What PCA Can’t See: Detecting Sector Rotation with Bivector Component Analysis

2026-01-11 · Source: Agus’s Substack · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Advanced, long

Summary

Bivector Component Analysis (BCA) is introduced as a geometric-algebraic extension of Principal Component Analysis (PCA) designed to detect temporal order in multivariate time series, a capability PCA lacks. While PCA analyzes the symmetric part of a lagged moment matrix, BCA focuses on the antisymmetric part, which captures directional lead-lag relationships. For financial time series, the antisymmetric component can account for a significant portion of the total Frobenius energy, specifically 58.2% in the analyzed sector ETF data. The method decomposes an antisymmetric matrix into rotation planes, yielding rotation strengths and complex eigenvectors whose real and imaginary parts indicate leading and lagging directions. Applied to 11 U.S. sector ETFs from November 2020 to October 2025, BCA identified a clear growth-to-defensive rotation pattern, where sectors like Technology and Industrials lead, and Consumer Staples and Healthcare lag, often moving counter-directionally. This lead-lag structure remains stable across lags from 1 to 20 days, and its strength varies over time, potentially serving as a market regime indicator.

Key takeaway

For quantitative analysts and portfolio managers seeking to understand directional market dynamics, BCA offers a critical tool beyond traditional PCA. It reveals which sectors lead or lag, providing actionable insights into market rotation patterns that are invisible to symmetric analyses. Your investment strategies could benefit from incorporating BCA's output to anticipate shifts between growth and defensive sectors, potentially improving timing for tactical asset allocation or risk management in varying market regimes.

Key insights

BCA extends PCA to uncover temporal lead-lag relationships in multivariate data by analyzing the antisymmetric component of lagged moment matrices.

Principles

• Symmetry in PCA obscures temporal order.
• Antisymmetric matrix components reveal directional relationships.
• Geometric algebra provides a framework for oriented planes.

Method

BCA involves splitting the lagged moment matrix $M_\tau$ into symmetric $C_\tau$ and antisymmetric $B_\tau$ parts, then performing eigendecomposition on $B_\tau$ to extract rotation strengths and leading/lagging direction vectors.

In practice

• Use BCA to identify sector rotation patterns.
• Apply BCA to detect lead-lag relationships in financial assets.
• Monitor rolling BCA rotation strength as a market regime indicator.

Topics

• Bivector Component Analysis
• Principal Component Analysis
• Sector Rotation
• Financial Time Series
• Geometric Algebra

Code references

• asudjianto-xml/substack

Best for: Data Scientist, AI Data Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by Agus’s Substack.