Gaussian Random Variables Research Articles

High-Order Spectral Method of Density Estimation for Stochastic Differential Equation Driven by Multivariate Gaussian Random Variables

There are some previous works on designing efficient and high-order numerical methods of density estimation for stochastic partial differential equation (SPDE) driven by multivariate Gaussian random variables. They mostly focus on proposing numerical methods of density estimation for SPDE with independent random variables and rarely research density estimation for SPDE is driven by multivariate Gaussian random variables. In this paper, we propose a high-order algorithm of gPC-based density estimation where SPDE driven by multivariate Gaussian random variables. Our main techniques are (1) we build a new multivariate orthogonal basis by adopting the Gauss–Schmidt orthogonalization; (2) with the newly constructed orthogonal basis in hand, we first assume the unknown function in the SPDE has the stochastic general polynomial chaos (gPC) expansion, second implement the stochastic gPC expansion for the SPDE in the multivariate Gaussian measure space, and third we obtain and numerical calculation deterministic differential equations for the coefficients of the expansion; (3) we used high-order algorithm of gPC-based for density estimation and moment estimation. We apply the newly proposed numerical method to a known random function, stochastic 1D wave equation, and stochastic 2D Schnakenberg model, respectively. All the presented stochastic equations are driven by bivariate Gaussian random variables. The efficiency is compared with the Monte-Carlo method based on the known random function.

Supremum Distribution of Weighted Sum of Random Processes from Orlicz Spaces of Exponential Type with Continuous Deviation

The paper studies distribution of sum of random processes from Orlicz spaces of exponential type weighted by continuous functions, in particular, processes from spaces Subϕ (Ω), SSubϕ (Ω) and class V (ϕ, ψ) are considered. Such spaces and classes of random variables and corresponding stochastic processes provide generalizations of Gaussian and sub-Gaussian random variables and processes and are important for various applications, for example, in queuing theory and financial mathematics. We derive the estimates for the distribution of supremum of weighted sum of such processes deviated by a continuous monotone function using the entropy method. As examples, weighted sum of Wiener and weighted sum of fractional Brownian motion processes with different Hurst indices from classes V (ϕ, ψ) are considered. Corresponding estimates of the probability of exceeding by trajectories of such weighted sums a positive level determined by a linear function are obtained. In the insurance risk theory, such aproblem arises during estimating a ruin probability of the corresponding risk process with a constant premium income, and in the communications theory, it appears for the buffer overflow probability for a single server with a constant service rate.

Bayesian semiparametric joint model of multivariate longitudinal and survival data with dependent censoring.

We consider a novel class of semiparametric joint models for multivariate longitudinal and survival data with dependent censoring. In these models, unknown-fashion cumulative baseline hazard functions are fitted by a novel class of penalized-splines (P-splines) with linear constraints. The dependence between the failure time of interest and censoring time is accommodated by a normal transformation model, where both nonparametric marginal survival function and censoring function are transformed to standard normal random variables with bivariate normal joint distribution. Based on a hybrid algorithm together with the Metropolis-Hastings algorithm within the Gibbs sampler, we propose a feasible Bayesian method to simultaneously estimate unknown parameters of interest, and to fit baseline survival and censoring functions. Intensive simulation studies are conducted to assess the performance of the proposed method. The use of the proposed method is also illustrated in the analysis of a data set from the International Breast Cancer Study Group.

RIN Reduction in Gain-Switched InAs-InP(113)B Quantum Dot Laser Based on Multi-Population Rate Equations

For the first time, relative intensity noise (RIN) of an InAs-InP(113)B quantum dot (Q-Dot) laser is examined theoretically under the gain-switching condition with the application of a Gaussian pulse beam (GPB) to excited state (Exs) of the laser. The multi-population rate equations considering nonlinear gain are solved by the Runge–Kutta method. Noises are added to rate equations as Langevin noise sources and a different method is applied here for the addition of these noise sources to equations. In this method, new rate equations are defined to eliminate the cross-correlations between the noise sources to make them independent and simulate with independent white Gaussian random variables. Obtained results showed that RIN decreases with the increasing ac peak current and the increasing inhomogeneous broadening. It was also observed that further RIN reduction of about 30 dB/Hz and gain-switched short pulses with a high power due to Exs emission being obtained with the application of GPB to Exs of the laser. Moreover, it was demonstrated that while carrier noise generated from Exs and ground state (Grs) affect the RIN spectrum, there is no effect of wetting layer (Wly) carrier noise.

