Target Probability Density Function Research Articles

Cross-fade sampling: extremely efficient Bayesian inversion for a variety of geophysical problems

SUMMARY This paper introduces cross-fade sampling, a computationally efficient Markov Chain Monte Carlo simulation method that uses a semi-analytical approach to quickly solve Bayesian inverse problems that do not themselves have an analytical solution. Cross-fading is efficient in two ways. First, it requires fewer samples to obtain the same quality simulation of the target probability density function (PDF). Secondly, it is much faster to evaluate the posterior probability of each sample than conventional sampling methods for simulating Bayesian posterior PDFs. Conventional methods require evaluating the prior probability (which describes your a priori constraints) and data likelihood (which describes the fit between the observations and the predictions of the model) for each sample model. However, cross-fading does not require evaluating the data likelihood, meaning that ‘big data’ can be fit with zero additional computational cost. Further, the cross-fading approach can be used to calculate the marginal likelihood associated with a model design, facilitating model comparison and Bayesian model averaging. Topics covered in this paper include derivation of the cross-fade approach and how it can be used to simulate Bayesian posterior PDFs and compute the marginal likelihood, discussion of the class of problems to which cross-fading can be applied (with examples from earthquake statistics, earthquake ground motion modelling, volcanic eruption forecasting, and finite fault slip modelling), demonstration of efficiency relative to existing sampling methods and discussion of how cross-fading can be used to account for prediction errors (i.e. epistemic errors) as part of the geophysical inverse problem.

Enhancing Spectroscopic Experiment Calibration through Differentiable Programming

In this work, we present an innovative calibration technique leveraging differentiable programming to enhance energy resolution and reduce the energy scale systematic uncertainty in X-ray spectroscopic experiments. This approach is demonstrated using synthetic data and is applicable in general to various spectroscopic measurements. This method extends the scope of differentiable programming for calibration, employing Kernel Density Estimation (KDE) to achieve a target Probability Density Function (PDF) for a fully differentiable model of the calibration. To assess the effectiveness of the calibration, we conduct a toy simulation replicating the entire detector response chain and compare it with a standard calibration. This ensures a robust and reliable calibration methodology, holding promise for improving energy resolution and providing a more versatile and efficient approach without the need for extensive fine-tuning.

Variance analysis of multiple importance sampling schemes

Multiple importance sampling (MIS) is an increasingly used methodology where several proposal densities are used to approximate integrals, generally involving target probability density functions. The use of several proposals allows for a large variety of sampling and weighting schemes. Then, the practitioner must choose a given scheme, i.e., sampling mechanism and weighting function. A variance analysis has been proposed in Elvira et al (2019, Statistical Science 34: 129-155), showing the superiority of the balanced heuristic estimator with respect to other competing schemes in some scenarios. However, some of their results are valid only for two proposals. In this paper, we extend and generalize these results, providing novel proofs that allow to determine the variance relations among MIS schemes.

Variation Approach-Based Nonlinear Companding Scheme for PAPR Reduction in OFDM Systems

Orthogonal frequency division multiplexing (OFDM) continues to find vast deployment in current and next generation wireless communication systems. However, high peak-to-average power ratio (PAPR) is one of the major drawbacks in the OFDM system. In this paper, we develop a variation approach based nonlinear companding (VANC) scheme to reduce the PAPR in OFDM systems. It aims to find the probability density function (PDF) for OFDM signal amplitudes with low PAPR and low distortion compared with their original Rayleigh distribution. The target PDF is found by solving a variation optimization problem model, which is built to find the lowest PDF distortion with both average signal power and probability conservation constraint. To the best of our knowledge, it is the first time that the variation method is used to find companding function to address the high PAPR problem. We then propose an algorithm with companding operation based on quantized levels. Simulation results validate the performance superiorities of the proposed scheme to that of other typical companding schemes.

