Generalized Gaussian Function Research Articles

The Analysis of WJ Distribution as an Extended Gaussian Function: Case Study

The double exponential WJ distribution has been shown to competently describe extreme events and critical phenomena, while the Gaussian function has celebrated rich applications in many other fields. Here we present the analysis that the WJ distribution may be properly treated as an extended Gaussian function. Based on the Taylor expansion, we propose three methods to formulate the WJ distribution in the form of Gaussian functions, with Method I and Method III being accurate and self-consistent, and elaborate the relationship among the parameters of the functions. Moreover, we derive the parameter scaling formula of the WJ distribution to express a general Gaussian function, with the work illustrated by a classical case of extreme events and critical phenomena and application to topical medical image processing to prove the effectiveness of the WJ distribution rather than the Gaussian function. Our results sturdily advocate that the WJ distribution can elegantly represent a Gaussian function of arbitrary parameters, whereas the latter usually is not able to satisfactorily describe the former except for specific parameter sets. Thus, it is conclusive that the WJ distribution offers applicability in extreme events and critical phenomena as well as processes describable by the Gaussian function, namely, implying plausibly a unifying approach to the pertinent data processing of those quite distinct areas and establishing a link between relevant extreme value theories and Gaussian processes.

Adaptive Robust Noise Modeling of Sparse Representation for Bearing Fault Diagnosis

The adoption of vibration signals in sparse representation (SR) modeling is very popular in realizing bearing fault diagnosis in recent years. However, it is still challenging since the feature information in the vibration signal is usually submerged by strong and complex noise. Thus, the noise distribution assumption in the SR model needs to be carefully studied, namely, noise modeling. In this article, we propose a new SR model under the assumption that the noise in the signal obeys a mixture of generalized Gaussian (MoGG) distribution. Also, the $L_{p}$ norm ( $0\leq p \leq 1$ ) is adopted as the regularization to keep the proposed method sufficiently sparse and adjustable. Thus, this new model is named a MoGG noise distribution enabled SR (MoGG-SR) model. The mixed distribution characteristic can make this model more adaptive, and the use of the generalized Gaussian function as the basis function makes it more robust to outliers. Then, an algorithm to solve MoGG-SR is developed based on expectation–maximization (EM) and alternating direction method of multipliers (ADMM). After a series of simulations, all parameters in MoGG-SR are determined, and it has a much better denoising performance compared with other classic sparse models under different noise distribution interferences. Finally, the effectiveness of MoGG-SR in bearing fault diagnosis is verified by extracting fault impulses from the vibration signal of two data sets. The simulation and experiment together demonstrate that MoGG-SR is not only innovative in theory but also valuable in practical bearing fault diagnosis.

Estimation of Parameters of Parathyroid Glands Using Particle Swarm Optimization and Multivariate Generalized Gaussian Function Mixture

The paper introduces a fitting method for Single-Photon Emission Computed Tomography (SPECT) images of parathyroid glands using generalized Gaussian function for quantitative assessment of preoperative parathyroid SPECT/CT scintigraphy results in a large patient cohort. Parathyroid glands are very small for SPECT acquisition and the overlapping of 3D distributions was observed. The application of multivariate generalized Gaussian function mixture allows modeling, but results depend on the optimization algorithm. Particle Swarm Optimization (PSO) with global best, ring, and random neighborhood topologies were compared. The obtained results show benefits of random neighborhood topology that gives a smaller error for 3D position and the position estimation was improved by about 3 % voxel size, but the most important is the reduction of processing time to a few minutes, compared to a few hours in relation to the random walk algorithm. Moreover, the frequency of obtaining low MSE values was more than two times higher for this topology. The presented method based on random neighborhood topology allows quantifying activity in a specific voxel in a short time and could be applied it in clinical practice.

Adaptive Generalized Gaussian Distribution Oriented Thresholding Function for Image De-Noising

In this paper, an Adaptive Generalized Gaussian Distribution (AGGD) oriented thresholding function for image de-noising is proposed. This technique utilizes a unique threshold function derived from the generalized Gaussian function obtained from the HH sub-band in the wavelet domain. Two-dimensional discrete wavelet transform is used to generate the decomposition. Having the threshold function formed by using the distribution of the high frequency wavelet HH coefficients makes the function data dependent, hence adaptive to the input image to be de-noised. Thresholding is performed in the high frequency sub-bands of the wavelet transform in the interval [-t, t], where t is calculated in terms of the standard deviation of the coefficients in the HH sub-band. After thresholding, inverse wavelet transform is applied to generate the final de-noised image. Experimental results show the superiority of the proposed technique over other alternative state-of-the-art methods in the literature.

