Abstract
One of the challenges with functional data is incorporating geometric structure, or local correlation, into the analysis. This structure is inherent in the output from an increasing number of biomedical technologies, and a functional linear model is often used to estimate the relationship between the predictor functions and scalar responses. Common approaches to the problem of estimating a coefficient function typically involve two stages: regularization and estimation. Regularization is usually done via dimension reduction, projecting onto a predefined span of basis functions or a reduced set of eigenvectors (principal components). In contrast, we present a unified approach that directly incorporates geometric structure into the estimation process by exploiting the joint eigenproperties of the predictors and a linear penalty operator. In this sense, the components in the regression are 'partially empirical' and the framework is provided by the generalized singular value decomposition (GSVD). The form of the penalized estimation is not new, but the GSVD clarifies the process and informs the choice of penalty by making explicit the joint influence of the penalty and predictors on the bias, variance and performance of the estimated coefficient function. Laboratory spectroscopy data and simulations are used to illustrate the concepts.
Highlights
The coefficient function, β, in a functional linear model represents the linear relationship between responses, y, and predictors, x, either of which may appear as a function
One obvious fix is to truncate the sum to d < r terms so that {σk}dk=1 is bounded away from zero. This leads to the truncated singular value or principal component regression (PCR) estimate: βPCR ≡ βPdCR = VdDd−1Ud′y where here, and subsequently, we use the notation Ad ≡ col[a1, . . . , ad] to denote the first d columns of a matrix A
There are many ways in which such constraints may be imposed, and we have focused on the algebraic aspects of a penalization process that imposes spatial structure directly into a regularized estimation. This approach fits into the classic framework of L2-penalized regression but with an emphasis on the algebraic role that a penalty operator plays to impart structure on the estimate
Summary
The coefficient function, β, in a functional linear model (fLM) represents the linear relationship between responses, y, and predictors, x, either of which may appear as a function. This is the essence of principal component regression (PCR) and these vectors form the basis for a ridge estimate This empirical basis does not technically impose local spatial structure (no order relation among the index parameter values is used), it may be justified by arguing that a few principal component vectors capture the “greatest part” of a set of predictors [17]. Properties of this approach for signal regression is the focus of [7] and [16].
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