Functional Regression

By Jeffrey S. Morris. 2014

512 citations recorded by the source corpus.

Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the development of this field, which has accelerated in the past 10 years to become one of the fastest growing areas of statistics, fueled by the growing number of applications yielding this type of data. One unique characteristic of FDA is the need to combine information both across and within functions, which Ramsay and Silverman called replication and regularization, respectively. This article will focus on functional regression, the area of FDA that has received the most attention in applications and methodological development. First will be an introduction to basis functions, key building blocks for regularization in functional regression methods, followed by an overview of functional regression methods, split into three types: [1] functional predictor regression (scalar-on-function), [2] functional response regression (function-on-scalar) and [3] function-on-function regression. For each, the role of replication and regularization will be discussed and the methodological development described in a roughly chronological manner, at times deviating from the historical timeline to group together similar methods. The primary focus is on modeling and methodology, highlighting the modeling structures that have been developed and the various regularization approaches employed. At the end is a brief discussion describing potential areas of future development in this field.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8bf620765e77c98424983de5d5b74b90d99e28fd

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Citations in the corpus

Functional Data Analysis (2016, background): “Reviews of FLMs are M¨uller (2005, 2011); Morris (2015).”
Distribution-on-Distribution Regression via Optimal Transport Maps (2021, methodology): “They used the tangent space structure to deﬁne a regression operation: using the log transform, the regressor and response are lifted to suitable tangent spaces, where a (linear) regression model is deﬁned in a more familiar Hilbertian setting [11, 4].”
Measuring Association on Topological Spaces Using Kernels and Geometric Graphs (2020, background): “This is particularly useful in functional regression models (see [73] for a survey), in reallife machine learning and human behavior recognition (see [28]), measuring association between two stochastic processes (see [27]), etc.”
Wasserstein Regression (2020, background): “2005; Cai and Hall 2006; Hall and Horowitz 2007) and reviewed (Morris 2015; Wang et al. 2016).”
A Unified Framework for Long Range and Cold Start Forecasting of Seasonal Profiles in Time Series (2017, background): “Most uses of FDA relevant to our approach are only developed for a very small number of covariates—however, our metadata features are on the order of tens of thousands of dimensions.”
Hierarchical generalized additive models in ecology: an introduction with mgcv (2019, background): “However, there is an extensive literature dedicated to the estimation of more complex functional regression models for any interested reader (see Morris (2015) and Greven & Scheipl (2017) for a good introduction and overview of more recent work in this ﬁ eld).”