Second-moment Properties Research Articles

Stochastic isogeometric analysis on arbitrary multipatch domains by spline dimensional decomposition

This paper presents a new stochastic method by integrating spline dimensional decomposition (SDD) of a high-dimensional random function and isogeometric analysis (IGA) on arbitrary multipatch geometries to solve stochastic boundary-value problems from linear elasticity. The method, referred to as SDD-mIGA, involves (1) analysis-suitable T-splines with significant approximating power for geometrical modeling, random field discretization, and stress analysis; (2) Bézier extraction operator for isogeometric mesh refinement; and (3) a novel Fourier-like expansion of a high-dimensional output function in terms of measure-consistent orthonormalized splines. The proposed method can handle arbitrary multipatch domains in IGA and uses standard least-squares regression to efficiently estimate the SDD expansion coefficients for uncertainty quantification applications. Analytical formulae have been derived to calculate the second-moment properties of an SDD-mIGA approximation for a general output random variable of interest. Numerical results, including those obtained for a 54-dimensional, industrial-scale problem, demonstrate that a low-order SDD-mIGA is capable of efficiently delivering accurate probabilistic solutions when compared with the benchmark results from crude Monte Carlo simulation.

A Spline Dimensional Decomposition for Uncertainty Quantification in High Dimensions

This study debuts a new spline dimensional decomposition (SDD) for uncertainty quantification analysis of high-dimensional functions, including those endowed with high nonlinearity and nonsmoothness, if they exist, in a proficient manner. The decomposition creates a hierarchical expansion for an output random variable of interest with respect to measure-consistent orthonormalized basis splines (B-splines) in independent input random variables. A dimensionwise decomposition of a spline space into orthogonal subspaces, each spanned by a reduced set of such orthonormal splines, results in SDD. Exploiting the modulus of smoothness, the SDD approximation is shown to converge in mean-square to the correct limit. The computational complexity of the SDD method is polynomial, as opposed to exponential, thus alleviating the curse of dimensionality to the extent possible. Analytical formulae are proposed to calculate the second-moment properties of a truncated SDD approximation for a general output random variable in terms of the expansion coefficients involved. Numerical results indicate that a low-order SDD approximation of nonsmooth functions calculates the probabilistic characteristics of an output variable with an accuracy matching or surpassing those obtained by high-order approximations from several existing methods. Finally, a 34-dimensional random eigenvalue analysis demonstrates the utility of SDD in solving practical problems.

Practical uncertainty quantification analysis involving statistically dependent random variables

This article presents a practical refinement of generalized polynomial chaos expansion for uncertainty quantification under dependent input random variables. Unlike the Rodrigues-type formula, which exists for select probability measures, a three-step computational algorithm is put forward to generate a sequence of any approximate measure-consistent multivariate orthonormal polynomials. For uncertainty quantification analysis under dependent random variables, two regression methods, comprising existing standard least-squares and newly developed partitioned diffeomorphic modulation under observable response preserving homotopy (D-MORPH), are proposed to estimate the coefficients of generalized polynomial chaos expansion for the very first time. In contrast to the existing regression devoted so far to the classical polynomial chaos expansion, no tensor-product structure is required or enforced. The partitioned D-MORPH regression is applicable to either an underdetermined or overdetermined system, thus substantially enhancing the ability of the original D-MORPH regression. Numerical results obtained for Gaussian and non-Gaussian probability measures with rectangular or non-rectangular domains point to highly accurate orthonormal polynomials produced by the three-step algorithm. More significantly, the generalized polynomial chaos approximations of mathematical functions and stochastic responses from solid-mechanics problems, in tandem with the partitioned D-MORPH regression, provide excellent estimates of the second-moment properties and reliability from only hundreds of function evaluations or finite element analyses.

