing the slip distribution. A popular objective method based on Akaike’s Bayesian information criterion (ABIC) is incorrect if positivity constraints are employed. We introduce a fully probabilistic inversion method to simultaneously estimate the slip distribution and smoothing parameter objectively and correctly in a Bayesian framework. The complete solution to the inverse problem, the joint posterior probability density function of slip and smoothing parameter, is formulated using Bayes’ theorem and sampled with a Markov chain Monte Carlo method. We apply the new method to coseismic displacement data from the 1999 Chi-Chi, Taiwan, earthquake and compare the results of our method with conventional methods. The slip distribution estimated with our method is significantly smoother than that estimated by the ABIC-based method, demonstrating that the ABIC-based method may in general generate significantly undersmoothed or oversmoothed slip distributions under positivity constraints. On the other hand, the estimated slip distribution is similar to that estimated using another conventional method based on cross validation. Multiple trade-off curve plots between slip roughness and data misfit show that trade-off curves provide no basis for selecting a single smoothing parameter. The new method is easily generalized to slip inversions of multiple data sets with unknown relative weighting and to slip inversions with unknown fault geometry parameters.