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Satarupa Bhattacharjee
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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Single index models provide an effective dimension reduction tool in regression, especially for high dimensional data, by projecting a general multivariate predictor onto a direction vector. We propose a novel single-index model for regression models where metric space-valued random object responses are coupled with multivariate Euclidean predictors. The responses in this regression model include complex, non-Euclidean data, including covariance matrices, graph Laplacians of networks, and univariate probability distribution functions among other complex objects that lie in abstract metric spaces. Frechet regression has provided an approach for modeling the conditional mean of such random objects given multivariate Euclidean vectors, but it does not provide for regression parameters such as slopes or intercepts, since the metric space-valued responses are not amenable to linear operations. We show here that for the case of multivariate Euclidean predictors, the parameters that define a single index and associated projection vector can be used to substitute for the inherent absence of parameters in Frechet regression. Specifically, we derive the asymptotic consistency of suitable estimates of these parameters subject to an identifiability condition. Consistent estimation of the link function of the single index Frechet regression model is obtained through local Frechet regression. We demonstrate the finite sample performance of estimation for the proposed single index Frechet regression model through simulation studies, including the special cases of probability distributions and graph adjacency matrices. The method is also illustrated for resting-state functional Magnetic Resonance Imaging (fMRI) data from the ADNI study.
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With the availability of more non-euclidean data objects, statisticians are faced with the task of developing appropriate statistical methods. For regression models in which the predictors lie in $R^p$ and the response variables are situated in a metric space, conditional Frechet means can be used to define the Frechet regression function. Global and local Frechet methods have recently been developed for modeling and estimating this regression function as extensions of multiple and local linear regression, respectively. This paper expands on these methodologies by proposing the Frechet Single Index (FSI) model and utilizing local Frechet along with $M$-estimation to estimate both the index and the underlying regression function. The method is illustrated by simulations for response objects on the surface of the unit sphere and through an analysis of human mortality data in which lifetable data are represented by distributions of age-of-death, viewed as elements of the Wasserstein space of distributions.
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