findAlphaVopt: Find the optimal smoothing parameters in FCP_TPA using GCV

In ClaraHapp/MFPCA: Multivariate Functional Principal Component Analysis for Data Observed on Different Dimensional Domains

Description

These functions find the optimal smoothing parameters α_v, α_w for the two image directions (v and w) in the FCP_TPA algorithm based on generalized cross-validation, which is nested in the tensor power algorithm. Given a range of possible values of α_v (or α_w, respectively), the optimum is found by optimizing the GCV criterion using the function optimize.

Usage

findAlphaVopt(alphaRange, data, u, w, alphaW, OmegaW, GammaV, lambdaV)

findAlphaWopt(alphaRange, data, u, v, alphaV, OmegaV, GammaW, lambdaW)

Arguments

alphaRange	A numeric vector with two elements, containing the minimal and maximal value for the smoothing parameter that is to be optimized.
data	The tensor containing the data, an array of dimensions N x S1 x S2.
u, v, w	The current value of the eigenvectors u_k, v_k, w_k (not normalized) of dimensions N, S1 and S2.
GammaV, GammaW	A matrix of dimension S1 x S1 (GammaV in findAlphaVopt) or S2 x S2 (GammaW in findAlphaWopt), containing the eigenvectors of the penalty matrix for the image direction for which the optimal smoothing parameter is to be found.
lambdaV,	lambdaW A numeric vector of length S1(lambdaV in findAlphaVopt) or S2 (lambdaW in findAlphaWopt), containing the eigenvalues of the penalty matrix for the image direction for which the optimal smoothing parameter is to be found.
alphaV, alphaW	The current value of the smoothing parameter for the other image direction (α_w for findAlphaVopt and α_v for findAlphaWopt), which is kept as fixed.
OmegaV,	OmegaW A matrix of dimension S1 x S1 (OmegaV in findAlphaWopt) or S2 x S2 (OmegaW in findAlphaVopt), the penalty matrix for other image direction.

Value

The optimal α_v (or α_w, respectively), found by optimizing the GCV criterion within the given range of possible values.

Functions

• findAlphaWopt:

References

G. I. Allen (2013), "Multi-way Functional Principal Components Analysis", IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing.

J. Z. Huang, H. Shen and A. Buja (2009), "The Analysis of Two-Way Functional Data Using Two-Way Regularized Singular Value Decomposition". Journal of the American Statistical Association, Vol. 104, No. 488, 1609 – 1620.

See Also

FCP_TPA, gcv

ClaraHapp/MFPCA documentation built on Nov. 9, 2021, 11:35 p.m.