# TCS: Treelet Covariance Smoothing In treelet: An Adaptive Multi-Scale Basis for High-Dimensional, Sparse and Unordered Data

## Description

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

## Usage

 1 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.

## 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.

## Value

 smooth the smoothed estimate of the covariance matrix.

## Author(s)

Trent Gaugler [email protected]

## References

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

Build_JTree, JTree_Basis, Run_JTree
  1 2 3 4 5 6 7 8 9 10 11 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.