Kalman Bayesian Neural Networks for Closed-Form Online Learning

Compared to point estimates calculated by standard neural networks, Bayesian neural networks (BNN) provide probability distributions over the output predictions and model parameters, i.e., the weights. Training the weight distribution of a BNN, however, is more involved due to the intractability of the underlying Bayesian inference problem and thus, requires efficient approximations. In this paper, we propose a novel approach for BNN learning via closed-form Bayesian inference. For this purpose, the calculation of the predictive distribution of the output and the update of the weight distribution are treated as Bayesian filtering and smoothing problems, where the weights are modeled as Gaussian random variables. This allows closed-form expressions for training the network's parameters in a sequential/online fashion without gradient descent. We demonstrate our method on several UCI datasets and compare it to the state of the art.

On the moments of the variance-gamma distribution

We obtain new closed-form formulas for the moments and absolute moments of the variance-gamma distribution. We thus deduce new formulas for the moments and absolute moments of the product of two correlated zero mean normal random variables.

Nonsymmetric examples for Gaussian correlation inequalities

In this paper, we compare two variances of maxima of N standard Gaussian random variables. One is a sequence of N i.i.d. standard Gaussians, and the other one is N standard Gaussians with covariances σ1,2=ρ∈(0,1) and σi,j=0, for other i≠j. It turns out that we need to discuss the covariance of two functions with respect to multivariate Gaussian distributions. Gaussian correlation inequalities hold for many symmetric (with respect to the origin) cases. However, in our case, the max function and its derivatives are not symmetric about the origin. We have two main results in this paper. First, we prove a specific case for a convex/log-concave correlation inequality for the standard multivariate Gaussian distribution. The other result is that the variance of maxima of standard Gaussians with σ1,2=ρ∈(0,1), while σi,j=0, for other i≠j, is larger than the variance of maxima of independent standard Gaussians. This implies that the variance of maxima of N i.i.d. standard Gaussians is decreasing in N.

A Generalization of Pisier Homogeneous Banach Algebra

In 1979, Pisier proved remarkably that a sequence of independent and identically distributed standard Gaussian random variables determines, via random Fourier series, a homogeneous Banach algebra P strictly contained in C(T), the class of continuous functions on the unit circle T and strictly containing the classical Wiener algebra A(T), that is, A(T)⫋P⫋C(T). This improved some previous results obtained by Zafran in solving a long-standing problem raised by Katznelson. In this paper, we extend Pisier’s result by showing that any probability measure on the unit circle defines a homogeneous Banach algebra contained in C(T). Thus Pisier algebra is not an isolated object but rather an element in a large class of Pisier-type algebras. We consider the case of spectral measures of stationary sequences of Gaussian random variables and obtain a sufficient condition for the boundedness of the random Fourier series ∑ n∈Zfˆ(n)ξnexp(2πint) in the general setting of dependent random variables (ξn).

Validity of Juang's Method to Predict the Liquefaction Potential of the Soils of Sablette Algiers

This work presents a deterministic and probabilistic analysis of the liquefaction potential of soils. Soil liquefaction is a process leading to the total loss of shear strength of loose sands under seismic stresses by increasing pore pressure. It is accompanied by deformations. Certain factors related to the characteristics of the ground and the seismic excitation can significantly influence the appearance or not of the phenomenon. Empirical methods of liquefaction potential assessment are widely used. The probabilistic method has been widely used for estimating the probability of liquefaction. Initially, the uncertain parameters are modeled by normal random variables. A parametric study of the coefficients of variation of the random variables showed that the number of strokes (N1)60 of the SPT test is the parameter that has the greatest influence on the probability of liquefaction. This study focuses on the liquefaction potential by methods based on the results of the in-situ "Standard Penetration Test" Through the results obtained in this study of the site of Sablettes in Algiers, in general, there is a risk of liquefaction of the soil in the sandy layers, and we can say that the site studied has an average liquefaction potential. According to the classification of liquefaction potential proposed by Juang et al., (2012).

