eigTest: Jointly Estimate and Test for Common Eigenvectors

Intro & Setup

This R package is developed for testing simultaneous diagonalizability.

To install, use the code in R: devtools::install_github('XycYuchenXu/eigTest', force = T, build_vignettes = T)

Usage

The package has the following functionalities:

1. Check whether the means of two random square matrices (matrixA and matrixB) with dimension d-by-d are simultaneously diagonalizable, either considering the commutator of the two matrices (use function commutatorTest), or using the log-likelihood ratio test framework (projTest), given asymptotic limiting covariance matrices (covMatA and covMatB) and a convergence rate (cn). For both functions, one needs to input an array of two matrices (A such that A[1,,] = matrixA and A[2,,] = matrixB), an array of two limiting covariance matrices (cov.arr such that cov.arr[1,,] = covMatA and cov.arr[2,,] = covMatB), and the convergence rate (cn).

2. Given an array of square random matrices (A), estimate a common eigenvector matrix V (by supplying the array of matrices A for function JDTE).

3. Test whether the array of means matrices (A with dimension p-by-d-by-d and p ≥ 2) for random matrices are simultaneously diagonalizable, by checking whether the eigenvector matrix estimated above (V) is indeed the common eigenvector matrix. Use function eigTest with input of two arrays (A and cov.arr) and the convergence rate cn.

4. Given an array of square random matrices (A with dimension p-by-d-by-d), estimate k (k ≤ d) common eigenvectors Vk (by supplying the array of matrices A and parameter k for function expmPartSchur).

5. Test whether the array of the random estimates (A with dimension p-by-d-by-d and p ≥ 2) for mean matrices share k of the all d eigenvectors (k ≤ d), by checking whether the k estimated common eigenvectors Vk from optimization above are indeed the common eigenvectors. Use function partialTest with input of two arrays (A and cov.arr), the convergence rate cn, and number of common eigenvectors k. Vk is optional here as partialTest can call the optimization function.

6. Generate Gaussian samples for simulations. The function generateMeans is used to generate p mean matrices with dimension d-by-d and noise levels snr of common eigenvectors V. Function simuSamples takes the output from generateMeans as input for mean matrices. For each mean matrix, it generates cn^2 Gaussian random matrices (with identity covariance matrix), and compute the sample mean and the sample covariance matrix as consistent estimators. It then combines those estimates into a list of sub-lists that include estimated mean mu.bar and covariance cov.bar.

7. For more details, read the vignette: browseVignettes('eigTest') and the paper 'Testing Simultaneous Diagonalizability'.