dctBasis2D function

Calculate a cosine basis representation for functional data on two- or three-dimensional domains

These functions calculate a tensor cosine basis representation for functional data on two- or three-dimensional domains based on a discrete cosine transformation (DCT) using the C-library fftw3

(http://www.fftw.org/). Coefficients under a given threshold are set to 0 to reduce complexity and for denoising.

dctBasis2D(funDataObject, qThresh, parallel = FALSE)

dctBasis3D(funDataObject, qThresh, parallel = FALSE)

Arguments

• funDataObject: An object of class funData

  containing the observed functional data samples and for which the basis representation is calculated.

• qThresh: A numeric with value in $[0,1]$, giving the quantile for thresholding the coefficients. See Details.

• parallel: Logical. If TRUE, the coefficients for the basis functions are calculated in parallel. The implementation is based on the foreach function and requires a parallel backend that must be registered before; see foreach for details. Defaults to FALSE.

Returns

• scores: A sparseMatrix of scores (coefficients) with dimension N x K, reflecting the weights $\theta_{mn}$ ($\theta_{mnk}$) for each basis function in each observation, where K is the total number of basis functions used. - B: A diagonal matrix, giving the norms of the different basis functions used (as they are orthogonal).

• ortho: Logical, set to FALSE, as basis functions are orthogonal, but in general not orthonormal.

• functions: NULL, as basis functions are known.

Details

Given the (discretized) observed functions $X_i$, the function dctBasis2D calculates a basis representation

of a two-dimensional function $X_i(s,t)$ in terms of (orthogonal) tensor cosine basis functions

with $c_m = \frac{1}{\sqrt{\pi}}$ for $m=0$ and $c_m = \sqrt{\frac{2}{\pi}}$ for $m=1,2,\ldots$

based on a discrete cosine transform (DCT).

If not thresholded (qThresh = 0), the function returns all non-zero coefficients $\theta_{mn}$ in the basis representation in a sparseMatrix (package Matrix) called scores. Otherwise, coefficients with

are set to zero, where $q$ is the qThresh-quantile of $|\theta_{mn}|$.

For functions $X_i(s,t,u)$ on three-dimensional domains, the function dctBasis3D calculates a basis representation

in terms of (orthogonal) tensor cosine basis functions

again with $c_m = \frac{1}{\sqrt{pi}}$ for $m=0$ and c("$c_m =\n$", "$\\sqrt{\\frac{2}{pi}}$") for $m=1,2,\ldots$ based on a discrete cosine transform (DCT). The thresholding works analogous as for the two-dimensional case.

Warning

If the C-library fftw3 is not available when the package MFPCA is installed, this function is disabled an will throw an error. For full functionality install the C-library fftw3 from http://www.fftw.org/ and reinstall MFPCA. This function has not been tested with ATLAS/MKL/OpenBLAS.

See Also

univDecomp, dct2D, dct3D