Real-time dynamics of matrix quantum mechanics beyond the classical approximation

We describe a numerical method which allows to go beyond the classical approximation for the real-time dynamics of many-body systems by approximating the many-body Wigner function by the most general Gaussian function with time-dependent mean and dispersion. On a simple example of a classically chaotic system with two degrees of freedom we demonstrate that this Gaussian state approximation is accurate for significantly smaller field strengths and longer times than the classical one. Applying this approximation to matrix quantum mechanics, we demonstrate that the quantum Lyapunov exponents are in general smaller than their classical counterparts, and even seem to vanish below some temperature. This behavior resembles the finite-temperature phase transition which was found for this system in Monte-Carlo simulations, and ensures that the system does not violate the Maldacena-Shenker-Stanford bound λL < 2πT, which inevitably happens for classical dynamics at sufficiently small temperatures.

A Vision-Based Real-Time Mobile Robot Controller Design Based on Gaussian Function for Indoor Environment

In this study, a visual servoing go-to-goal behavior controller is designed to control a differential drive mobile robot for a static target. Inputs for the controller method are based on a weighted graph or a triangle trigonometry kinematic model. The controller is designed with general Gaussian function by adapting the differential drive mobile robot dynamics. State parameters of dynamics are obtained by processing images in real time. It is aimed to develop an efficient internal sensor-independent visual-based control method. The single-head camera takes image frames from indoor environment. A real-time tracking process tracks the robot and target in sequential frames. The distances between graph nodes or the angles between edges are assigned as main control inputs according to utilized kinematic model. The velocity of wheels is computed for both models by using the general Gaussian function. We compare our method with two classical control methods that are PID and fuzzy-PID. Control of mobile robot has been made with high accuracy by using the designed visual-based controller.

The Waveform Model of Laser Altimeter System with Flattened Gaussian Laser

The current waveform model of a laser altimeter is based on a Gaussian laser beam of fundamental mode, while the flattened Gaussian beam has many advantages such as nearly constant energy distribution on the center of the cross-section. Following the theory of the flattened Gaussian beam and the waveform theory of the laser altimeter, some of the primary parameters of the received waveform were derived, and a laser altimetry waveform simulator and waveform processing software were programmed and improved under the circumstance of a flattened Gaussian beam. The result showed that the bias between theoretical and simulated waveforms was less than 3% for every order mode, the waveform width and range error would increase as target slope or order number rose. Under higher order mode, the shapes of the received waveforms were no longer Gaussian, and could be fitted more precisely as a generalized Gaussian function with power bigger than 2. The flattened beam got much better performance for a multi-surface target, especially when the small surface is far from the center of the laser footprint. This article provides the waveform theoretical basis for the use of a flattened Gaussian beam in a laser altimeter.

Sensitivity of breeding birds to the “human footprint” in western Great Lakes forest landscapes

Breeding birds in forest ecosystems are generally diverse, habitat selective, and easily sampled. Because they must integrate environmental variables over space and time, local populations of forest birds (like other animal and plant taxa) may provide meaningful signals of local forest health or degradation. We evaluated 949 breeding bird surveys in areas ranging from degraded urban/suburban forest remnants to relatively pristine old growth forests in the western Laurentian Great Lakes region of North America. The “human footprint” across this landscape was represented by a one‐dimensional numeric gradient derived from land cover variables, forest fragmentation metrics, and publicly available data on housing density and transportation corridors. We used an iterative, maximum likelihood approach to quantify species‐specific responses to this human disturbance gradient. Many species showed significant directional responses, consistent with known life history attributes. Other species were most commonly detected at intermediate levels of anthropogenic disturbance, yielding unimodal responses. Relationships between the “human footprint” and occurrences of 38 bird species were illustrated by general Gaussian functions that represented both unidirectional and unimodal patterns. These biotic response (BR) functions were combined into a bird‐based index of ecological condition (IEC) ranging from 0 (maximally degraded) to 10 (minimally degraded). We described a successful application of the IEC method at the Wild Rivers Legacy Forest (WRLF), a >260 km2 conservation landscape in northeastern Wisconsin, USA, managed primarily under a working forest conservation easement established in 2006. In general, areas within the WRLF yielded high IEC values (7.0–9.0), but nearby forest areas not under the conservation easement were characterized by significantly lower IEC values based on breeding bird assemblages.

