| Type: | Package |
| Title: | Eigenvectors from Eigenvalues Sparse Principal ComponentAnalysis (EESPCA) |
| Version: | 0.8.0 |
| Maintainer: | H. Robert Frost <rob.frost@dartmouth.edu> |
| Description: | Contains logic for computing sparse principal components via the EESPCA method, which is based on an approximation of the eigenvector/eigenvalue identity. Includes logic to support execution of the TPower and rifle sparse PCA methods, as well as logic to estimate the sparsity parameters used by EESPCA, TPower and rifle via cross-validation to minimize the out-of-sample reconstruction error. H. Robert Frost (2021) <doi:10.1080/10618600.2021.1987254>. |
| Depends: | R (≥ 3.6.0), rifle (≥ 1.0.0), MASS, PMA |
| License: | GPL-2 |GPL-3 [expanded from: GPL (≥ 2)] |
| Copyright: | Dartmouth College |
| Encoding: | UTF-8 |
| NeedsCompilation: | no |
| Packaged: | 2025-07-21 17:34:17 UTC; d37329b |
| Author: | H. Robert Frost [aut, cre] |
| Repository: | CRAN |
| Date/Publication: | 2025-07-21 17:50:08 UTC |
Eigenvectors
Description
Implementation of Eigenvectors from Eigenvalues Sparse Principal Component Analysis (EESPCA).
Details
| Package: | EESPCA |
| Type: | Package |
| Version: | 0.7.0 |
| Date: | 2021 |
| License: | GPL-2 |
Note
This work was supported by the National Institutes of Health grants K01LM012426, R21CA253408, P20GM130454,and P30CA023108.
Author(s)
H. Robert Frost
References
Frost, H. R. (2022). Eigenvectors from Eigenvalues Sparse Principal Component Analysis (EESPCA). Journal of Computational and Graphical Statistics.
Approximates the normed squared eigenvector loadings
Description
Approximates the normed squared eigenvector loadings using a simplified version of the formulaassociating normed squared eigenvector loadings with the eigenvalues of the full matrix and sub-matrices.
Usage
computeApproxNormSquaredEigenvector(cov.X, v1, lambda1, max.iter=5, lambda.diff.threshold=1e-6, trace=FALSE)Arguments
cov.X | Covariance matrix. |
v1 | Principal eigenvector of |
lambda1 | Largest eigenvalue of |
max.iter | Maximum number of iterations for power iteration method when computing sub-matrix eigenvalues.See description |
lambda.diff.threshold | Threshold for exiting the power iteration calculation. See description |
trace | True if debugging messages should be displayed during execution. |
Value
Vector of approximate normed squared eigenvector loadings.
See Also
Examples
set.seed(1) # Simulate 10x5 MVN data matrix X=matrix(rnorm(50), nrow=10) # Estimate covariance matrix cov.X = cov(X) # Compute eigenvectors/values eigen.out = eigen(cov.X) v1 = eigen.out$vectors[,1] lambda1 = eigen.out$values[1] # Print true squared loadings v1^2 # Compute approximate normed squared eigenvector loadings computeApproxNormSquaredEigenvector(cov.X=cov.X, v1=v1, lambda1=lambda1)Calculates the residual matrix from the reduced rank reconstruction
Description
Utility function for computing the residual matrix formed by subtracting fromX a reduced rank approximation of matrixX generated from the top k principal components contained in matrixV.
Usage
computeResidualMatrix(X,V,center=TRUE)Arguments
X | An n-by-p data matrix whose top k principal components are contained in the p-by-k matrix |
V | A p-by-k matrix containing the loadings for the top k principal components of |
center | If true (the default), |
Value
Residual matrix.
Examples
set.seed(1) # Simulate 10x5 MVN data matrix X=matrix(rnorm(50), nrow=10) # Perform PCA prcomp.out = prcomp(X) # Get rank 2 residual matrix computeResidualMatrix(X=X, V=prcomp.out$rotation[,1:2])Eigenvectors from Eigenvalues Sparse Principal Component Analysis (EESPCA)
Description
Computes the first sparse principal component of the specified data matrix using the Eigenvectors from Eigenvalues Sparse Principal Component Analysis (EESPCA) method.
