Functional regression: Difference between revisions

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=== Functional linear models with functional response ===

For a functionfunctional response <math>Y(\cdot)</math> on <math>\mathcal{T}_Y</math> and a functional covariate <math>X(\cdot)</math> on <math>\mathcal{T}_X</math>, two primary models have been considered<ref name=wang:16/><ref>Ramsay and [[Bernard Silverman|Silverman]] (2005). ''Functional data analysis'', 2nd ed., New York&#160;: Springer, [[Special:BookSources/038740080X|ISBN 0-387-40080-X]]</ref>. One functional linear model regressing <math>Y(\cdot)</math> on <math>X(\cdot)</math> is given by

<math display="block">Y(s) = \beta_0(s) + \int_{\mathcal{T}_X} \beta(s,t) X^c(t)dt + \epsilon(s)</math>

where <math>s\in\mathcal{T}_Y</math>, <math>t\in\mathcal{T}_X</math>, <math>X^c(\cdot) = X(\cdot) - \mathbb{E}(X(\cdot))</math> is still the centered functional covariate, <math>\beta_0(\cdot)</math> and <math>\beta(\cdot,\cdot)</math> are coefficient functions, and <math>\epsilon(\cdot)</math> is usually assumed to be a Gaussian process with mean zero. In this case, at any given time <math>s\in\mathcal{T}_Y</math>, the value of <math>Y</math>, i.e. <math>Y(s)</math>, depends on the entire trajectory of <math>X</math>. This model, for any given time <math>s</math>, is an extension of the traditional multivariate linear regression model by simply replacing the inner product in Euclidean space by that in <math>L^2</math> space. Thus,An estimationestimating ofequation this model can be givenmotivated by analogy to multivariate linear regression is

<math display="block">r_{XY} = R_{XX}\beta, \text{ for } \beta\in L^2(\mathcal{T}_X\times\mathcal{T}_X)</math>

where <math>r_{XY}(s,t) = \text{cov}(X(s),Y(t))</math>, <math>R_{XX}: L^2\times L^2 \rightarrow L^2\times L^2</math> is defined as <math>(R_{XX}\beta)(s,t) = \int r_{XX}(s,w)\beta(w,t)dw</math> with <math>r_{XX}(s,t) = \text{cov}(X(s),X(t))</math>. Furthermore, regularizationRegularization is needed because <math>R_{XX}</math> is a compact operator and its inverse is not bounded<ref name=wang:16>Wang, Chiou and M&uuml;ller (2016). "Functional data analysis". ''Annual Review of Statistics and Its Application''. '''3''':257&ndash;295. [[Digital object identifier|doi]]:[http://dx.doi.org/10.1146/annurev-statistics-041715-033624 10.1146/annurev-statistics-041715-033624]</ref>.<br />

In particular, taking <math>X(\cdot)</math> as a constant function gives a special case of this model

<math display="block">Y(s) = \sum_{j=1}^p X_j \beta_j(s) + \epsilon(s)</math>

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Assuming that <math>\mathcal{T}_X = \mathcal{T}_Y := \mathcal{T}</math>, another model called varying-coefficient model is of the form

<math display="block">Y(s) = \alpha_0(s) + \alpha(s)X(s)+\epsilon(s)</math>

Note that this model assumes the value of <math>Y</math> at time <math>s</math>, i.e. <math>Y(s)</math>, only depends on that of <math>X</math> at the same time, <math>X(s)</math>, and thus is a concurrent regression model. AFor possibleestimation, wayone tomay estimateuse <math>\alpha</math>the isfact athat two-step procedure: (i) Forfor any <math>s\in\mathcal{T}</math> fixed, an estimate of <math>\alpha(s)</math> can be computed by applying [[Ordinary least squares|ordinary least squares]] to a neighborhood of <math>s</math>. Let the corresponding estimate be denoted by <math>\tilde\alpha(s)</math>. (ii) The final estimate <math>\hat\alpha</math> is then obtained by smoothing <math>\tilde\alpha(s)</math> with respect to <math>s</math><ref name=wang:16/>.

== Functional nonlinear models ==