Exploiting probability density function of deep convolutional autoencoders’ latent space for reliable COVID-19 detection on CT scans

We present a probabilistic method for classifying chest computed tomography (CT) scans into COVID-19 and non-COVID-19. To this end, we design and train, in an unsupervised manner, a deep convolutional autoencoder (DCAE) on a selected training data set, which is composed only of COVID-19 CT scans. Once the model is trained, the encoder can generate the compact hidden representation (the hidden feature vectors) of the training data set. Afterwards, we exploit the obtained hidden representation to build up the target probability density function (PDF) of the training data set by means of kernel density estimation (KDE). Subsequently, in the test phase, we feed a test CT into the trained encoder to produce the corresponding hidden feature vector, and then, we utilise the target PDF to compute the corresponding PDF value of the test image. Finally, this obtained value is compared to a threshold to assign the COVID-19 label or non-COVID-19 to the test image. We numerically check our approach’s performance (i.e. test accuracy and training times) by comparing it with those of some state-of-the-art methods.

Developing an innovative method to control the probability density function shape of the state response for nonlinear stochastic systems

AbstractFor nonlinear stochastic systems, controlling low‐order moment indexes, such as mean and variance, may not effectively achieve actual control requirements since they do not reflect the complete statistical characteristics of random variables. Instead, the probability density function (PDF), which provides complete characterization of statistical properties, can be used to control each order moment more efficiently. Existing PDF shape control methods are mainly suited for nonlinear systems with polynomial forms, which leads to limitations in their applications. In this article, we propose a PDF shape control method using the Fokker–Planck–Kolmogorov (FPK) equation for nonlinear stochastic systems with arbitrary expression. In the proposed method, the exact stationary solution of the FPK equation is derived first. If the nonlinear function of the stochastic system is not polynomial, it is first converted into a polynomial form using the approximation method. The controller is then devised for PDF shape control, and the parameters of the controller are optimized to obtain the optimal controller. We conducted simulation experiments to validate the effectiveness of the proposed method. The results show that the proposed method can effectively control the PDF shape of the state response for nonlinear stochastic systems, regardless of whether the target PDF is unimodal, symmetric bimodal, or asymmetric bimodal. Compared with the Fourier transform method, the control effect of the proposed method is much better. The results suggest that the proposed method provides a scientific basis for wider applications of the PDF shape control method.

Survey on stochastic distribution systems: A full probability density function control theory with potential applications

AbstractComplex systems seen either in general engineering practice or economics are subjected to ever increased uncertainties that are mostly represented as random variables or parameters, and the characteristics of random variables are represented by their probability density functions (PDFs). Controlling their PDFs means to shape their stochastic distributions and in general it would provide a full treatment for system analysis and operational control and optimization. This leads to the development of stochastic distribution control (SDC) systems theory in the past decades, where the original aim of the controller design is to realize a shape control of the distributions of certain random variables in their PDFs sense for some engineering processes. Indeed, once the PDFs of these random variables or parameters are used to describe their distribution characters, the control task is to obtain control signals so that the output PDFs of stochastic systems are made to follow their target PDFs. The subject of SDC was initially originated for non‐Gaussian stochastic control systems design but has found a wide spectrum of applications in general systems in terms of data‐driven modeling, analysis, signal processing (filtering), data mining via multivariable statistics, decision‐making (optimization) for systems subjected to uncertainties and even in economics. In this context, SDC constitutes an effective primer tool for complex system analysis, control and operational optimizations. In this review paper, a detailed survey of the developments on the research of SDC systems will be made together with their wide spectrum applications and future perspectives.

