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On sufficient dimension reduction for functional data: Inverse moment‐based methods

Advanced Review

Jun Song

Published Online: Mar 27 2019

DOI: 10.1002/wics.1459

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In this article, we present methods of sufficient dimension reduction (SDR) for functional data and organize them in a unified framework. The current existing methods for multivariate, functional data, functional response, and nonlinear SDR for functional data are illustrated in a way that they are naturally generalized. If a covariance operator is defined on an infinite‐dimensional space, the inverse of it is not bounded. Thus it is difficult to estimate and even impossible to apply the conventional results. We present solutions to resolve the inverse issue for the methods of functional SDR. Then we explain the functional SDR methods in three different scenarios: scalar‐on‐function, function‐on‐function, nonlinear SDR for function‐on‐function problem. This article is categorized under: Data: Types and Structure > Streaming data Statistical and Graphical Methods of Data Analysis > Dimension Reduction Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data

An example of multivariate functional data: The data are recorded as (x, y)‐coordinates along with time, which is an observation of two‐dimensional vector‐valued function defined on an interval

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