This function calculates a smooth PCA representation based on the FCP_TPA
algorithm (see References) 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 data
for N observations can be represented as third order tensor of
dimension N x S1 x S2.
fcptpaBasis(
funDataObject,
npc,
smoothingDegree = rep(2, 2),
alphaRange,
orderValues = TRUE,
normalize = FALSE
)A matrix of scores (coefficients) with dimension N
x npc, reflecting the weights for principal component in each observation.
A matrix containing the scalar product of all pairs of basis functions.
Logical, indicating whether the eigenfunctions are
orthonormal. Set to normalize, as this influences whether a
normalization is done or not.
A functional data object, representing the functional principal component basis functions.
A vector of length npc, containing the eigenvalues in
decreasing order.
An object of class funData
containing the observed functional data samples (here: images) for which the
smooth PCA is to be calculated.
An integer, giving the number of principal components to be calculated.
A numeric vector of length 2, specifying the degree of
the difference penalties inducing smoothness in both directions of the
image. Defaults to 2 for each direction (2nd differences).
A list of length 2 with entries v and w
containing the range of smoothness parameters to test for each direction.
Logical. If TRUE, the eigenvalues are ordered
decreasingly, together with their associated eigenimages and scores.
Defaults to TRUE.
Logical. If TRUE the eigenfunctions are normalized to
have norm 1. Defaults to FALSE.
The smooth PCA of the tensor data is calculated via the FCP_TPA
function. Smoothness is induced by difference penalty matrices for both
directions of the images, weighted by smoothing parameters \(\alpha_v,
\alpha_w\). The resulting eigenvectors can be interpreted in terms of
eigenfunctions and individual scores for each observation. See
FCP_TPA for details.
G. I. Allen, "Multi-way Functional Principal Components Analysis", In IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2013.
univDecomp, FCP_TPA