Usage
eespca(X, max.iter=20, sparse.threshold, lambda.diff.threshold=1e-6, compute.sparse.lambda=FALSE, sub.mat.max.iter=5, trace=FALSE)Arguments
X | An n-by-p data matrix for which the first sparse PC will be computed. |
max.iter | Maximum number of iterations for power iteration method. See |
sparse.threshold | Threshold on loadings used to induce sparsity. Loadings below this value are set to 0. If not specified, defaults to |
lambda.diff.threshold | Threshold for exiting the power iteration calculation. If the absolute relative difference in lambda is less than this threshold between subsequent iterations,the power iteration method is terminated. See |
compute.sparse.lambda | If true, the sparse loadings will be used to compute the sparse eigenvalue. |
sub.mat.max.iter | Maximum iterations for computation of sub-matrix eigenvalues usingthe power iteration method. To maximize performance, set to 1. Uses the same lambda.diff.threshold. |
trace | True if debugging messages should be displayed during execution. |
Value
Alist with the following elements:
"v1": The first non-sparse PC as calculated via power iteration.
"lambda1": The variance of the first non-sparse PC as calculated via power iteration.
"v1.sparse": First sparse PC.
"lambda1.sparse": Variance of the first sparse PC. NA if compute.sparse.lambda is FALSE.
"ratio": Vector of ratios of the sparse to non-sparse PC loadings.
References
Frost, H. R. (2021). Eigenvectors from Eigenvalues Sparse Principal Component Analysis (EESPCA). arXiv e-prints. https://arxiv.org/abs/2006.01924
See Also
eespcaForK,computeApproxNormSquaredEigenvector,powerIteration
Examples
set.seed(1) # Simulate 10x5 MVN data matrix X=matrix(rnorm(50), nrow=10) # Compute first sparse PC loadings using default threshold eespca(X=X)Cross-validation for Eigenvectors from Eigenvalues Sparse Principal Component Analysis (EESPCA)
Description
Performs cross-validation of EESPCA to determine the optimal sparsity threshold. Selection is based on the minimization of reconstruction error.Based on the cross-validation approach of Witten et al. as implemented by theSPC.cv method in the PMA package.
Usage
eespcaCV(X, max.iter=20, sparse.threshold.values, nfolds=5, lambda.diff.threshold=1e-6, compute.sparse.lambda=FALSE, sub.mat.max.iter=5, trace=FALSE)Arguments
X | See description for |
max.iter | See description for |
sparse.threshold.values | Vector of threshold values to evaluate via cross-validation. See description for |
nfolds | Number of cross-validation folds. |
lambda.diff.threshold | See description for |
compute.sparse.lambda | See description for |
sub.mat.max.iter | See description for |
trace | See description for |
Value
Alist with the following elements:
"cv": The mean of the out-of-sample reconstruction error computed for each threshold.
"cv.error": The standard deviations of the means of the out-of-sample reconstruction error computed for eachthreshold.
"best.sparsity": Threshold value with the lowest mean reconstruction error.
"best.sparsity.1se": Threshold value whose mean reconstruction error is within 1 standard error of the lowest.
"nonzerovs": Mean number of nonzero values for each threshold.
"sparse.threshold.values": Tested threshold values.
"nfolds": Number of cross-validation folds.
References
Frost, H. R. (2021). Eigenvectors from Eigenvalues Sparse Principal Component Analysis (EESPCA). arXiv e-prints. https://arxiv.org/abs/2006.01924
Witten, D. M., Tibshirani, R., and Hastie, T. (2009). A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics, 10(3), 515-534.
See Also
Examples
set.seed(1) # Simulate 10x5 MVN data matrix X=matrix(rnorm(50), nrow=10) # Generate range of threshold values to evaluate default.threshold = 1/sqrt(5) threshold.values = seq(from=.5*default.threshold, to=1.5*default.threshold, length.out=10) # Use 5-fold cross-validation to estimate optimal sparsity threshold eespcaCV(X=X, sparse.threshold.values=threshold.values)Multi-PC version of Eigenvectors from Eigenvalues Sparse Principal Component Analysis (EESPCA)
Description
Computes multiple sparse principal components of the specified data matrix via sequential application ofthe Eigenvectors from Eigenvalues Sparse Principal Component Analysis (EESPCA) algorithm.After computing the first sparse PC via theeespca function, subsequent sparse PCs are computing by repeatedly applyingeespca to the residual matrix formedby subtracting the reconstruction ofX from the originalX. Multiple sparse PCs are not guaranteed to be orthogonal.
Note that the accuracy of the sparse approximation declines substantially for PCs with very smallvariances. To avoid this issue,k should not be set higher than the number of statistically significant PCs according to a Tracey-Widom test.
