2016 | 17 | 3 | 449-466
Article title

An Extension of the Classical Distance Correlation Coefficient for Multivariate Functional Data with Applications

Title variants
Languages of publication
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.
Physical description
  • Adam Mickiewicz University
  • 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 interpolation, Ph.D. thesis, Université Paul Sabatier, Toulouse III.
  • BEUTLER, F. J., (1970). Alias-free randomly timed sampling of stochastic process. IEEE Transactions on Information Theory 16, 147–152.
  • ESCOUFIER, Y., (1970). Echantillonnage dans une population de variables aléatoires 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 optimizing 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., KRZYŚKO, M.,WASZAK, Ł.,WOŁYŃSKI,W., (2014). Methods of reducing dimension for functional data. Statistics in Transition new series 15(2), 231–242.
  • GÓRECKI, T., KRZYŚKO, M., WOŁYŃSKI,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 Applications, Springer.
  • JAMES, G. M., (2002). Generalized linear models with functional predictors. Journal of the Royal Statistical Society 64(3), 411–432.
  • JACQUES, J., PREDA, C., (2014). Model-based clustering for multivariate functional data. Computational Statistics & Data Analysis 71(C), 92–106.
  • JOSSE, J., HOLMES, S.,(2014). Tests of independence and beyond. arXiv:1307.7383v3.
  • KRZY´SKO, M., WASZAK, Ł., (2013). Canonical correlation analysis for functional data. Biometrical Letters 50(2), 95–105.
  • LEE, A. J., (1976). On band-limited stochastic processes. SIAM Journal on Applied 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.
  • MÜLLER, H. G., STADMÜLLER, U., (2005). Generalized functional linear models. Annals of Statistics 33, 774–805.
  • R CORE TEAM (2015). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.
  • RAMSAY, J. O., SILVERMAN, B. W. (2005). Functional Data Analysis, Second Edition, Springer.
  • RAMSAY, J. O., WICKHAM, H., GRAVES, S., HOOKER, G., (2014). fda: Functional Data Analysis. R package version 2.4.4.
  • REISS, P. T., OGDEN, R.T., (2007). Functional principal component regression and functional partial least squares. Journal of the American Statistcal Assosiation 102(479), 984–996.
  • RIZZO, M. L., SZÉKELY, G. J., (2014). energy: E-statistics (energy statistics). R package version 1.6.2.
  • 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 classification. 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 independence in high dimension. Journal of Multivariate Analysis 117, 193–213.
Document Type
Publication order reference
YADDA identifier
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.