No effect tests in regression on functional variable and some applications to spectrometric studies

Recent advances in structural tests for regression on functional variable are used to construct test of no effect. Various bootstrap procedures are considered and compared in a simulation study. These tests are finally applied on real world datasets dealing with spectrometric studies using the information collected during this simulation study. The results obtained for the Tecator dataset are relevant and corroborated by former studies. The study of a smaller dataset concerning corn samples shows the efficiency of our method on small size samples. Getting information on which derivatives (or which parts) of the spectrometric curves have a significant effect allows to get a better understanding of the way spectrometric curves influence the quantity to predict. In addition, a better knowledge of the structure of the underlying regression model may be useful to construct a relevant predictor.

[1]  James Stephen Marron,et al.  Semiparametric Comparison of Regression Curves , 1990 .

[2]  L. Ferré,et al.  Multilayer Perceptron with Functional Inputs: an Inverse Regression Approach , 2006, 0705.0211.

[3]  Gareth M. James,et al.  Functional Adaptive Model Estimation , 2005 .

[4]  Ingrid Van Keilegom,et al.  On the Validity of the Bootstrap in Non‐Parametric Functional Regression , 2009 .

[5]  André Mas,et al.  Functional linear regression with derivatives , 2006 .

[6]  Valentin Patilea,et al.  Breaking the curse of dimensionality in nonparametric testing , 2008 .

[7]  Hans-Georg Müller,et al.  Functional Data Analysis , 2016 .

[8]  Frédéric Ferraty,et al.  Additive prediction and boosting for functional data , 2009, Comput. Stat. Data Anal..

[9]  Philippe Vieu,et al.  Structural test in regression on functional variables , 2011, J. Multivar. Anal..

[10]  Winfried Stute,et al.  Nonparametric model checks for regression , 1997 .

[11]  Isneri Talavera-Bustamante,et al.  Support Vector Regression Methods for Functional Data , 2007, CIARP.

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

[13]  Michel Verleysen,et al.  Representation of functional data in neural networks , 2005, Neurocomputing.

[14]  Ingrid Van Keilegom,et al.  A goodness-of-fit test for parametric and semi-parametric models in multiresponse regression , 2010, 1001.1667.

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

[16]  P. Vieu,et al.  Estimating Some Characteristics of the Conditional Distribution in Nonparametric Functional Models , 2006 .

[17]  Thomas Laloë.  A k-nearest neighbor approach for functional regression , 2008 .

[18]  P. Sarda,et al.  Testing for No Effect in Functional Linear Regression Models, Some Computational Approaches , 2004 .

[19]  T. Auton.  Applied Functional Data Analysis: Methods and Case Studies , 2004 .

[20]  R. Cao-Abad,et al.  Rate of Convergence for the Wild Bootstrap in Nonparametric Regression , 1991 .

[21]  E. Mammen,et al.  Comparing Nonparametric Versus Parametric Regression Fits , 1993 .

[22]  Wenceslao González-Manteiga,et al.  Statistics for Functional Data , 2007, Comput. Stat. Data Anal..

[23]  Frédéric Ferraty,et al.  Factor-based comparison of groups of curves , 2007, Comput. Stat. Data Anal..

[24]  A. Cuevas,et al.  On the Bootstrap Methodology for Functional Data , 2004 .

[25]  D. Bosq.  Linear Processes in Function Spaces: Theory And Applications , 2000 .

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

[27]  Ricardo Fraiman,et al.  On the use of the bootstrap for estimating functions with functional data , 2006, Comput. Stat. Data Anal..

[28]  H. Muller,et al.  Generalized functional linear models , 2005, math/0505638.

[29]  L. Ferré,et al.  Smoothed Functional Inverse Regression , 2005 .

[30]  Frédéric Ferraty,et al.  Dimension fractale et estimation de la régression dans des espaces vectoriels semi-normés , 2000 .

[31]  Sophie Lambert-Lacroix,et al.  Editorial of the Special Issue "Functional data" , 2014 .

[32]  I. Gijbels,et al.  Bandwidth Selection in Nonparametric Kernel Testing , 2008 .

[33]  Philippe Vieu,et al.  Semi-functional partial linear regression , 2006 .

[34]  Frédéric Ferraty,et al.  ON THE VALIDITY OF THE BOOTSTRAP IN NONPARAMETRIC FUNCTIONAL REGRESSION , 2008 .

[35]  B. K. Alsberg.  Representation of spectra by continuous functions , 1993 .

[36]  María Dolores Martínez Miranda,et al.  The choice of smoothing parameter in nonparametric regression through Wild Bootstrap , 2004, Comput. Stat. Data Anal..

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

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

[39]  F. Ferraty,et al.  The Oxford Handbook of Functional Data Analysis , 2011, Oxford Handbooks Online.

[40]  Winfried Stute,et al.  Bootstrap Approximations in Model Checks for Regression , 1998 .

[41]  R. Ignaccolo,et al.  Test of no-effect hypothesis by nonparametric regression , 2004 .

[42]  André Mas,et al.  Testing hypotheses in the functional linear model , 2003 .

[43]  Peter Hall,et al.  Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems , 1990 .

[44]  Wenceslao González-Manteiga,et al.  A note on variable selection in nonparametric regression with dependent data , 2002 .

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

[46]  Belén Fernández de Castro,et al.  Functional Samples and Bootstrap for Predicting Sulfur Dioxide Levels , 2005, Technometrics.

[47]  P. Hall.  On Bootstrap Confidence Intervals in Nonparametric Regression , 1992 .

[48]  B. Silverman,et al.  Canonical correlation analysis when the data are curves. , 1993 .

[49]  E. Mammen.  Bootstrap and Wild Bootstrap for High Dimensional Linear Models , 1993 .

[50]  David Hinkley,et al.  Bootstrap Methods: Another Look at the Jackknife , 2008 .

[51]  Laurent Delsol,et al.  Régression sur variable fonctionnelle : estimation, tests de structure et applications , 2008 .

[52]  Peter Hall,et al.  Bootstrap test for difference between means in nonparametric regression , 1990 .

[53]  Riccardo Leardi,et al.  Nature-Inspired Methods in Chemometrics: Genetic Algorithms and Artificial Neural Networks , 2005 .

[54]  Frédéric Ferraty,et al.  High-dimensional data: a fascinating statistical challenge , 2010, J. Multivar. Anal..

[55]  P. Sarda,et al.  Functional linear model , 1999 .