The Mutual Nearest Neighbor Method in Functional Nonparametric Regression

Abstract

In recent decades, functional data have become a commonly encountered type of data. Its ideal units of observation are functions defined on some continuous domain and the observed data are sampled on a discrete grid. An important problem in functional data analysis is how to fit regression models with scalar responses and functional predictors (scalar-on-function regression). This paper focuses on the nonparametric approaches to this problem. First there is a review of the classical k-nearest neighbors (kNN) method for functional regression. Then the mutual nearest neighbors (MNN) method, which is a variant of kNN method, is applied to functional regression. Compared with the classical kNN approach, the MNN method takes use of the concept of mutual nearest neighbors to construct regression model and the pseudo nearest neighbors will not be taken into account during the prediction process. In addition, any nonparametric method in the functional data cases is affected by the curse of infinite dimensionality. To prevent this curse, it is legitimate to measure the proximity between two curves via a semi-metric. The effectiveness of MNN method is illustrated by comparing the predictive power of MNN method with kNN method first on the simulated datasets and then on a real chemometrical example. The comparative experimental analyses show that MNN method preserves the main merits inherent in kNN method and achieves better performances with proper proximity measures.