Statistics and Its Interface

Volume 12 (2019)

Number 4

A multivariate coefficient of variation for functional data

Pages: 647 – 658



Mirosław Krzysko (Interfaculty Institute of Mathematics and Statistics, Stanisław Wojciechowski State University of Applied Sciences, Kalisz, Poland)

Łukasz Smaga (Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland)


This paper considers an adaptation of the multivariate coefficient of variation to functional data. Similarly to the coefficient of variation and its multivariate generalizations, the functional multivariate coefficient of variation (FMCV) is useful in practical applications. Namely, it may be helpful for comparing the relative variation in different populations or the performance of different equipment characterized by univariate or multivariate functional data. Some theoretical properties of the new functional data analysis method are discussed. Using the basis function representation of the data, it is shown that the FMCV reduces to the multivariate coefficient of variation of a vector of coefficients of that representation. This enables effective computation of the FMCV. The performance of classical and robust estimators of the FMCV is compared in a finite sample setting using simulation studies. The new methods are illustrated on electrocardiography (ECG) data. These data are divided into two groups: normal and abnormal (representative of some cardiac pathology). The variability in the abnormal group is shown to be significantly greater than that in the normal group.


dispersion measure, functional data analysis, multivariate coefficient of variation, robust estimation, variability measure

2010 Mathematics Subject Classification

Primary 62H05. Secondary 62M99.

Received 25 September 2018

Received revised 24 May 2019

Accepted 30 May 2019

Published 18 July 2019