Steady-state solutions for subsurface chlorophyll maximum in stratified water columns with a bell-shaped vertical profile of chlorophyll

Abstract. A bell-shaped vertical profile of chlorophyll a (Chl a) concentration, conventionally referred to as a subsurface chlorophyll maximum (SCM) phenomenon, has frequently been observed in stratified oceans and lakes. This profile is assumed to be a general Gaussian distribution in this study. By substituting the general Gaussian function into ecosystem dynamical equations, the steady-state solutions for SCM characteristics (i.e., SCM layer depth, thickness, and intensity) in various scenarios are derived. These solutions indicate that (1) the maximum concentration of Chl a occurs at or below the depth of maximum growth rates of phytoplankton located at the transition from nutrient limitation to light limitation, and the depth of SCM layer deepens logarithmically with an increase in surface light intensity; (2) thickness and intensity of the SCM layer are mainly affected by nutrient supply, but independent of surface light intensity; and (3) intensity of the SCM layer is proportional to the diffusive flux of nutrients from below, which becomes stronger as a result of this layer being shrunk by a higher light attenuation coefficient or a larger sinking velocity of phytoplankton. In addition, the limitation and potential application of the analytical solutions are also presented.

Electroactive self-assembled monolayers: A versatile function to fit symmetric voltammetric peak

We propose a versatile function to fit adsorption voltammetric peak in order to extract the characteristic parameters such as the full width at half maximum, the peak potential, the peak current and the surface coverage. Based on the generalized Gaussian function and supported by generalized lateral interaction model, this method has been tested on different electroactive self-assembled monolayers (i.e. ferrocene, bithiophene, tetrathiafulvalene and radical aminoxyl units). • A generalized Gaussian function fits adsorption voltammetric peak. • Drastic experimental conditions are required to such calculations. • This approach is not only dedicated to SAMs but also to a few electroactive layers.

Complexity of random smooth functions on the high-dimensional sphere

We analyze the landscape of general smooth Gaussian functions on the sphere in dimension $N$, when $N$ is large. We give an explicit formula for the asymptotic complexity of the mean number of critical points of finite and diverging index at any level of energy and for the mean Euler characteristic of level sets. We then find two possible scenarios for the bottom landscape, one that has a layered structure of critical values and a strong correlation between indexes and critical values and another where even at levels below the limiting ground state energy the mean number of local minima is exponentially large. We end the paper by discussing how these results can be interpreted in the language of spin glasses models.

Position-momentum Heisenberg uncertainty in Gaussian enfoldments of Euclidian space

Gaussian enfoldments have been derived from the form of general Gaussian functions centered at arbitrary positions in Euclidian space. Every point of Euclidian space acts in this way as the center of a Gaussian function defined in position space. Using the position-momentum Fourier transform in the quantum mechanical way and applied into the position functions of the Gaussian Euclidian enfoldment, the transform result provides a unique momentum Gaussian function, centered at the origin. In this way, the Euclidian enfoldment disappears in momentum space. Further analysis of the position-momentum relationship indicates that the product of the variances of the enfoldment in position and the corresponding momentum Fourier transform produces some kind of Heisenberg’s uncertainty relation.

A Gaussian holographic theorem and the projection of electronic density functions into the surface of a sphere

A theorem, associated to the holographic nature of general Gaussian functions, is employed to describe holographic projections of electronic density model functions, both in position and momentum spaces, over the surface of a sphere.

Efficient methods for calculating the number of states, levels and lines in atomic configurations

Exact or statistical methods for determining the distribution of the M J values (projection of total angular momentum J) in an electron configuration are presented. This distribution, noted P( M J ), is used to calculate the allowed values of J and the number of electric-dipolar (E1) lines between two configurations. First, the difficulty to account for the Pauli exclusion principle for equivalent electrons is stressed. Showing the limit of the usual exact approach, a very efficient recursive technique is proposed for determining exactly the distribution P( M J ). Second, the statistical approach of Bauche and Bauche-Arnoult [J. Phys. B Atom. Mol. Opt. Phys. 20 (1987) 1659] is extended in order to account for configurations with a high- ℓ spectator. In this case, identical consecutive values may exist in the center of P( M J ), which can neither be modeled by a Gaussian nor by a Gram–Charlier type function. It is shown that the Generalized Gaussian function, with the exponent constrained by the kurtosis (reduced fourth-order centered moment) of P( M J ), is more suited in these situations. A new analytical formula for the evaluation of the number of E1 lines with a larger range of applicability is then proposed.

Secondary-ion-mass spectrometry study on near-stoichiometric LiNbO3 strip waveguide fabricated by vapour transport equilibration and Ti co-diffusion

