Diagonally Dominant Principal Component Analysis

Description

Efficient procedures for fitting the DD-PCA (Ke et al., 2019, <arXiv:1906.00051>) by decomposing a large covariance matrix into a low-rank matrix plus a diagonally dominant matrix. The implementation of DD-PCA includes the convex approach using the Alternating Direction Method of Multipliers (ADMM) and the non-convex approach using the iterative projection algorithm. Applications of DD-PCA to large covariance matrix estimation and global multiple testing are also included in this package.

Details

The DESCRIPTION file:

Index of help topics:

DDHC                    DD-HC test
DDPCA_convex            Diagonally Dominant Principal Component
                        Analysis using Convex approach
DDPCA_nonconvex         Diagonally Dominant Principal Component
                        Analysis using Nonconvex approach
HCdetection             Higher Criticism for detecting rare and weak
                        signals
IHCDD                   IHC-DD test
ProjDD                  Projection onto the Diagonally Dominant Cone
ProjSDD                 Projection onto the Symmetric Diagonally
                        Dominant Cone
ddpca-package           Diagonally Dominant Principal Component
                        Analysis

This package contains DDPCA_nonconvex and DDPCA_convex function, which decomposes a positive semidefinite matrix into a low rank component, and a diagonally dominant component using either nonconvex approach or convex approach.

Note

Please cite the reference paper to cite this R package.

Author(s)

Tracy Ke [aut], Lingzhou Xue [aut], Fan Yang [aut, cre]

Maintainer: Fan Yang <fyang1@uchicago.edu>

References

Ke, Z., Xue, L. and Yang, F., 2019. Diagonally Dominant Principal Component Analysis. Journal of Computational and Graphic Statistics, under review.

DD-HC test

Description

Combining DDPCA with orthodox Higher Criticism for detecting sparse mean effect.

Usage

DDHC(X, known_Sigma = NA, method = "nonconvex", K = 1, lambda = 3, 
max_iter_nonconvex = 15 ,SDD_approx = TRUE, max_iter_SDD = 20, eps = NA, 
rho = 20, max_iter_convex = 50, alpha = 0.5, pvalcut = NA)

Arguments

Details

See reference paper for more details.

Value

Returns a list containing the following items

Author(s)

Fan Yang <fyang1@uchicago.edu>

References

Ke, Z., Xue, L. and Yang, F., 2019. Diagonally Dominant Principal Component Analysis. Journal of Computational and Graphic Statistics, under review.

See Also

IHCDD, HCdetection, DDPCA_convex, DDPCA_nonconvex

Examples

library(MASS)
n = 200
p = 200
k = 3
rho = 0.5
a = 0:(p-1)
Sigma_mu = rho^abs(outer(a,a,'-'))
Sigma_mu = (diag(p) + Sigma_mu)/2 # Now Sigma_mu is a symmetric diagonally dominant matrix
B = matrix(rnorm(p*k),nrow = p)
Sigma = Sigma_mu + B %*% t(B)
X = mvrnorm(n,rep(0,p),Sigma)
results = DDHC(X,K = k)
print(results$H)
print(results$HCT)

Diagonally Dominant Principal Component Analysis using Convex approach

Description

This function decomposes a positive semidefinite matrix into a low rank component, and a diagonally dominant component by solving a convex relaxation using ADMM.

Usage

DDPCA_convex(Sigma, lambda, rho = 20, max_iter_convex = 50)

Arguments

Details

This function decomposes a positive semidefinite matrix Sigma into a low rank component L and a symmetric diagonally dominant component A, by solving the following convex program

\textrm{minimize} \quad 0.5*\|\Sigma - L - A\|^2 + \lambda \|L\|_{*}

\textrm{subject to} \quad A\in SDD

where \|L\|_{*} is the nuclear norm of L (sum of singular values) and SDD is the symmetric diagonally dominant cone.

