Gauss Hermite Research Articles

A heuristic optimization considering probabilistic constraints via an equivalent single variable Pearson distribution system

The computational cost of an optimization considering probabilistic constraints is often expensive due to the reliability analysis performed inside the optimization loop. This study computes reliability using an equivalent single variable Pearson’s distribution system. To retain the accuracy without sacrificing efficiency, dimension reduction method is used to reduce the required number of points in Gaussian–Hermite integration. In addition, four advance machine learning techniques are used to construct a hyper-probability density function to solve the singularity issue in the Pearson distribution system. The ε-constrained is utilized in a metaheuristic optimization algorithm (PSO/SOS) to consider the probabilistic constraints. The proposed algorithm is verified with several literature studies. Results shown that it is able to find an optimal design for problems having linear, highly nonlinear, and implicit probabilistic constraint functions with normal or non-normal variables. It is found that increasing number of variates in dimension reduction can provide a more accurate estimation of moments of the performance functions and enhance the solution accuracy. To demonstrate the applicability of the proposed algorithm to a practical problem, a platform of the SAP2000 Open Application Programming Interface (OAPI) is developed to find an optimal design for a three story steel frame under nonlinear time history analysis.

Enhanced biologically inspired model for image recognition based on a novel patch selection method with moment

Biologically inspired model (BIM) for image recognition is a robust computational architecture, which has attracted widespread attention. BIM can be described as a four-layer structure based on the mechanisms of the visual cortex. Although the performance of BIM for image recognition is robust, it takes the randomly selected ways for the patch selection, which is sightless, and results in heavy computing burden. To address this issue, we propose a novel patch selection method with oriented Gaussian–Hermite moment (PSGHM), and we enhanced the BIM based on the proposed PSGHM, named as PBIM. In contrast to the conventional BIM which adopts the random method to select patches within the feature representation layers processed by multi-scale Gabor filter banks, the proposed PBIM takes the PSGHM way to extract a small number of representation features while offering promising distinctiveness. To show the effectiveness of the proposed PBIM, experimental studies on object categorization are conducted on the CalTech05, TU Darmstadt (TUD) and GRAZ01 databases. Experimental results demonstrate that the performance of PBIM is a significant improvement on that of the conventional BIM.

Log-Burr XII Gamma–Weibull Regression Model with Random Effects and Censored Data

It may happen in some applications that the assumption of independence of survival times does not hold. Thus, we propose a new log-Burr XII regression model with log-gamma–Weibull distributions for the random effects. The maximum likelihood method is used to estimate the model parameters based on the Gauss–Hermite numerical integration technique. For different parameter settings, sample sizes, censoring percentages and correlated data, various simulations are performed. Some global-influence measurements are also investigated. In order to assess the robustness of the maximum likelihood estimators, we evaluate local influence on the estimates of the parameters under different perturbation schemes. We illustrate the importance of the new model by means of a real data set in analysis of experiments.

Asymptotic Performance of Composite Lognormal-X Fading Channels

It is challenging to derive closed-form performance expressions for composite channels considering both the large-scale lognormal fading and small-scale fading, thus most existing analyses on the composite channels are based on numerical approaches such as the Gauss–Hermite quadrature. In this paper, we develop a simple framework to achieve closed-form asymptotic expressions for the outage probabilities and the error rates of the composite lognormal-X fading channels, where the “X” can be Rayleigh, Rician, Nakagami- $m$ or any other fading channels that have finite diversity orders. It is proved that the lognormal fading contributes an exponential factor to the asymptotic expressions, and the exponential factor is related to the diversity order of the X channel and the lognormal parameters. The new analytical tool is shown to be versatile in evaluating the performance of radio-frequency communications suffering both the large-scale and small-scale fading, including diversity reception systems, relaying systems, and distributed antenna systems. It is also shown to be useful in evaluating free-space optical communication systems with pointing error, and in deriving asymptotic signal-to-noise ratio gaps between different modulation formats over lognormal fading channels. The elegant asymptotic expressions reveal insights into the composite fading channels and can be a criterion for various system designs.

