ShrinkCovMat: Shrinkage Covariance Matrix Estimators

Installation

You can install the release version of ShrinkCovMat:

install.packages("ShrinkCovMat")

The source code for the release version of ShrinkCovMat is available on CRAN at:

• https://CRAN.R-project.org/package=ShrinkCovMat

Or you can install the development version of ShrinkCovMat:

# install.packages('devtools')
devtools::install_github("AnestisTouloumis/ShrinkCovMat")

The source code for the development version of ShrinkCovMat is available on github at:

• https://github.com/AnestisTouloumis/ShrinkCovMat

To use ShrinkCovMat, you should first load the package as follows:

library("ShrinkCovMat")

Usage

This package provides estimates of the covariance matrix and in particular, it implements the nonparametric Stein-type shrinkage covariance matrix estimators proposed in Touloumis (2015). These estimators are suitable and statistically efficient regardless of the dimensionality.

Each of the three implemented shrinkage covariance matrix estimates is a convex linear combination of the sample covariance matrix and of a target matrix. The core function is called shrinkcovmat and the argument target defines one of the following three options for the target matrix:

• the identity matrix (target = "identity"),
• the scaled identity matrix (target = "spherical"),
• the diagonal matrix with diagonal elements the corresponding sample variances (target = "diagonal").

Calculation of the corresponding optimal shrinkage intensities is discussed in Touloumis (2015).

The utility function targetselection is designed to ease the selection of the target matrix. This is based on empirical observation by inspecting the estimated optimal intensities and the range and average of the sample variances.

Example

Consider the colon cancer data example analyzed in Touloumis (2015). The data consists of two tissue groups: the normal tissue group and the tumor tissue group.

data(colon)
normal_group <- colon[, 1:40]
tumor_group <- colon[, 41:62]

To decide the target matrix for covariance matrix of the normal group, inspect the following output:

targetselection(normal_group)
#> ESTIMATED SHRINKAGE INTENSITIES WITH TARGET MATRIX THE 
#> Spherical matrix : 0.1401 
#> Identity  matrix : 0.1125 
#> Diagonal  matrix : 0.14 
#> 
#> SAMPLE VARIANCES 
#> Range   : 0.4714 
#> Average : 0.0882

The estimated optimal shrinkage intensity for the spherical matrix is slightly larger than the other two. In addition the sample variances appear to be of similar magnitude and their average is smaller than 1. Thus, the spherical matrix seems to be the most appropriate target for the covariance matrix. The resulting covariance matrix estimate is:

estimated_covariance_normal <- shrinkcovmat(normal_group, target = "spherical")
estimated_covariance_normal
#> SHRINKAGE ESTIMATION OF THE COVARIANCE MATRIX 
#> 
#> Estimated Optimal Shrinkage Intensity = 0.1401 
#> 
#> Estimated Covariance Matrix [1:5,1:5] =
#>        [,1]   [,2]   [,3]   [,4]   [,5]
#> [1,] 0.0396 0.0107 0.0101 0.0214 0.0175
#> [2,] 0.0107 0.0499 0.0368 0.0171 0.0040
#> [3,] 0.0101 0.0368 0.0499 0.0147 0.0045
#> [4,] 0.0214 0.0171 0.0147 0.0523 0.0091
#> [5,] 0.0175 0.0040 0.0045 0.0091 0.0483
#> 
#> Target Matrix [1:5,1:5] =
#>        [,1]   [,2]   [,3]   [,4]   [,5]
#> [1,] 0.0882 0.0000 0.0000 0.0000 0.0000
#> [2,] 0.0000 0.0882 0.0000 0.0000 0.0000
#> [3,] 0.0000 0.0000 0.0882 0.0000 0.0000
#> [4,] 0.0000 0.0000 0.0000 0.0882 0.0000
#> [5,] 0.0000 0.0000 0.0000 0.0000 0.0882

We follow a similar procedure for the tumor group:

targetselection(tumor_group)
#> ESTIMATED SHRINKAGE INTENSITIES WITH TARGET MATRIX THE 
#> Spherical matrix : 0.1956 
#> Identity  matrix : 0.1705 
#> Diagonal  matrix : 0.1955 
#> 
#> SAMPLE VARIANCES 
#> Range   : 0.4226 
#> Average : 0.0958

As before, we may choose the spherical matrix as the target matrix. The resulting covariance matrix estimate for the tumor group is:

estimated_covariance_tumor <- shrinkcovmat(tumor_group, target = "spherical")
estimated_covariance_tumor
#> SHRINKAGE ESTIMATION OF THE COVARIANCE MATRIX 
#> 
#> Estimated Optimal Shrinkage Intensity = 0.1956 
#> 
#> Estimated Covariance Matrix [1:5,1:5] =
#>        [,1]   [,2]   [,3]   [,4]   [,5]
#> [1,] 0.0490 0.0179 0.0170 0.0195 0.0052
#> [2,] 0.0179 0.0450 0.0265 0.0092 0.0034
#> [3,] 0.0170 0.0265 0.0465 0.0084 0.0031
#> [4,] 0.0195 0.0092 0.0084 0.0498 0.0036
#> [5,] 0.0052 0.0034 0.0031 0.0036 0.0361
#> 
#> Target Matrix [1:5,1:5] =
#>        [,1]   [,2]   [,3]   [,4]   [,5]
#> [1,] 0.0958 0.0000 0.0000 0.0000 0.0000
#> [2,] 0.0000 0.0958 0.0000 0.0000 0.0000
#> [3,] 0.0000 0.0000 0.0958 0.0000 0.0000
#> [4,] 0.0000 0.0000 0.0000 0.0958 0.0000
#> [5,] 0.0000 0.0000 0.0000 0.0000 0.0958

How to cite

To cite 'ShrinkCovMat' in publications, please use:

  Touloumis A. (2015). "Nonparametric Stein-type Shrinkage Covariance
  Matrix Estimators in High-Dimensional Settings." _Computational
  Statistics & Data Analysis_, *83*, 251-261.
  <https://www.sciencedirect.com/science/article/pii/S0167947314003107>.

A BibTeX entry for LaTeX users is

  @Article{,
    title = {Nonparametric Stein-type Shrinkage Covariance Matrix Estimators in High-Dimensional Settings},
    author = {{Touloumis A.}},
    year = {2015},
    journal = {Computational Statistics & Data Analysis},
    volume = {83},
    pages = {251-261},
    url = {https://www.sciencedirect.com/science/article/pii/S0167947314003107},
  }

References

Touloumis, A. (2015) Nonparametric Stein-type Shrinkage Covariance Matrix Estimators in High-Dimensional Settings. Computational Statistics & Data Analysis, 83, 251–261.