Shrinkage Covariance Matrix Estimators

Description

Provides nonparametric Stein-type shrinkage estimators of the covariance matrix that are suitable and statistically efficient when the number of variables is larger than the sample size. These estimators are non-singular and well-conditioned regardless of the dimensionality.

Details

Each of the implemented shrinkage covariance matrix estimators is a convex linear combination of the sample covariance matrix and of a target matrix. Three options are considered for the target matrix: (a) the diagonal matrix with diagonal elements the average of the sample variances (shrinkcovmat.equal), (b) the diagonal matrix with diagonal elements the corresponding sample variances (shrinkcovmat.unequal), and (c) the identity matrix (shrinkcovmat.identity). The optimal shrinkage intensity determines how much the sample covariance matrix will be shrunk towards the selected target matrix. Estimation of the corresponding optimal shrinkage intensities is discussed in Touloumis (2015). The function targetselection is designed to ease the selection of the target matrix.

Author(s)

Anestis Touloumis

Maintainer: Anestis Touloumis <A.Touloumis@brighton.ac.uk>

References

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

See Also

Useful links:

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

• Report bugs at http://github.com/AnestisTouloumis/ShrinkCovMat/issues

Colon Cancer Dataset

Description

The dataset describes a colon cancer study (Alon et al., 1999) in which gene expression levels were measured on 40 normal tissues and on 22 tumor colon tissues. Note that a logarithmic (base 10) transformation has been applied to the gene expression levels.

Format

A data frame in which the rows correspond to 2000 genes and the columns to 62 tissues. The first 40 columns belong to the normal tissue group while the last 22 columns to the tumor colon tissue group.

Source

http://genomics-pubs.princeton.edu/oncology/affydata # nolint [Last Assessed: 2016-05-21]

References

Alon, U., Barkai, N., Notterman, D.A., Gish, K., Ybarra, S., Mack, D. and Levine, A.J. (1999) Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays. Proceedings of the National Academy of Sciences of the United States of America 96, 6745–6750.

Examples

data(colon)
summary(colon)

Shrinking the Sample Covariance Matrix Towards a Sphericity Matrix

Description

Provides a nonparametric Stein-type shrinkage estimator of the covariance matrix that is a linear combination of the sample covariance matrix and of a diagonal matrix with the average of the sample variances on the diagonal and zeros elsewhere.

Usage

shrinkcovmat.equal(data, centered = FALSE)

Arguments

Details

The rows of the data matrix data correspond to variables and the columns to subjects.

Value

Returns an object of the class 'shrinkcovmathat' that has components:

Author(s)

Anestis Touloumis

References

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

See Also

shrinkcovmat.unequal and shrinkcovmat.identity.

Examples

data(colon)
normal_group <- colon[, 1:40]
tumor_group <- colon[, 41:62]
sigma_hat_normal_group <- shrinkcovmat.equal(normal_group)
sigma_hat_normal_group
sigma_hat_tumor_group <- shrinkcovmat.equal(tumor_group)
sigma_hat_tumor_group

Shrinking the Sample Covariance Matrix Towards the Identity Matrix

Description

Provides a nonparametric Stein-type shrinkage estimator of the covariance matrix that is a linear combination of the sample covariance matrix and of the identity matrix.

Usage

shrinkcovmat.identity(data, centered = FALSE)

Arguments

Details

The rows of the data matrix data correspond to variables and the columns to subjects.

Value

Returns an object of the class 'shrinkcovmathat' that has components:

Author(s)

Anestis Touloumis

References

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

See Also

shrinkcovmat.equal and shrinkcovmat.unequal.

Examples

data(colon)
normal_group <- colon[, 1:40]
tumor_group <- colon[, 41:62]
sigma_hat_normal_group <- shrinkcovmat.identity(normal_group)
sigma_hat_normal_group
sigma_hat_tumor_group <- shrinkcovmat.identity(tumor_group)
sigma_hat_tumor_group

Shrinking the Sample Covariance Matrix Towards a Diagonal Matrix with Diagonal Elements the Sample Variances.

Description

Provides a nonparametric Stein-type shrinkage estimator of the covariance matrix that is a linear combination of the sample covariance matrix and of the diagonal matrix with elements the corresponding sample variances on the diagonal and zeros elsewhere.

Usage

shrinkcovmat.unequal(data, centered = FALSE)

Arguments

Details

The rows of the data matrix data correspond to variables and the columns to subjects.

Value

Returns an object of the class 'shrinkcovmathat' that has components:

Author(s)

Anestis Touloumis

References

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

See Also

shrinkcovmat.equal and shrinkcovmat.identity.

Examples

data(colon)
normal_group <- colon[, 1:40]
tumor_group <- colon[, 41:62]
sigma_hat_normal_group <- shrinkcovmat.unequal(normal_group)
sigma_hat_normal_group
sigma_hat_tumor_group <- shrinkcovmat.unequal(tumor_group)
sigma_hat_tumor_group

Target Matrix Selection

Description

Implements the rule of thumb proposed by Touloumis (2015) for target matrix selection. If the estimated optimal shrinkage intensities of the three target matrices are of similar magnitude, then the average and the range of the sample variances should be inspected in order to adopt the most plausible target matrix.

Usage

targetselection(data, centered = FALSE)

Arguments

Details

The rows of the data matrix data correspond to variables and the columns to subjects.

Value

Prints the estimated optimal shrinkage intensities, the range and average of the sample variances and returns an object of the class 'targetsel' that has components:

Author(s)

Anestis Touloumis

References

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

Examples

data(colon)
normal_group <- colon[, 1:40]
targetselection(normal_group)
## Similar intensities, the range of the sample variances is small and the
## average is not close to one. The scaled identity matrix seems to be the
## most suitable target matrix for the normal group.

tumor_group <- colon[, 41:62]
targetselection(tumor_group)
## Similar intensities, the range of the sample variances is small and the
## average is not close to one. The scaled identity matrix seems to be the
## most suitable target matrix for the colon group.