Limits {RMT4DS}

Limits in High-dimensional Sample Covariance

Description

Some limits of eigenvalues and eigenvectors in high-dimensional sample covariance.

Usage

MP_vector_dist(k, v, ndf=NULL, pdim, svr=ndf/pdim, cov=NULL)
cov_spike(spikes, eigens, ndf, svr)
quadratic(k, cov, svr, spikes, type=1)

Arguments

k

k-th eigenvector. In MP_vector_dist, k can be a serie.

v

vector to be projected on.

ndf

the number of degrees of freedom for the Wishart matrix.

pdim

the number of dimensions (variables) for the Wishart matrix.

svr

samples to variables ratio; the number of degrees of freedom per dimension.

cov

population covariace matrix. If it is null, it will be regarded as identity.

eigens

input eigenvalues of population covariance matrix without spikes.

spikes

spikes in population covariance matrix.

type

transformation of eigenvalues. n for n-th power. 0 for logarithm.

Details

In MP_vector_dist, the variance computed is for \sqrt{\code{pdim}}u_k^T v, where u_k is the k-th eigenvector.

Note in quadratic, k should be within the spikes.

Value

MP_vector_dist gives asymptotic variance of projection of eigenvectors of non-spiked Wishart matrix,

cov_spike gives spikes in sample covariance matrix and their asymptotic variance.

quadratic gives mean of certain quadratic forms of k-th sample eigenvector of spiked models. Note k should be within the spikes.

Author(s)

Xiucai Ding, Yichen Hu

References

[1] Knowles, A., & Yin, J. (2017). Anisotropic local laws for random matrices. Probability Theory and Related Fields, 169(1), 257-352.

[2] Jolliffe, I. (2005). Principal component analysis. Encyclopedia of statistics in behavioral science.

Examples

k = 1
n = 200
p = 100
v = runif(p)
v = v/sqrt(sum(v^2))
MP_vector_dist(k,v,n,p,cov=diag(p))
cov_spike(c(10),rep(1,p),n,n/p)
quadratic(k,diag(p),n/p,c(30))