Independent Standard Normal Variables Research Articles

Non-integrability and chaos for natural Hamiltonian systems with a random potential

Consider the ensemble of Gaussian random potentials {VL(q)}L=1∞ on the d-dimensional torus where, essentially, VL(q) is a real-valued trigonometric polynomial of degree L whose coefficients are independent standard normal variables. Our main result ensures that, with a probability tending to 1 as L→∞, the dynamical system associated with the natural Hamiltonian function defined by this random potential, HL:=12|p|2+VL(q), exhibits a number of chaotic regions which coexist with a positive-volume set of invariant tori. In particular, these systems are typically neither integrable with non-degenerate first integrals nor ergodic. An analogous result for random natural Hamiltonian systems defined on the cotangent bundle of an arbitrary compact Riemannian manifold is presented too.

An orthogonal normal transformation of correlated non-normal random variables for structural reliability

In this paper, an efficient and explicit technique is proposed for transforming correlated non-normal random variables into independent standard normal variables based on the three-parameter (3P) lognormal distribution. In contrast with the classic Nataf transformation, the derived equivalent correlation coefficient in non-orthogonal standard normal space of the proposed transformation is expressed as an explicit formula, thereby avoiding tedious iteration algorithm or multifarious empirical formulas. Meanwhile, the applicable range of the original correlation coefficient is determined based on fundamental properties of the proposed expression of correlation distortion and the definition of correlation coefficient. The proposed transformation requires only the first three moments (i.e., mean, standard deviation, and skewness) of basic random variables, as well as their correlation matrix. Therefore, the proposed transformation can also be applied even when the joint distribution or marginal distributions of the basic random variables are unknown. Several numerical examples are presented to demonstrate the user-friendliness, efficiency, and accuracy of the proposed transformation applied in structural reliability analysis involving correlated non-normal random variables.

Hermite polynomial normal transformation for structural reliability analysis

PurposeNormal transformation is often required in structural reliability analysis to convert the non-normal random variables into independent standard normal variables. The existing normal transformation techniques, for example, Rosenblatt transformation and Nataf transformation, usually require the joint probability density function (PDF) and/or marginal PDFs of non-normal random variables. In practical problems, however, the joint PDF and marginal PDFs are often unknown due to the lack of data while the statistical information is much easier to be expressed in terms of statistical moments and correlation coefficients. This study aims to address this issue, by presenting an alternative normal transformation method that does not require PDFs of the input random variables.Design/methodology/approachThe new approach, namely, the Hermite polynomial normal transformation, expresses the normal transformation function in terms of Hermite polynomials and it works with both uncorrelated and correlated random variables. Its application in structural reliability analysis using different methods is thoroughly investigated via a number of carefully designed comparison studies.FindingsComprehensive comparisons are conducted to examine the performance of the proposed Hermite polynomial normal transformation scheme. The results show that the presented approach has comparable accuracy to previous methods and can be obtained in closed-form. Moreover, the new scheme only requires the first four statistical moments and/or the correlation coefficients between random variables, which greatly widen the applicability of normal transformations in practical problems.Originality/valueThis study interprets the classical polynomial normal transformation method in terms of Hermite polynomials, namely, Hermite polynomial normal transformation, to convert uncorrelated/correlated random variables into standard normal random variables. The new scheme only requires the first four statistical moments to operate, making it particularly suitable for problems that are constraint by limited data. Besides, the extension to correlated cases can easily be achieved with the introducing of the Hermite polynomials. Compared to existing methods, the new scheme is cheap to compute and delivers comparable accuracy.

