Estimation Of Probability Density Function Research Articles

Higher-order averaging procedure for performance analysis of a mono-stable Duffing piezoelectric energy harvesting system under white noise excitation

This work is concerned with the energy harvesting of mono-stable Duffing oscillators with piezoelectric coupling under white noise excitation. This is the first time that the higher-order averaging procedure is extended to find the first-, second- and third-order approximate stationary joint probability density functions of amplitude and full phase by solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation. Then the mean-square electric voltage and mean output power are derived through the approximate relation between the electric voltage and the mechanical states. The analytical results are verified by Monte Carlo simulation. The effects of system parameters, such as excitation intensity, mechanical damping term, nonlinear stiffness coefficients, and the resistance ratio of the resistive–capacitive circuit on the mean-square displacement and mean output power are discussed in detail. It has been shown that the averaging procedure of the averaging method of coefficients in FPK equation can be considered as a particular case of the proposed higher-order averaging procedure and the higher-order approximate solutions are much more accurate than the one obtained by the averaging method of coefficients in FPK equation.© 2022 The Authors. Published by Elsevier Ltd.

Probability Density Estimation through Nonparametric Adaptive Partitioning and Stitching

We present a novel nonparametric adaptive partitioning and stitching (NAPS) algorithm to estimate a probability density function (PDF) of a single variable. Sampled data is partitioned into blocks using a branching tree algorithm that minimizes deviations from a uniform density within blocks of various sample sizes arranged in a staggered format. The block sizes are constructed to balance the load in parallel computing as the PDF for each block is independently estimated using the nonparametric maximum entropy method (NMEM) previously developed for automated high throughput analysis. Once all block PDFs are calculated, they are stitched together to provide a smooth estimate throughout the sample range. Each stitch is an averaging process over weight factors based on the estimated cumulative distribution function (CDF) and a complementary CDF that characterize how data from flanking blocks overlap. Benchmarks on synthetic data show that our PDF estimates are fast and accurate for sample sizes ranging from 29 to 227, across a diverse set of distributions that account for single and multi-modal distributions with heavy tails or singularities. We also generate estimates by replacing NMEM with kernel density estimation (KDE) within blocks. Our results indicate that NAPS(NMEM) is the best-performing method overall, while NAPS(KDE) improves estimates near boundaries compared to standard KDE.

Polynomial Propagation of Moments in Stochastic Differential Equations

We address the problem of approximating the moments of the solution, , of an Itô stochastic differential equation (SDE) with drift and diffusion terms over a time grid . In particular, we assume an explicit numerical scheme for the generation of sample paths and then obtain recursive equations that yield any desired noncentral moment of as a function of the initial condition . The core of the methodology is the decomposition of the numerical solution into a “central part” and an “effective noise” term. The central term is computed deterministically from the ordinary differential equation (ODE) that results from eliminating the diffusion term in the SDE, while the effective noise accounts for the stochastic deviation from the numerical solution of the ODE. For simplicity, we describe the proposed methodology based on an Euler–Maruyama integrator, but other explicit numerical schemes can be exploited in the same way. We also apply the moment approximations to construct estimates of the 1-dimensional marginal probability density functions of based on a Gram–Charlier expansion. Both for the approximation of moments and 1-dimensional densities, we describe how to handle the cases in which the initial condition is fixed (i.e., for some deterministic and known ) or random. In the latter case, we resort to polynomial chaos expansion (PCE) schemes in order to approximate the target moments. The methodology has been inspired by the PCE and differential algebra methods used for uncertainty propagation in astrodynamics problems. Hence, we illustrate its application for the quantification of uncertainty in a 2-dimensional Keplerian orbit perturbed by a Wiener noise process.

Practical Assessment of Quartz Crystallographic Preferred Orientation Strength

The ordering strength of crystallographic preferred orientation data can be assessed in different ways, however, the appropriateness of the methodology used to do so can depend on the geometry of the distributions. Eigenvector-based ordering evaluation methods, for example, may not be appropriate for data that comprise multiple, variably oriented distributions such as those commonly found in quartz c-axis crystallographic preferred orientations. Examination of artificial data that represent a variety of different potential c-axis distributions shows a significant correlation between the relative orientations of those distributions (i.e., the opening angle in a cross-girdled quartz c-axis pole figure) and the strengths calculated using eigen-vector based evaluation methods; larger pole figure opening angles correlate with decreasing distribution ordering strength. The same correlation does not exist when strength is evaluated using the l2 - norm of the estimated probability density function (JPF) of the same data. The direct correlations between pole figure c-axis opening angles and ordering strength noted in the artificial distributions are also demonstrated in the evaluation of real-world data, though significant complications related to heterogeneous nature, and/or deformation, of the natural specimens can partially obfuscate the relationship. Regardless, given the potential effect of geometry on eigenvector-based evaluation methods we recommend that the ordering strength of pole figure data be evaluated using JPF.

