treelet (version 1.1)

TCS: Treelet Covariance Smoothing

Description

This function thresholds values in the treelet estimated covariance and returns a smoothed estimate of a covariance matrix.

Usage

TCS(basis, cov, lambda)

Arguments

basis

the orthonormal treelet basis calculated at a specific level $\ell$ of the tree.

cov

the corresponding covariance matrix calculated at level $\ell$ of the tree. The covariances in this matrix are those between the weights (orthogonal projections onto local basis vectors) in the basis expansion of the data vector.

lambda

a positive thresholding coefficient. Any element of the matrix cov that is less than this coefficient in absolute value will be set to zero.

Value

• smooth
  the smoothed estimate of the covariance matrix.

Details

This function implements the TCS method presented in the Crossett et al arXiv paper. The arguments basis and cov should be obtained from the Run_JTree function. The TCS function is written so that it does not calculate the treelet basis within the function but asks for it as an argument so that the subsampling method presented in the arXiv paper, or another method to obtain a reasonable value of lambda, can be implemented.

References

http://arxiv.org/abs/1208.2253{arXiv:1208.2253v1 [stat.AP]}

Lee, AB, Nadler, B, Wasserman, L (2008). Treelets - an adaptive multi-scale basis for sparse unordered data. The Annals of Applied Statistics 2: 435-471. http://www.stat.cmu.edu/~annlee/AOAS137.pdf

See Also

Build_JTree, JTree_Basis, Run_JTree

Examples

data(Ahat)
out=Run_JTree(Ahat,49,49)
basis=out$basis[[49]]
cov=out$TreeCovs[[49]]

temp=TCS(basis,cov,.04)
#The value .04 above is arbitrary, and the user 
#should carefully select this value.  One approach
#is the subsampling method outlined in the Crossett et al
#arXiv paper.  The value in 'temp' is the smoothed estimate
#of the relationship matrix.