We study a system of semilinear hyperbolic equations passively advected by smooth white noise in time random velocity fields. Such a system arises in modelling non-premixed isothermal turbulent flames under single-step kinetics of fuel and oxidizer. We derive closed equations for one-point and multi-point probability distribution functions (PDFs) and closed-form analytical formulae for the one-point PDF function, as well as the two-point PDF function under homogeneity and isotropy. Exact solution formulae allow us to analyse the ensemble-averaged fuel/oxidizer concentrations and the motion of their level curves. We recover the empirical formulae of combustion in the thin reaction zone limit and show that these approximate formulae can either underestimate or overestimate average concentrations when the reaction zone is not tending to zero. We show that the averaged reaction rate slows down locally in

Variable selection in high-dimensional space characterizes many contemporary problems in scientific discovery and decision making. Many frequently-used techniques are based on independence screening; examples include correlation ranking (Fan and Lv, 2008) or feature selection using a two-sample t-test in high-dimensional classification (Tibshirani et al., 2003). Within the context of the linear model, Fan and Lv (2008) showed that this simple correlation ranking possesses a sure independence screening property under certain conditions and that its revision, called iteratively sure independent screening (ISIS), is needed when the features are marginally unrelated but jointly related to the response variable. In this paper, we extend ISIS, without explicit definition of residuals, to a general pseudo-likelihood framework, which includes generalized linear models as a special case. Even in the least-squares setting, the new method improves ISIS by allowing variable deletion in the iterative process. Our technique allows us to select important features in high-dimensional classification where the popularly used two-sample t-method fails. A new technique is introduced to reduce the false discovery rate in the feature screening stage. Several simulated and two real data examples are presented to illustrate the methodology.

We propose several modifications to the grid based particle method (GBPM) [21] for moving interface modeling. There are several nice features of the proposed algorithm. The new method can significantly improve the distribution of sampling particles on the evolving interface. Unlike the original GBPM where footpoints (sampling points) tend to cluster to each other, the sampling points in the new method tend to be better separated on the interface. Moreover, by replacing the grid-based discretization using the cell-based discretization, we naturally decompose the interface into segments so that we can easily approximate surface integrals. As a possible alternative to the local polynomial least square approximation, we also study a geometric basis for local reconstruction in the resampling step. We will show that such modification can simplify the overall implementations. Numerical examples in two- and three

We propose a semiparametric latent Gaussian copula model for modelling mixed multivariate data, which contain a combination of both continuous and binary variables. The model assumes that the observed binary variables are obtained by dichotomizing latent variables that satisfy the Gaussian copula distribution. The goal is to infer the conditional independence relationship between the latent random variables, based on the observed mixed data. Our work has two main contributions: we propose a unified rankbased approach to estimate the correlation matrix of latent variables; we establish the concentration inequality of the proposed rankbased estimator. Consequently, our methods achieve the same rates of convergence for precision matrix estimation and graph recovery, as if the latent variables were observed. The methods proposed are numerically assessed through extensive simulation studies, and real

Quadratic regression functionals are important for bandwidth selection of nonparametric regression techniques and for nonparametric goodness-of-fit test. Based on local polynomial regression, we propose estimators for weighted integrals of squared derivatives of regression functions. The rates of convergence in mean square error are calculated under various degrees of smoothness and appropriate values of the smoothing parameter. Asymptotic distributions of the proposed quadratic estimators are considered with the Gaussian noise assumption. It is shown that when the estimators are pseudo-quadratic (linear components dominate quadratic components), asymptotic normality with rate n-1/2 can be achieved.