Time‐varying reliability method based on linearized Nataf transform

AbstractIn actual engineering, material properties, load effects, and other factors of mechanical structures change due to long‐term use. In order to understand the operation of a mechanical structure in real time, it is crucial to obtain the dynamic trajectory of its reliability. Considering the time variability of a mechanical structure over time, uncertain random variables are introduced to express the uncertainty of various parameters of structures, and the Wiener process is used to describe the strength degradation process of structures so as to solve the calculation problem of time‐varying reliability of mechanical structures. Based on the advanced first‐order and second moment method (AFOSM), the proposed linearized Nataf change is used to complete the transformation from related nonnormal variables to independent standard normal variables in order to simplify the calculation process of reliability solution and solve the reliability calculation problem of random parameters subjected to arbitrary distribution. The deduced random variable sensitivity factor indicates the degree of influence of different random variables on the reliability of the mechanical structure, providing a theoretical basis for the optimal design and maintenance of a mechanical structure. The proposed method is analyzed using the cantilever beam and compared with the nonlinear Nataf transform and verified by the Monte Carlo simulation results. The results show that the proposed method can effectively solve the reliability sensitivity problem of structural system strength degradation.

Reliability analysis of strip footing under rainfall using KL-FORM

Spatial variability is an inherent uncertainty of soil properties. Current reliability analyses generally incorporate random field theory and Monte Carlo simulation (MCS) when dealing with spatial variability, in which the computational efficiency is a significant challenge. This paper proposes a KL-FORM algorithm to improve the computational efficiency. In the proposed KL-FORM, Karhunen-Loeve (KL) expansion is used for discretizing random fields, and first-order reliability method (FORM) is employed for reliability analysis. The KL expansion and FORM can be used in conjunction, through adopting independent standard normal variables in the discretization of KL expansion as the basic variables in the FORM. To illustrate the effectiveness of this KL-FORM, it is applied to a case study of a strip footing in spatially variable unsaturated soil under rainfall, in which the bearing capacity of the footing is computed by numerical simulation. This case study shows that the KL-FORM is accurate and efficient. The parametric analyses suggest that ignoring the spatial variability of the soil may lead to an underestimation of the reliability index of the footing.

Approximation to multivariate normal integral and its application in time-dependent reliability analysis

It is common to evaluate high-dimensional normal probabilities in many uncertainty-related applications such as system and time-dependent reliability analysis. An accurate method is proposed to evaluate high-dimensional normal probabilities, especially when they reside in tail areas. The normal probability is at first converted into the cumulative distribution function of the extreme value of the involved normal variables. Then the series expansion method is employed to approximate the extreme value with respect to a smaller number of mutually independent standard normal variables. The moment generating function of the extreme value is obtained using the Gauss-Hermite quadrature method. The saddlepoint approximation method is finally used to estimate the cumulative distribution function of the extreme value, thereby the desired normal probability. The proposed method is then applied to time-dependent reliability analysis where a large number of dependent normal variables are involved with the use of the First Order Reliability Method. Examples show that the proposed method is generally more accurate and robust than the widely used randomized quasi Monte Carlo method and equivalent component method.

Three-Parametric Marcenko–Pastur Density

The complex Wishart ensemble is the statistical ensemble of $M \times N$ complex random matrices with $M \geq N$ such that the real and imaginary parts of each element are given by independent standard normal variables. The Marcenko--Pastur (MP) density $\rho(x; r), x \geq 0$ describes the distribution for squares of the singular values of the random matrices in this ensemble in the scaling limit $N \to \infty$, $M \to \infty$ with a fixed rectangularity $r=N/M \in (0, 1]$. The dynamical extension of the squared-singular-value distribution is realized by the noncolliding squared Bessel process, and its hydrodynamic limit provides the two-parametric MP density $\rho(x; r, t)$ with time $t \geq 0$, whose initial distribution is $\delta(x)$. Recently, Blaizot, Nowak, and Warchol studied the time-dependent complex Wishart ensemble with an external source and introduced the three-parametric MP density $\rho(x; r, t, a)$ by analyzing the hydrodynamic limit of the process starting from $\delta(x-a), a > 0$. In the present paper, we give useful expressions for $\rho(x; r, t, a)$ and perform a systematic study of dynamic critical phenomena observed at the critical time $t_{\rm c}(a)=a$ when $r=1$. The universal behavior in the long-term limit $t \to \infty$ is also reported. It is expected that the present system having the three-parametric MP density provides a mean-field model for QCD showing spontaneous chiral symmetry breaking.