Adjustable Nonlinear Companding Transform Based on Scaling of Probability Density Function for PAPR Reduction in OFDM Systems

Orthogonal frequency division multiplexing (OFDM) systems suffer from the inherent problem of high peak-to-average power ratio (PAPR). In this article, a novel adjustable nonlinear companding transform is proposed, which is based on scaling of original probability density function (PDF) of OFDM signal amplitudes. The target PDF consists of two parts. The first part with amplitudes no more than the transition point is same to that of original PDF. The second part with amplitudes larger than the transition point and smaller than the cutoff point is obtained by scaling original PDF in dimensions of ordinate and abscissa concurrently, for guaranteeing the constant average power. Companding and decompanding functions are derived, and constraint equation for solving parameters is formulated. Parameters of the proposed algorithm can be adjusted flexibly to achieve desired tradeoffs among algorithmic complexity, bit error rate (BER), out-of-band (OOB) rejection and PAPR reduction performance. Some key properties of the proposed algorithm are theoretically analyzed and proved, which may be much instructive for practical implementations. To further reduce algorithmic complexity, a piecewise curve fitting scheme employing quadratic polynomials is first proposed to simplify companding and decompanding functions. Simulation results confirm analysis results and verify the superiorities of proposed algorithms to other existing algorithms.

Prediction and Optimal Feedback Steering of Probability Density Functions for Safe Automated Driving

We propose a stochastic prediction-control framework to promote safety in automated driving by directly controlling the joint state probability density functions (PDFs) subject to the vehicle dynamics via trajectory-level state feedback. To illustrate the main ideas, we focus on a multi-lane highway driving scenario although the proposed framework can be adapted to other contexts. The computational pipeline consists of a PDF prediction layer, followed by a PDF control layer. The prediction layer performs moving horizon nonparametric forecasts for the ego and the non-ego vehicles' stochastic states, and thereby derives safe target PDF for the ego. The latter is based on the forecasted collision probabilities, and promotes the probabilistic safety for the ego. The PDF control layer designs a feedback that optimally steers the joint state PDF subject to the controlled ego dynamics while satisfying the endpoint PDF constraints. Our computation for the PDF prediction layer leverages the structure of the controlled Liouville PDE to evolve the joint PDF values, as opposed to empirically approximating the PDFs. Our computation for the PDF control layer leverages the differential flatness structure in vehicle dynamics. We harness recent theoretical and algorithmic advances in optimal mass transport, and the Schrödinger bridge. The numerical simulations illustrate the efficacy of the proposed framework.

Joint Adaptive Sampling Interval and Power Allocation for Maneuvering Target Tracking in a Multiple Opportunistic Array Radar System.

In this paper, a joint adaptive sampling interval and power allocation (JASIPA) scheme based on chance-constraint programming (CCP) is proposed for maneuvering target tracking (MTT) in a multiple opportunistic array radar (OAR) system. In order to conveniently predict the maneuvering target state of the next sampling instant, the best-fitting Gaussian (BFG) approximation is introduced and used to replace the multimodal prior target probability density function (PDF) at each time step. Since the mean and covariance of the BFG approximation can be computed by a recursive formula, we can utilize an existing Riccati-like recursion to accomplish effective resource allocation. The prior Cramér-Rao lower boundary (prior CRLB-like) is compared with the upper boundary of the desired tracking error range to determine the adaptive sampling interval, and the Bayesian CRLB-like (BCRLB-like) gives a criterion used for measuring power allocation. In addition, considering the randomness of target radar cross section (RCS), we adopt the CCP to package the deterministic resource management model, which minimizes the total transmitted power by effective resource allocation. Lastly, the stochastic simulation is embedded into a genetic algorithm (GA) to produce a hybrid intelligent optimization algorithm (HIOA) to solve the CCP optimization problem. Simulation results show that the global performance of the radar system can be improved effectively by the resource allocation scheme.

