non-Gaussian Statistical Models Research Articles

Bayesian Estimation of Inverted Beta Mixture Models With Extended Stochastic Variational Inference for Positive Vector Classification.

The finite inverted beta mixture model (IBMM) has been proven to be efficient in modeling positive vectors. Under the traditional variational inference framework, the critical challenge in Bayesian estimation of the IBMM is that the computational cost of performing inference with large datasets is prohibitively expensive, which often limits the use of Bayesian approaches to small datasets. An efficient alternative provided by the recently proposed stochastic variational inference (SVI) framework allows for efficient inference on large datasets. Nevertheless, when using the SVI framework to address the non-Gaussian statistical models, the evidence lower bound (ELBO) cannot be explicitly calculated due to the intractable moment computation. Therefore, the algorithm under the SVI framework cannot directly use stochastic optimization to optimize the ELBO, and an analytically tractable solution cannot be derived. To address this problem, we propose an extended version of the SVI framework with more flexibility, namely, the extended SVI (ESVI) framework. This framework can be used in many non-Gaussian statistical models. First, some approximation strategies are applied to further lower the ELBO to avoid intractable moment calculations. Then, stochastic optimization with noisy natural gradients is used to optimize the lower bound. The excellent performance and effectiveness of the proposed method are verified in real data evaluation.

On a new class of non-Gaussian molecular-based constitutive models with limiting chain extensibility for incompressible rubber-like materials

In constitutive modelling of rubber-like materials, the strain-hardening effect at large deformations has traditionally been captured successfully by non-Gaussian statistical molecular-based models involving the inverse Langevin function, as well as the phenomenological limiting chain extensibility models. A new model proposed by Anssari-Benam and Bucchi (Int. J. Non Linear Mech. 2021; 128; 103626. DOI: 10.1016/j.ijnonlinmec.2020.103626), however, has both a direct molecular structural basis and the functional simplicity of the limiting chain extensibility models. Therefore, this model enjoys the benefits of both approaches: mathematical versatility, structural objectivity of the model parameters, and preserving the physical features of the network deformation such as the singularity point. In this paper we present a systematic approach to constructing the general class of this type of model. It will be shown that the response function of this class of models is defined as the [1/1] rational function of , the first principal invariant of the Cauchy–Green deformation tensor. It will be further demonstrated that the model by Anssari-Benam and Bucchi is a special case within this class as a rounded [3/2] Padé approximant in (the chain stretch) of the inverse Langevin function. A similar approach for devising a general term as an adjunct to the part of the model will also be presented, for applications where the addition of an term to the strain energy function improves the fits or is otherwise required. It is concluded that compared with the Gent model, which is a [0/1] rational approximation in and has no direct connection to Padé approximations of any order in , the presented new class of the molecular-based limiting chain extensibility models in general, and the proposed model by Anssari-Benam and Bucchi in specific, are more accurate representations for modelling the strain-hardening behaviour of rubber-like materials in large deformations.

Insights Into Multiple/Single Lower Bound Approximation for Extended Variational Inference in Non-Gaussian Structured Data Modeling.

For most of the non-Gaussian statistical models, the data being modeled represent strongly structured properties, such as scalar data with bounded support (e.g., beta distribution), vector data with unit length (e.g., Dirichlet distribution), and vector data with positive elements (e.g., generalized inverted Dirichlet distribution). In practical implementations of non-Gaussian statistical models, it is infeasible to find an analytically tractable solution to estimating the posterior distributions of the parameters. Variational inference (VI) is a widely used framework in Bayesian estimation. Recently, an improved framework, namely, the extended VI (EVI), has been introduced and applied successfully to a number of non-Gaussian statistical models. EVI derives analytically tractable solutions by introducing lower bound approximations to the variational objective function. In this paper, we compare two approximation strategies, namely, the multiple lower bounds (MLBs) approximation and the single lower bound (SLB) approximation, which can be applied to carry out the EVI. For implementation, two different conditions, the weak and the strong conditions, are discussed. Convergence of the EVI depends on the selection of the lower bound, regardless of the choice of weak or strong condition. We also discuss the convergence properties to clarify the differences between MLB and SLB. Extensive comparisons are made based on some EVI-based non-Gaussian statistical models. Theoretical analysis is conducted to demonstrate the differences between the weak and strong conditions. Experimental results based on real data show advantages of the SLB approximation over the MLB approximation.

