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Functional linear regression has been widely used to model the relationship between a scalar response and functional predictors. If the original data do not satisfy the linear assumption, an intuitive solution is to perform some transformation such that transformed data will be linearly related. The problem of finding such transformations has been rather neglected in the development of functional data analysis tools. In this paper, we consider transformation on the response variable in functional linear regression and propose a nonparametric transformation model in which we use spline functions to construct the transformation function. The functional regression coefficients are then estimated by an innovative procedure called mixed data canonical correlation analysis (MDCCA). MDCCA is analogous to the canonical correlation analysis between two multivariate samples, but is between a multivariate sample and a set of functional data. Here, we apply the MDCCA to the projection of the transformation function on the BB-spline space and the functional predictors. We then show that our estimates agree with the regularized functional least squares estimate for the transformation model subject to a scale multiplication. The dimension of the space of spline transformations can be determined by a model selection principle. Typically, a very small number of BB-spline knots is needed. Real and simulation data examples are further presented to demonstrate the value of this approach.

Functional data, such as data in the form of curves and images, are becoming more popularly seen. There has been a rapidly increasing level of interest in developing statistical tools for functional data analysis (FDA). An extensive review of FDA can be found in Ramsay and Silverman, 2002 and Ramsay and Silverman, 2005, Ferraty and Vieu (2006) and Ferraty and Romain (2010). Many authors have made an effort to adapt existing multivariate regression methods to the functional regression model with a scalar response equation(1) y=r(x)+ε,y=r(x)+ε, Turn MathJax on where xx is a random variable valued in L2([a,b])L2([a,b]), the space of square integrable functions from [a,b][a,b] into RR, yy and εε are real scalar variables, r:L2([a,b])→Rr:L2([a,b])→R is a regression operator, and the inner product in L2([a,b])L2([a,b]) is defined by View the MathML source〈f,g〉=∫abf(t)g(t)dt. The regression operator rr in this model can be modeled either as a parametric or nonparametric function. We refer the reader to the books of Ferraty and Vieu (2006) and Ferraty and Romain (2010) for an overview of the nonparametric approach, with more advanced results that can be found in Ferraty and Vieu (2002) and Ferraty et al. (2007). In the parametric way, the most extensively studied model is the functional linear model equation(2) y=〈x,β〉+ε,y=〈x,β〉+ε, Turn MathJax on where r(⋅)=〈β,⋅〉r(⋅)=〈β,⋅〉 for a certain functional regression coefficient β∈L2([a,b])β∈L2([a,b]), xx is a zero-mean random variable such that E(‖x‖2)<∞E(‖x‖2)<∞, and εε satisfies E(ε)=0E(ε)=0, E(ε2)<∞E(ε2)<∞ and E(xε)=0E(xε)=0. There are two major methods to estimating ββ. One is the regularization method (see Cardot et al., 2003, Ramsay and Silverman, 2002, Ramsay and Silverman, 2005 and Yuan and Cai, 2010), and the other is based on functional principal component analysis (see Cai and Hall, 2006, Ferraty et al., 2011 and Manteiga and Calvo, 2011). When interpretability of the model is a more important study, the parametric approach, especially functional linear regression model (2), is generally preferable. However, the linearity assumption may not be satisfied and some transformation on the response variable yy becomes necessary. For example, consider the presence of heteroscedasticity where the error term View the MathML sourceε=σ(x)ε̃ with View the MathML sourceε̃ independent of xx and View the MathML sourceE(ε̃)=0 and View the MathML sourcevar(ε̃)=1, and σ(x)σ(x) is a real function satisfying E(σ(x)2)<∞E(σ(x)2)<∞. Using model (1) and (2) can lead to invalid results (see Crambes et al., 2008 and Delaigle et al., 2009). Currently, when transformation on the response yy is necessary, the transformation function needs to be prespecified with a known form. For example, in functional nonparametric regression, transformation models have been studied by Crambes et al. (2008) and Ferraty et al. (2010) with the transformation function being prespecified with a known form. Crambes et al. (2008) focused on robust nonparametric estimation of rr in model (1), and Ferraty et al. (2010) proved the uniform almost complete convergence rate of the nonparametric estimate. In functional linear regression analysis, Ramsay and Dalzell (1991) used logarithm transformation when analyzing the Canadian climate data. However, for complex data, a proper transformation is generally difficult to pre-conceive and the users will have to rely on a cumbersome trial-and-error process, and such efforts are easily futile if the transformation is not of a simple parametric form. In this article, our goal is to develop a systematic and flexible approach to decide such transformations in the context of functional linear regression by simultaneously estimating the transformation function and the functional regression coefficients. Motivated by He and Shen (1997), we propose a nonparametric approach called the functional linear regression after spline transformation (FLIRST). That is, we extend (2) as equation(3) h(y)=〈x,β〉+ε,h(y)=〈x,β〉+ε, Turn MathJax on where h(⋅)h(⋅) is an unknown