Approximation and sampling of multivariate probability distributions in the tensor train decomposition

General multivariate distributions are notoriously expensive to sample from, particularly the high-dimensional posterior distributions in PDE-constrained inverse problems. This paper develops a sampler for arbitrary continuous multivariate distributions that is based on low-rank surrogates in the tensor train format, a methodology that has been exploited for many years for scalable, high-dimensional density function approximation in quantum physics and chemistry. We build upon recent developments of the cross approximation algorithms in linear algebra to construct a tensor train approximation to the target probability density function using a small number of function evaluations. For sufficiently smooth distributions, the storage required for accurate tensor train approximations is moderate, scaling linearly with dimension. In turn, the structure of the tensor train surrogate allows sampling by an efficient conditional distribution method since marginal distributions are computable with linear complexity in dimension. Expected values of non-smooth quantities of interest, with respect to the surrogate distribution, can be estimated using transformed independent uniformly-random seeds that provide Monte Carlo quadrature or transformed points from a quasi-Monte Carlo lattice to give more efficient quasi-Monte Carlo quadrature. Unbiased estimates may be calculated by correcting the transformed random seeds using a Metropolis–Hastings accept/reject step, while the quasi-Monte Carlo quadrature may be corrected either by a control-variate strategy or by importance weighting. We show that the error in the tensor train approximation propagates linearly into the Metropolis–Hastings rejection rate and the integrated autocorrelation time of the resulting Markov chain; thus, the integrated autocorrelation time may be made arbitrarily close to 1, implying that, asymptotic in sample size, the cost per effectively independent sample is one target density evaluation plus the cheap tensor train surrogate proposal that has linear cost with dimension. These methods are demonstrated in three computed examples: fitting failure time of shock absorbers; a PDE-constrained inverse diffusion problem; and sampling from the Rosenbrock distribution. The delayed rejection adaptive Metropolis (DRAM) algorithm is used as a benchmark. In all computed examples, the importance weight-corrected quasi-Monte Carlo quadrature performs best and is more efficient than DRAM by orders of magnitude across a wide range of approximation accuracies and sample sizes. Indeed, all the methods developed here significantly outperform DRAM in all computed examples.

Accelerating MCMC via Kriging-based adaptive independent proposals and delayed rejection

Markov Chain Monte Carlo (MCMC) simulation has a considerable computational burden when the target probability density function (PDF) evaluation involves a black-box, potentially computationally-expensive, numerical model. A novel framework to accelerate MCMC is developed here for such applications. It leverages a Kriging surrogate model approximation to the target PDF to improve computational efficiency, while preserves convergence properties to the exact target PDF, avoiding potential accuracy problems introduced through the surrogate model error. The approach relies on the delayed-rejection (DR) scheme and combines the rapid exploration characteristics of global (independent) proposals with the local search robustness of random walk proposals under the MCMC setting. The global proposal is chosen as the Kriging-based target PDF approximation. This proposal may resemble, depending on the surrogate model error, the actual target PDF very well, and therefore can provide significant computational benefits, including fast convergence of the Markov chain to its stationary distribution and, more importantly, low correlation between samples. At each MCMC step a candidate sample is generated from the independent proposal. If rejected, DR allows an extra random walk around the current sample. The DR step guarantees convergence to the actual target PDF, circumventing robustness problems that may arise when the Kriging-based independent proposal offers a poor approximation to (i.e., underestimates) the actual target PDF. The overall computational efficiency is further improved through an adaptive updating of the Kriging surrogate model during the MCMC sampling phase by extracting information from candidate samples whose inclusion in the training database can substantially enhance Kriging’s accuracy. The computational efficiency and robustness of the established algorithm, termed Adaptive Kriging Delayed Rejection Adaptive Metropolis algorithm(AK-DRAM), are demonstrated in two analytical benchmark problems and two engineering problems focusing on conditional failure sample simulation and Bayesian inference.