Accelerated Entropic Path Sampling with a Bidirectional Generative Adversarial Network.

The role of entropy in mediating the dynamic outcomes of chemical reactions remains largely unknown. To evaluate the change of entropy along post-transition state paths, we have previously developed entropic path sampling that computes configurational entropy from an ensemble of reaction trajectories. However, one major caveat of this approach lies in its high computational demand: about 2000 trajectories are needed to converge the computation of an entropic profile. Here, by leveraging a deep generative model, we developed an accelerated entropic path sampling approach that evaluates entropic profiles using merely a few hundred reaction dynamic trajectories. The new method, called bidirectional generative adversarial network-entropic path sampling, can enhance the estimation of probability density functions of molecular configurations by generating pseudo-molecular configurations that are statistically indistinguishable from the true data. The method was established using cyclopentadiene dimerization, in which we reproduced the reference entropic profiles (derived from 2480 trajectories) using merely 124 trajectories. The method was further benchmarked using three reactions with symmetric post-transition-state bifurcation, including endo-butadiene dimerization, 5-fluoro-1,3-cyclopentadiene dimerization, and 5-methyl-1,3-cyclopentadiene dimerization. The results indicate the existence of a "hidden entropic intermediate", which is a dynamic species that binds to a local entropic maximum where no free energy minimum is formed.

Recurrent Estimation of Phases Probability Density Function of Modulated Signals

Recurrent Estimation of Phases Probability Density Function of Modulated Signals

Recurrent Estimation of Phases Probability Density Function of Modulated Signals

The problem of constructing an empirical probability density distribution of the phases of a signal with phase shift keying modulation, as well as nonparametric estimation methods, is considered. The transition from the approximation method using basis functions to the method of windowed kernel functions based on harmonic functions is shown. The results of numerical experiments are presented.

Density Estimation in RKHS with Application to Korobov Spaces in High Dimensions

A kernel method for estimating a probability density function from an independent and identically distributed sample drawn from such density is presented. Our estimator is a linear combination of kernel functions, the coefficients of which are determined by a linear equation. An error analysis for the mean integrated squared error is established in a general reproducing kernel Hilbert space setting. The theory developed is then applied to estimate probability density functions belonging to weighted Korobov spaces, for which a dimension-independent convergence rate is established. Under a suitable smoothness assumption, our method attains a rate arbitrarily close to the optimal rate. Numerical results support our theory.

Uncertainty Propagation for the Structures with Fuzzy Variables and Uncertain-but-Bounded Variables.

Various uncertain factors exist in the practical systems. Random variables, uncertain-but-bounded variables and fuzzy variables are commonly employed to measure these uncertain factors. Random variables are usually employed to define uncertain factors with sufficient samples to accurately estimate probability density functions (PDFs). Uncertain-but-bounded variables are usually employed to define uncertain factors with limited samples that cannot accurately estimate PDFs but can precisely decide variation ranges of uncertain factors. Fuzzy variables can commonly be employed to define uncertain factors with epistemic uncertainty relevant to human knowledge and expert experience. This paper focuses on the practical systems subjected to epistemic uncertainty measured by fuzzy variables and uncertainty with limited samples measured by uncertain-but-bounded variables. The uncertainty propagation of the systems with fuzzy variables described by a membership function and uncertain-but-bounded variables defined by a multi-ellipsoid convex set is investigated. The combination of the membership levels method for fuzzy variables and the non-probabilistic reliability index for uncertain-but-bounded variables is employed to solve the uncertainty propagation. Uncertainty propagation is sued to calculate the membership function of the non-probabilistic reliability index, which is defined by a nested optimization problem at each membership level when all fuzzy variables degenerate into intervals. Finally, three methods are employed to seek the membership function of the non-probabilistic reliability index. Various examples are utilized to demonstrate the applicability of the model and the efficiency of the proposed method.