Stochastic isogeometric analysis in linear elasticity

A new stochastic method, integrating spline dimensional decomposition (SDD) and isogeometric analysis (IGA), is proposed for solving stochastic boundary-value problems from linear elasticity. The method, referred to as SDD–IGA, involves Galerkin isogeometric analysis as a deterministic solver for governing partial differential equations and a novel Fourier-like orthogonal spline expansion generated from the analysis-of-variance decomposition of a high-dimensional function. For the stochastic part of the SDD–IGA method, an innovative dimension-reduction integration technique is presented for efficiently calculating the expansion coefficients. Analytical formulae have been derived to calculate the second-moment properties of an SDD–IGA approximation for a general output random variable of interest. Numerical examples demonstrate the capability of a low-order SDD–IGA in efficiently delivering probabilistic solutions with an approximation quality as good as, if not better than, that obtained from a high-order polynomial dimensional decomposition. The proposed SDD–IGA method is most suitable in the presence of locally nonlinear or nonsmooth behavior commonly found in applications.

Uncertainty quantification under dependent random variables by a generalized polynomial dimensional decomposition

This paper is concerned with uncertainty quantification analysis of complex systems subject to dependent input random variables. The analysis focuses on a new, generalized version of polynomial dimensional decomposition (PDD), referred to as GPDD, entailing hierarchically ordered measure-consistent multivariate orthogonal polynomials in dependent variables. Under a few prescribed assumptions, GPDD exists for any square-integrable output random variable and converges in mean-square to the correct limit. New analytical formulae are proposed to calculate the mean and variance of a GPDD approximation of a general output variable in terms of the expansion coefficients and second-moment properties of multivariate orthogonal polynomials. However, unlike in PDD, calculating the coefficients of GPDD requires solving a coupled system of linear equations. Besides, the variance formula of GPDD contains extra terms due to statistical dependence among input variables. The extra terms disappear when the input variables are statistically independent, reverting GPDD to PDD. Two numerical examples, the one derived from a stochastic boundary-value problem and the other entailing a random eigenvalue problem, illustrate second-moment error analysis and estimation of the probabilistic characteristics of eigensolutions.

Wiener–Hermite polynomial expansion for multivariate Gaussian probability measures

This paper introduces a new, transformation-free, generalized polynomial chaos expansion (PCE) comprising multivariate Hermite orthogonal polynomials in dependent Gaussian random variables. The second-moment properties of Hermite polynomials reveal a weakly orthogonal system when obtained for a general Gaussian probability measure. Still, the exponential integrability of norm allows the Hermite polynomials to constitute a complete set and hence a basis in a Hilbert space. The completeness is vitally important for the convergence of the generalized PCE to the correct limit. The optimality of the generalized PCE and the approximation quality due to truncation are discussed. New analytical formulae are proposed to calculate the mean and variance of a generalized PCE approximation of a general output variable in terms of the expansion coefficients and statistical properties of Hermite polynomials. However, unlike in the classical PCE, calculating the coefficients of the generalized PCE requires solving a coupled system of linear equations. Besides, the variance formula of the generalized PCE contains additional terms due to statistical dependence among Gaussian variables. The additional terms vanish when the Gaussian variables are statistically independent, reverting the generalized PCE to the classical PCE. Numerical examples illustrate the generalized PCE approximation in estimating the statistical properties of various output variables.

Reliability of Linear Systems Subjected to Wind Loads

AbstractProbabilistic models for wind loads are developed and used to estimate the properties of the responses of linear systems. This analysis involves four steps. First, along-wind, across-wind forces, and torque are represented as polynomials of turbulence fluctuations and wake excitations, which are assumed to be stationary Gaussian processes. Second, two types of models are provided, namely, the empirical model and the mathematical model, for the second-moment properties of the turbulence fluctuations and wake excitations so that the probability law of the wind loads is characterized completely. Proposed models are then calibrated to the experimental observations. Third, the mathematical model provides an efficient method to estimate the response properties relative to Monte Carlo simulation. The responses are modeled by translation processes that match the target second-moment properties and marginal distributions of the responses. Fourth, the response properties that are of interest, e.g., the mean...