Value

A list containing the following items

Author(s)

Fan Yang <fyang1@uchicago.edu>

References

Ke, Z., Xue, L. and Yang, F., 2019. Diagonally Dominant Principal Component Analysis. Journal of Computational and Graphic Statistics, under review.

See Also

DDPCA_nonconvex

Examples

library(MASS)
p = 30
n = 30
k = 3
rho = 0.5
a = 0:(p-1)
Sigma_mu = rho^abs(outer(a,a,'-'))
Sigma_mu = (diag(p) + Sigma_mu)/2 # Now Sigma_mu is a symmetric diagonally dominant matrix
mu = mvrnorm(n,rep(0,p),Sigma_mu)
B = matrix(rnorm(p*k),nrow = p)
F = matrix(rnorm(k*n),nrow = k)
Y = mu + t(B %*% F) 
Sigma_sample = cov(Y)
result = DDPCA_convex(Sigma_sample,lambda=3)

Diagonally Dominant Principal Component Analysis using Nonconvex approach

Description

This function decomposes a positive semidefinite matrix into a low rank component, and a diagonally dominant component using an iterative projection algorithm.

Usage

DDPCA_nonconvex(Sigma, K, max_iter_nonconvex = 15, SDD_approx = TRUE, 
max_iter_SDD = 20, eps = NA)

Arguments

Details

This function performs iterative projection algorithm to decompose a positive semidefinite matrix Sigma into a low rank component L and a symmetric diagonally dominant component A. The projection onto the set of low rank matrices is done via eigenvalue decomposition, while the projection onto the symmetric diagonally dominant (SDD) cone is done via function ProjSDD, unless SDD_approx = TRUE where an approximation is used to speed up the algorithm.

Value

A list containing the following items

Author(s)

Fan Yang <fyang1@uchicago.edu>

References

Ke, Z., Xue, L. and Yang, F., 2019. Diagonally Dominant Principal Component Analysis. Journal of Computational and Graphic Statistics, under review.

See Also

DDPCA_convex

Examples

library(MASS)
p = 200
n = 200
k = 3
rho = 0.5
a = 0:(p-1)
Sigma_mu = rho^abs(outer(a,a,'-'))
Sigma_mu = (diag(p) + Sigma_mu)/2 # Now Sigma_mu is a symmetric diagonally dominant matrix
mu = mvrnorm(n,rep(0,p),Sigma_mu)
B = matrix(rnorm(p*k),nrow = p)
F = matrix(rnorm(k*n),nrow = k)
Y = mu + t(B %*% F) 
Sigma_sample = cov(Y)
result = DDPCA_nonconvex(Sigma_sample,K=k)

Higher Criticism for detecting rare and weak signals

Description

This function takes a bunch of p-values as input and ouput the Higher Criticism statistics as well as the decision (rejection or not).

Usage

HCdetection(p, alpha = 0.5, pvalcut = NA)

Arguments

Details

This function is an adaptation of the Matlab code here http://www.stat.cmu.edu/~jiashun/Research/software/HC/

Value

Returns a list containing the following items

Author(s)

Fan Yang <fyang1@uchicago.edu>

References

Donoho, D. and Jin, J., Higher criticism for detecting sparse heterogeneous mixtures. Ann. Statist. 32 (2004), no. 3, 962–994.

Ke, Z., Xue, L. and Yang, F., 2019. Diagonally Dominant Principal Component Analysis. Journal of Computational and Graphic Statistics, under review.

Examples

n = 1e5
data = rnorm(n)
p = 2*(1 - pnorm(abs(data)))
result = HCdetection(p)
print(result$H)
print(result$HCT)

IHC-DD test

Description

Combining Innovated Higher Criticism with DDPCA for detecting sparse mean effect.

Usage

IHCDD(X, method = "nonconvex", K = 1, lambda = 3, max_iter_nonconvex = 15, 
SDD_approx = TRUE, max_iter_SDD = 20, eps = NA, rho = 20, max_iter_convex = 50, 
alpha = 0.5, pvalcut = NA)

Arguments

Details

See reference paper for more details.