Usage
eespcaForK(X, k=2, max.iter=20, sparse.threshold, lambda.diff.threshold=1e-6, compute.sparse.lambda=FALSE, sub.mat.max.iter=5, trace=FALSE)Arguments
X | An n-by-p data matrix for which the first |
k | The number of sparse PCs to compute. The specified k must be 2 or greater (for k=1, usethe |
max.iter | See description for |
sparse.threshold | See description for |
lambda.diff.threshold | See description for |
compute.sparse.lambda | See description for |
sub.mat.max.iter | See description for |
trace | See description for |
Value
Alist with the following elements:
"V": Matrix of sparse loadings for the first k PCs.
"lambdas": Vector of variances of the first k sparse PCs.
References
Frost, H. R. (2021). Eigenvectors from Eigenvalues Sparse Principal Component Analysis (EESPCA). arXiv e-prints. https://arxiv.org/abs/2006.01924
See Also
Examples
set.seed(1) # Simulate 10x5 MVN data matrix X=matrix(rnorm(50), nrow=10) # Get first two sparse PCs eespcaForK(X=X, sparse.threshold=1/sqrt(5), k=2)Power iteration method for calculating principal eigenvector and eigenvalue.
Description
Computes the principal eigenvector and eigenvalue of the specified matrix using the power iteration method.Includes support for truncating the estimated eigenvector on each iteration to retain just thek eigenvector loadings with the largest absolute values with all other values set to 0,i.e., the the TPower method by Yuan & Zhang.
Usage
powerIteration(X, k, v1.init, max.iter=10, lambda.diff.threshold=1e-6, trace=FALSE)Arguments
X | Matrix for which the largest eigenvector and eigenvalue will be computed. |
k | If specified, the estimated eigenvector is truncated on each iteration to retainonly the k loadings with the largest absolute values, all other loadings are set to 0. Must be an integer between 1 and ncol(X). |
v1.init | If specified, the power iteration calculation will be initialized using this vector, otherwise,the calculation will be initialized using a unit vector with equal values. |
max.iter | Maximum number of iterations for power iteration method. |
lambda.diff.threshold | Threshold for exiting the power iteration calculation. If the absolute relative difference in computed eigenvalue is less than this threshold between subsequent iterations,the power iteration method is terminated. |
trace | True if debugging messages should be displayed during execution. |
Value
Alist with the following elements:
"v1": The principal eigenvector of
X."lambda": The largest eigenvalue of
X."num.iter": Number of iterations of the power iteration method before termination.
References
Yuan, X.-T. and Zhang, T. (2013). Truncated power method for sparse eigenvalue problems. J. Mach. Learn. Res., 14(1), 899-925.
See Also
Examples
set.seed(1) # Simulate 10x5 MVN data matrix X=matrix(rnorm(50), nrow=10) # Compute sample covariance matrix cov.X = cov(X) # Use power iteration to get first PC loadings using default initial vector powerIteration(X=cov.X)Calculates the reduced rank reconstruction
Description
Utility function for computing the reduced rank reconstruction ofX using the PC loadings inV.
Usage
reconstruct(X,V,center=TRUE)Arguments
X | An n-by-p data matrix whose top k principal components are contained the p-by-k matrix |
V | A p-by-k matrix containing the loadings for the top k principal components of |
center | If true (the default), |
Value
Reduced rank reconstruction of X.
Examples
set.seed(1) # Simulate 10x5 MVN data matrix X=matrix(rnorm(50), nrow=10) # Perform PCA prcomp.out = prcomp(X) # Get rank 2 reconstruction reconstruct(X, prcomp.out$rotation[,1:2])Calculates the reduced rank reconstruction error
Description
Utility function for computing the squared Frobenius norm of the residual matrix formed by subtracting fromX a reduced rank approximation of matrixX generated from the top k principal components contained in matrixV.
Usage
reconstructionError(X,V,center=TRUE)Arguments
X | An n-by-p data matrix whose top k principal components are contained the p-by-k matrix |
V | A p-by-k matrix containing the loadings for the top k principal components of |
center | If true (the default), |
Value
The squared Frobenius norm of the residual matrix.
Examples
set.seed(1) # Simulate 10x5 MVN data matrix X=matrix(rnorm(50), nrow=10) # Perform PCA prcomp.out = prcomp(X) # Get rank 2 reconstruction error, which will be the minimum since the first 2 PCs are used reconstructionError(X, prcomp.out$rotation[,1:2]) # Use all PCs to get approximately 0 reconstruction error reconstructionError(X, prcomp.out$rotation)Computes the initial eigenvector for the rifle method of Tan et al.
Description
Computes the initial eigenvector for the rifle method of Tan et al. (as implemented by therifle method in the rifle R package) using theinitial.convex method from the rifle package with lambda=sqrt(log(p)/n) and K=1.