Image Classification Combined with Fusion Gaussian–Hermite Moments Feature and Improved Nonlinear SVM Classifier

With the development of computer technology, data mining, artificial intelligence, and image-processing technology have been applied to medical diagnosis. Image classification is one of the main technologies of medical image processing, which can be used to determine whether a patient suffers from breast cancer according to x-ray images of the breast. To achieve reliable classification of breast images, an image classification method combined with a fusion Gaussian–Hermite moments feature and improved nonlinear support vector machine (SVM) classifier is proposed. The proposed fusion Gaussian–Hermite moments features can improve the robustness and distinguish the ability of features by constructing Gaussian–Hermite invariant moments according to invariant moment theory and constructing a Gaussian–Hermite Fisher moment according to Fisher’s idea. The proposed improved nonlinear SVM classifier can improve the efficiency and accuracy of the classifier through eigen decomposition and sample learning. Experimental results demonstrate that the proposed method has a high accuracy rate for breast x-ray image classification.

Approximate composite marginal likelihood inference in spatial generalized linear mixed models

ABSTRACTNon-Gaussian spatial responses are usually modeled using spatial generalized linear mixed model with spatial random effects. The likelihood function of this model cannot usually be given in a closed form, thus the maximum likelihood approach is very challenging. There are numerical ways to maximize the likelihood function, such as Monte Carlo Expectation Maximization and Quadrature Pairwise Expectation Maximization algorithms. They can be applied but may in such cases be computationally very slow or even prohibitive. Gauss–Hermite quadrature approximation only suitable for low-dimensional latent variables and its accuracy depends on the number of quadrature points. Here, we propose a new approximate pairwise maximum likelihood method to the inference of the spatial generalized linear mixed model. This approximate method is fast and deterministic, using no sampling-based strategies. The performance of the proposed method is illustrated through two simulation examples and practical aspects are investigated through a case study on a rainfall data set.

A unified performance analysis of energy detector over [formula omitted]/lognormal and [formula omitted]/lognormal composite fading channels with diversity and cooperative spectrum sensing

In this paper, our objective is to formulate a generalized composite distribution which includes both α-η-μ and the α-κ-μ distributions for modelling small scale fading while shadowing effect is captured by the lognormal distribution. An approximate closed-form of generalized/lognormal (GL) distribution is derived using Gauss-Hermite (GH) integration. Subsequently, the distribution is utilized to formulate analytical expressions for the average probability of energy detection (PD) as well as average area under the receiver operating characteristic curve (AUC) with and without maximal ratio combining (MRC) diversity reception technique. The proposed solutions are expressed in terms of single and bivariate Fox’s H functions, which are currently being used in open literature. Further, the derived average PD is utilized for analysing the performance of energy detector (ED) over square-law selection (SLS) and square-law combining (SLC) diversity schemes. Optimization of detection threshold and number of cooperative sensing nodes to minimize the total error rate has been discussed for CSS network with erroneous feedback channels. Finally, the simplified closed-form asymptotic expressions of the average PD and AUC are also derived. The analytical expressions are validated by comparing them with the exact numerical results and Monte Carlo simulation.

Computer aided detection of mammographic mass using exact Gaussian–Hermite moments

Breast cancer is one of the common cancer deaths in women worldwide. Early detection is the key to reduce the mortality rate. Clinical trials have shown that computer aided systems (CAD) have improved the accuracy of breast cancer detection. This paper proposed a highly accurate CAD system based on extracting highly significant features using exact Gaussian–Hermite moments. The obtained feature vector is presented to K-NN, random forests and AdaBoost classifiers. The proposed system is evaluated using two different datasets namely IRMA and MIAS. The evaluation metrics of accuracy, TP, FP and area under ROC curve using 10-fold cross-validation are calculated. The results indicate the usefulness of the proposed exact Gaussian–Hermite moments features for distinguishing between normal and abnormal lesions and the superiority of the moments features compared with the conventional methods.

Dynamic asset allocation with event risk, transaction costs and predictable returns

We examine the interplay between event risk, transaction costs and predictability on the dynamic asset allocation of an investor with discrete trading opportunities. The model is calibrated to the U.S. stock market and a Gauss–Hermite quadrature approach is used to solve the investor’s dynamic optimization problem. Numerical scenarios are examined to show the impact of event risk on asset allocations, hedging demands, no-trading regions, and certainty equivalent returns. It is found that event risk shrinks hedging demand. Neglecting event risk can also lead to sizeable certainty equivalent return losses.

