Stochastic Expansions Research Articles

Asymptotic loss of the MLE of a truncation parameter in the presence of a nuisance parameter for a one-sided truncated family of distributions

For a truncated family of distributions with a truncation parameter γ and a parameter θ as a nuisance parameter, we derive the stochastic expansions of bias-adjusted maximum likelihood estimators γ ̂ M L ∗ θ and γ ̂ M L ∗ of γ based on a sample of size n when θ is known and when θ is unknown, respectively. The asymptotic loss of γ ̂ M L ∗ relative to γ ̂ M L ∗ θ is obtained up to the second order, that is the order n −1. The results are a generalization of those for a one-sided truncated exponential family of distributions. Its application to truncated t-distributions is also given.

On the relationship between higher-order stochastic expansions, influence functions and U-statistics for M-estimators

It is shown that higher-order influence functions for M-estimators are mathematically equivalent to higher-order stochastic approximations to these estimators. The stochastic expansions are also shown to have corresponding higher-order U-statistic representations, providing an alternative approach for deriving and analyzing the approximate properties of M-estimators.

Impact of Turbulence Models and Objective Function on Three-Dimensional Robust Aerodynamic Optimization

The objective of this paper was to investigate the impact of two commonly used turbulence models in Reynolds-averaged Navier–Stokes simulations (Spalart–Allmaras and Menter’s shear stress transport models) on the three-dimensional optimum wing design obtained with the gradient-based deterministic and robust aerodynamic shape optimization in transonic, viscous, turbulent flow. In particular, the main contribution of this study to aerodynamic design area is to evaluate the impact of turbulence models and different weight distributions in the multi-objective function (equal, mean biased, and variance biased) on the computational cost, optimal shape, and its performance under Mach-number uncertainty obtained with robust optimization. The results of the study show that the effect of weight distribution in the objective function is more significant than the effect of turbulence model on the final shape obtained with robust design at lower off-design Mach numbers. Robust design tends to mitigate the impact of the turbulence model selection on the optimum shape and performance over the uncertain Mach-number range, whereas the choice of the turbulence model becomes significant at off-design conditions for the optimal shapes obtained with deterministic design. This study also demonstrates the effectiveness of using stochastic expansions in robust aerodynamic shape optimization of three-dimensional wings.

Higher-Order Stochastic Expansions and Approximate Moments for Non-linear Models with Heterogeneous Observations

Higher-Order Stochastic Expansions and Approximate Moments for Non-linear Models with Heterogeneous Observations

Second order expansions of estimators in nonparametric moment conditions models with weakly dependent data

This paper considers estimation of nonparametric moment conditions models with weakly dependent data. The estimator is based on a local linear version of the generalized empirical likelihood approach, and is an alternative to the popular local linear generalized method of moment estimator. The paper derives uniform convergence rates and pointwise asymptotic normality of the resulting local linear generalized empirical likelihood estimator. The paper also develops second order stochastic expansions (under a standard undersmoothing condition) that explain the better finite sample performance of the local linear generalized empirical likelihood estimator compared to that of the efficient local linear generalized method of moments estimator, and can be used to obtain (second order) bias corrected estimators. Monte Carlo simulations and an empirical application illustrate the competitive finite sample properties and the usefulness of the proposed estimators and second order bias corrections.

Analysis and comparison of turbulence model coefficient uncertainty for canonical flow problems

