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Journal of the American Statistical Association
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The aim of this work is to introduce a new nonparametric regression technique in the context of functional covariate and scalar response. We propose a local linear regression estimator and study its asymptotic behaviour. Its finite-sample performance is compared with a Nadayara-Watson type kernel regression estimator and with the linear regression estimator via a Monte Carlo study and the analysis of two real data sets. In all the scenarios considered, the local linear regression estimator performs better than the kernel one, in the sense that the mean squared prediction error is lower.
Increasingly, statisticians are faced with the task of analyzing complex data that are non-Euclidean and specifically do not lie in a vector space. To address the need for statistical methods for such data, we introduce the concept of Fréchet regression. This is a general approach to regression when responses are complex random objects in a metric space and predictors are in Rp, achieved by extending the classical concept of a Fréchet mean to the notion of a conditional Fréchet mean. We develop generalized versions of both global least squares regression and local weighted least squares smoothing. The target quantities are appropriately defined population versions of global and local regression for response objects in a metric space. We derive asymptotic rates of convergence for the corresponding fitted regressions using observed data to the population targets under suitable regularity conditions by applying empirical process methods. For the special case of random objects that reside in a Hilbert space, such as regression models with vector predictors and functional data as responses, we obtain a limit distribution. The proposed methods have broad applicability. Illustrative examples include responses that consist of probability distributions and correlation matrices, and we demonstrate both global and local Fréchet regression for demographic and brain imaging data. Local Fréchet regression is also illustrated via a simulation with response data which lie on the sphere.
For multivariate functional data recorded from a sample of subjects on a common domain, one is often interested in the covariance
between pairs of the component functions, extending the notion of a covariance matrix for multivariate data to the functional
case. A straightforward approach is to integrate the pointwise covariance matrices over the functional time domain. We generalize
this approach by defining the Fréchet integral, which depends on the metric chosen for the space of covariance matrices, and
demonstrate that ordinary integration is a special case where the Frobenius metric is used. As the space of covariance matrices
is nonlinear, we propose a class of power metrics as alternatives to the Frobenius metric. For any such power metric, the
calculation of Fréchet integrals is equivalent to transforming the covariance matrices with the chosen power, applying the
classical Riemann integral to the transformed matrices, and finally using the inverse transformation to return to the original
scale. We also propose data-adaptive metric selection with respect to a user-specified target criterion, such as fastest decline
of the eigenvalues, establish consistency of the proposed procedures, and demonstrate their effectiveness in a simulation.
The proposed functional covariance approach through Fréchet integration is illustrated by a comparison of connectivity between
brain voxels for normal subjects and Alzheimer's patients based on fMRI data.
This paper introduces local distance-based generalized linear models. These models extend (weighted) distance-based linear models first to the generalized linear model framework. Then, a nonparametric version of these models is proposed by means of local fitting. Distances between individuals are the only predictor information needed to fit these models. Therefore, they are applicable, among others, to mixed (qualitative and quantitative) explanatory variables or when the regressor is of functional type. An implementation is provided by the R package dbstats, which also implements other distance-based prediction methods. Supplementary material for this article is available online, which reproduces all the results of this article.
The problem of nonparametrically predicting a scalar response variable from a functional predictor is considered. A sample of pairs (functional predictor and response) is observed. When predicting the response for a new functional predictor value, a semi-metric is used to compute the distances between the new and the previously observed functional predictors. Then each pair in the original sample is weighted according to a decreasing function of these distances. A Weighted (Linear) Distance-Based Regression is fitted, where the weights are as above and the distances are given by a possibly different semi-metric. This approach can be extended to nonparametric predictions from other kinds of explanatory variables (e.g., data of mixed type) in a natural way.