Modeling error PDF optimization based wavelet neural network modeling of dynamic system and its application in blast furnace ironmaking

Abstract In general, the modeling errors of dynamic system model are a set of random variables. The traditional performance index of modeling such as means square error (MSE) and root means square error (RMSE) cannot fully express the connotation of modeling errors with stochastic characteristics both in the dimension of time domain and space domain. Therefore, the probability density function (PDF) is introduced to completely describe the modeling errors in both time scales and space scales. Based on it, a novel wavelet neural network (WNN) modeling method is proposed by minimizing the two-dimensional (2D) PDF shaping of modeling errors. First, the modeling error PDF by the traditional WNN is estimated using data-driven kernel density estimation (KDE) technique. Then, the quadratic sum of 2D deviation between the modeling error PDF and the target PDF is utilized as performance index to optimize the WNN model parameters by gradient descent method. Since the WNN has strong nonlinear approximation and adaptive capability, and all the parameters are well optimized by the proposed method, the developed WNN model can make the modeling error PDF track the target PDF, eventually. Simulation example and application in a blast furnace ironmaking process show that the proposed method has a higher modeling precision and better generalization ability compared with the conventional WNN modeling based on MSE criteria. Furthermore, the proposed method has more desirable estimation for modeling error PDF that approximates to a Gaussian distribution whose shape is high and narrow.

Adaptive independent sticky MCMC algorithms

Monte Carlo methods have become essential tools to solve complex Bayesian inference problems in different fields, such as computational statistics, machine learning, and statistical signal processing. In this work, we introduce a novel class of adaptive Monte Carlo methods, called adaptive independent sticky Markov Chain Monte Carlo (MCMC) algorithms, to sample efficiently from any bounded target probability density function (pdf). The new class of algorithms employs adaptive non-parametric proposal densities, which become closer and closer to the target as the number of iterations increases. The proposal pdf is built using interpolation procedures based on a set of support points which is constructed iteratively from previously drawn samples. The algorithm’s efficiency is ensured by a test that supervises the evolution of the set of support points. This extra stage controls the computational cost and the convergence of the proposal density to the target. Each part of the novel family of algorithms is discussed and several examples of specific methods are provided. Although the novel algorithms are presented for univariate target densities, we show how they can be easily extended to the multivariate context by embedding them within a Gibbs-type sampler or the hit and run algorithm. The ergodicity is ensured and discussed. An overview of the related works in the literature is also provided, emphasizing that several well-known existing methods (like the adaptive rejection Metropolis sampling (ARMS) scheme) are encompassed by the new class of algorithms proposed here. Eight numerical examples (including the inference of the hyper-parameters of Gaussian processes, widely used in machine learning for signal processing applications) illustrate the efficiency of sticky schemes, both as stand-alone methods to sample from complicated one-dimensional pdfs and within Gibbs samplers in order to draw from multi-dimensional target distributions.

Adaptive rejection sampling with fixed number of nodes

ABSTRACTThe adaptive rejection sampling (ARS) algorithm is a universal random generator for drawing samples efficiently from a univariate log-concave target probability density function (pdf). ARS generates independent samples from the target via rejection sampling with high acceptance rates. Indeed, ARS yields a sequence of proposal functions that converge toward the target pdf, so that the probability of accepting a sample approaches one. However, sampling from the proposal pdf becomes more computational demanding each time it is updated. In this work, we propose a novel ARS scheme, called Cheap Adaptive Rejection Sampling (CARS), where the computational effort for drawing from the proposal remains constant, decided in advance by the user. For generating a large number of desired samples, CARS is faster than ARS.

Minimum Entropy Active Fault Tolerant Control of the Non-Gaussian Stochastic Distribution System Subjected to Mean Constraint

Stochastic distribution control (SDC) systems are a group of systems where the outputs considered is the measured probability density function (PDF) of the system output whilst subjected to a normal crisp input. The purpose of the active fault tolerant control of such systems is to use the fault estimation information and other measured information to make the output PDF still track the given distribution when the objective PDF is known. However, if the target PDF is unavailable, the PDF tracking operation will be impossible. Minimum entropy control of the system output can be considered as an alternative strategy. The mean represents the center location of the stochastic variable, and it is reasonable that the minimum entropy fault tolerant controller can be designed subjected to mean constraint. In this paper, using the rational square-root B-spline model for the shape control of the system output probability density function (PDF), a nonlinear adaptive observer based fault diagnosis algorithm is proposed to diagnose the fault. Through the controller reconfiguration, the system entropy subjected to mean restriction can still be minimized when fault occurs. An illustrative example is utilized to demonstrate the use of the minimum entropy fault tolerant control algorithms.

