R: Eigenfunctions via weighted, regularized SVD

fpc_wsvd {tf}

R Documentation

Eigenfunctions via weighted, regularized SVD

Description

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

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()

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.

Author(s)

Trevor Hastie, Rahul Mazumder, Cheng Meng, Fabian Scheipl

References

code adapted from / inspired by mogsa::wsvd() by Cheng Meng and softImpute::softImpute() by Trevor Hastie and Rahul Mazumder.
Meng C (2023). mogsa: Multiple omics data integrative clustering and gene set analysis. 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. R package version 1.4-1, https://CRAN.R-project.org/package=softImpute.