The purpose of this paper is to present results of an uncertainty and sensitivity analysis study of commonly used turbulence models in Reynolds-Averaged Navier–Stokes codes due to the epistemic uncertainty in closure coefficients for a set of turbulence model validation cases that represent the structure of several canonical flow problems. The study focuses on the analysis of a 2D Zero Pressure Gradient Flat Plate, a 2D NASA Wall Mounted Hump, and an Axisymmetric Shock Wave Boundary Layer Interaction, all of which are well documented on the NASA Langley Research Center Turbulence Modeling Resource website. The Spalart–Allmaras (SA), the Wilcox (2006) κ−ω (W2006), and the Menter Shear-Stress Transport (SST) turbulence models are considered in the stochastic analyses of these flow problems, and the FUN3D code was utilized as the flow solver. The uncertainty quantification approach involves stochastic expansions based on non-intrusive polynomial chaos to efficiently propagate the uncertainty. Sensitivity analysis is performed with Sobol indices to rank the relative contribution of each closure coefficient to the total uncertainty for several output flow quantities. The results generalize a set of closure coefficients which have been identified as contributing most to the uncertainty in various output quantities of interest for the set of canonical flow problems considered in this study. Mainly, the SA turbulence model is most sensitive to the uncertainties in the diffusion constant (σ), the log layer calibration constant (κ), and the turbulent destruction constant (cw2). The predictive capability of the W2006 model is most sensitive to the uncertainties in a dissipation rate constant (σw), the shear stress limiter (Clim), and a turbulence-kinetic energy constant (β*). Likewise, the SST turbulence model was found to be most sensitive to the diffusion constants (σw1 and σw2), the log layer calibration constant (κ), and the shear stress limiter (a1).

Marginal posterior distributions for regression parameters in the Cox model using Dirichlet and gamma process priors

A Bayesian treatment of the proportional hazard (PH) model is revisited using Dirichlet process and gamma process priors for the baseline survival and cumulative hazard functions respectively. Such priors, due to their discrete support, conflict with the absolutely continuous nature of survival responses expressed through the hazard function structure that defines the PH model. We resolve this conflict through the use of a proposed ε-grid likelihood approach which we apply to the PH model to consider marginal inference for the regression parameter β using expansions as ε↓0. Using Dirichlet process priors, the ε-grid likelihood approach leads to a new explicit marginal posterior density expansion for β which accommodates the most general setting with arbitrary ties and right censoring. In the previously considered case of gamma process priors, the approach extends the results of Kalbfleisch (1978) to deal with arbitrary arrangements of ties and also provides a rigorous justification for his posterior expressions. As in Kalbfleisch (1978), we show that the leading terms of both expansions approximate Cox’s partial likelihood when there are no ties and the process priors are diffuse. The ε-grid likelihood approach is similar in concept to the grouped data likelihood approach of Sinha et al. (2003) but differs in the likelihood approximation. Whereas our expansions are Poincaré expansions (with relative error O(ε) as ε↓0), those of Sinha et al. (2003) are stochastic expansions and not proper Poincaré expansion (as we show), so that they lack relative error O(ε). Using a flat improper prior for β, the two marginal posterior expressions for β are shown to be integrable for general arrangements of ties and censoring under weak conditions on the design and failures.

Maximum likelihood estimation for a one-sided truncated family of distributions

For a one-sided truncated family of distributions with an interest parameter $$\theta $$ and a truncation parameter $$\gamma $$ as a nuisance parameter, we consider the maximum likelihood estimators (MLEs) $${\hat{\theta }}_\mathrm{ML}^{\gamma} $$ and $${\hat{\theta }}_{\mathrm{ML}}$$ of $$\theta $$ for known $$\gamma $$ and unknown $$\gamma $$ , respectively. In this paper, the stochastic expansions of $${\hat{\theta }}_{\mathrm{ML}}^{\gamma} $$ and $${\hat{\theta }}_\mathrm{ML}$$ are derived, and their second-order asymptotic variances are obtained. The second-order asymptotic loss of a bias-adjusted $${\hat{\theta }}_{\mathrm{ML}^*}$$ relative to $${\hat{\theta }}_{\mathrm{ML}}^{\gamma} $$ is also given. The results are a generalization of those for a one-sided truncated exponential family of distributions. Examples on a one-sided truncated Cauchy distribution, a general truncated exponential family, etc. are also given.