Recent advances in machine learning for non-Gaussian data processing

With the widespread explosion of sensing and computing, an increasing number of industrial applications and an ever-growing amount of academic research generate massive multi-modal data from multiple sources. Gaussian distribution is the probability distribution ubiquitously used in statistics, signal processing, and pattern recognition. However, in reality data are neither always Gaussian nor can be safely assumed to be Gaussian disturbed. In many real-life applications, the distribution of data is, e.g., bounded, asymmetric, and, therefore, is not Gaussian distributed. It has been found in recent studies that explicitly utilizing the non-Gaussian characteristics of data (e.g., data with bounded support, data with semi-bounded support, and data with L1/L2-norm constraint) can significantly improve the performance of practical systems. Hence, it is of particular importance and interest to make thorough studies of the non-Gaussian data and the corresponding non-Gaussian statistical models (e.g., beta distribution for bounded support data, gamma distribution for semi-bounded support data, and Dirichlet/vMF distribution for data with L1/L2-norm constraint). In order to analyze and understand such kind of non-Gaussian distributed data, the developments of related learning theories, statistical models, and efficient algorithms become crucial. The scope of this special issue of the Elsevier's Journal on Neurocomputing is to provide theoretical foundations and ground-breaking models and algorithms to solve this challenge.

Extremes of Nonlinear Vibration: Comparing Models Based on Moments, L-Moments, and Maximum Entropy

Wind and wave loads on offshore structures show nonlinear effects, which require non-Gaussian statistical models. Here we critically review the behavior of various non-Gaussian models. We first survey moment-based models; in particular, the four-moment “Hermite” model, a cubic transformation often used in wind and wave applications. We then derive an “L-Hermite” model, an alternative cubic transformation calibrated by the response “L-moments” rather than its ordinary statistical moments. These L-moments have recently found increasing use, in part because they show less sensitivity to distribution tails than ordinary moments. We find here, however, that these L-moments may not convey sufficient information to accurately estimate extreme response statistics. Finally, we show that four-moment maximum entropy models, also applied in the literature, may be inappropriate to model broader-than-Gaussian cases (e.g., responses to wind and wave loads).

Inflation of elastomeric circular membranes using network constitutive equations

The present paper deals with the use of network-based hyperelastic constitutive equations in the context of thin membranes inflation. The study focus on the inflation of plane circular membranes and the materials are assumed to obey Gaussian and non-Gaussian statistical chains network models. The governing equations of the inflation of axisymmetric thin rubber-like membranes are briefly recalled. The material models are implemented in a numerical tool that incorporates an efficient B-spline interpolation method and a coupled Newton–Raphson/arc-length solving algorithm. Two numerical examples are studied: the homogeneous inflation of spherical balloons and the inflation of initially plane circular membranes. In the second example, the inflation profiles and the distributions of extension ratios along the membrane are extensively analysed during the inflation process. Both examples highlight the need of an accurate modelling of the strain-hardening phenomenon in elastomers.

An Algebraic Formalism for Computing the Moments of Distributions of Quadratic Forms

Distribution moments for quadratic forms are computed. Formulas for moments are derived with the use of the algebra of symmetric polynomials and invariants. The results are applied to parametrization and polynomial approximation of distributions in non-Gaussian statistical models of multialternative classification.

Identification of the Non-Gaussian Chaotic Dynamics of the Radioemission Back Scattering Processes

Regularitys of combinational resonant scattering of signals at the boundary of two optically different media are studied. Conditions are obtained under which a sea-clutter signal should be described by Non-Gaussian statistical models or chaotic dynamics equations. Mathematical models of systems for identifying parameters of mechanical trajectory of a small-size object are suggested. It is shown that quantum filters and small-size object detectors can be constructed.