Approximation of probability density functions by the Multilevel Monte Carlo Maximum Entropy method

We develop a complete convergence theory for the Maximum Entropy method based on moment matching for a sequence of approximate statistical moments estimated by the Multilevel Monte Carlo method. Under appropriate regularity assumptions on the target probability density function, the proposed method is superior to the Maximum Entropy method with moments estimated by the Monte Carlo method. New theoretical results are illustrated in numerical examples.

Feature Transformation Detection Method with Best Spectral Band Selection Process for Hyper-spectral Imaging

We present a newly developed feature transformation (FT) detection method for hyper-spectral imagery (HSI) sensors. In essence, the FT method, by transforming the original features (spectral bands) to a different feature domain, may considerably increase the statistical separation between the target and background probability density functions, and thus may significantly improve the target detection and identification performance, as evidenced by the test results in this paper. We show that by differentiating the original spectral, one can completely separate targets from the background using a single spectral band, leading to perfect detection results. In addition, we have proposed an automated best spectral band selection process with a double-threshold scheme that can rank the available spectral bands from the best to the worst for target detection. Finally, we have also proposed an automated cross-spectrum fusion process to further improve the detection performance in lower spectral range (<1000 nm) by selecting the best spectral band pair with multivariate analysis. Promising detection performance has been achieved using a small background material signature library for concept-proving, and has then been further evaluated and verified using a real background HSI scene collected by a HYDICE sensor.

Predictive PDF control in shaping of molecular weight distribution based on a new modeling algorithm

The aims of this work are to develop an efficient modeling method for establishing dynamic output probability density function (PDF) models using measurement data and to investigate predictive control strategies for controlling the full shape of output PDF rather than the key moments. Using the rational square-root (RSR) B-spline approximation, a new modeling algorithm is proposed in which the actual weights are used instead of the pseudo weights in the weights dynamic model. This replacement can reduce computational load effectively in data-based modeling of a high-dimensional output PDF model. The use of the actual weights in modeling and control has been verified by stability analysis. A predictive PDF model is then constructed, based on which predictive control algorithms are established with the purpose to drive the output PDF towards the desired target PDF over the control process. An analytical solution is obtained for the non-constrained predictive PDF control. For the constrained predictive control, the optimal solution is achieved via solving a constrained nonlinear optimization problem. The integrated method of data-based modeling and predictive PDF control is applied to closed-loop control of molecular weight distribution (MWD) in an exemplar styrene polymerization process, through which the modeling efficiency and the merits of predictive control over standard PDF control are demonstrated and discussed.

Fault diagnosis and minimum entropy fault tolerant control for non-Gaussian singular stochastic distribution systems using square-root approximation

The fault diagnosis (FD) and minimum entropy fault tolerant control (FTC) algorithms are proposed for non-Gaussian singular stochastic distribution control (SDC) systems in this paper. Different from general SDC systems, in singular SDC systems, the relationship between the weight vector and the control input is expressed by a singular state space model, which increases the difficulty in the design of fault diagnosis and fault tolerant control. A non-singular state transformation is made to transform the singular dynamic system into a differential-algebraic system. An iterative learning observer (ILO) is designed for fault estimation. The concept of entropy is introduced to fault tolerant control of the non-Gaussian stochastic distribution system when the target probability density function (PDF) is not known in advance. Based on the estimated fault information, the controller is reconfigured by minimising the performance function with regard to the entropy subjected to mean constraint. The reconfigured controller can make the output of the post-fault SDC system still have minimum uncertainty, leading to minimum entropy FTC of the non-Gaussian singular SDC system. Computer simulations are given to demonstrate the validity of the fault diagnosis and minimum entropy FTC algorithms.