Value

Returns a list containing the following items

Author(s)

Fan Yang <fyang1@uchicago.edu>

References

Ke, Z., Xue, L. and Yang, F., 2019. Diagonally Dominant Principal Component Analysis. Journal of Computational and Graphic Statistics, under review.

See Also

DDHC, HCdetection, DDPCA_convex, DDPCA_nonconvex

Examples

library(MASS)
n = 200
p = 200
k = 3
rho = 0.5
a = 0:(p-1)
Sigma_mu = rho^abs(outer(a,a,'-'))
Sigma_mu = (diag(p) + Sigma_mu)/2 # Now Sigma_mu is a symmetric diagonally dominant matrix
B = matrix(rnorm(p*k),nrow = p)
Sigma = Sigma_mu + B %*% t(B)
X = mvrnorm(n,rep(0,p),Sigma)
results = IHCDD(X,K = k)
print(results$H)
print(results$HCT)

Projection onto the Diagonally Dominant Cone

Description

Given a matrix C, this function outputs the projection of C onto the cones of diagonally domimant matrices.

Usage

ProjDD(C)

Arguments

Details

This function projects the input matrix C of size n\times n onto the cones of diagonally domimant matrices defined as

\{A = (a_{ij})_{1\le i\le n, 1\le j\le n} : a_{jj} \ge \sum_{k\not=j} |a_{jk}| \quad \textrm{for all} \quad 1\le j\le n \}

The algorithm is described in Mendoza, M., Raydan, M. and Tarazaga, P., 1998. Computing the nearest diagonally dominant matrix.

Value

A n\times n diagonally dominant matrix

Author(s)

Fan Yang <fyang1@uchicago.edu>

References

Mendoza, M., Raydan, M. and Tarazaga, P., 1998. Computing the nearest diagonally dominant matrix. Numerical linear algebra with applications, 5(6), pp.461-474.

Ke, Z., Xue, L. and Yang, F., 2019. Diagonally Dominant Principal Component Analysis. Journal of Computational and Graphic Statistics, under review.

See Also

ProjSDD

Examples

ProjDD(matrix(runif(100),nrow=10))

Projection onto the Symmetric Diagonally Dominant Cone

Description

Given a matrix C, this function outputs the projection of C onto the cones of symmetric diagonally domimant matrices using Dykstra's projection algorithm.

Usage

ProjSDD(A, max_iter_SDD = 20, eps = NA)

Arguments

Details

This function projects the input matrix C of size n\times n onto the cones of symmetric diagonally domimant matrices defined as

\{A = (a_{ij})_{1\le i\le n, 1\le j\le n} : a_{ij} = a_{ji}, a_{jj} \ge \sum_{k\not=j} |a_{jk}| \quad \textrm{for all} \quad 1\le j\le n, 1\le i\le n \}

It makes use of Dykstra's algorithm, which is a variation of iterative projection algorithm. The two key steps are projection onto the diagonally domimant cone by calling function ProjDD and projection onto the symmetric matrix cone by simple symmetrization.

More details can be found in Mendoza, M., Raydan, M. and Tarazaga, P., 1998. Computing the nearest diagonally dominant matrix.

Value

A n\times n symmetric diagonally dominant matrix

Author(s)

Fan Yang <fyang1@uchicago.edu>

References

Mendoza, M., Raydan, M. and Tarazaga, P., 1998. Computing the nearest diagonally dominant matrix. Numerical linear algebra with applications, 5(6), pp.461-474.

Ke, Z., Xue, L. and Yang, F., 2019. Diagonally Dominant Principal Component Analysis. Journal of Computational and Graphic Statistics, under review.

See Also

ProjDD

Examples

ProjSDD(matrix(runif(100),nrow=10))