Statistical properties of the method of regularization with periodic Gaussian reproducing kernel

The method of regularization with the Gaussian reproducing kernel is popular in the machine learning literature and successful in many practical applications. In this paper we consider the periodic version of the Gaussian kernel regularization. We show in the white noise model setting, that in function spaces of very smooth functions, such as the infinite-order Sobolev space and the space of analytic functions, the method under consideration is asymptotically minimax; in finite-order Sobolev spaces, the method is rate optimal, and the efficiency in terms of constant when compared with the minimax estimator is reasonably high. The smoothing parameters in the periodic Gaussian regularization can be chosen adaptively without loss of asymptotic efficiency. The results derived in this paper give a partial explanation of the success of the Gaussian reproducing kernel in practice. Simulations are carried out to study the finite sample properties of the periodic Gaussian regularization.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Statistics (http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000045

Nonparametric regression analysis

textNonparametric regression uses nonparametric and flexible methods in analyzing complex data with unknown regression relationships by imposing minimum assumptions on the regression function. The theory and applications of nonparametric regression methods with an emphasis on kernel regression, smoothing spines and Gaussian process regression are reviewed in this report. Two datasets are analyzed to demonstrate and compare the three nonparametric regression models in R.Statistic

Nonparametric likelihood based estimation of linear filters for point processes

We consider models for multivariate point processes where the intensity is given nonparametrically in terms of functions in a reproducing kernel Hilbert space. The likelihood function involves a time integral and is consequently not given in terms of a finite number of kernel evaluations. The main result is a representation of the gradient of the log-likelihood, which we use to derive computable approximations of the log-likelihood and the gradient by time discretization. These approximations are then used to minimize the approximate penalized log-likelihood. For time and memory efficiency the implementation relies crucially on the use of sparse matrices. As an illustration we consider neuron network modeling, and we use this example to investigate how the computational costs of the approximations depend on the resolution of the time discretization. The implementation is available in the R package ppstat.Comment: 10 pages, 3 figure