Switched and Iterated Square-Root Gauss–Hermite Filter for Passive Target Tracking

In this paper, a novel algorithm referred to as the switched and iterated square-root Gauss–Hermite filter is developed for the nonlinear state estimation. The algorithm is derived by embedding the matrix triangularization, iteration update, and switching control into the Gauss–Hermite filter architecture. This new filtering approach is used and analyzed in passive localization and tracking system, which is troublesome to other filtering algorithms because of its intrinsic disadvantages such as large initial error and weak observability. Comparative simulation results are demonstrated that a more accurate state estimation can be obtained rapidly by the proposed algorithm with lower computational burden and stronger robustness.

Adaptive sparse-grid Gauss–Hermite filter

In this paper, a new nonlinear filter based on sparse-grid quadrature method has been proposed. The proposed filter is named as adaptive sparse-grid Gauss–Hermite filter (ASGHF). Ordinary sparse-grid technique treats all the dimensions equally, whereas the ASGHF assigns a fewer number of points along the dimensions with lower nonlinearity. It uses adaptive tensor product to construct multidimensional points until a predefined error tolerance level is reached. The performance of the proposed filter is illustrated with two nonlinear filtering problems. Simulation results demonstrate that the new algorithm achieves a similar accuracy as compared to sparse-grid Gauss–Hermite filter (SGHF) and Gauss–Hermite filter (GHF) with a considerable reduction in computational load. Further, in the conventional GHF and SGHF, any increase in the accuracy level may result in an unacceptably high increase in the computational burden. However, in ASGHF, a little increase in estimation accuracy is possible with a limited increase in computational burden by varying the error tolerance level and the error weighting parameter. This enables the online estimator to operate near full efficiency with a predefined computational budget. • A novel nonlinear filter based on sparse-grid quadrature is proposed. • Accuracy of the GHF and SGHF is retained. • Considerable reduction in computational burden of GHF and SGHF. • A better trade-off between the accuracy and computational burden is achieved.

Design and analysis of a generalized DLL/FLL discriminator for GPS receivers

High-performance discriminators are of a significant interest in GPS receiver tracking-loop design. Main lobes of both the auto-correlation function of the band-limited pseudorandom noise code and the sinc function resulting from the carrier frequency tracking, resemble the shape of an inverted parabola. This interesting observation led to the proposal of a generalized DLL/FLL discriminator based on parabola fitting, which is the topic of this research. To provide better insight into the mechanism of this discriminator, a semi-analytical noise performance prediction method is presented based on the Gauss–Hermite cubature transformation, and simple loss functions are defined to comprehensively measure the S-curve-related performances. On this basis, design parameter optimization for the generalized DLL/FLL discriminator is addressed by trade-off analysis. Controlled experiments with real and simulated signals show that the overall performance of the generalized discriminator is significantly better than the commonly used DLL and FLL discriminators and is more suitable for use in applications that need to handle weak signal, high dynamics and/or large offsets in frequency and code phase estimation during the acquisition to tracking transition.

Investigation on different discrete velocity quadrature rules in gas-kinetic unified algorithm solving Boltzmann model equation

Different discrete velocity quadrature rules are analyzed and applied in gas-kinetic unified algorithm (GKUA) for rarefied free-molecule transition to continuum flows, including the original and modified Gauss–Hermite quadrature rules, the composite Newton–Cotes integration methods, the multi-subinterval Gauss–Legendre numerical quadrature rule and the Gauss–Chebyshev integral rule. Mathematical description and basic procedures of GKUA are presented briefly in solving the gas flow problems covering various flow regimes on the basis of the direction solution to the unified kinetic model of the Boltzmann equation. The numerical analyses and the application conditions of these quadrature rules are given in detail, as well as the evaluating method for the discretized velocity ordinate (DVO) points and their corresponding weights of each rule. According to the comparisons for some Gauss-type function integrations, the original and modified Gauss–Hermite quadrature rules can obtain the integration results with enough accuracy by using least DVO points among these rules, while the impossibility of more nodes limits the application scope of these two quadrature rules. Other four rules can conduct the problems with a needful wide integral interval, even if they use more DVO points than the Gauss–Hermite rules. Besides, the multi-subinterval Gauss–Legendre and the Gauss–Chebyshev rules can use less DVO points to obtain accurate integration results than the other two rules, which demonstrate their advantage and are recommended to be applied in GKUA to solve the Boltzmann model equation efficiently.Numerical simulations for the Sod and Lax shock-tube problems and the shock-density wave disturbing interaction problems are conducted by using GKUA with the above quadrature rules to make comparison. All quadrature rules can obtain nice accurate results with different number of the DVO points. The original and modified Gauss–Hermite quadrature rules, using the least DVO nodes to obtain results with enough computed precision, are the best options for GKUA to simulate low mach number flow regimes, while the Gauss–Chebyshev quadrature rule, which can obtain results with adequate and controllable precision for a wide integral interval using little DVO nodes, is the most appropriate quadrature rule for GKUA solving some complex and high mach number flows covering various flow regimes.