The cyclical properties of disaggregated capital flows

We analyze the second-moment properties of the components of international capital flows and their relationship to business cycle variables (output, investment, and the real interest rate) in 22 industrial and emerging countries. Total inward flows are procyclical with respect to all three macro variables. Net outward flows are countercyclical with respect to output and investment in most industrial and emerging countries. Disaggregated inward flows positively comove with output in industrial countries and with investment and the real interest rate in the G7 economies. Inward foreign direct investment is the only non-procyclical type of inward capital flows (with respect to output) in the developing economies. Formal statistical tests based on nonparametric bootstrap techniques detect significant variance increases in all G7 countries' disaggregated capital flows over exogenous and endogenously estimated breaks.

Changes in the second-moment properties of disaggregated capital flows

Using formal statistical tests, we detect (i) significant volatility increases for various types of capital flows for a period of changes in business cycle comovement among the G7 countries, and (ii) mixed evidence of changes in covariances and correlations with a set of macroeconomic variables.

Changes in the Second-Moment Properties of Disaggregated Capital Flows

Using formal statistical tests, we detect (i) significant volatility increases for various types of capital flows for a period of changes in business cycle comovement among the G7 countries, and (ii) mixed evidence of changes in covariances and correlations with a set of macroeconomic variables.

Haar-Based Multiresolution Stochastic Processes

Modifying a Haar wavelet representation of Brownian motion yields a class of Haar-based multiresolution stochastic processes in the form of an infinite series $$X_t = \sum_{n=0}^\infty\lambda_n\varDelta _n(t)\epsilon_n,$$ where λnΔn(t) is the integral of the nth Haar wavelet from 0 to t, and en are i.i.d. random variables with mean 0 and variance 1. Two sufficient conditions are provided for Xt to converge uniformly with probability one. Each stochastic process Open image in new window, the collection of all almost sure uniform limits, retains the second-moment properties and the same roughness of sample paths as Brownian motion, yet lacks some of the features of Brownian motion, e.g., does not have independent and/or stationary increments, is not Gaussian, is not self-similar, or is not a martingale. Two important tools are developed to analyze elements of Open image in new window, the nth-level self-similarity of the associated bridges and the tree structure of dyadic increments. These tools are essential in establishing sample path results such as Holder continuity and fractional dimensions of graphs of the processes.

The Cyclical Properties of Disaggregated Capital Flows

We analyze the second-moment properties of the components of international capital flows and their relationship to business cycle variables (output, investment, and real interest rate) in 22 industrial and emerging countries. Total inward flows are procyclical with respect to all three macro variables. Net outward flows are countercyclical with respect to output and investment in most industrial and emerging countries. Disaggregated inward flows positively comove with output in industrial countries and with investment and the real interest rate in the G7 economies. Inward foreign direct investment is the only non-procyclical type of inward capital flows (with respect to output) in the developing economies. Formal statistical tests based on nonparametric bootstrap techniques detect significant variance increases in all G7 countries' disaggregated capital flows over exogenous and endogenously estimated breaks.

Statistical Moments of Polynomial Dimensional Decomposition

This technical note presents explicit formulas for calculating the response moments of stochastic systems by polynomial dimensional decomposition entailing independent random input with arbitrary probability measures. The numerical results indicate that the formulas provide accurate, convergent, and computationally efficient estimates of the second-moment properties.

Offshoring and Volatility: Evidence from Mexico's Maquiladora Industry

This paper studies the second-moment properties of offshoring, the arrangement whereby firms carry out particular stages of production abroad. It documents a new empirical regularity: maquiladora industries in Mexico that are associated with US offshoring experience fluctuations in employment that are twice as volatile as the corresponding industries in the United States. This finding is not attributable simply to higher volatility in the overall Mexican economy, nor to the smaller size of Mexico's industries compared to US counterparts. (JEL F14, F23, L24, L25, L60, O14)

Income effects and indeterminacy in a calibrated one-sector growth model

This note analyzes how the indeterminacy of competitive equilibrium in one-sector growth models depends on the magnitude of the households' income effect on the demand for leisure. Since I am interested in quantitatively characterizing regions of indeterminacy, I use the Jaimovich and Rebelo [N. Jaimovich, S. Rebelo, Can news about the future drive the business cycle? Mimeo, Northwestern University, 2007] preferences that span a wide range of income effect values. I find that indeterminacy can occur for levels of aggregate-returns-to-scale that are well within recent empirical estimates. For these regions of indeterminacy, the model, when driven solely by sunspot shocks, generates second-moment properties that are consistent with the U.S. data at the business cycle frequency.

