Additive regression for non-Euclidean responses and predictors

Abstract

Additive regression is studied in a very general setting where both the response and predictors are allowed to be non-Euclidean. The response takes values in a general separable Hilbert space, whereas the predictors take values in general semimetric spaces, which covers a very wide range of nonstandard response variables and predictors. A general framework of estimating additive models is presented for semimetric space-valued predictors. In particular, full details of implementation and the corresponding theory are given for predictors taking values in Hilbert spaces and/or Riemannian manifolds. The existence of the estimators, convergence of a backfitting algorithm, rates of convergence and asymptotic distributions of the estimators are discussed. The finite sample performance of the estimators is investigated by means of two simulation studies. Finally, three data sets covering several types of non-Euclidean data are analyzed to illustrate the usefulness of the proposed general approach.

Random Forest Weighted Local Fréchet Regression

“…And some advanced statistical tools designed for responses lying on the Riemannian manifold were developed, like the ILPR [58] and MAM [39]. For metric space being a specific Hilbert space, vector operations and an inner product structure are available, which inspires several promising nonparametric Hilbertian regression such as [31,32,33]. For the above two types of responses, we can consider nonparametric regression framework based on random forest kernels like our proposed RFWFR and RFWLLFR in future research.…”

“…It is of great interest to incorporate random forests for regression with specific structure information of non-Euclidean responses. Another interesting direction is to follow [14,33] and study random forests weighted Fréchet regression when the predictors are also non-Euclidean. We leave these to future research.…”

Density estimation and regression analysis on S^d in the presence of measurement error

“…In this section, we discuss the asymptotic distributions of our density and regression estimators. Recently, [32] derived some asymptotic distributions for their deconvolution estimators on compact and connected Lie groups. We note that S 1 and S 3 are such Lie groups.…”

“…[55], [26]). Previous works on error-free circular, spherical or general hyperspherical data include density estimation ( [27], [23]), regression analysis ( [8], [51], [52], [32]) and statistical testing ( [11], [5], [24]). Among them, [8] did not cover a measurement error problem and simply considered the case where both response and predictor are spherical variables and the response is symmetrically distributed around the product of an unknown orthogonal matrix and the predictor.…”

Density estimation and regression analysis on hyperspheres in the presence of measurement error

“…The latter estimator, denoted by , is based on the geodesic distance. For the implementation of , we refer to Jeon et al (2021). In the second simulation study, we compared the two asymptotic confidence intervals for that we constructed in Theorems 5 and 7, namely the confidence interval based on the asymptotic normality (AN) and the confidence interval based on the empirical likelihood (EL), respectively.…”