Sharp lower error bounds for strong approximation of SDEs with discontinuous drift coefficient by coupling of noise

In the past decade, an intensive study of strong approximation of stochastic differential equations (SDEs) with a drift coefficient that has discontinuities in space has begun. In the majority of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and that the diffusion coefficient is globally Lipschitz continuous and nondegenerate at the discontinuities of the drift coefficient. Under this type of assumptions the best Lp-error rate obtained so far for approximation of scalar SDEs at the final time is 3/4 in terms of the number of evaluations of the driving Brownian motion. In the present article, we prove for the first time in the literature sharp lower error bounds for such SDEs. We show that for a huge class of additive noise driven SDEs of this type the Lp-error rate 3/4 can not be improved. For the proof of this result we employ a novel technique by studying equations with coupled noise: we reduce the analysis of the Lp-error of an arbitrary approximation based on evaluation of the driving Brownian motion at finitely many times to the analysis of the Lp-distance of two solutions of the same equation that are driven by Brownian motions that are coupled at the given time-points and independent, conditioned on their values at these points. To obtain lower bounds for the latter quantity, we prove a new quantitative version of positive association for bivariate normal random variables (Y,Z) by providing explicit lower bounds for the covariance Cov(f(Y),g(Z)) in case of piecewise Lipschitz continuous functions f and g. In addition it turns out that our proof technique also leads to lower error bounds for estimating occupation time functionals ∫01f(Wt)dt of a Brownian motion W, which substantially extends known results for the case of f being an indicator function.

Distribution Transformer Loading: Probabilistic Modeling and Diversity Factor

For planning purposes, the loading of assets in electric power distribution circuits is often modelled in terms of a diversified load. The temporal and probabilistic nature of the load is a sum of several individual loads of various levels and characteristics. The diversity factor of such circuits is the ratio of the sum of the individual non-simultaneous load asset ratings energized by that circuit to the total peak simultaneous demand. The diversity factor is temporal and probabilistic in nature. In this paper, the probability density of the load factor is described and modelled for power distribution loads. The model used is the ratio of two dependent bivariate normal random variables. The resulting probability density is termed a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ratio density</i> and a recent algebraic formulation of the ratio density is used. The probabilistic model is compared to results obtained by others. The potential impact of electric vehicle charging loads is applied to the sizing of distribution power supply assets using diversity loading strategies for Level 1 and single phase Level 2 EV chargers. The probabilistic method is offered as an added tool to the conventional, fixed diversity factor method in distribution engineering.

Numerical convergence and error analysis for the truncated iterative generalized stochastic perturbation-based finite element method

The main aim of this work is to present a numerical analysis of convergence and accuracy of the selected generalized perturbation-based schemes in linear and nonlinear problems of solid mechanics. An algorithm for determining the basic probabilistic characteristics has been developed using the iterative generalized stochastic higher-order perturbation method adjacent to symmetrically truncated Gaussian random variables. It has been confirmed that usage of a sufficiently high order of the Truncated Iterative Stochastic Perturbation Technique (TISPT) allows for achieving any desired accuracy in determining up to the fourth-order probabilistic characteristics of static structural response. The semi-analytical probabilistic approach is the reference solution in this study, which is based upon the same composite response functions determined with the use of the Least Squares Method created using specific series of FEM experiments. The entire methodology has been provided for the given extreme value of the coefficient of variation αmax of the input uncertainty source. On the other hand, a selection procedure of the stochastic perturbation method order to achieve 1% numerical accuracy in all up to the fourth-order probabilistic moments has been proposed. Computational experiments include simply supported elastic Euler–Bernoulli beam, a set of steel diagrid structures, nonlinear tension of steel round bar as well as homogenization procedure of some particulate composite.

Characterization of defects in composite materials by the intrusive stochastic method using the Gaussian type random variable

Characterization of defects in composite materials by the intrusive stochastic method using the Gaussian type random variable

Stochastic Convergence of Sobol-Based Mahalanobis Shell Sampling Collision Probability Computation

Sample-based computation of the joint-time probability of collision motivates developing the Mahalanobis Shell Sampling (MSS) algorithm, which samples nondegenerate normal random variables, enabling rare event simulation without unduly penalizing sample size. The MSS method has unbiased estimators in sample mean and covariance, and it may achieve arbitrary precision when approximating probability measures. For Clohessy–Wiltshire relative orbital dynamics, computational MSS exponential rates of error convergence (in the mean-square-error sense) are shown to improve by one order of magnitude (for sample mean and covariance) over Monte Carlo; when reproducing the instantaneous probability of collision, MSS has a comparable mean-square-error convergence rate performance to Monte Carlo.