BDIS: Bayesian Dense Inverse Searching Method for Real-Time Stereo Surgical Image Matching

In stereoscope-based minimally invasive surgeries (MISs), dense stereo matching plays an indispensable role in 3-D shape recovery, AR, VR, and navigation tasks. Although numerous deep neural network (DNN) approaches are proposed, the conventional prior-free approaches are still popular in the industry because of the lack of open-source annotated dataset and the limitation of the task-specific pretrained DNNs. Among the prior-free stereo matching algorithms, there is no successful real-time algorithm in none GPU environment for MIS. This article proposes the first CPU-level real-time prior-free stereo matching algorithm for general MIS tasks. We achieve an average <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$14-17$</tex-math></inline-formula> Hz on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$640 \times 480$</tex-math></inline-formula> images with a single-core CPU (i5-9400) for surgical images. Meanwhile, it achieves slightly better accuracy than the popular efficient large-scale stereo matching (ELAS) method. The patch-based fast disparity searching algorithm is adopted for the rectified stereo images. A coarse-to-fine Bayesian probability and a spatial Gaussian mixed model were proposed to evaluate the patch probability at different scales. An optional probability density function estimation algorithm was adopted to quantify the prediction variance. Extensive experiments demonstrated the proposed method's capability to handle ambiguities introduced by the textureless surfaces and the photometric inconsistency from the non-Lambertian reflectance and dark illumination. The estimated probability managed to balance the confidences of the patches for stereo images at different scales. It has similar or higher accuracy and fewer outliers than the baseline ELAS in MIS, while it is 4–5 times faster.

Further exploration into the valid regions of Gram–Charlier densities

Density estimation plays an important and fundamental role in finance, pattern recognition, machine learning, and statistics. Based on derivatives of a Gaussian density, a Gram–Charlier series presents an infinite expansion. Its truncated series is often used in many fields to approximate probability density functions. Although the expansions are convenient, there are constrained regions on the value of the cumulants (or moments) that admit a valid (nonnegative) probability density function. Lin and Zhang (2022)’s paper focuses on Gram–Charlier densities to show how the valid region of higher cumulants can be numerically implemented by semidefinite programming, which ensures that a series truncated at a cumulant of an arbitrary even order represents a valid probability density. This paper is the further exploration into the same problem. First, we use the representation theorem of such polynomials as sum of squares on the Gram–Charlier density to show how to develop the corresponding convex optimization problem for its valid region. Second, we provide the valid skewness–kurtosis regions of Gram–Charlier densities only up to the sixteenth-order because the semidefinite programming fails to calculate these regions when the order is above that. Third, we explore the valid region of the fourth-order Gram–Charlier defined on an arbitrary finite domain [−q,q] but not the field R of real numbers. Our analysis proves that the ranges of skewness and kurtosis can be broadened with the finite domains, which earn a wider application. Furthermore, the impact of the length of finite domains 2q on valid regions has been shown.

Harmonic transform-based non-parametric density estimation method for forward uncertainty propagation and reliability analysis

Reconstructing the probability density function (PDF) of the limit state function (LSF) is regarded as a non-intrusive method for forward uncertainty propagation and reliability analysis. In this paper, a novel non-parametric density estimation method is proposed to derive the unknown distribution of the LSF for this purpose, where the efficiency and accuracy are ensured. The main innovation of the paper is to derive the PDF of the LSF via a novel concept, i.e., the harmonic transform, regardless of the complexity of the problem. First, the non-parametric method for PDF estimation based on complex fractional moments/Mellin transform is briefly revisited, where the limitations are also pointed out. Inspired from Mellin transform, the concept of harmonic transform is put forward, which overcomes the disadvantages of Mellin transform. Then, the unknown PDF is recovered via the inverse harmonic transform, where the analytical formula is derived accordingly. Numerical estimation method is also given for practical implementations, where some critical issues are also pointed out. Detailed step-by-step procedures of the proposed method are also outlined. The proposed method is a non-parametric one since it does not require a distribution model in advance. Some commonly-used distributions are first applied to validate the efficacy of the proposed method. Then, four numerical examples involving different types of problems are extensively investigated to validate the proposed method for forward uncertainty propagation and reliability analysis. The results demonstrate that the proposed method is effective for deriving the PDFs for both static and dynamic reliability analyses with nonlinear explicit or implicit LSFs.

