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== Functional linear models (FLMs) ==
Functional linear models (FLMs) are an extension of [[Linear regression|traditional multivariate linear models]] with scalar response <math>Y\in\mathbb{R}</math> and scalar covariates <math>X\in\mathbb{R}^p</math>, which can be written as
<math display="block">Y = \beta_0 + \langle X,\beta\rangle + \epsilon,</math>
where <math>\langle\cdot,\cdot\rangle</math> denotes the [[Inner product space|inner product]] in [[Euclidean space|Euclidean space]], <math>\beta_0\in\mathbb{R}</math> and <math>\beta\in\mathbb{R}^p</math> denote the regression coefficients, and <math>\epsilon</math> is a random error with [[Expected value|mean]] zero and [[Variance|variance]] finite. FLMs can be divided into three types based on responses and covariates.
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where in implementation the infinite sum is replaced by a finite sum truncated at <math>K</math>
<math display="block">Y = \beta_0 + \sum_{k=1}^K \beta_k x_k +\epsilon</math>
where <math>K\in\mathbb{N}</math> is finite<
Adding multiple functional and scalar covariates, the FLR can be extended as
<math display="block">Y = \langle\mathbf{Z},\alpha\rangle + \sum_{j=1}^p \int_{\mathcal{T}_j} X_j^c(t) \beta_j(t) dt + \epsilon</math>
where <math>\mathbf{Z}=(Z_1,\cdots,Z_q)^T</math> with <math>Z_1=1</math> is a vector of scalar covariates, <math>\alpha=(\alpha_1,\cdots,\alpha_q)^T</math> is a vector of coefficients corresponding to <math>\mathbf{Z}</math>, <math>\langle\cdot,\cdot\rangle</math> denotes the inner product in Euclidean space, <math>X^c_1,\cdots,X^c_p</math> are multiple centered functional covariates given by <math>X_j^c(\cdot) = X_j(\cdot) - \mathbb{E}(X_j(\cdot))</math>, and <math>\mathcal{T}_j</math> is the interval <math>X_j(\cdot)</math> is defined on. However, due to the parametric component <math>\alpha</math>, the estimation of this model is different from that of the FLR. A possible approach to estimating <math>\alpha</math> is through [[Generalized estimating equation|generalized estimating equation]] with the nonparametric part <math> \sum_{j=1}^p \int_{\mathcal{T}_j} X_j^c(t) \beta_j(t) dt</math> replaced by its estimate for a given <math>\alpha</math>
=== Functional linear models with functional response ===
For a function <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<
<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, estimation of this model can be given by analogy to multivariate linear regression
<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, regularization is needed because <math>R_{XX}</math> is a compact operator and its inverse is not bounded<
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. A possible way to estimate <math>\alpha</math> is a two-step procedure: (i) For 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><
== Functional nonlinear models ==
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A functional additive model can be given by replacing the linear function of <math>x_k</math> by a general smooth function <math>f_k</math>
<math display="block">\mathbb{E}(Y|X)=\mathbb{E}(Y) + \sum_{k=1}^\infty f_k(x_k)</math>
where <math>f_k</math> satisfies <math>\mathbb{E}(f_k(x_k))=0</math> for <math>k\in\mathbb{N}</math><
== Extensions ==
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* [[Stochastic processes|Stochastic processes]]
* [[Lp space|Lp space]]
== References ==▼
<references/>
== Further reading ==
* Morris (2015). Functional regression. ''Annual Review of Statistics and Its Application''. '''2''':321–359. [[Digital object identifier|doi]]:[http://doi.org/10.1146/annurev-statistics-010814-020413 10.1146/annurev-statistics-010814-020413].
▲== References ==
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