Jay Damask | Searcharxiv

arXiv2024

A Consistently Oriented Basis for Eigenanalysis: Improved Directional Statistics

The algorithm derived in this article, which builds upon the original paper, takes a holistic view of the handedness of an orthonormal eigenvector matrix so as to transfer what would have been labeled as a reflection in the original algorithm into a rotation through a major arc in the new algorithm. In so doing, the angular wrap-around on the interval {\pi} that exists in the original is extended to a 2{\pi} interval for primary rotations, which in turn provides clean directional statistics. The modified algorithm is detailed in this article and an empirical example is shown. The empirical example is analyzed in the context of random matrix theory, after which two methods are discussed to stabilize eigenvector pointing directions as they evolve in time. The thucyd Python package and source code, reported in the original paper, has been updated to include the new algorithm and is freely available.

math.NA↗

arXiv2019

A Consistently Oriented Basis for Eigenanalysis

Repeated application of machine-learning, eigen-centric methods to an evolving dataset reveals that eigenvectors calculated by well-established computer implementations are not stable along an evolving sequence. This is because the sign of any one eigenvector may point along either the positive or negative direction of its associated eigenaxis, and for any one eigen call the sign does not matter when calculating a solution. This work reports an algorithm that creates a consistently oriented basis of eigenvectors. The algorithm postprocesses any well-established eigen call and is therefore agnostic to the particular implementation of the latter. Once consistently oriented, directional statistics can be applied to the eigenvectors in order to track their motion and summarize their dispersion. When a consistently oriented eigensystem is applied to methods of machine-learning, the time series of training weights becomes interpretable in the context of the machine-learning model. Ordinary linear regression is used to demonstrate such interpretability. A reference implementation of the algorithm reported herein has been written in Python and is freely available, both as source code and through the thucyd Python package.

math.NA↗