Usage
rifleInit(X)Arguments
X | n-by-p data matrix to be evaluated via PCA. |
Value
Initial eigenvector to use with rifle method.
References
Tan, K. M., Wang, Z., Liu, H., and Zhang, T. (2018). Sparse generalized eigenvalue problem: optimal statistical rates via truncated rayleigh flow. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80(5), 1057-1086.
See Also
riflePCACV,rifle,initial.convex
Examples
set.seed(1) # Simulate 10x5 MVN data matrix X=matrix(rnorm(50), nrow=10) # Compute initial eigenvector to use with rifle method v1.init = rifleInit(X) # Use with rifle method to get first PC loadings with 2 non-zero elements rifle(A=cov(X), B=diag(5), init=v1.init, k=2)Sparsity parameter selection via cross-validation for rifle method of Tan et al.
Description
Sparsity parameter selection for PCA-based rifle (as implemented by therifle method in the rifle package) using the cross-validation approach of Witten et al. as implemented by theSPC.cv method in the PMA package.
Usage
riflePCACV(X, k.values, nfolds=5)Arguments
X | n-by-p data matrix being evaluated via PCA. |
k.values | Set of truncation parameter values to evaluate via cross-validation. Values must be between 1 and p. |
nfolds | Number of folds for cross-validation |
Value
k value that generated the smallest cross-validation error.
References
Tan, K. M., Wang, Z., Liu, H., and Zhang, T. (2018). Sparse generalized eigenvalue problem: optimal statistical rates via truncated rayleigh flow. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80(5), 1057-1086.
Witten, D. M., Tibshirani, R., and Hastie, T. (2009). A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics, 10(3), 515-534.
See Also
Examples
set.seed(1) # Simulate 10x5 MVN data matrix X=matrix(rnorm(50), nrow=10) # Generate range of k values to evaluate k.values = 1:5 # Use 5-fold cross-validation to estimate optimal k value riflePCACV(X=X, k.values=k.values)Implementation of the Yuan and Zhang TPower method.
Description
Implements the TPower method by Yuan and Zhang. Specifically, it computes the sparse principal eigenvector using power iteration method where the estimated eigenvector is truncated on each iteration to retain just thek eigenvector loadings with the largest absolute values with all other values set to 0.
Usage
tpower(X, k, v1.init, max.iter=10, lambda.diff.threshold=1e-6, trace=FALSE)Arguments
X | Matrix for which the largest eigenvector and eigenvalue will be computed. |
k | Must be an integer between 1 and ncol(X). The estimated eigenvector is truncated on each iteration to retainonly the k loadings with the largest absolute values, all other loadings are set to 0. |
v1.init | If specified, the power iteration calculation will be initialized using this vector, otherwise,the calculation will be initialized using a unit vector with equal values. |
max.iter | Maximum number of iterations for power iteration method. |
lambda.diff.threshold | Threshold for exiting the power iteration calculation. If the absolute relative difference in computed eigenvalues is less than this threshold between subsequent iterations,the power iteration method is terminated. |
trace | True if debugging messages should be displayed during execution. |
Value
The estimated sparse principal eigenvector.
References
Yuan, X.-T. and Zhang, T. (2013). Truncated power method for sparse eigenvalue problems. J. Mach. Learn. Res., 14(1), 899-925.
See Also
Examples
set.seed(1) # Simulate 10x5 MVN data matrix X=matrix(rnorm(50), nrow=10) # Compute first sparse PC loadings with 2 non-zero elements tpower(X=cov(X), k=2)Sparsity parameter selection for the Yuan and Zhang TPower method using cross-validation.
Description
Sparsity parameter selection for PCA-based TPower using the cross-validation approach of Witten et al. as implemented by theSPC.cv method in the PMA package.
Usage
tpowerPCACV(X, k.values, nfolds=5)Arguments
X | n-by-p data matrix being evaluated via PCA. |
k.values | Set of truncation parameter values to evaluate via cross-validation. Values must be between 1 and p. |
nfolds | Number of folds for cross-validation |
Value
k value that generated the smallest cross-validation error.
References
Yuan, X.-T. and Zhang, T. (2013). Truncated power method for sparse eigenvalue problems. J. Mach. Learn. Res., 14(1), 899-925.
Witten, D. M., Tibshirani, R., and Hastie, T. (2009). A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics, 10(3), 515-534.
See Also
Examples
set.seed(1) # Simulate 10x5 MVN data matrix X=matrix(rnorm(50), nrow=10) # Generate range of k values to evaluate k.values = 1:5 # Use 5-fold cross-validation to estimate optimal k value tpowerPCACV(X=X, k.values=k.values)