Multiple combined gamma kernel estimations for nonnegative data with Bayesian adaptive bandwidths

The modified (or second version) gamma kernel of Chen [Probability density function estimation using gamma kernels, Annals of the Institute of Statistical Mathematics 52 (2000), pp. 471–480] should not be automatically preferred to the standard (or first version) gamma kernel, especially for univariate convex densities with a pole at the origin. In the multivariate case, multiple combined gamma kernels, defined as a product of univariate standard and modified ones, are here introduced for nonparametric and semiparametric smoothing of unknown orthant densities with support \([0,\infty )^d\). Asymptotical properties of these multivariate associated kernel estimators are established. Bayesian estimation of adaptive bandwidth vectors using multiple pure combined gamma smoothers, and in semiparametric setup, are exactly derived under the usual quadratic function. The simulation results and four illustrations on real datasets reveal very interesting advantages of the proposed combined approach for nonparametric smoothing, compare to both pure standard and pure modified gamma kernel versions, and under integrated squared error and average log-likelihood criteria.

Two-stage conditional density estimation based on Bernstein polynomials

Two-stage conditional probability density function estimators are proposed and studied. Specifically, the Nadaraya-Watson (NW) and local linear (LL) conditional distribution function estimators have been smoothed using Bernstein polynomials in the first stage. Second, the proposed estimators are obtained by differentiating NW and LL estimators. The asymptotic properties of these estimators are established such as asymptotic bias, variance, and normality. Finally, a simulation study is carried out to assess the relative advantage of our estimators compared to other competitors. In addition, the well-known Old Faithful Geyser data is analyzed using the proposed estimators.

The generalized EPC method for the non-stationary probabilistic response of nonlinear dynamical system

It is well known that the non-stationary response probability density function (PDF) plays an important role in the reliability and failure analysis. With current approximate or numerical methods, the non-stationary approximate PDF is generally obtained in terms of the ones at the discrete time instants. Repeated computation must be conducted for the ones at other time instants since the response PDF at only one time instant can be obtained after performing the solution procedure. Thus, the computational efficiency suffers a major setback. In this paper, the exponential polynomial closure (EPC) method is further improved and enhanced to obtain the completely non-stationary PDF solution, which is distributed continuously in the time domain. It takes the temporal base function into the PDF approximation. The unknown coefficients in the EPC solution are generalized to be explicit functions of the time parameter. With the least squares method, the explicit time functions can be determined based on the few simulated response PDF values. This implementation makes the continues PDF distribution in the time domain available, which greatly improves the computational efficiency without the repeating computation. Two typical nonlinear systems under stationary and non-stationary random excitations are taken as examples to illustrate the efficiency of the proposed method. Numerical results show that the results obtained by the proposed method agree well with the simulated results. In addition, the relationship between the explicit time function and the modulate function is discussed.

Efficient conditional probability theorem and importance sampling-based methods for global reliability sensitivity analysis

Global reliability sensitivity (GRS) analysis can measure the effect of random inputs on failure probability (FP). To efficiently solve GRS, two conditional probability theorem (CPT)-based methods are proposed by combining adaptive Kriging (AK) with importance sampling (IS) (CPT-AK-IS) and combining AK with Meta-IS (CPT-AK-Meta-IS) respectively. Firstly, differentiation approximation and CPT are used to convert the estimation of conditional probability density function (PDF), which is required by the existing Bayes theorem-based methods, into that of a series of probabilities. Secondly, GRS can be directly estimated by the failure samples of IS, while the existing Bayes theorem methods based on IS need to transform the failure samples of IS into those of original PDF. Both the first and second strategies can reduce the computational complexity of solving GRS. Thirdly, by selecting a suitable differentiation interval with a proposed adaptive strategy, the estimation of a series of probabilities can be accurately completed as a byproduct of one IS based simulation for solving FP without additional computational cost. Finally, by introducing AK into IS and Meta-IS, it can reduce the number of evaluating performance function and the size of candidate sample pool simultaneously. These novelties are sufficiently verified by the presented examples.