A computational framework of kinematic accuracy reliability analysis for industrial robots

A new computational method to evaluate comprehensively the positional accuracy reliability for single coordinate, single point, multipoint and trajectory accuracy of industrial robots is proposed using the sparse grid numerical integration method and the saddlepoint approximation method. A kinematic error model of end-effector is constructed in three coordinate directions using the sparse grid numerical integration method considering uncertain parameters. The first-four order moments and the covariance matrix for three coordinates of the end-effector are calculated by extended Gauss–Hermite integration nodes and corresponding weights. The eigen-decomposition is conducted to transform the interdependent coordinates into independent standard normal variables. An equivalent extreme value distribution of response is applied to assess the reliability of kinematic accuracy. The probability density function and probability of failure for extreme value distribution are then derived through the saddlepoint approximation method. Four examples are given to demonstrate the effectiveness of the proposed method.

Kinematic Reliability Analysis of Robotic Manipulator

AbstractKinematic reliability of robotic manipulators is the linchpin for restraining the positional errors within acceptable limits. This work develops an efficient reliability analysis method to account for random dimensions and joint angles of robotic mechanisms. It aims to proficiently predict the kinematic reliability of robotic manipulators. The kinematic reliability is defined by the probability that the actual position of an end-effector falls into a specified tolerance sphere, which is centered at the target position. The motion error is indicated by a compound function of independent standard normal variables constructed by three co-dependent coordinates of the end-effector. The saddle point approximation is then applied to compute the kinematic reliability. Exemplification demonstrates satisfactory accuracy and efficiency of the proposed method due to the construction and the saddle point since random simulation is spared.

Dimension-free bounds for largest singular values of matrix Gaussian series

The matrix Gaussian series refers to a sum of fixed matrices weighted by independent standard normal variables and plays an important in various fields related to probability theory. In this paper, we present the dimension-free tail bounds and expectation bounds for the largest singular value (LSV) of matrix Gaussian series, respectively. By using the resulting bounds, we compute the expectation bounds for LSVs of Gaussian Wigner matrix and Gaussian Toeplitz matrix, respectively.

Investigation of equivalent correlation coefficient based on the Mehler’s formula

PurposeConversion of the correlated random variables into independent variables, especially into independent standard normal variables, is the common technology for estimating the statistical moments of response and evaluating reliability of random system, in which calculating the equivalent correlation coefficient is an important component. The purpose of this paper is to investigate an accurate, efficient and easy to implement estimation method for the equivalent correlation coefficient of various incomplete probability systems.Design/methodology/approachFirst, an approach based on the Mehler’s formula for evaluating the equivalent correlation coefficient is introduced, then, by combining with polynomial normal transformations, this approach is improved to be valid for various incomplete probability systems, which is named as the direct method. Next, with the convenient linear reference variables for eight frequently used random variables and the approximation of the Rosenblatt transformation introduced, a further improved implementation without iteration process is developed, which is named as the simplified method. Finally, several examples are investigated to verify the characteristics of the proposed methods.FindingsThe results of the examples in this paper show that both the proposed two methods are of high accuracy, by comparison, the proposed simplified method is more effective and convenient.Originality/valueBased on the Mehler’s formula, two practical implementations for evaluating the equivalent correlation coefficient are proposed, which are accurate, efficient, easy to implement and valid for various incomplete probability systems.

Developing an Instantaneous Response Surface Method t-IRS for Time-Dependent Reliability Analysis

In practical engineering, many uncertain factors in loading or degradation of material properties may vary with time. Stochastic process modeling constitutes a suitable approach for describing these time-dependent uncertainties. By adopting this approach, however, the time-dependent reliability calculation is a great challenge owing to the complexity and the huge computational burden. This paper presents a new instantaneous response surface method t-IRS for time-dependent reliability analysis. Different from the adaptive extreme response surface approach, the proposed method does not need to build and update surrogate models separately at each time node. It first uses the expansion optimal linear estimation method to discretize the stochastic processes into a set of independent standard normal variables together with some deterministic functions of time. Time is then treated as an independent one-dimensional variable. Next, initial samples are generated by Latin hypercube sampling, and the corresponding response values are calculated and utilized to construct an instantaneous response surrogate model of the Kriging type. The active learning method is applied to update the Kriging surrogate model until satisfactory accuracy is achieved. Finally, the instantaneous response surrogate model is used to compute the time-dependent reliability via Monte Carlo simulation. Four case studies are utilized to demonstrate the effectiveness of the t-IRS method for time-dependent reliability analysis.

