Locally modelled regression and functional data

The general framework of this paper deals with the nonparametric regression of a scalar response on a functional variable (i.e. one observation can be a curve, surface, or any other object lying into an infinite-dimensional space). This paper proposes to model local behaviour of the regression operator (i.e. the link between a scalar response and an explanatory functional variable). To this end, one introduces a functional approach in the same spirit as local linear ideas in nonparametric regression. The main advantage of this functional local method is to propose an explicit expression of a kernel-type estimator which makes its computation easy and fast while keeping good predictive performance. Asymptotic properties are stated, and a functional data set illustrates the good behaviour of this functional locally modelled regression method.

[1]  B. Silverman,et al.  Functional Data Analysis , 1997 .

[2]  B. Presnell,et al.  Nonparametric estimation of the mode of a distribution of random curves , 1998 .

[3]  P. Vieu,et al.  k-Nearest Neighbour method in functional nonparametric regression , 2009 .

[4]  Mariano J. Valderrama,et al.  An overview to modelling functional data , 2007, Comput. Stat..

[5]  Amparo Baíllo,et al.  Local linear regression for functional predictor and scalar response , 2009, J. Multivar. Anal..

[6]  J. Dauxois,et al.  Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference , 1982 .

[7]  Z. Q. John Lu,et al.  Nonparametric Functional Data Analysis: Theory And Practice , 2007, Technometrics.

[8]  R. Fraiman,et al.  Kernel-based functional principal components ( , 2000 .

[9]  W. González-Manteiga,et al.  Local linear regression estimation of the variogram , 2003 .

[10]  H. H. Thodberg,et al.  Optimal minimal neural interpretation of spectra , 1992 .

[11]  E. Nadaraya.  On Non-Parametric Estimates of Density Functions and Regression Curves , 1965 .

[12]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[13]  Peter Hall,et al.  A Functional Data—Analytic Approach to Signal Discrimination , 2001, Technometrics.

[14]  Nicolas W. Hengartner,et al.  Bandwidth selection for local linear regression smoothers , 2002 .

[15]  J. Friedman,et al.  A Statistical View of Some Chemometrics Regression Tools , 1993 .

[16]  Anestis Antoniadis,et al.  Estimation and inference in functional mixed-effects models , 2007, Comput. Stat. Data Anal..

[17]  Felix Abramovich,et al.  Testing in mixed-effects FANOVA models , 2006 .

[18]  M. Forina,et al.  Multivariate calibration. , 2007, Journal of chromatography. A.

[19]  J. Fortiana,et al.  Local Linear Functional Regression Based on Weighted Distance-based Regression , 2008 .

[20]  Sophie Dabo-Niang,et al.  Functional and operatorial statistics , 2008 .

[21]  E. A. Sylvestre,et al.  Principal modes of variation for processes with continuous sample curves , 1986 .

[22]  J. Ramsay,et al.  Some Tools for Functional Data Analysis , 1991 .

[23]  T. Hastie,et al.  [A Statistical View of Some Chemometrics Regression Tools]: Discussion , 1993 .

[24]  Jianqing Fan.  Local Linear Regression Smoothers and Their Minimax Efficiencies , 1993 .

[25]  Germán Aneiros‐Pérez,et al.  Functional methods for time series prediction: a nonparametric approach , 2011 .

[26]  Thomas C. M. Lee,et al.  Bandwidth selection for local linear regression: A simulation study , 1999, Comput. Stat..

[27]  James O. Ramsay,et al.  Applied Functional Data Analysis: Methods and Case Studies , 2002 .

[28]  Jorge Barrientos-Marín.  Some practical problems of recent nonparametric procedures: testing, estimation and application , 2007 .

[29]  Jianqing Fan,et al.  Variable Bandwidth and Local Linear Regression Smoothers , 1992 .

[30]  ERROR-DEPENDENT SMOOTHING RULES IN LOCAL LINEAR REGRESSION , 2002 .

[31]  N. Altman,et al.  Variation in height acceleration in the Fels growth data , 1994 .

[32]  M. Wand,et al.  Multivariate Locally Weighted Least Squares Regression , 1994 .

[33]  J. Lafferty,et al.  Rodeo: Sparse, greedy nonparametric regression , 2008, 0803.1709.

[34]  Frédéric Ferraty,et al.  The Functional Nonparametric Model and Application to Spectrometric Data , 2002, Comput. Stat..

[35]  G. S. Watson,et al.  Smooth regression analysis , 1964 .

[36]  Q. Shao,et al.  Gaussian processes: Inequalities, small ball probabilities and applications , 2001 .

[37]  E. Nadaraya.  On Estimating Regression , 1964 .

[38]  H. Müller.  Weighted Local Regression and Kernel Methods for Nonparametric Curve Fitting , 1987 .

[39]  Mustapha Rachdi,et al.  Local smoothing regression with functional data , 2007, Comput. Stat..

[40]  Anestis Antoniadis,et al.  Bandwidth selection for functional time series prediction , 2009 .

[41]  Frédéric Ferraty,et al.  Nonparametric Functional Data Analysis: Theory and Practice (Springer Series in Statistics) , 2006 .

[42]  D. G. Simpson,et al.  Robust principal component analysis for functional data , 2007 .

[43]  G. Pflug.  Kernel Smoothing. Monographs on Statistics and Applied Probability - M. P. Wand; M. C. Jones. , 1996 .

[44]  Brani Vidakovic,et al.  Optimal Testing in a Fixed-Effects Functional Analysis of Variance Model , 2004, Int. J. Wavelets Multiresolution Inf. Process..