Type: Package
Encoding: UTF-8
Title: Robust Singular Value Decomposition using Density Power Divergence
Version: 1.0.0
Date: 2021-10-23
Description: Computing singular value decomposition with robustness is a challenging task. This package provides an implementation of computing robust SVD using density power divergence (<doi:10.48550/arXiv.2109.10680>). It combines the idea of robustness and efficiency in estimation based on a tuning parameter. It also provides utility functions to simulate various scenarios to compare performances of different algorithms.
License: MIT + file LICENSE
Imports: Rcpp (≥ 1.0.5), MASS, stats, utils, matrixStats
LinkingTo: Rcpp, RcppArmadillo
RoxygenNote: 7.1.1
Suggests: knitr, rmarkdown, microbenchmark, pcaMethods
VignetteBuilder: knitr
URL: https://github.com/subroy13/rsvddpd
BugReports: https://github.com/subroy13/rsvddpd/issues
NeedsCompilation: yes
Packaged: 2021-10-27 10:45:18 UTC; subroy13
Author: Subhrajyoty Roy [aut, cre]
Maintainer: Subhrajyoty Roy <subhrajyotyroy@gmail.com>
Repository: CRAN
Date/Publication: 2021-10-27 14:30:02 UTC

Add outlier to matrix

Description

AddOutlier returns a matrix with outliers randomly added to a matrix given certain proportion of contamination

Usage

AddOutlier(X, proportion, value, seed = NULL, method = "element")

Arguments

X

matrix, to which outliers are added

proportion

numeric, proportion of elements, rows or columns to be contaminated. Must be between 0 and 1.

value

numeric, the outlying value to be used for contamination

seed

numeric, a seed to reproduce the randomization behaviour

method

character, must be one of the following:

  • "element" - For contaminating at random positions of the matrix

  • "row" - For contaminating an entire row of the matrix

  • "col" - For contaminating an entire column of the matrix

Value

A matrix with elements / rows / columns contaminated.

Note

Due to randomization, it is possible that the none of the entries of the matrix become contaminated. In that case, it is recommended to use different seed value.

Examples

X = matrix(1:20, nrow = 4, ncol = 5)
AddOutlier(X, 0.5, 10, seed = 1234)

Calculate optimal robustness parameter

Description

cv.alpha returns the optimal robustness parameter

Usage

cv.alpha(X, alphas = 10)

Arguments

X

matrix, whose singular value decomposition is required

alphas

numeric vector, vector of robustness parameters to try.

Value

A list containing

References

S. Roy, A. Basu and A. Ghosh (2021), A New Robust Scalable Singular Value Decomposition Algorithm for Video Surveillance Background Modelling https://arxiv.org/abs/2109.10680

Robust Singular Value Decomposition using Density Power Divergence

Description

rSVDdpd returns the singular value decomposition of a matrix with robust singular values in presence of outliers

Usage

rSVDdpd(
  X,
  alpha,
  nd = NA,
  tol = 1e-04,
  eps = 1e-04,
  maxiter = 100L,
  initu = NULL,
  initv = NULL
)

Arguments

X

matrix, whose singular value decomposition is required

alpha

numeric, robustness parameter between 0 and 1. See details for more.

nd

integer, must be lower than nrow(X) and ncol(X) both. If NA, defaults to min(nrow(X), ncol(X))

tol

numeric, a tolerance level. If the residual matrix has lower norm than this, then subsequent singular values will be taken as 0.

eps

numeric, a tolerance level for the convergence of singular vectors. If in subsequent iterations the singular vectors do not change its norm beyond this, then the iteration will stop.

maxiter

integer, upper limit to the maximum number of iterations.

initu

matrix, initializing vectors for left singular values. Must be of dimension nrow(X) \times min(nrow(X), ncol(X)). If NULL, defaults to random initialization.

initv

matrix, initializing vectors for right singular values. Must be of dimension ncol(X) \times min(nrow(X), ncol(X)). If NULL, defaults to random initialization.

Details

The usual singular value decomposition is highly prone to error in presence of outliers, since it tries to minimize the L_2 norm of the errors between the matrix X and its best lower rank approximation. While there is considerable effort to impose robustness using L_1 norm of the errors instead of L_2 norm, such estimation lacks efficiency. Application of density power divergence bridges the gap.

DPD(f|g) = \int f^{(1+\alpha)} - (1 + \frac{1}{\alpha}) \int f^{\alpha}g + \frac{1}{\alpha} \int g^{(1 + \alpha)}

The parameter alpha should be between 0 and 1, if not, then a warning is shown. Lower alpha means less robustness but more efficiency in estimation, while higher alpha means high robustness but less efficiency in estimation. The recommended value of alpha is 0.3. The function tries to obtain the best rank one approximation of a matrix by minimizing this density power divergence of the true errors with that of a normal distribution centered at the origin.

Value

A list containing different components of the decomposition X = U D V'

References

S. Roy, A. Basu and A. Ghosh (2021), A New Robust Scalable Singular Value Decomposition Algorithm for Video Surveillance Background Modelling https://arxiv.org/abs/2109.10680

See Also

svd

Examples

X = matrix(1:20, nrow = 4, ncol = 5)
rSVDdpd(X, alpha = 0.3)

Simulate SVD and measure performances of various algorithms

Description

simSVD simulates various models for the errors in the data matrix, and summarize performance of a singular value decomposition algorithm under presence or absence of outlying data introduced through various outlying schemes, using Monte Carlo approach.

Usage

simSVD(
  trueSVD,
  svdfun,
  B = 100,
  seed = NULL,
  dist = "normal",
  tau = 0.95,
  outlier = FALSE,
  out_method = "element",
  out_value = 10,
  out_prop = 0.1,
  return_details = FALSE,
  ...
)

Arguments

trueSVD

list, containing three different named components.

  • d - a vector containing the singular values.

  • u - a matrix with left singular vectors, each column being a singular vector.

  • v - a matrix with right singular vectors, each column being a singular vector.

svdfun

function which takes a numeric matrix as first argument and returns singular value decomposition of it as a list, with three components d, u and v as indicated before.

B

numeric, denoting the number of Monte Carlo simulation.

seed

numeric, a seed value used for reproducibility.

dist

character string, denoting the distribution from which errors will be generated. It must be equal to one of the following: normal, cauchy, exp, logis, lognormal

tau

numeric, a value between 0 and 1, see details for more.

outlier

logical, if TRUE, simulates the situation by adding outliers.

out_method

character, the method to add outliers. Must be one of "element", "row" or "col". See AddOutlier for details.

out_value

numeric, the outlying observation. See AddOutlier for details.

out_prop

a numeric, between 0 and 1 denoting the proportion of contamination. See AddOutlier for details.

return_details

logical, whether to return detailed results for each Monte Carlo simulation. See value for details.

...

extra arguments to be passed to svdfun function.

Value

Based on whether return_details is TRUE or FALSE, returns a list with two or one components.

If return_details is FALSE, only Summary component of the larger list is returned.