A Consistently Oriented Basis for Eigenanalysis: Improved Directional Statistics

TL;DR

This work tackles consistent orientation of eigenbases produced by $V$ from SVD/eigendecompositions, enabling robust directional statistics by embedding the rotation into a near-identity matrix $\\mathcal{V} = V S$. It introduces a modified arctan2-based method that allows $N-1$ rotation angles to span a full $2\\pi$ range, delaying reflections to the last irreducible subspace and using a cascade of Givens rotations. The approach yields a trade-off with the original arcsin method, offering improved angular disambiguation for time-evolving systems and better interpretability for directionality analyses, with distinct use cases for regression versus directional statistics. Empirical validation on financial data demonstrates that unwrapping angular wrap-around aligns informative eigenmodes with Random Matrix Theory predictions and enables both dynamic and static stabilization of eigenvectors, contributing practical tools for eigenvector analysis and correlation matrix cleaning. The work is implemented in the thucyd package and connected to broader themes in random matrix theory, including MP distributions and eigenvector stability, with clear guidance for practitioners on when to apply arcsin versus modified arctan2.

Abstract

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 π that exists in the original is extended to a 2π 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.

Figures (11)

• Figure 1: Two methods to orient a lefthanded basis in $\mathbb{R}^3$ onto the righthanded identity matrix $I$. The top row shows the orientation sequence reported in the original article wherein $v_2$ as it appears in pane (b) is reflected in order to have it point in the direction of $\pi_2$. The bottom rows show the new orientation sequence wherein rotation through a major angle replaces reflection in reducible subspaces while reflection is reserved for the last subspace, which is irreducible.
• Figure 2: The orientation of a 4D vector via the three Givens rotations in (\ref{['eigen::eq: Givens cascade to orient a-vec to 1-vec']}). The blue vector in pane (a) shows the orientation of $(a_1, a_2)$ in the $(\pi_1, \pi_2)$ plane. This vector is rotated into $\pi_1$, annihilating the $\pi_2$ component. Subsequent rotations, panes (b) and (c), annihilate the $\pi_3$ and $\pi_4$ components. Only in pane (a) does the vector point into the back hemisphere with respect to $\pi_1$.
• Figure 3: Rotation of a vector $\mathbf{a}$ through a major angle, as projected onto the $(\pi_1, \pi_2)$ plane, changes the hyper-/hemisphere in which the vector points. The same rotation also reorients the next vector in the system, here labeled $\mathbf{b}$. This effect can be summarized as modifying two adjacent entries in reflection matrix $S$.
• Figure 4: Hemispherical angles for quote and trade eigenvectors over 23 days and the six reducible modes. The angles were calculated using the arcsin method, and therefore they are all minor angles. The radii indicate the participation scores of the associated eigenvectors for each day: larger radii indicate better participation among the elements of an eigenvector.
• Figure 5: Overlay of empirical eigenvalue spectra with averaged Marčenko–Pastur distributions. (The colors here are used to distinguish the eigenmodes but do not correspond to the colors in the previous figures.) Top, modes 1–3 fall outside of the averaged MP distribution, indicating that their content has not been corrupted by sample noise. Bottom, only mode 1 falls well outside of the MP distribution while the rest lies firmly within the eigenvalue range that is indistinguishable from noise.