Secondary-ion-mass spectrometry was applied to study Ti-concentration profiles in the depth direction and on the surface of near-stoichiometric (NS) Ti:LiNbO3 strip waveguides fabricated using vapour transport equilibration (VTE) and co-diffusion of Ti-metal strips. Results show that the profile of Ti concentration along the width direction on the waveguide surface can be well fitted by a sum of two error functions, while, in the depth direction, the Ti concentration follows either a complementary error function or a generalized Gaussian function. Surface Ti concentration, 1/e width and depth, mean diffusivities along the width and depth directions of the guide are, respectively, 1.04 × 1021 cm–3, 8.5 µm, 6.3 µm, 0.18 and 0.1 µm2/h. Two-dimensional refractive index profile in the NS waveguides is indirectly constructed by assuming linearity between Ti-induced index change and Ti concentration. The surface refractive index increments at 1545 nm, Δno and Δne , are evaluated to be 3.132 × 10–3 and 1.186 × 10–2, respectively. All of the above-mentioned diffusion and optical parameters are compared with the corresponding data of the common congruent Ti:LiNbO3 waveguide or NS waveguide fabricated starting from a NS substrate. The rationality of the assumed linear relationship between the index change and the Ti concentration is discussed. The results show that the assumed linearity remains controversial, and all expressions and data with regard to the refractive index are the approximate results and need to be verified by direct measurement on refractive index.

Min-max hyperellipsoidal clustering for anomaly detection in network security

A novel hyperellipsoidal clustering technique is presented for an intrusion-detection system in network security. Hyperellipsoidal clusters toward maximum intracluster similarity and minimum intercluster similarity are generated from training data sets. The novelty of the technique lies in the fact that the parameters needed to construct higher order data models in general multivariate Gaussian functions are incrementally derived from the data sets using accretive processes. The technique is implemented in a feedforward neural network that uses a Gaussian radial basis function as the model generator. An evaluation based on the inclusiveness and exclusiveness of samples with respect to specific criteria is applied to accretively learn the output clusters of the neural network. One significant advantage of this is its ability to detect individual anomaly types that are hard to detect with other anomaly-detection schemes. Applying this technique, several feature subsets of the tcptrace network-connection records that give above 95% detection at false-positive rates below 5% were identified.

A Moving Least Squares weighting function for the Element-free Galerkin Method which almost fulfills essential boundary conditions

The Element-free Galerkin Method has become a very popular tool for the simulation of mechanical problems with moving boundaries. The internally applied Moving Least Squares interpolation uses in general Gaussian or cubic weighting functions and has compact support. Due to the approximative character of this interpolation the obtained shape functions do not fulfill the interpolation conditions, which causes additional numerical effort for the application of the boundary conditions. In this paper a new weighting function is presented, which was designed for meshless shape functions to fulfill these essential conditions with very high accuracy without any additional effort. Furthermore this interpolation gives much more stable results for varying size of the influence radius and for strongly distorted nodal arrangements than existing weighting function types.

Image probability distribution based on generalized gamma function

In this letter, we propose results of distribution tests that indicate that for many natural images, the statistics of the discrete cosine transform (DCT) coefficients are best approximated by a generalized gamma function (G/spl Gamma/F), which includes the conventional Gaussian, Laplacian, and gamma probability density functions. The major parameter of the G/spl Gamma/F is estimated according to the maximum likelihood (ML) principle. Experimental results on a number of /spl chi//sup 2/ tests indicate that the G/spl Gamma/F can be used effectively for modeling the DCT coefficients compared to the conventional Laplacian and generalized Gaussian function (GGF).

The use of Gaussian trend surfaces for modelling glacio-isostatic crustal rebound

Synopsis The behaviour of the Earth's continental crust and mantle may be modelled as a buoyant floating plate on a viscous liquid, and the variations of load imposed by an ice sheet may be modelled as a time-dependent force. In recent work it has been shown that the mathematical solution of this problem can be subdivided into a propagating far-field forebulge term and a non-propagating near-field term. The latter often dominates and can be approximated satisfactorily by a generalized Gaussian function. Here we fit empirical data from the Main Postglacial Shoreline of northern Britain to a Gaussian trend surface. We show that the fit is significantly better than that of a polynomial trend surface previously published, and that the method has the potential to predict the likely sea surface level offshore at the zero isobase for the shoreline.

Linear thermo-viscoelasticity of isotactic polypropylene

Observations are reported on injection-molded isotactic polypropylene in torsion oscillation tests at various temperatures from room temperature to T=150 °C, as well as in torsion relaxation and bending creep tests in the range of temperatures between 30 and 80 °C. Constitutive equations are derived for the viscoelastic response of a semicrystalline polymer at three-dimensional deformations with small strains. A polymer is treated as an equivalent transient network of strands connected by junctions (entanglements, physical cross-links on the surfaces of crystallites and lamellar blocks). The network is thought of as an ensemble of meso-regions with various activation energies for rearrangement of strands. The distribution of meso-domains is described by a generalized Gaussian function. The time-dependent behavior of the ensemble reflects separation of active strands from their junctions and merging of dangling strands with the network. The rearrangement events occur at random times, when the strands are excited by thermal fluctuations. Stress–strain relations involve five material constants that are found by fitting the experimental data. Good agreement in demonstrated between the observations and the results of numerical simulation. The study concentrates on the effect of temperature on the adjustable parameters.