A Reproducing Kernel Hilbert Space Approach to Functional Linear Regression

By Ming Yuan, Tommaso Cai. 2010

309 citations recorded by the source corpus.

We study in this paper a smoothness regularization method for functional linear regression and provide a unified treatment for both the prediction and estimation problems. By developing a tool on simultaneous diagonalization of two positive definite kernels, we obtain shaper results on the minimax rates of convergence and show that smoothness regularized estimators achieve the optimal rates of convergence for both prediction and estimation under conditions weaker than those for the functional principal components based methods developed in the literature. Despite the generality of the method of regularization, we show that the procedure is easily implementable. Numerical results are obtained to illustrate the merits of the method and to demonstrate the theoretical developments.

Subfields: Computational Mechanics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/25506126f64a4b8c67c3ca570a5fa40a2cac5263

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Citations in the corpus

Wasserstein Regression (2020, methodology): “This model can also be viewed as function-to-scalar regression, which has been well studied in functional data analysis (Cardot et al. 1999, 2003; Cai and Hall 2006; Hall and Horowitz 2007; Yuan and Cai 2010; Lin and Yao 2017).”
Nonparametric inference in generalized functional linear models (2014, methodology): “We remark that Assumption A3 is the price we need to pay for making valid statistical inference in addition to those required for minimax estimation, as in, for example, [4, 35].”
Minimax adaptive tests for the Functional Linear model (2012, methodology): “The theoretical properties of these classes of estimators have been investigated from different viewpoints: prediction [6, 9, 13, 36] (estimation of 〈Xn+1, θ〉 where Xn+1 follows the same distribution as X), pointwise prediction [4] (estimation of 〈x, θ〉 for a fixed x ∈ H) or the inverse problem [11,”
Truncated linear models for functional data (2014, methodology): “…Mas and Pumo proposed an alternative formulation of the functional linear model; He et al. (2010) introduced techniques based on canonical analysis; Yuan and Cai (2010) suggested a method founded on reproducing kernel Hilbert space analysis; Ferraty et al. (2012) discussed presmoothing methods;…”
Estimation in functional linear quantile regression (2012, methodology): “basis in functional data analysis, which is the reason why the PCA basis is used in this paper. Other estimation methods such as smoothing splines [14] and a reproducing kernel Hilbert space approach [39] could be adapted in the quantile regression case, which is left as a future topic. The interpol”
Adaptive Global Testing for Functional Linear Models (2014, methodology): “Minimax rates of estimation were established by Hall and Horowitz (2007), Cai and Zhou (2008); and Meister (2011), using functional principal components regression, and by Yuan and Cai (2010) using a reproducing kernel hilbert Space approach.”
Locally Sparse Estimator for Functional Linear Regression Models (2017, background): “Preda (2007) and Yuan and Cai (2010) explored FLR from the point of view of reproducing kernel Hilbert space.”
Shrinkage estimation and selection for multiple functional regression (2011, background): “ suggests new approaches to the problems that have not been discovered before. Some recent developments in functional regression include Yao et al. (2005); Cai and Hall (2006); Crambes et al. (2009); Yuan and Cai (2010+). The literature contains an impressive range of functional analysis tools for v”
Regression imputation in the functional linear model with missing values in the response (2019, background): “Among this, the functional linear model has received a special attention (see [24, 4, 5, 3, 15, 10, 6, 30] for main references).”
Functional Regression (2014, unknown): “Yuan and Cai (2010) introduced a general reproducing kernel Hilbert space approach to functional linear regression regularized by roughness penalties, implemented using smoothing spline-like kernels.”