The investigator will study the relation between inconstant penalization and local adaptivity. The goal is to achieve location adaptive functional estimation when the underlying model (of the relation between a response and input variables) has inhomogeneous roughness. The investigator will (1) quantify the connection between a variable penalty function and the resulting local adaptivity, (2) generalize the methods for high-dimensional input variables, (3) study the selection of algorithmic parameters, (4) derive theoretical properties of the resulting estimators, and (5) design fast computational strategies.
Functional estimation under inhomogeneous roughness is a fundamental problem in many statistical applications. This study will contribute to the fundamental understanding to these problems, as well as providing deployable tools. Especially for large size datasets, better performance than the current state-of-the-art methodology is anticipated.
This project studied the relation between inconstant penalization and local adaptation. In practice, there are various types of functions with varying smoothness, where the functions change rapidly in some regions while being smooth in other regions. Smoothing splines are widely used for estimating an unknown function in the nonparametric regression. If data have large spatial variations, however, the standard smoothing splines (which adopt a global smoothing parameter) perform poorly. If one chooses a global smoothing parameter to be relatively small, the resulting spline estimate will describe the function well in the regions of large local variations, however it will under-smooth in other regions. On the other hand, if the global smoothing parameter is chosen to be relatively large, then the estimated function will be over-smoothed in the regions of large local variations. This indicates that using a global smoothing parameter is not sufficient in fitting functions with varying roughness. We derive an asymptotically optimal local penalty function. The key in the derivation is to use the connection between the smoothing spline and kernel smoothing. It is well known that a smoothing spline is asymptotically equivalent to a variant of kernel smoothing. Based on this result, together with the locally optimal bandwidth in kernel smoothing, we derive an asymptotically optimal local penalty function. Using the derived locally optimal penalty function, we propose a locally optimal adaptive smoothing spline estimator using the RKHS (reproducing kernel Hilbert space) framework. In the numerical study, we show that our estimator performs very well using several simulated and real data sets. We have also studied a spatially adaptive (i.e., inconstant) penalty multiplier, where the univariate smoothing parameter in the classical smoothing spline is converted into an adaptive multivariate parameter. Under this framework, we derived the closed-form solution and showed that this estimator achieves the optimal rate of convergence. We proposed a strategy of minimizing a multivariate version of the generalized cross validation function; then the resulting estimator is shown to be consistent and asymptotically optimal under some general conditions. Numerical examples demonstrated the advantage of our method against ordinary smoothing splines, as well as other contemporary adaptive splines methods. Functional estimation under inhomogeneous roughness is a fundamental problem in many statistical applications. Our work contributes to the relevant literature.
