Skip to contents

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 \(\phi_j'\phi_k = \sum_i \phi_j(t_i)\phi_k(t_i) = 0\).
For incomplete data, this uses a soft-impute iterative-SVD scheme (see references). Note that this 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

the soft-impute SVD algorithm for incomplete data is described in Mazumder, Rahul, Hastie, Trevor, Tibshirani, Robert (2010). “Spectral Regularization Algorithms for Learning Large Incomplete Matrices.” The Journal of Machine Learning Research, 11, 2287–2322.

See also

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

Other tfb_fpc-class: tfb_fpc(), tfb_mfpc()

Author

Fabian Scheipl

Examples

arg <- seq(0, 1, length.out = 41)
data <- t(sapply(seq(0, 2 * pi, length.out = 10),
                 function(p) sin(2 * pi * arg + p)))
fpc <- fpc_wsvd(data, arg = arg, pve = 0.99)
fpc$npc
#> [1] 2