Smoothing splines estimators for functional linear regression

By Christophe Crambes, Alois Kneip, Pascal Sarda. 2009

322 citations recorded by the source corpus.

The paper considers functional linear regression, where scalar responses $Y_1,...,Y_n$ are modeled in dependence of random functions $X_1,...,X_n$. We propose a smoothing splines estimator for the functional slope parameter based on a slight modification of the usual penalty. Theoretical analysis concentrates on the error in an out-of-sample prediction of the response for a new random function $X_{n+1}$. It is shown that rates of convergence of the prediction error depend on the smoothness of the slope function and on the structure of the predictors. We then prove that these rates are optimal in the sense that they are minimax over large classes of possible slope functions and distributions of the predictive curves. For the case of models with errors-in-variables the smoothing spline estimator is modified by using a denoising correction of the covariance matrix of discretized curves. The methodology is then applied to a real case study where the aim is to predict the maximum of the concentration of ozone by using the curve of this concentration measured the preceding day.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7afd67c647c0432b056c82058d6ebda989434a4b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Statistics and Probability

1. Heng Lian
2. H. Muller
3. Peter Hall
4. Philippe Vieu
5. Jing Lei
6. Fang Yao
7. Han Lin Shang
8. Guang Cheng

Citations in the corpus

• Curse of dimensionality and related issues in nonparametric functional regression (2011, background): “[20, 23, 2, 22, 6, 3, 4, 7, 13] or [14] for a few paths in that direction.”
• Functional linear regression with functional response (2017, background): “Some recent references include Cardot, Ferraty, and Sarda (2003), Hall and Horowitz (2007), Horowitz and Lee (2007), Darolles, Fan, Florens and Renault (2011), Crambes, Kneip, and Sarda (2009), and Florens and Van Bellegem (2015).”
• Single and multiple index functional regression models with nonparametric link (2011, methodology): “We refer to the papers by [4, 5] and [7] for further discussion of these objectives in the context of the functional linear model.”
• Robust nonparametric estimation for functional data (2008, background): “6 for more precise works on the estimation of the functional parameter in the functional linear model.”
• Presmoothing in Functional Linear Regression (2012, methodology): “Common methods use basis systems such as Fourier series, wavelets, or splines (see a general review in Ramsay and Silverman (2005), and see Cardot, Ferraty, and Sarda (2003) or Crambes, Kneip, and Sarda (2009) for the popular splines basis).”
• On the Validity of the Bootstrap in Non‐Parametric Functional Regression (2009, methodology): “It is an interesting and complementary alternative to the functional linear model (see Cardot et al., 1999; Cai & Hall, 2006; Crambes et al., 2009).”
• Nonparametric inference in generalized functional linear models (2014, background): “[6], Yuan and Cai [35] and Cai and Yuan [4], among others, have proposed roughness regularization methods that circumvent the aforementioned perfect alignment requirement and allow one to regularize the model complexity in a continuous manner.”
• Truncated linear models for functional data (2014, methodology): “Cardot et al. (2003), and Zhang and Chen (2007), discussed the effect of smoothing on inference in the functional linear model; Crambes et al. (2008, 2009), and Maronna and Yohai (2013), introduced methods based on smoothing splines; Baı́llo (2009) suggested kernel techniques and made comparisons…”
• Continuously additive models for nonlinear functional regression (2013, methodology): “For continuous responses, the functional linear model is a standard tool and has been thoroughly investigated (see, e.g. Cai & Hall 2006; Cardot et al. 2007; Crambes et al. 2009).”
• Using basis expansions for estimating functional PLS regression Applications with chemometric data (2010, background): “Different spline estimators of the parameter function have been proposed in [5,6,8].”
• Shrinkage estimation and selection for multiple functional regression (2011, methodology): “To address these shortcomings of Lasso, Fan and Li (2001) proposed the smoothly clipped absolute deviation (SCAD) penalty and Zou (2006) proposed the adaptive lasso in the fixed p case using a weighted L1 penalty with weights determined by an initial estimator, and there are many other extensions of”
• Locally Sparse Estimator for Functional Linear Regression Models (2017, unknown): “Smooth estimators of β(t ) based on a direct roughness penalty onβ(t )were studied inCardot, Ferraty, and Sarda (2003), Li and Hsing (2007), and Crambes, Kneip, and Sarda (2009).”
• Local linear regression for functional data (2007, background): “Mas (2008) proved that � depends only on the sequence a(�) that de…nes X in ( 10 ).”