Quantiles for finite and infinite dimensional data

A new projection-based definition of quantiles in a multivariate setting is proposed. This approach extends in a natural way to infinite-dimensional Hilbert spaces. The directional quantiles we define are shown to satisfy desirable properties of equivariance and, from an interpretation point of view, the resulting quantile contours provide valuable information when plotting them. Sample quantiles estimating the corresponding population quantiles are defined and consistency results are obtained. The new concept of principal quantile directions, closely related in some situations to principal component analysis, is found specially attractive for reducing the dimensionality and visualizing important features of functional data. Asymptotic properties of the empirical version of principal quantile directions are also obtained. Based on these ideas, a simple definition of robust principal components for finite and infinite-dimensional spaces is also proposed. The presented methodology is illustrated with examples throughout the paper.

[1]  Ricardo Fraiman,et al.  On depth measures and dual statistics. A methodology for dealing with general data , 2009, J. Multivar. Anal..

[2]  P. Hall,et al.  Properties of principal component methods for functional and longitudinal data analysis , 2006, math/0608022.

[3]  P. Bickel,et al.  Uniform Convergence of Probability Measures on Classes of Functions , 2008 .

[4]  J. Romo,et al.  On the Concept of Depth for Functional Data , 2009 .

[5]  Linglong Kong,et al.  Quantile tomography: using quantiles with multivariate data , 2008, Statistica Sinica.

[6]  Robert Serfling,et al.  Quantile functions for multivariate analysis: approaches and applications , 2002 .

[7]  Ricardo Fraiman,et al.  Robust estimation and classification for functional data via projection-based depth notions , 2007, Comput. Stat..

[8]  Juan Romo,et al.  Depth-based inference for functional data , 2007, Comput. Stat. Data Anal..

[9]  R. Koenker,et al.  Regression Quantiles , 2007 .

[10]  G. Boente.  Asymptotic theory for robust principal components , 1987 .

[11]  R. Serfling.  Multivariate Symmetry and Asymmetry , 2006 .

[12]  Rob J Hyndman,et al.  Rainbow Plots, Bagplots, and Boxplots for Functional Data , 2010 .

[13]  R. Serfling,et al.  General notions of statistical depth function , 2000 .

[14]  P. Hall,et al.  On properties of functional principal components analysis , 2006 .

[15]  M. Febrero,et al.  Outlier detection in functional data by depth measures, with application to identify abnormal NOx levels , 2008 .

[16]  Manuel Febrero-Bande,et al.  Statistical Computing in Functional Data Analysis: The R Package fda.usc , 2012 .

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

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

[19]  B. Ripley,et al.  Robust Statistics , 2018, Encyclopedia of Mathematical Geosciences.

[20]  Heng Tao Shen,et al.  Principal Component Analysis , 2009, Encyclopedia of Biometrics.

[21]  Regina Y. Liu,et al.  Multivariate analysis by data depth: descriptive statistics, graphics and inference, (with discussion and a rejoinder by Liu and Singh) , 1999 .

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

[23]  Stáephane Girard,et al.  A nonlinear PCA based on manifold approximation , 2000, Comput. Stat..

[24]  Rob J. Hyndman,et al.  Robust forecasting of mortality and fertility rates: A functional data approach , 2007, Comput. Stat. Data Anal..

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

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

[27]  D. Paindaveine,et al.  Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth , 2010, 1002.4486.

[28]  R. Fraiman,et al.  Trimmed means for functional data , 2001 .

[29]  J. A. Cuesta-Albertos,et al.  Random projections and goodness-of-fit tests in infinite-dimensional spaces , 2006 .

[30]  P. Chaudhuri.  On a geometric notion of quantiles for multivariate data , 1996 .

[31]  Wenceslao González-Manteiga,et al.  A functional analysis of NOx levels: location and scale estimation and outlier detection , 2007, Comput. Stat..

[32]  Patrick Billingsley,et al.  Uniformity in weak convergence , 1967 .