Endogenous Entry, Product Variety, and Business Cycles

This paper builds a framework for the analysis of macroeconomic fluctuations that incorporates the endogenous determination of the number of producers and products over the business cycle. Economic expansions induce higher entry rates by prospective entrants subject to irreversible investment costs. The sluggish response of the number of producers (due to sunk entry costs and a time-to-build lag) generates a new and potentially important endogenous propagation mechanism for real business cycle models. The return to investment (corresponding to the creation of new productive units) determines household saving decisions, producer entry, and the allocation of labor across sectors. The model performs at least as well as the benchmark real business cycle model with respect to the implied second-moment properties of key macroeconomic aggregates. In addition, our framework jointly predicts procyclical product variety and procyclical profits even for preference specifications that imply countercyclical markups. When we include physical capital, the model can simultaneously reproduce most of the variance of GDP, hours worked, and total investment found in the data.

Parametric translation models for stationary non-Gaussian processes and fields

Parametric representations, that is, deterministic functions with time and/or space argument depending on finite families of random variables, are defined and used to describe approximately stationary non-Gaussian processes and fields specified partially by their marginal distribution and second-moment properties. The proposed parametric representations are memoryless transformations of sequences of parametric stationary Gaussian processes and fields, and are referred to as parametric translation models. Conditions are established for the convergence of statistics of sequences of parametric translation models to target statistics. Two numerical examples are presented to illustrate some properties of parametric translation models and demonstrate their use in random vibration.

Spectral Representation for a Class of Non-Gaussian Processes

A new model is developed for non-Gaussian processes. The model is based on the spectral representation theorem for weakly stationary processes, can match the second moment properties and several higher order moments of any non-Gaussian process, and consists of a superposition of harmonics with uncorrelated but dependent random amplitudes. The calibration of the model to a target non-Gaussian process may require iterations. The Ito⁁ formula is used to calculate higher-order moments of the proposed model. These moments can be used to tune the model such that it matches some of the higher-order moments of a target non-Gaussian process in addition to its second-moment properties. The proposed model is useful for both Monte Carlo simulation and analytical studies on the response of linear and nonlinear systems to non-Gaussian noise. Examples are presented to illustrate the use of the proposed model.

Investigation of spatial clustering from individually matched case-control studies.

The paper demonstrates how existing theory to assess spatial clustering based on second-moment properties of a labelled point process can be adapted to matched case-control studies. The null hypothesis that cases are a random sample from the superposition of cases and controls is replaced by the hypothesis that each case is a random sample from the set consisting of itself and its k matched controls. We compare the proposed test with other tests of spatial clustering, and describe an application to data on childhood diabetes in Yorkshire, England.

Identification of rain cells from radar and stochastic modelling of space-time rainfall

We discuss two problems related to the utilization of weather radar: the automated identification of rain cells and the extension of cluster models to include the random variability of space-time rainfall within and between cells. The need for such extension emerges from visual inspection and formal analysis of radar reflectivity images. The algorithm proposed for the identification of cells is based on statistical techniques for the estimation of probability density mixtures. The algorithm does not assign pixels to cells deterministically; rather, it calculates the probability with which each pixel belongs to the different cells. Through an iterative procedure, the cell parameters and pixel probabilities are updated until the final identification of cells is reached. The second part of the paper deals with a generalization of cluster rainfall models in space and time. The models studied here combine an arbitrary birth point process with arbitrary random fields generated by the cells. Second-moment properties of these processes are derived.