Functional correlation
In statistics, functional correlation is a dimensionality reduction technique used to quantify the correlation and dependence between two variables when the data is functional. Several approaches have been developed to quantify the relation between two functional variables. Overview A pair of real valued random functions and with , a compact interval, can be viewed as realizations of square-integrable stochastic process in a Hilbert space. Since both and are infinite dimensional, some kind of dimension reduction is required to explore their relationship. Notions of correlation for functional data include the following. Functional canonical correlation coefficient (FCCA) FCCA is a direct extension of multivariate canonical correlation. For a pair of random functions and the first canonical coefficient is defined as: where denotes the inner product in Lp space (p=2) i.e. The canonical coefficient , given is defined as: where is uncorrelated with all previous pairs . Thus FCCA implements projections in the directions of and for and respectively, such that their linear combinations (inner products) are maximally correlated. and are uncorrelated if all their canonical correlations are zero, equivalently, if and only if . Alternative formulation The cross-covariance operator for two random functions and defined as and analogously the auto covariance operators