# umpcaBasis: Calculate an uncorrelated multilinear principal component... In MFPCA: Multivariate Functional Principal Component Analysis for Data Observed on Different Dimensional Domains

## Description

This function calculates an uncorrelated multilinear principal component analysis (UMPCA) representation for functional data on two-dimensional domains. In this case, the data can be interpreted as images with ```S1 x S2``` pixels (assuming `nObsPoints(funDataObject) = (S1, S2)`), i.e. the total observed data are represented as third order tensor of dimension `N x S1 x S2`. The UMPCA of a tensor of this kind is calculated via the UMPCA function, which is an `R`-version of the analogous functions in the `UMPCA` MATLAB toolbox by Haiping Lu (Link: https://www.mathworks.com/matlabcentral/fileexchange/35432-uncorrelated-multilinear-principal-component-analysis-umpca, see also references).

## Usage

 `1` ```umpcaBasis(funDataObject, npc) ```

## Arguments

 `funDataObject` An object of class `funData` containing the observed functional data samples (here: images) for which the UMPCA is to be calculated. `npc` An integer, giving the number of principal components to be calculated.

## Value

 `scores` A matrix of scores (coefficients) with dimension `N x k`, reflecting the weight of each principal component in each observation. `B` A matrix containing the scalar product of all pairs of basis functions. `ortho` Logical, set to `FALSE`, as basis functions are not orthonormal. `functions` A functional data object, representing the functional principal component basis functions.

## Warning

As this algorithm aims more at uncorrelated features than at an optimal reconstruction of the data, hence it might give poor results when used for the univariate decomposition of images in MFPCA. The function therefore throws a warning.

## References

Haiping Lu, K.N. Plataniotis, and A.N. Venetsanopoulos, "Uncorrelated Multilinear Principal Component Analysis for Unsupervised Multilinear Subspace Learning", IEEE Transactions on Neural Networks, Vol. 20, No. 11, Page: 1820-1836, Nov. 2009.

`univDecomp`