Methodology and convergence rates for functional linear regression
By Peter Hall, Joel L. Horowitz. 2007
717 citations recorded by the source corpus.
In functional linear regression, the slope "parameter" is a function. Therefore, in a nonparametric context, it is determined by an infinite number of unknowns. Its estimation involves solving an ill-posed problem and has points of contact with a range of methodologies, including statistical smoothing and deconvolution. The standard approach to estimating the slope function is based explicitly on functional principal components analysis and, consequently, on spectral decomposition in terms of eigenvalues and eigen-functions. We discuss this approach in detail and show that in certain circumstances, optimal convergence rates are achieved by the PCA technique. An alternative approach based on quadratic regularisation is suggested and shown to have advantages from some points of view.
Subfields: Artificial Intelligence, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/b5b4c3d12b0a85a0adfc1811145106f190a2b21d
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Citations in the corpus
- A survey of functional principal component analysis (2014, neutral): “( 2005b ) , Hall & Horowitz ( 2007 ) , Hall & Hosseini - Nasab ( 2006 ) , Hall & Vial ( 2006 ) , Hall et al .”
- Shrinkage estimation and selection for multiple functional regression (2011, methodology): “Since ξijk = ∫ (Xij − µj)φjk and ξ̂ijk = ∫ (Xij − X̄j)φ̂jk, we have |ξ̂ijk − ξijk|2 = Op(||X̄j − µj ||2 + ||φ̂jk − φjk||2) = Op(K2α+2/n), using assumption (c2) and equation (5.2) in Hall and Horowitz (2007).”
- Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains (2015, methodology): “For fully observed data, Hall and Horowitz (2007) show rΓ N = N −1/2, while the approach of Yao et al.”
- Functional Principal Component Regression and Functional Partial Least‐squares Regression: An Overview and a Comparative Study (2017, methodology): “Finally, Hall and Horowitz (2007) give optimal convergence rates of β̂S , assuming that kn tends to ∞ at a certain rate and certain conditions on the spacings of a1, a2, .”
- Truncated linear models for functional data (2014, background): “…Hall (2006) and Apanasovich and Goldstein (2008) addressed mean-squared prediction error in functional linear regression; and Cardot et al. (2007), Hall and Horowitz (2007), Li and Hsing (2007), Cai and Yuan (2012) and Johannes and Schenk (2013) discussed convergence rates of estimators of a and…”
- Asymptotics of prediction in functional linear regression with functional outputs (2009, methodology): “It remains to show that ensure switching T ∗ m,n and Tm,n as announced just below display (19) is possible.”
- Smoothing splines estimators for functional linear regression (2009, neutral): “In a recent work, Hall and Horowitz [18] derive optimal rates of convergence of ‖α̂ − α‖2.”
- Distribution-on-Distribution Regression via Optimal Transport Maps (2021, methodology): “They used the tangent space structure to define a regression operation: using the log transform, the regressor and response are lifted to suitable tangent spaces, where a (linear) regression model is defined in a more familiar Hilbertian setting [11, 4].”
- Peter Hall, functional data analysis and random objects (2016, background): “Convergence rates of estimates of the regression function β itself were derived in Hall and Horowitz (2007).”
- Partially functional linear regression in high dimensions (2016, methodology): “We impose mild conditions on the decay rates of the eigenvalues {w jk} and regression coefficients {b jk0}, similar to those adopted by Hall & Horowitz (2007) and Lei (2014).”
- Adaptive functional linear regression (2011, background): “For example, Hall and Horowitz [2007] and Crambes et al. [2009] consider differentiable slope functions and a polynomial decay of the eigenvalues of Γ .”
- A Goodness-of-Fit Test for the Functional Linear Model with Scalar Response (2012, positive): “The so-called functional principal component (FPC) regression estimation, proposed by Cardot, Ferraty, and Sarda (1999), provides an orthogonal data-driven basis that gives the most rapidly convergent representation of the functional dataset predictor in an L2 sense (see Hall and Horowitz 2007).”
- Functional Additive Models (2008, methodology): “It has been implemented through, for example, splines or ridge regression (Hall and Horowitz 2007).”
- On rates of convergence in functional linear regression (2007, methodology): “Another popular approach in functional linear model is the principal component regression (PCR) approach, which was considered by Hall and Horowitz [11], Cai and Hall [3], Müller and Stadtmüller [14], among others.”