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Eigenfunctions via weighted, regularized SVD

Source: R/tfb-fpc-utils.R
fpc_wsvd.Rd

Compute (truncated) orthonormal eigenfunctions and scores for (partially missing) data on a common (potentially non-equidistant) grid.

Usage

fpc_wsvd(data, arg, pve = 0.995)

# S3 method for class 'matrix'
fpc_wsvd(data, arg, pve = 0.995)

# S3 method for class 'data.frame'
fpc_wsvd(data, arg, pve = 0.995)

Arguments

data

numeric matrix of function evaluations (each row is one curve, no NAs).

arg

numeric vector of argument values.

pve

percentage of variance explained.

Value

a list with entries

  • mu estimated mean function (numeric vector)

  • efunctions estimated FPCs (numeric matrix, columns represent FPCs)

  • scores estimated FPC scores (one row per observed curve)

  • npc how many FPCs were returned for the given pve (integer)

  • scoring_function a function that returns FPC scores for new data and given eigenfunctions, see tf:::.fpc_wsvd_scores for an example.

Details

Performs a weighted SVD with trapezoidal quadrature weights s.t. returned vectors represent (evaluations of) orthonormal eigenfunctions \(\phi_j(t)\), not eigenvectors \(\phi_j = (\phi_j(t_1), \dots, \phi_j(t_n))\), specifically:
\(\int_T \phi_j(t)^2 dt \approx \sum_i \Delta_i \phi_j(t_i)^2 = 1\) given quadrature weights \(\Delta_i\), not \(\phi_j'\phi_j = \sum_i \phi_j(t_i)^2 = 1\);
\(\int_T \phi_j(t) \phi_k(t) dt = 0\) not \(\phi_j'\phi_k = \sum_i \phi_j(t_i)\phi_k(t_i) = 0\), c.f. mogsa::wsvd().
For incomplete data, this uses an adaptation of softImpute::softImpute(), see references. Note that will not work well for data on a common grid if more than a few percent of data points are missing, and it breaks down completely for truly irregular data with no/few common timepoints, even if observed very densely. For such data, either re-evaluate on a common grid first or use more advanced FPCA approaches like refund::fpca_sc(), see last example for tfb_fpc()

References

code adapted from / inspired by mogsa::wsvd() by Chen Meng and softImpute::softImpute() by Trevor Hastie and Rahul Mazumder.
Meng C (2023). mogsa: Multiple omics data integrative clustering and gene set analysis. doi:10.18129/B9.bioc.mogsa , https://bioconductor.org/packages/mogsa.

Mazumder, Rahul, Hastie, Trevor, Tibshirani, Robert (2010). “Spectral Regularization Algorithms for Learning Large Incomplete Matrices.” The Journal of Machine Learning Research, 11, 2287–2322.

Hastie T, Mazumder R (2021). softImpute: Matrix Completion via Iterative Soft-Thresholded SVD. doi:10.32614/CRAN.package.softImpute , R package version 1.4-1, https://CRAN.R-project.org/package=softImpute.

See also

Other tfb-class: tfb, tfb_fpc(), tfb_spline()

Other tfb_fpc-class: tfb_fpc()

Author

Trevor Hastie, Rahul Mazumder, Chen Meng, Fabian Scheipl