Abstract
In the case of vector data, Gretton et al. (Algorithmic learning theory. Springer, Berlin, pp 63–77, 2005) defined Hilbert–Schmidt independence criterion, and next Cortes et al. (J Mach Learn Res 13:795–828, 2012) introduced concept of the centered kernel target alignment (KTA). In this paper we generalize these measures of dependence to the case of multivariate functional data. In addition, based on these measures between two kernel matrices (we use the Gaussian kernel), we constructed independence test and nonlinear canonical variables for multivariate functional data. We show that it is enough to work only on the coefficients of a series expansion of the underlying processes. In order to provide a comprehensive comparison, we conducted a set of experiments, testing effectiveness on two real examples and artificial data. Our experiments show that using functional variants of the proposed measures, we obtain much better results in recognizing nonlinear dependence.
Highlights
The theory and practice of statistical methods in situations where the available data are functions is often referred to as Functional Data Analysis (FDA)
We describe how Hilbert–Schmidt Independence Criterion (HSIC) can be used as an independence measure, and as the basis for an independence test
– KTA—centered kernel target alignment, – HSIC—Hilbert–Schmidt Independence Criterion, – FCCA—classical functional canonical correlation analysis (Ramsay and Silverman 2005; Horváth and Kokoszka 2012), – HSIC.FCCA—functional canonical correlation analysis based on HSIC, – HSIC.KTA—functional canonical correlation analysis based on KTA
Summary
The theory and practice of statistical methods in situations where the available data are functions (instead of real numbers or vectors) is often referred to as Functional Data Analysis (FDA). Based on these measures we constructed independence test and nonlinear canonical correlation variables for multivariate functional data. These results are based on the assumption that the applied kernel function is Gaussian. They are defined by such concepts as: kernel function alignment, kernel matrix alignment, and Hilbert–Schmidt Independence Criterion (HSIC) and associations between them have been shown. 5 we present kernel-based independence test for multivariate functional data. 5, based on the concept of alignment between kernel matrices, nonlinear canonical variables were constructed It is a generalization of the results of Chang et al (2013) for random vectors.
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