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Nonlinear dynamical systems are encountered in many areas of social science, natural science, and engineering, and are of particular interest for complex biological processes like the spiking activity of neural ensembles in the brain. To describe such spiking activity, we adapt the Volterra series expansion of an analytic function to account for the point-process nature of multiple inputs and a single output (MISO) in a neural ensemble. Our model describes the transformed spiking probability for the output as the sum of kernel-weighted integrals of the inputs. The kernel functions need to be identified and estimated, and both local sparsity (kernel functions may be zero on part of their support) and global sparsity (some kernel functions may be identically zero) are of interest. The kernel functions are approximated by B-splines and a penalized likelihood-based approach is proposed for estimation. Even for moderately complex brain functionality, the identification and estimation of this sparse fun...

Variational approximations provide fast, deterministic alternatives to Markov Chain Monte Carlo for Bayesian inference on the parameters of complex, hierarchical models. Variational approximations are often limited in practicality in the absence of conjugate posterior distributions. Recent work has focused on the application of variational methods to models with only partial conjugacy, such as in semiparametric regression with heteroskedastic errors. Here, both the mean and log variance functions are modeled as smooth functions of covariates. For this problem, we derive a mean field variational approximation with an embedded Laplace approximation to account for the non-conjugate structure. Empirical results with simulated and real data show that our approximate method has significant computational advantages over traditional Markov Chain Monte Carlo; in this case, a delayed rejection adaptive Metropolis algorithm. The variational approximation is much faster and eliminates the need for tuning parameter selection, achieves good fits for both the mean and log variance functions, and reasonably reflects the posterior uncertainty. We apply the methods to log-intensity data from a small angle X-ray scattering experiment, in which properly accounting for the smooth heteroskedasticity leads to significant improvements in posterior inference for key physical characteristics of an organic molecule.

Geographic information systems (GIS) organize spatial data in multiple two-dimensional arrays called layers. In many applications, a response of interest is observed on a set of sites in the landscape, and it is of interest to build a regression model from the GIS layers to predict the response at unsampled sites. Model selection in this context then consists not only of selecting appropriate layers, but also of choosing appropriate neighborhoods within those layers. We formalize this problem as a linear model and propose the use of Lasso to simultaneously select variables, choose neighborhoods, and estimate parameters. Spatially dependent errors are accounted for using generalized least squares and spatial smoothness in selected coefficients is incorporated through use of a priori spatial covariance structure. This leads to a modification of the Lasso procedure, called spatial Lasso. The spatial Lasso can be implemented by a fast algorithm and it performs well in numerical examples, including an applicat...

It is common in practice to estimate the quantiles of a complicated distribution by using the order statistics of a simulated sample. If the distribution of interest has known population mean, then it is often possible to improve the mean square error of the standard quantile estimator substantially through the simple device of mean-correction: subtract off the sample mean and add on the known population mean. Asymptotic results for the meancorrected quantile estimator are derived and compared to the standard sample quantile. Simulation results for a variety of distributions and processes illustrate the asymptotic theory. Application to Markov chain Monte Carlo and to simulation-based uncertainty analysis is described.