smooth function approximated by a quadratic BB-spline, and estimated together with ββ. Since the functional coefficient ββ in model (3) is identifiable only up to a scale multiplication, the real interest will be in estimating its normalized version. We propose an innovative algorithm called mixed-data canonical correlation analysis (MDCCA) to consistently estimate the functional coefficient and the transformation function. MDCCA is analogous to the canonical correlation analysis between two multivariate samples, but is between a multivariate sample and a set of functional data. Knots selection for the BB-spline is then solved as a model selection problem. Our approach is closely related to the regularized functional sliced inverse regression (RFSIR) of Ferré and Villa (2006). They considered the following model equation(4) y=g(〈x,β1〉,…,〈x,βK〉,ε),y=g(〈x,β1〉,…,〈x,βK〉,ε), Turn MathJax on where gg is an unknown link function, and βjβj’s are linearly independent functions in H2H2, the subspace of L2([a,b])L2([a,b]) that contains the functions with a squared integrable second derivative. It is worth noting that, though FLIRST and RFSIR are related, their goals are different. FLIRST focuses more on finding the proper transformation on the response variable, while RFSIR focuses more on dimension reduction and does not provide an estimate of the link function gg. Our FLIRST model (3) can be viewed as a special case of (4) when K=1K=1 plus some mild constraints on g(⋅)g(⋅). We will show that, given a linear condition, FLIRST and RFSIR are equivalent under model (4) with K=1K=1. When the linear condition does not hold, FLIRST and RFSIR no longer provide consistent estimates of the functional parameter, but the FLIRST estimate still gives the projection of the functional predictor that best correlates with the transformed response. The rest of this paper is organized as follows. In Section 2, we first describe the MDCCA algorithm and then propose the FLIRST methodology as an application of MDCCA to functional linear regression transformation model. Theoretical properties of the FLIRST method are also given. In Section 3, we perform simulation studies to compare FLIRST with other methods. Real data examples are then presented in Section 4, and we conclude the paper and provide some discussions in Section 5.

In this paper, we consider in functional data analysis finding a nonparametric transformation of the response after which the functional linear regression applies. Motivated by the linear regression after spline transformation in He and Shen (1997), we propose FLIRST in the context of functional linear regression. FLIRST is based on a functional counterpart to the multivariate canonical correlation analysis which we call mixed-data canonical correlation analysis (MDCCA). We show that, under the transformation model, the leading canonical variate from MDCCA provides consistent estimates of the functional regression coefficient and the spline transformation simultaneously. We also show that the proposed method is closely related to the regularized functional sliced inverse regression and the constrained regularized functional least squares estimate. Compared with the Box–Cox transformation, FLIRST allows more flexible transformation and does not require the response to be positive. In several simulation and real data examples, FLIRST is shown to provide better transformation than the commonly used Box–Cox transformation. In FLIRST, we used a particular type of penalization as the bilinear form in (7), but our results should also hold for other regularization functionals satisfying assumption (C4). For example, we may use a penalty term similar to the one used in Ridge-PDA (Hastie et al., 1995), which will lead the roughness penalty terms in (7) to View the MathML source〈(Γ+λI)β,β〉=var(bTy)=1. We have chosen quadratic BB-spline to model the transformation function in FLIRST for its convenience. In general, our results remain valid when splines of other order are used. Denote ℋqℋq to be the collection of all functions on [0, 1] whose (q−1)(q−1)th order derivative satisfies the Hölder condition. That is, there exists a constant W0W0 such that |h(q−1)(s)−h(q−1)(t)|≤W0|t−s|.|h(q−1)(s)−h(q−1)(t)|≤W0|t−s|. Turn MathJax on If we assume that the transformation function h(⋅)∈ℋqh(⋅)∈ℋq, we can use BB-spline with degree qq as it uniformly approximates any function in ℋqℋq (see Schumaker, 1981, p. 224). In the present paper, we focused on the functional linear transformation model, which needs only the first eigenvalue and the corresponding eigenvector in MDCCA. This is because the eigenvector corresponding to the first eigenvalue gives the largest linear correlation between the transformation function h(y)h(y) and 〈x,β〉〈x,β〉, which means that the FLIRST estimate gives the projection of the functional predictor 〈x,β〉〈x,β〉 best correlated with the transformed response. In general, MDCCA can serve as an effective dimensional reduction tool for functional regression, where the dimensionality is determined by selecting a proper number of eigenvalues. Another possible extension of FLIRST is to functional semi-parametric models, such as the functional partial linear model (Aneiros and Vieu, 2008) and the functional single-index model (Aitsïdi et al., 2008). However, when the regression operator involves some nonparametric component, careful considerations should made towards the identifiability of the model when the transformation function is also estimated nonparametrically. We will study these extensions in our future research.