Canonical correlation analysis for functional data

Abstract

Classical canonical correlation analysis seeks the associations between two data sets, i.e. it searches for linear combinations of the original variables having maximal correlation. Our task is to maximize this correlation, and is equivalent to solving a generalized eigenvalue problem. The maximal correlation coefficient (being a solution of this problem) is the first canonical correlation coefficient. In this paper we propose a new method of constructing canonical correlations and canonical variables for a pair of stochastic processes represented by a finite number of orthonormal basis functions.

An Application of Functional Multivariate Regression Model to Multiclass Classification

“…More precisely, c imn are the random variables with finite variance (see Ramsay and Silverman, 2005). To estimate the coefficients c imn (for each predictor separately), the least squares method can be used (see, for instance, Krzyśko and Waszak, 2013). The selection method of the values B m may depend on the aim of the research.…”

“…For this reason, all discrete functional variables in a given data set were extended to the same length of the longest one by the method described and used, for example, in Górecki et al (2015) (see also Rodriguez et al, 2005). To obtain the basis functions representation (3) of the observations, the orthonormal Fourier basis and the least squares method of estimating the coefficients were used (see Krzyśko and Waszak, 2013). As we noted in Section 2, the quantities B m , m = 1, .…”

A note on variable selection in functional regression via random subspace method

“…. , p) can be represented as a linear combination of a finite number of basis functions, i.e., Krzyśko and Waszak 2013) or by the regularized maximum likelihood method (see Matsui et al 2008) or by the roughness penalty approach (see Ramsay and Silverman 2005, Chapter 5). The basis functions ϕ jl as well as the values k j may be selected depending on the data.…”

A comparison of tests for the one-way ANOVA problem for functional data

“…The vectors c i j can be estimated by the least squares method, and the optimum value of K in the sense of smoothness can be selected for each process (X i j (t), t ∈ T ) using the Bayesian Information Criterion (BIC); then from the values of K corresponding to all processes a modal value is selected as the common value for all Krzyśko and Waszak 2013;Górecki et al 2014, for more details). We should prefer K to be large, particularly when the stochastic processes (X i j (t), t ∈ T ) are observed at high frequency with little noise.…”

“…The orthonormal basis functions chosen in performing the FP, LH, R, P and W tests are the Fourier system (see Krzyśko and Waszak 2013). The optimal values of K as given in (4) were selected using the BIC from the set K = {3, 5, .…”