Analysis of Turbulence Model Uncertainty for Shock-Wave/Boundary-Layer Interaction Simulations

The purpose of this paper is to present results of an uncertainty and sensitivity analysis study of commonly used turbulence models in Reynolds-averaged Navier–Stokes codes due to the epistemic uncertainty in closure coefficients for shock-wave/turbulent boundary-layer interaction simulations. The study focuses on the analysis of an axisymmetric shock-wave/boundary-layer interaction (ASWBLI) turbulence model validation case including an axisymmetric cylindrical body with a trailing-edge flare of 20 deg at an upstream Mach number of 7.11. The Spalart–Allmaras, the Wilcox 2006 , and the Menter shear-stress transport turbulence models are considered in the stochastic analyses of the ASWBLI, which used the FUN3D flow solver. The uncertainty quantification approach involves stochastic expansions based on nonintrusive polynomial chaos to efficiently propagate the uncertainty. Sensitivity analysis is performed with Sobol indices to rank the relative contribution of each closure coefficient to the total uncertainty for several output flow quantities. The results of the current study identify a set of closure coefficients that contribute most to the uncertainty in various output quantities of interest, which are also consistent with the similar shock-induced separation studies.

Stochastic Expansions Maintain the Clonal Stability of CD8+ T Cell Populations Undergoing Memory Inflation Driven by Murine Cytomegalovirus

CMV is an obligate and persistent intracellular pathogen that continually drives the production of highly differentiated virus-specific CD8+ T cells in an Ag-dependent manner, a phenomenon known as memory inflation. Extensive proliferation is required to generate and maintain inflationary CD8+ T cell populations, which are counterintuitively short-lived and typically exposed to limited amounts of Ag during the chronic phase of infection. An apparent discrepancy therefore exists between the magnitude of expansion and the requirement for ongoing immunogenic stimulation. To address this issue, we explored the clonal dynamics of memory inflation. First, we tracked congenically marked OT-I cell populations in recipient mice infected with murine CMV (MCMV) expressing the cognate Ag OVA. Irrespective of numerical dominance, stochastic expansions were observed in each population, such that dominant and subdominant OT-I cells were maintained at stable frequencies over time. Second, we characterized endogenous CD8+ T cell populations specific for two classic inflationary epitopes, M38 and IE3. Multiple clonotypes simultaneously underwent Ag-driven proliferation during latent infection with MCMV. In addition, the corresponding CD8+ T cell repertoires were stable over time and dominated by persistent clonotypes, many of which also occurred in more than one mouse. Collectively, these data suggest that stochastic encounters with Ag occur frequently enough to maintain oligoclonal populations of inflationary CD8+ T cells, despite intrinsic constraints on epitope display at individual sites of infection with MCMV.

Uncertainty and sensitivity analysis of inflow parameters for HyShot II scramjet numerical simulaiton

The flowfield inside a combustor of scramjet such as Hyshot II system is highly complicated and many factors have effect on its numerical prediction. Uncertainty and sensitivity analysis of aerodynamics prediction on Hyshot II intake configuration is performed in this work. The inflow conditions of intake configuration are of vital crucial since they directly determine the inflow conditions of combustor configuration, which in turn have a remarkable influence on the combustion and mixing process. Meanwhile, the varied flight conditions may also introduce great and direct impact on the performance in actual flight tests. It is significant to evaluate the uncertainties and sensitivities and to identify the critical factors of inflow conditions for the intake configuration. Five input parameters are selected as the epistemic uncertain variables. The flowfield prediction is conducted via using Reynolds-averaged Navier-Stokes simulations as the numerical study approach with standard Menter's shear stress transport turbulence model and species transport process for Hyshot II configuration. Stochastic expansions based on point collocation non-intrusive polynomial chaos, which require 42 deterministic simulations in this study, are employed to propagate and estimate the uncertainties. Meanwhile, Sobol indices are adopted to rank and indicate the individual contribution and to identify the critical inflow parameters. The results demonstrate that freestream Mach number and temperature are the dominated contributing parameters among the performance parameters. In addition, as observed from the wall-normal profile of related variables extracted at 355 mm in x coordinate direction in the simulation results, the freestream pressure plays a dominant role in determining the pressure distribution and the freestream temperature is the most prominent contributor to the distribution of temperature and velocity.

