Functional regression: Difference between revisions

Revision as of 08:41, 11 December 2024

Revision as of 08:42, 11 December 2024

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

where <math>r_{XY}(s,t) = \text{cov}(X(s),Y(t))</math>, <math>R_{XX}: L^2(\mathcal{S}\times\mathcal{S}) \rightarrow L^2(\mathcal{S}\times\mathcal{T})</math> is defined as <math>(R_{XX}\beta)(s,t) = \int_\mathcal{S} r_{XX}(s,w)\beta(w,t)dw</math> with <math>r_{XX}(s,w) = \text{cov}(X(s),X(w))</math> for <math>s,w\in\mathcal{S}</math>.<ref name=wang:16/> Regularization is needed and can be done through truncation, <math>L^2</math> penalization or <math>L^1</math> penalization.<ref name=morr:15/> Various estimation methods for model ({{EquationNote|4}}) are available.<ref>{{Cite journal |last=Ramsay |first=J. O. |last2=Dalzell |first2=C. J. |date=1991 |title=Some Tools for Functional Data Analysis |url=https://www.jstor.org/stable/2345586 |journal=Journal of the Royal Statistical Society. Series B (Methodological) |volume=53 |issue=3 |pages=539–572 |issn=0035-9246}}</ref><ref>{{Cite journal |last=Yao |first=Fang |last2=Müller |first2=Hans-Georg |last3=Wang |first3=Jane-Ling |date= |title=Functional linear regression analysis for longitudinal data |url=https://projecteuclid.org/journals/annals-of-statistics/volume-33/issue-6/Functional-linear-regression-analysis-for-longitudinal-data/10.1214/009053605000000660.full |journal=The Annals of Statistics |volume=33 |issue=6 |pages=2873–2903 |doi=10.1214/009053605000000660 |issn=0090-5364}}</ref><br />

When <math>X</math> and <math>Y</math> are concurrently observed, i.e., <math>\mathcal{S}=\mathcal{T}</math>,<ref>{{Cite journal |last=Grenander (1950).|first=Ulf "|date= |title=Stochastic processes and statistical inference". ''Arkiv Matematik''. '''1''' (3):195&ndash;277. [[Digital object identifier|doi]]:[httpurl=https://doiprojecteuclid.org/journals/arkiv-for-matematik/volume-1/issue-3/Stochastic-processes-and-statistical-inference/10.1007/BF02590638.full |journal=Arkiv för Matematik |volume=1 |issue=3 |pages=195–277 |doi=10.1007/BF02590638]. |issn=0004-2080}}</ref> it is reasonable to consider a historical functional linear model, where the current value of <math>Y</math> only depends on the history of <math>X</math>, i.e., <math>\beta(s,t)=0</math> for <math>s>t</math> in model ({{EquationNote|4}}).<ref name=wang:16/><ref>Malfait and Ramsay (2003). "The historical functional linear model". ''Canadian Journal of Statistics''. '''31''' (2):115&ndash;128. [[Digital object identifier|doi]]:[http://doi.org/10.2307/3316063 10.2307/3316063].</ref> A simpler version of the historical functional linear model is the functional concurrent model (see below).<br />

Adding multiple functional covariates, model ({{EquationNote|4}}) can be extended to

{{NumBlk|::|<math display="block">Y(t) = \beta_0(t) + \sum_{j=1}^p\int_{\mathcal{S}_j} \beta_j(s,t) X^c_j(s)\,ds + \varepsilon(t),\ \text{for}\ t\in\mathcal{T},</math>|{{EquationRef|5}}}}