Adaptive estimation for statistical moments of response based on the exact dimension reduction method in terms of vector

Obtaining the statistical moments of system responses remains one of the main topics of stochastic analysis. This paper presents a new adaptive point estimate method (PEM) based on the exact dimension reduction method in terms of vector. Firstly, the rigorous and exact dimension reduction method in terms of vector is derived theoretically. Secondly, by introducing the nonnormal-to-normal transformation, the original variables are transformed into independent standard normal variables which are classified into several sub-vectors based on the delineation of the cross terms, and then the response moments can be described by an explicit function of the moments of multiple component functions. Thirdly, by combining with the two different approaches for estimating the moments of component sub-vector function, an adaptive PEM based on the exact dimension reduction method in terms of vector, which comprises two sub-methods, is proposed. Finally, several examples illustrate the accuracy and efficiency of the proposed method.

An Improved Stochastic Response Surface Method Based Probabilistic Load Flow for Studies on Correlated Wind Speeds in the AC/DC Grid

This paper proposed a probabilistic load flow technique of AC/VSC-MTDC (Alternate Current/Voltage Source Control-Multiple Terminal Direct Current) hybrid grids based on an improved stochastic response surface method. The applied traditional stochastic response surface method is inherent with the capability to tackle correlated normal variables; however, the accuracy is poor in the case of correlated diverse distributions. To address this issue, NATAF transformation was adopted to transform the correlated wind speeds and loads following arbitrary distributions into the variables that are subject to standard normal distributions. The collection points could be selected to establish the polynomial relationship among the independent standard normal variables and the output responses. Then, the probability distributions and statistics of the responses could be accurately and efficiently estimated. The modified IEEE 14-bus system, involving an additional VSC-MTDC system, wind speeds following various distributions, and diverse consumer behaviors, was used to demonstrate the validity and capability of the proposed method.

Reliability Analysis of Nonlinear Vibratory Systems Under Non-Gaussian Loads

A general methodology is presented for time-dependent reliability and random vibrations of nonlinear vibratory systems with random parameters excited by non-Gaussian loads. The approach is based on polynomial chaos expansion (PCE), Karhunen–Loeve (KL) expansion, and quasi Monte Carlo (QMC). The latter is used to estimate multidimensional integrals efficiently. The input random processes are first characterized using their first four moments (mean, standard deviation, skewness, and kurtosis coefficients) and a correlation structure in order to generate sample realizations (trajectories). Characterization means the development of a stochastic metamodel. The input random variables and processes are expressed in terms of independent standard normal variables in N dimensions. The N-dimensional input space is space filled with M points. The system differential equations of motion (EOM) are time integrated for each of the M points, and QMC estimates the four moments and correlation structure of the output efficiently. The proposed PCE–KL–QMC approach is then used to characterize the output process. Finally, classical MC simulation estimates the time-dependent probability of failure using the developed stochastic metamodel of the output process. The proposed methodology is demonstrated with a Duffing oscillator example under non-Gaussian load.

Adaptive estimation of statistical moments of the responses of random systems

Estimation of statistical moments of structural response is one of the main topics for the analysis of random systems; balancing between accuracy and efficiency is still a challenge. In this work, a new point estimate method (PEM) based on adaptive dimensional decomposition is presented. Firstly, by introducing the normal-to-nonnormal transformation, a system with general variables is changed into one with independent standard normal variables. Secondly, the criterion for delineating the existence of cross terms is derived theoretically, together with its practical implementation. Thirdly, by combining with the dimension-reduction method (DRM), an adaptive dimensional decomposition for moment function is proposed and the PEM based on adaptive dimensional decomposition, which comprises two types of sub-methods, is developed. Finally, several examples are investigated to verify the accuracy, efficiency, convergence and rationale of the proposed method.

