Economical Models for Functional Covariation - SAMSI

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Overview <strong>Functional</strong> PCA finite element basis Estimation Examples Extensions and discussion The functional factor analysis model Matrix Σ is now a bivariate function σ such that σ(s, t) is the covariance between the values of a sample or population of functional observations at times s and t. The factor analysis model is now ∫ σ(s, t) = λ(s, w)λ(t, w) dw + ψ(s, t) It may also be that the second term represents independent white noise contributions, so that ψ(s, t) = 0, s ≠ t
Overview <strong>Functional</strong> PCA finite element basis Estimation Examples Extensions and discussion The principal component function λ ∫ σ(s, t) = λ(s, w)λ(t, w) dw + ψ(s, t) Let’s assume observations defined over [0, T ]. But λ need not vary over [0, T ] as a function of w. In fact, we require that λ(s, w) = 0, w > s corresponding to the lower triangular matrix L ′ in the Choleski decomposition, Σ = L ′ L.
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Economical Models for Functional Covariation - SAMSI

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