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