On the concentration of measure and the [formula omitted]-norm

We seek a function in an N-dimensional subspace of L1 that attains a half (more generally, a given part) of its L1-norm on a set of least possible measure. We prove that such a measure is asymptotically maximal when the space is spanned by independent standard normal variables. This answers a question of Y. Benyamini, A. Kroó, A. Pinkus.

Improved knowledge-based clustered partitioning approach and its application to slope reliability analysis

A knowledge-based clustered partitioning (KCP) approach is improved to determine the reliability index and probability of failure of a rock slope. The Nataf transformation is adopted to transform the correlated non-normal random variables involved in the KCP approach into independent standard normal variables. An improved KCP technique is proposed to search the design point and calculate the reliability index. Two illustrative examples are presented to demonstrate the capability and validity of the proposed approach. The results indicate that the improved KCP-based reliability method can be applied to evaluate the reliability of rock slopes involving multiple correlated non-normal variables accurately and efficiently. Its accuracy is shown to be higher than that of the traditional KCP using the bisection method, and it is much more efficient than Monte Carlo simulation. The improved KCP-based reliability method is especially suitable for dealing with an implicit performance function with a large number of random variables, which is often involved in slope reliability analysis.

Probabilistic analysis of underground rock excavations using response surface method and SORM

Probabilistic analysis of underground rock excavations is performed using response surface method and SORM, in which the quadratic polynomial with cross terms is used to approximate the implicit limit state surface at the design point. The response surface is found using an iterative algorithm and the probability of failure is evaluated using the first-order and the second-order reliability method (FORM/SORM). Independent standard normal variables in U-space are chosen as basic random variables and transformed into correlated non-normal variables in the original space of random variables for constructing the response surface. The proposed method is first illustrated for a circular tunnel with analytical solutions considering Mohr–Coulomb (M–C) and Hoek–Brown (H–B) yield criteria separately. The failure probability with respect to the plastic zone criterion and the tunnel convergence criterion are estimated from FORM/SORM and compared to those obtained from Monte Carlo Simulations. The results show that the support pressure has great influence on the failure probability of the two failure modes. For the M–C model, the hypothesis of uncorrelated friction angle and cohesion will generate higher non-performance probability in comparison to the case of negatively correlated shear strength parameters. Reliability analyses involving non-normal distributions are also investigated. Finally, an example of a horseshoe-shaped highway tunnel is presented to illustrate the feasibility and validity of the proposed method for practical applications where numerical procedures are needed to calculate the performance function values.

Stochastic response surface method for reliability analysis of rock slopes involving correlated non-normal variables

This paper proposes a stochastic response surface method for reliability analysis involving correlated non-normal random variables, in which the Nataf transformation is adopted to effectively transform the correlated non-normal variables into independent standard normal variables. Transformations of random variables that are often used in reliability analyses in terms of standard normal variables are summarized. The closed-form expressions for fourth to sixth order Hermite polynomial chaos expansions involving any number of random variables are formulated. The proposed method will substantially extend the application of stochastic response surface method for reliability problems. An example of reliability analysis of rock slope stability with plane failure is presented to demonstrate the validity and capability of the proposed stochastic response surface method. The results indicate that the proposed stochastic response surface method can evaluate the reliability of rock slope stability involving correlated non-normal variables accurately and efficiently. Its accuracy is shown to be higher than that for the first-order reliability method, and it is much more efficient than direct Monte-Carlo simulation. The results also show that the number of collocation points selected should ensure that the Hermite polynomial matrix has a full rank so that different order SRSMs can produce a robust estimation of probability of failure for a specified performance function. Generally, the accuracy of SRSM increases as the order of SRSM increases.