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 X ( t ) {\displaystyle \textstyle X(t)} and Y ( t ) {\displaystyle \textstyle Y(t)} with t ∈ T {\displaystyle t\in T} , a compact interval, can be viewed as realizations of square-integrable stochastic process in a Hilbert space. Since both X {\displaystyle X} and Y {\displaystyle Y} 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 X ∈ L 2 ( I X ) {\displaystyle X\in {\mathcal {L}}^{2}({\mathcal {I_{X}}})} and Y ∈ L 2 ( I Y ) {\displaystyle Y\in {\mathcal {L}}^{2}({\mathcal {I_{Y}}})} the first canonical coefficient ρ 1 {\displaystyle \rho _{1}} is defined as:
where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } denotes the inner product in Lp space (p=2) i.e.
⟨ f 1 , f 2 ⟩ = ∫ I f 1 ( t ) f 2 ( t ) d t , {\displaystyle \langle f_{1},f_{2}\rangle =\int _{\mathcal {I}}f_{1}(t)f_{2}(t)\,dt,} f 1 , f 2 ∈ L 2 ( I ) {\displaystyle \quad f_{1},f_{2}\in {\mathcal {L}}^{2}({\mathcal {I}})}
The k t h {\displaystyle k^{th}} canonical coefficient ρ k {\displaystyle \rho _{k}} , given ρ 1 , ρ 2 , … , ρ k − 1 {\displaystyle \rho _{1},\rho _{2},\ldots ,\rho _{k-1}} is defined as:
… excerpt ends here. Continue reading the full article.