Spatial modeling of rainfall accumulated over short periods of time

This article proposes a new random field model to describe the spatial variation of rainfall amounts accumulated over short periods of time. The model is intended to satisfy a set of desiderata motivated by the understanding of rainfall generating mechanisms and exploratory data analysis of datasets of this type. First and second order properties of the proposed model are derived, including the mean and covariance functions, as well as the families of marginal and bivariate distributions. Properties of the proposed model are shown by a mix of analytical derivations and numerical exploration that use Gauss–Hermite quadrature to approximate the required integrals. The proposed model also satisfies a stochastic dominance property, which is argued to be sensible and consistent with most rainfall data of this type. A study of identifiability is carried out, which strongly suggests all model parameters are identifiable. The generalized method of moments is proposed to estimate the parameters, and the properties of these estimators are explored based on simulated data.

Compressed sparse tensor based quadrature for vibrational quantum mechanics integrals

A new method for fast evaluation of high dimensional integrals arising in quantum mechanics is proposed. The method is based on sparse approximation of a high dimensional function followed by a low-rank compression. In the first step, we interpret the high dimensional integrand as a tensor in a suitable tensor product space and determine its entries by a compressed sensing based algorithm using only a few function evaluations. Secondly, we implement a rank reduction strategy to compress this tensor in a suitable low-rank tensor format using standard tensor compression tools. This allows representing a high dimensional integrand function as a small sum of products of low dimensional functions. Finally, a low dimensional Gauss–Hermite quadrature rule is used to integrate this low-rank representation, thus alleviating the curse of dimensionality. Numerical tests on synthetic functions, as well as on energy correction integrals for water and formaldehyde molecules demonstrate the efficiency of this method using very few function evaluations as compared to other integration strategies.

Multi-UAV Doppler Information Fusion for Target Tracking Based on Distributed High Degrees Information Filters

Multi-Unmanned Aerial Vehicle (UAV) Doppler-based target tracking has not been widely investigated, specifically when using modern nonlinear information filters. A high-degree Gauss–Hermite information filter, as well as a seventh-degree cubature information filter (CIF), is developed to improve the fifth-degree and third-degree CIFs proposed in the most recent related literature. These algorithms are applied to maneuvering target tracking based on Radar Doppler range/range rate signals. To achieve this purpose, different measurement models such as range-only, range rate, and bearing-only tracking are used in the simulations. In this paper, the mobile sensor target tracking problem is addressed and solved by a higher-degree class of quadrature information filters (HQIFs). A centralized fusion architecture based on distributed information filtering is proposed, and yielded excellent results. Three high dynamic UAVs are simulated with synchronized Doppler measurement broadcasted in parallel channels to the control center for global information fusion. Interesting results are obtained, with the superiority of certain classes of higher-degree quadrature information filters.