A Multi-Hop QoS-Aware and Predicting Link Quality Estimation (PLQE) Routing Protocol for Reliable WBSN

Technological advancements and miniaturization of sensor have made it possible to use tiny size sensors nodes to monitor patients’ physiological data in real-time at very low cost. The network of such small size sensor nodes is often called wireless body sensor networks (WBSNs). The patient’s data is highly sensitive which required to reach reliably and timely at the destination node. However, short range wireless nodes and postural body movements cause frequent topological changes and recurrent network partitioning. The network partitioning leads to delayed and significant loss of critical patient’s data such as Electrocardiogram (ECG), Electroencephalogram (EEG) and Electromyogram (EMG). Therefore, QoS requirement of WBSNs in terms of reliability, end-to-end delay and throughput are highly compromise. We in this paper propose a Predicting routing protocol PLQE to address reliable data transmission and network partitioning due to node mobility. The PLQE dynamically determines the efficient links by using beta probability density function, link quality and link delay estimation (LDE). The PLQE is composed of link reliability factor (LRF) and expected probability Indicator (EPI) to obtain most often updated status of the links to figure out stable and reliable end to end route to the destination node. The simulation results confirm the higher performance of the proposed scheme against state of the art routing protocols in terms of packet delivery ratio, end to end delay, throughput and normalized routing load.

GMSK Demodulation Combining 1D‐CNN and Bi‐LSTM Network Over Strong Solar Wind Turbulence Channel

AbstractTo solve the problem of bit error rate (BER) performance degradation over strong solar wind turbulence channel, this paper addresses to Gaussian minimum frequency shift keying (GMSK) demodulation using machine learning. First, by analyzing the scintillation characteristics of the telemetry signal caused by solar wind turbulence during the solar superior conjunction, the K distribution channel model is innovatively established over strong solar wind turbulence channel. Then, the approximate probability density function of the K distribution of the established channel is studied by using Laguerre orthogonal polynomial for convenience. Additionally, the BER performance of GMSK over strong solar wind turbulence channel is analyzed. Second, using a one‐dimensional convolutional neural network (1D‐CNN) and bidirectional long short‐term memory network (Bi‐LSTM), two demodulators for GMSK are proposed. By integrating the 1D‐CNN and LSTM to extract local features and process the dynamic information of GMSK signals, we propose a novel GMSK demodulator over strong solar wind turbulence channel with combining 1D‐CNN and Bi‐LSTM. The simulation results show that, even if the solar wind turbulence has serious effects on the signal at a low signal‐to‐noise ratio, the proposed combining neural network demodulation can obtain better BER performance than the classical Viterbi algorithm, demodulation using only a single 1D‐CNN or Bi‐LSTM. The proposed GMSK demodulator is more suitable for the deep space exploration of the solar system.

A Novel Methodology to Estimate Probability Density Function of Voltage Sag Duration and Failure Rates on Power Distribution Systems

Voltage sags and power interruptions are important power quality problems that affect sensitive customers, mainly because they cause annual massive economical losses to the industrial sector as a result of unexpected production process disruptions. In this sense, to propose corrective and preventive measures and improve the power quality of distribution systems, stochastic methodologies have been proposed in the literature to estimate annual voltage sags and power interruptions. However, these methodologies generally use typical cumulative distribution functions of voltage sag duration (P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SgD</sub> ), which can not reflect the real estate of the network under study. To solve this constraint, this paper proposes a novel methodology to estimate a proper P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SgD</sub> considering the information of the distribution network (i.e., topology and coordination schemes of the protection system) and the stochastic behaviors of short-circuits that can affect the distribution system. Moreover, the proposed methodology allows estimating permanent failure rates and average repair time considering known or expected values of reliability indicators. The results show that the proposed methodology is capable to adapt from an initial P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SgD</sub> curve to another one with fidelity to achieve real values of expected annual power interruptions.

ABC for model selection and parameter estimation of drill-string bit-rock interaction models and stochastic stability

The bit-rock interaction considerably affects the dynamics of a drill string, thus choosing a suitable model and identifying its parameters considering uncertainties is crucial. This paper proposes a procedure to estimate the parameters of four bit-rock interaction models, one of which is new, and at the same time select the most suitable model, given the available field data. The approximate Bayesian computation (ABC) is used for this purpose. An approximate posterior probability density function is obtained for the parameters of each model, which allows uncertainty to be analyzed. Furthermore, the impact of the uncertainties of the selected models on the torsional stability map (varying the nominal top drive angular speed and the weight on bit) of the system is evaluated.