fcptpaBasis function

Calculate a smooth PCA representation for functional data on two-dimensional domains

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
)

Arguments

• funDataObject: An object of class funData

  containing the observed functional data samples (here: images) for which the smooth PCA is to be calculated.

• npc: An integer, giving the number of principal components to be calculated.

• smoothingDegree: 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).

• alphaRange: A list of length 2 with entries v and w

  containing the range of smoothness parameters to test for each direction.

• orderValues: Logical. If TRUE, the eigenvalues are ordered decreasingly, together with their associated eigenimages and scores. Defaults to TRUE.

• normalize: Logical. If TRUE the eigenfunctions are normalized to have norm 1. Defaults to FALSE.

Returns

• scores: A matrix of scores (coefficients) with dimension N x npc, reflecting the weights for principal component in each observation.

• B: A matrix containing the scalar product of all pairs of basis functions. - ortho: Logical, indicating whether the eigenfunctions are orthonormal. Set to normalize, as this influences whether a normalization is done or not. - functions: A functional data object, representing the functional principal component basis functions.

• values: A vector of length npc, containing the eigenvalues in decreasing order.

Details

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 c("$\\alpha_v, \n$", "$\\alpha_w$"). The resulting eigenvectors can be interpreted in terms of eigenfunctions and individual scores for each observation. See FCP_TPA for details.

References

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

See Also

univDecomp, FCP_TPA