cv.svd.gabriel: Cross-Validation for choosing the rank of an SVD approximation.

Description

Perform Wold- or Gabriel-style cross-validation for determining the appropriate rank SVD approximation of a matrix.

Usage

cv.svd.gabriel(
  x,
  krow = 2,
  kcol = 2,
  maxrank = floor(min(n - n/krow, p - p/kcol))
)
cv.svd.wold(x, k = 5, maxrank = 20, tol = 1e-04, maxiter = 20)

Value

call: the function call
msep: the mean square error of prediction (MSEP); this is a matrix whose columns contain the mean square errors in the predictions of the holdout sets for ranks 0, 1, ..., maxrank across the different replicates.
maxrank: the maximum rank for which prediction error is estimated; this is equal to nrow(msep)+1.
krow: the number of row folds (for Gabriel-style only).
kcol: the number of column folds (for Gabriel-style only).
rowsets: the partition of rows into krow holdout sets (for Gabriel-style only).
colsets: the partition of the columns into kcol holdout sets (for Gabriel-style only).
k: the number of folds (for Wold-style only).
sets: the partition of indices into k holdout sets (for Wold-style only).

Arguments

x: the matrix to cross-validate.
krow: the number of row folds (for Gabriel-style CV).
kcol: the number of column folds (for Gabriel-style CV).
maxrank: the maximum rank to cross-validate up to.
k: the number of folds (for Wold-style CV).
tol: the convergence tolerance for impute.svd.
maxiter: the maximum number of iterations for impute.svd.

Author

Patrick O. Perry

Details

These functions are for cross-validating the SVD of a matrix. They assume a model $X = U D V' + E$ with the terms being signal and noise, and try to find the best rank to truncate the SVD of x at for minimizing prediction error. Here, prediction error is measured as sum of squares of residuals between the truncated SVD and the signal part.

For both types of cross-validation, in each replicate we leave out part of the matrix, fit an SVD approximation to the left-in part, and measure prediction error on the left-out part.

In Wold-style cross-validation, the holdout set is "speckled", a random set of elements in the matrix. The missing elements are predicted using impute.svd.

In Gabriel-style cross-validation, the holdout set is "blocked". We permute the rows and columns of the matrix, and leave out the lower-right block. We use a modified Schur-complement to predict the held-out block. In Gabriel-style, there are krow*kcol total folds.

References

Gabriel, K.R. (2002). Le biplot - outil d'explaration de données multidimensionelles. Journal de la Société française de statistique, Volume 143 (2002) no. 3-4, pp. 5-55. B.

Owen, A.B. and Perry, P.O. (2009). Bi-cross-validation of the SVD and the non-negative matrix factorization. Annals of Applied Statistics 3(2) 564--594.

Wold, S. (1978). Cross-validatory estimation of the number of components in factor and principal components models. Technometrics 20(4) 397--405.

Examples

Run this code


  # generate a rank-2 matrix plus noise
  n <- 50; p <- 20; k <- 2
  u <- matrix( rnorm( n*k ), n, k )
  v <- matrix( rnorm( p*k ), p, k )
  e <- matrix( rnorm( n*p ), n, p )
  x <- u %*% t(v) + e
  
  # perform 5-fold Wold-style cross-validtion
  (cvw <- cv.svd.wold( x, 5, maxrank=10 ))
  
  # perform (2,2)-fold Gabriel-style cross-validation
  (cvg <- cv.svd.gabriel( x, 2, 2, maxrank=10 ))

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