Second order concentration via logarithmic Sobolev inequalities

We show sharpened forms of the concentration of measure phenomenon centered at first order stochastic expansions. The bound are based on second order difference operators and second order derivatives. Applications to functions on the discrete cube and stochastic Hoeffding type expansions in mathematical statistics are studied as well as linear eigenvalue statistics in random matrix theory.

Quantification of parametric uncertainty in k–ω–γ transition model for hypersonic flow heat transfer

The uncertain empirical values of closure coefficients in k–ω–γ transition model flaws the predictive capability. The current work aims to examine the uncertainty of k–ω–γ transition model due to uncertain closure coefficients for hypersonic flow over a flat plate and a straight cone, with introducing the stochastic expansions based on the point-collocation non-intrusive polynomial chaos method into computational fluid dynamics simulations for efficient and accurate uncertainty propagation. Specifically, five model coefficients are regarded as epistemic uncertain probabilistic variables on intervals. The output quantities of interests include the surface heat flux, skin friction coefficient, transition onsets and transition lengths. To better identify the key uncertainty contributor, Sobol indices are utilized as a variable-based global sensitivity analysis approach. Results show that the performance of the k–ω–γ transition model is highly susceptible to its closure coefficients evidenced by that small changes (10%) of model coefficients could yield more than 30% uncertainty in the transition onsets and roughly 17% uncertainty in the transition lengths. The largest uncertainty source contributing to the uncertainty in heat flux, Cf and transition onsets is C3 for hypersonic transitional flows. Besides, C4 dominates the uncertainty in transition lengths. In the future model improvement, attention should be paid to improve apprehension of rational values or constitutions of C2, C3 and C4.

Higher order concentration of measure

We study sharpened forms of the concentration of measure phenomenon typically centered at stochastic expansions of order [Formula: see text] for any [Formula: see text]. The bounds are based on [Formula: see text]th order derivatives or difference operators. In particular, we consider deviations of functions of independent random variables and differentiable functions over probability measures satisfying a logarithmic Sobolev inequality, and functions on the unit sphere. Applications include concentration inequalities for [Formula: see text]-statistics as well as for classes of symmetric functions via polynomial approximations on the sphere (Edgeworth-type expansions).

Uncertainty and sensitivity analysis of flow parameters for transition models on hypersonic flows

The impossibility to reproduce the experimental conditions, which is difficult to perfectly control and inevitably fluctuate, raises the uncertainties in numerical simulation. The objective of this work is to quantify the uncertainty and sensitivity of hypersonic transition modeling due to uncertainty in the flow parameters. Two representative transition models, e.g. γ-Reθ and k-ω-γ, are considered. And five flow parameters including freesteam Mach number, pressure, temperature, turbulence intensity and wall temperature are selected as aleatory uncertain variables uniformly distributing among the bound. Then the stochastic expansions based on point-collocation non-intrusive polynomial chaos (NIPC) method is utilized to represent and propagate the uncertainties in the output quantities of interests (QoIs) for hypersonic flow over the flat plate and the straight cone. Benefiting from the efficiency of the NIPC method, accurate uncertainty results are obtained with only 42 evaluations for each problem. Furthermore, Sobol indices are utilized to provide a relative ranking of each input uncertainty to the overall uncertainty in the output. The uncertainty results quantitatively verify that the transition zone is great sensitive to changes in flow parameters. Besides, for both the heat flux and skin friction coefficient, the importance ranking of the flow parameters and their trends are roughly resemble between the γ-Reθ and k-ω-γ model. This signifies that their sensitivity with respect to the input parameters is mainly associated with flow features rather than transition models. However, the contrary is the case for uncertainty in transition onset and length whose sensitivity to uncertainties of input parameters depends strongly on transition models. The present investigation shows that the freestream pressure is the greatest contributor to uncertainty in most cases. The freestream turbulence intensity, whose effect can be nearly negligible in heat flux and Cf, plays a predominant role on transition length.

