Principal Component Analysis of Functional Data based on Constant Numerical Characteristics
In
1 2022, vol. RNTI-E-38, pp.138-154
Résumé
A new approach to principal component analysis (PCA) is proposed
for functional data. In prevailing methods of functional principal component
analysis (FPCA), the definition of a mean is in the form of a function. However,
data centralisation based on this kind of mean actually obtains a residual
function. The result of FPCA, given its matrix of residual functions, may thus
fail to present the essential variation of the original data. Besides, applications in
FPCA are mainly for types of one sample problems. Numerical characteristics of
functional data are defined as real constants. Centralisation in terms of constant
numerical characteristics implies the relocation of the entire matrix of functional
variances in order to obtain original curves whose centres of gravity are settled
on the origin. Furthermore, based on the covariance matrix obtained from constant
numerical characteristics, functional principal components for multivariate
sample problems are proposed. Conclusions are validated by simulation in a real
situation.