Performance of an energy detector over η-μ/lognormal composite channel with cooperative sensing

ABSTRACTThe η-μ/lognormal composite distribution corresponds to a physical realistic scenario where the multipath effect is characterised by well-known η-μ generalised model and shadowing effect is captured by the lognormal (LN) distribution. In this work, an approximate closed-form of η-μ/LN distribution is derived using Gauss-Hermite (GH) integration. Further, the derived probability density function is utilised to develop the closed-form expression as well as infinite series representation of the average probability of energy detection (PD) and average area under the receiver operating characteristics curve (AUC) along with their respective upper bound of the truncation error. Maximal ratio combining (MRC) microdiversity technique is applied over η-μ/LN composite fading model and then its average PD is derived. In order to reduce the total probability of error, the optimisation of the detection threshold parameter is done. The result clearly shows the significant improvement in the detection probability when the optimised threshold parameter is used. Further, an extensive analysis of the optimisation of the cooperative spectrum sensing (CSS) over η-μ/LN distribution with ‘v-out-of-U’ rule at fusion center assuming erroneous feedback channels is also carried out. The closed-form expressions are validated by comparing them with exact results and Monte Carlo simulation.

The stepwise accuracy-improvement strategy based on the Kriging model for structural reliability analysis

For structural reliability analysis with time-consuming performance functions, an innovative design of experiment (DoE) strategy of the Kriging model is proposed, which is named as the stepwise accuracy-improvement strategy. The epistemic randomness of the performance value at any point provided by the Kriging model is used to derive an accuracy measure of the Kriging model. The basic idea of the proposed strategy is to enhance the accuracy of the Kriging model with the best next point that has the largest improvement with regard to the accuracy measure. An optimization problem is developed to define the best next point. The objective function is the expectation that quantifies how much an untried point could enhance the accuracy of the Kriging model. Markov chain Monte Carlo sampling and Gauss–Hermite quadrature are employed to make several approximations to solve the optimization problem and get the best next point. A structural reliability analysis method is also constructed based on the proposed strategy and the accuracy measure employed. Several examples are studied. The results validate the advantages of the proposed DoE strategy.

Shape of LOSVDs in Barred Disks: Implications for Future IFU Surveys

Abstract The shape of line-of-sight velocity distributions (LOSVDs) carries important information about the internal dynamics of galaxies. The skewness of LOSVDs represents their asymmetric deviation from a Gaussian profile. Correlations between the skewness parameter (h 3) and the mean velocity ( ) of a Gauss–Hermite series reflect the underlying stellar orbital configurations of different morphological components. Using two self-consistent N-body simulations of disk galaxies with different bar strengths, we investigate correlations at different inclination angles. Similar to previous studies, we find anticorrelations in the disk area, and positive correlations in the bar area when viewed edge-on. However, at intermediate inclinations, the outer parts of bars exhibit anticorrelations, while the core areas dominated by the boxy/peanut-shaped (B/PS) bulges still maintain weak positive correlations. When viewed edge-on, particles in the foreground/background disk (the wing region) in the bar area constitute the main velocity peak, whereas the particles in the bar contribute to the high-velocity tail, generating the correlation. If we remove the wing particles, the LOSVDs of the particles in the outer part of the bar only exhibit a low-velocity tail, resulting in a negative correlation, whereas the core areas in the central region still show weakly positive correlations. We discuss implications for IFU observations on bars, and show that the variation of the correlation in the disk galaxy may be used as a kinematic indicator of the bar and the B/PS bulge.

Development of a fast reactor multigroup cross section generation code EXUS-F capable of direct processing of evaluated nuclear data files

Abstract The methods and performance of a fast reactor multigroup cross section (XS) generation code EXUS-F are described that is capable of directly processing Evaluated Nuclear Data File format nuclear data files. RECONR of NJOY is used to generate pointwise XS data, and Doppler broadening is incorporated by the Gauss–Hermite quadrature method. The self-shielding effect is incorporated in the ultrafine group XSs in the resolved and unresolved resonance ranges. Functions to generate scattering transfer matrices and fission spectrum matrices are realized. The extended transport approximation is used in zero-dimensional calculations, whereas the collision probability method and the method of characteristics are used for one-dimensional cylindrical geometry and two-dimensional hexagonal geometry problems, respectively. Verification calculations are performed first for various homogeneous mixtures and cylindrical problems. It is confirmed that the spectrum calculations and the corresponding multigroup XS generations are performed adequately in that the reactivity errors are less than 50 pcm with the McCARD Monte Carlo solutions. The nTRACER core calculations are performed with the EXUS-F–generated 47 group XSs for the two-dimensional Advanced Burner Reactor 1000 benchmark problem. The reactivity error of 160 pcm and the root mean square error of the pin powers of 0.7% indicate that EXUF-F generates properly the broad-group XSs.