An extension of the  classical distance correlation coefficient for multivariate functional data with  applications

Tomasz Górecki; Mirosław  Krzyśko; Waldemar  Ratajczak; Waldemar  Wołyński

doi:https://doi.org/10.59170/stattrans-2016-026

An extension of the classical distance correlation coefficient for multivariate functional data with applications

Tomasz Górecki Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poland. Department of Statistics, Colorado State University, USA , Mirosław Krzyśko Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poland. , Waldemar Ratajczak Faculty of Geographical and Geological Sciences, Adam Mickiewicz University, Poland. , Waldemar Wołyński Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poland. Statistics in Transition new series, vol. 17, 2016, 3, pages: 449-466 Published online: 1 September 2016 https://doi.org/10.59170/stattrans-2016-026

190 Views 15 Downloads

ARTICLE

(English) PDF

ABSTRACT

The relationship between two sets of real variables defined for the same individuals can be evaluated by a few different correlation coefficients. For the functional data we have one important tool: canonical correlations. It is not immediately straightforward to extend other similar measures to the context of functional data analysis. In this work we show how to use the distance correlation coefficient for a multivariate functional case. The approaches discussed are illustrated with an application to some socio-economic data.

KEYWORDS

multivariate functional data, functional data analysis, correlation analysis

REFERENCES

ANDO, T., (2009). Penalized optimal scoring for the classification of multi-dimensional functional data. Statistcal Methodology 6, 565–576.

BERRENDERO, J. R., JUSTEL, A., SVARC, M., (2011). Principal components for multivariate functional data. Computational Statistics & Data Analysis 55(9),2619–2634.

BESSE, P., (1979). Étude descriptive d’un processus: Aproximation et interpola tion, Ph.D. thesis, Université Paul Sabatier, Toulouse III.

BEUTLER, F. J., (1970). Alias-free randomly timed sampling of stochastic pro cess. IEEE Transactions on Information Theory 16, 147–152.

ESCOUFIER, Y., (1970). Echantillonnage dans une population de variables aléa toires réelles, Ph.D thesis, Université des sciences et techniques du Languedoc,Montpellier.

ESCOUFIER, Y., (1973). Le traitement des variables vectorielles. Biometrics 29(4), 751–760.

ESCOUFIER, Y., ROBERT, P., (1979). Choosing variables and metrics by opti mizing the RV coefficient. In Optimizing Methods in Statistics, Rustagi, J.S.,Ed., Academic: New York, 205–219.

FERRATY, F., VIEU, P., (2003). Curve discrimination. A nonparametric functional approach. Computational Statistics & Data Analysis 44 161–173.

FERRATY, F., VIEU, P., (2009). Additive prediction and boosting for functional data. Computational Statistics & Data Analysis 53(4), 1400–1413.

GÓRECKI, T., KRZYSKO, M., WASZAK, Ł., WOŁY ´ NSKI, W., (2014). Methods ´of reducing dimension for functional data. Statistics in Transition new series 15(2), 231–242.

GÓRECKI, T., KRZYSKO, M., WOŁY ´ NSKI, W., (2015). Classification problems ´based on regression models for multi-dimensional functional data. Statistics in Transition new series 16(1), 97–110.

HASTIE, T. J., TIBSHIRANI, R. J., BUJA, A., (1995). Penalized discriminant analysis. Annals of Statistics 23, 73–102.

HE, G., MULLER, H. G., WANG, J. L., (2004). Methods of canonical analysis for functional data. Journal of Statistical Planning and Inference 122(1-2),141–159.

HOTELLING, H., (1936). Relation between two sets of variables. Biometrika 28(3/4), 321–377.

HORVÁTH, L., KOKOSZKA, P., (2012). Inference for Functional Data with Ap plications, Springer.

JAMES, G. M., (2002). Generalized linear models with functional predictors. Jour nal of the Royal Statistical Society 64(3), 411–432.

JACQUES, J., PREDA, C., (2014). Model-based clustering for multivariate func tional data. Computational Statistics & Data Analysis 71(C), 92–106.

JOSSE, J., HOLMES, S.,(2014). Tests of independence and beyond.arXiv:1307.7383v3.

KRZYSKO, M., WASZAK, Ł., (2013). Canonical correlation analysis for func- ´tional data. Biometrical Letters 50(2), 95–105.

LEE, A. J., (1976). On band-limited stochastic processes. SIAM Journal on Ap plied Mathematics 30, 169–177.

LEURGANS, S. E., MOYEED, R. A., SILVERMAN, B. W., (1993). Canonical correlation analysis when the data are curves. Journal of the Royal Statistical Society. Series B (Methodological) 55(3), 725–740.

MASRY, E., (1978). Poisson sampling and spectral estimation of continuous time processes. IEEE Transactions on Information Theory 24, 173–183.

MATSUI, H., ARAKI, Y., KONISHI, S., (2008). Multivariate regression modeling for functional data. Journal of Data Science 6, 313–331.

MENZEL, U., (2012). CCP: Significance Tests for Canonical Correlation Analysis (CCA). R package version 1.1. http://CRAN.R-project.org/package=CCP.

MÜLLER, H. G., STADMÜLLER, U., (2005). Generalized functional linear mod els. Annals of Statistics 33, 774–805.

R CORE TEAM (2015). R: A language and environment for statistical comput ing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R project.org/.

RAMSAY, J. O., SILVERMAN, B. W. (2005). Functional Data Analysis, Second Edition, Springer.

RAMSAY, J. O., WICKHAM, H., GRAVES, S., HOOKER, G., (2014). fda: Func tional Data Analysis. R package version 2.4.4.http://CRAN.R-project.org/package=fda.

REISS, P. T., OGDEN, R.T., (2007). Functional principal component regression and functional partial least squares. Journal of the American Statistcal Assosi ation 102(479), 984–996.

RIZZO, M. L., SZÉKELY, G. J., (2014). energy: E-statistics (energy statistics).R package version 1.6.2. http://CRAN.R-project.org/package=energy.

ROBERT, P., ESCOUFIER, Y., (1976). A unifying tool for linear multivariate statistical methods: the RV coefficient. Journal of the Royal Statistical Society.Series C (Applied Statistics) 25(3), 257–265.

ROSSI, F., DELANNAYC, N., CONAN-GUEZA, B., VERLEYSENC, M., (2005).Representation of functional data in neural networks. Neurocomputing 64,183–210.

ROSSI, F., VILLA, N., (2006). Support vector machines for functional data classi fication. Neural Computing 69, 730–742.

SZÉKELY, G. J., RIZZO, M. L., BAKIROV, N.K., (2007). Measuring and testing dependence by correlation of distances. The Annals of Statistics 35(6), 2769–2794.

SZÉKELY, G. J., RIZZO, M. L., (2009). Brownian distance covariance. Annals of Applied Statistics 3(4), 1236–1265.

SZÉKELY, G. J., RIZZO, M. L., (2012). On the uniqueness of distance covariance.Statistical Probability Letters 82(12), 2278–2282.

SZÉKELY, G. J., RIZZO, M. L., (2013). The distance correlation t-test of indepen dence in high dimension. Journal of Multivariate Analysis 117, 193–213