A Doubly Corrected Robust Variance Estimator for Linear GMM

We propose a new finite sample corrected variance estimator for the linear generalized method of moments (GMM) including the one-step, two-step, and iterated estimators. Our formula additionally corrects for the over-identification bias in variance estimation on top of the commonly used finite sample correction of Windmeijer (2005) which corrects for the bias from estimating the efficient weight matrix, so is doubly corrected. Formal stochastic expansions are derived to show the proposed double correction estimates the variance of some higher-order terms in the expansion. In addition, the proposed double correction provides robustness to misspecification of the moment condition. In contrast, the conventional variance estimator and the Windmeijer correction are inconsistent under misspecification. That is, the proposed double correction formula provides a convenient way to obtain improved inference under correct specification and robustness against misspecification at the same time.

Grid Influence on Turbulence Model Coefficient Uncertainties in Transonic Wall-Bounded Flows

The goal of this work is to determine the effect of grid topology and flow solver discretization on the uncertainty quantification and sensitivity analysis of commonly used turbulence models in Reynolds-averaged Navier–Stokes codes due to uncertainty in the values of turbulence model coefficients as well as to rank the contribution of each coefficient to uncertainty in various output quantities of interest for transonic flow over an RAE 2822 airfoil. Three turbulence models were considered: Spalart–Allmaras, 2006 Wilcox , and Menter shear-stress transport. Several structured and unstructured grid topologies were employed in the analysis. The Fun3D code of NASA Langley Research Center and the BCFD code of the Boeing Company were used as the flow solvers. The uncertainty analysis employed stochastic expansions based on nonintrusive polynomial chaos for efficient uncertainty propagation. Sobol indices were used to rank the relative contributions of each coefficient to the total uncertainty in the output quantities of interest. The solutions obtained with all grids and computational-fluid-dynamics codes identified a common set of coefficients for each turbulence model that influence the output uncertainty significantly for transonic wall-bounded flows.

Robustness of stochastic expansions for the stability of uncertain nonlinear dynamical systems – Application to brake squeal

Abstract This paper is devoted to the prediction and analysis of brake squeal under random uncertainty. This problem, which is a particular application of a wider issue, namely the stability of random parameter-dependent (RPD) nonlinear dynamical systems, is undertaken by using the non-intrusive generalized polynomial chaos (GPC) and Wiener-Haar expansions. The main objective is to assess the capacities of these meta-models within this framework. A reduced nonlinear non-equally damped, iso-damped and non-damped, disc/pad models are considered in this perspective in order to analyze the robustness of the proposed meta-models with respect to perfect and non-perfect mode coalescence. It turns out that the Wiener-Haar meta-model shows a more robust performance than GPC expansion and consequently offers a more reliable tool for the nonlinear stability analysis and thus for the prediction of brake squeal under parameter uncertainty.

The ABC of simulation estimation with auxiliary statistics

The frequentist method of simulated minimum distance (SMD) is widely used in economics to estimate complex models with an intractable likelihood. In other disciplines, a Bayesian approach known as Approximate Bayesian Computation (ABC) is far more popular. This paper connects these two seemingly related approaches to likelihood-free estimation by means of a Reverse Sampler that uses both optimization and importance weighting to target the posterior distribution. Its hybrid features enable us to analyze an ABC estimate from the perspective of SMD. We show that an ideal ABC estimate can be obtained as a weighted average of a sequence of SMD modes, each being the minimizer of the deviations between the data and the model. This contrasts with the SMD, which is the mode of the average deviations. Using stochastic expansions, we provide a general characterization of frequentist estimators and those based on Bayesian computations including Laplace-type estimators. Their differences are illustrated using analytical examples and a simulation study of the dynamic panel model.

Estimation of the error distribution in a varying coefficient regression model

This paper deals with the estimation of the error distribution function in a varying coefficient regression model. We propose two estimators and study their asymptotic properties by obtaining uniform stochastic expansions. The first estimator is a residual-based empirical distribution function. We study this estimator when the varying coefficients are estimated by under-smoothed local quadratic smoothers. Our second estimator which exploits the fact that the error distribution has mean zero is a weighted residual-based empirical distribution whose weights are chosen to achieve the mean zero property using empirical likelihood methods. The second estimator improves on the first estimator. Bootstrap confidence bands based on the two estimators are also discussed.