Reprint of: Formulation and estimation of stochastic frontier production function models

Previous studies of the so-called frontier production function have not utilized an adequate characterization of the disturbance term for such a model. In this paper we provide an appropriate specification, by defining the disturbance term as the sum of symmetric normal and (negative) half-normal random variables. Various aspects of maximum-likelihood estimation for the coefficients of a production function with an additive disturbance term of this sort are then considered.

Entire Gaussian Functions: Probability of Zeros Absence

In this paper, we consider a random entire function of the form f(z,ω)=∑n=0+∞εn(ω1)×ξn(ω2)fnzn, where (εn) is a sequence of independent Steinhaus random variables, (ξn) is the a sequence of independent standard complex Gaussian random variables, and a sequence of numbers fn∈C is such that lim¯n→+∞|fn|n=0 and #{n:fn≠0}=+∞. We investigate asymptotic estimates of the probability P0(r)=P{ω:f(z,ω) has no zeros inside rD} as r→+∞ outside of some set E of finite logarithmic measure, i.e., ∫E∩[1,+∞)dlnr&lt;+∞. The obtained asymptotic estimates for the probability of the absence of zeros for entire Gaussian functions are in a certain sense the best possible result. Furthermore, we give an answer to an open question of A. Nishry for such random functions.

On the distribution-tail behaviour of the product of normal random variables

In this paper we consider the product Pi _{n}=prod_{k=1}^{n}xi _{k} of n independent normally distributed zero mean random variables xi _{1},dots ,xi _{n}. We derive an asymptotic formula for the survival probability {mathbb{P}}(Pi _{n}>x), as xto infty , with the first remaining term.

Near-optimal approximation methods for elliptic PDEs with lognormal coefficients

This paper studies numerical methods for the approximation of elliptic PDEs with lognormal coefficients of the form − div ⁡ ( a ∇ u ) = f -\operatorname {div}(a\nabla u)=f where a = exp ⁡ ( b ) a=\exp (b) and b b is a Gaussian random field. The approximant of the solution u u is an n n -term polynomial expansion in the scalar Gaussian random variables that parametrize b b . We present a general convergence analysis of weighted least-squares approximants for smooth and arbitrarily rough random field, using a suitable random design, for which we prove optimality in the following sense: their convergence rate matches exactly or closely the rate that has been established in Bachmayr, Cohen, DeVore, and Migliorati [ESAIM Math. Model. Numer. Anal. 51 (2017), pp. 341–363] for best n n -term approximation by Hermite polynomials, under the same minimial assumptions on the Gaussian random field. This is in contrast with the current state of the art results for the stochastic Galerkin method that suffers the lack of coercivity due to the lognormal nature of the diffusion field. Numerical tests with b b as the Brownian bridge confirm our theoretical findings.

Solving a multi-objective chance constrained hierarchical optimization problem under intuitionistic fuzzy environment with its application

Various optimization approaches have been developed and used for generating optimal solutions for different industry related optimization problems. The semantic representation of imprecise coefficients and various types of uncertainties arising in real life optimization problems are still a challenging task and require attention of academicians as well as professionals. Decision makers (DMs) often face problems in presenting the linguistically expressed constraints. The impreciseness of information also plays a key role in modelling and thus in decision making. The impreciseness of coefficients is usually handled by using various elements of fuzzy set theory. The uncertainty associated with constraints in optimization problem is handled by using stochastic programming approaches. These coefficients are usually revealed by a logical DM, who takes into account the scope of hesitation along with the imprecise nature of data. Such coefficients can be very well represented by using Intuitionistic fuzzy numbers. In this work, an approach for solving multi-objective bi-level chance constrained optimization problem in an intuitionistic fuzzy environment has been presented. The technological coefficients and the coefficients of objective functions are represented by Triangular Intuitionistic fuzzy numbers whereas normal random variables are used to represent the coefficient of resources. The concept of component wise optimization is used for the reduction of intuitionistic fuzzy objective into their equivalent deterministic form. The concept of alpha-beta cut, chance constrained programming and interval programming are then used to convert the problem into a deterministic multi-objective bi-level linear programming problem. The resultant deterministic optimization problem is then solved by using TOPSIS method. The production planning problem of a firm has been modelled and solved by using the proposed approach.

SPDE bridges with observation noise and their spatial approximation

This paper introduces SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. The SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. A general framework for the spatial discretization of these bridge processes is introduced. Explicit convergence rates are derived for a spectral and a finite element based method. It is shown that for sufficiently rough observation noise, the rates are essentially the same as those of the corresponding discretization of the original SPDE.