Eigenvalue Distribution of Empirical Correlation Matrices for Multiscale Complex Systems and Application to Financial Data

ArXiv ID: 2507.14325 “View on arXiv”

Authors: Luan M. T. de Moraes, Antônio M. S. Macêdo, Giovani L. Vasconcelos, Raydonal Ospina

Abstract

We introduce a method for describing eigenvalue distributions of correlation matrices from multidimensional time series. Using our newly developed matrix H theory, we improve the description of eigenvalue spectra for empirical correlation matrices in multivariate financial data by considering an informational cascade modeled as a hierarchical structure akin to the Kolmogorov statistical theory of turbulence. Our approach extends the Marchenko-Pastur distribution to account for distinct characteristic scales, capturing a larger fraction of data variance, and challenging the traditional view of noise-dressed financial markets. We conjecture that the effectiveness of our method stems from the increased complexity in financial markets, reflected by new characteristic scales and the growth of computational trading. These findings not only support the turbulent market hypothesis as a source of noise but also provide a practical framework for noise reduction in empirical correlation matrices, enhancing the inference of true market correlations between assets.

Keywords: Correlation Matrices, Marchenko-Pastur, Matrix H Theory, Turbulence, Noise Reduction, Equities

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper is densely packed with advanced mathematics, including random matrix theory, Wishart and inverse-Wishart distributions, Meijer G functions, and multivariate integrals, justifying a high math score. Empirically, it applies the theory to S&P 500 data to fit eigenvalue distributions, aiming to improve noise reduction in correlation matrices, which aligns with practical backtesting and implementation concerns, though it lacks explicit algorithmic code or trading strategies.

  flowchart TD
    A["Research Goal<br>Understand & model eigenvalue distribution of<br>empirical correlation matrices in complex systems<br>e.g., financial markets"] --> B["Methodology<br>Matrix H Theory<br>Extended Marchenko-Pastur (MP) distribution"]
    B --> C["Input Data<br>Multivariate Financial Time Series<br>Asset Returns"]
    C --> D["Computational Process<br>1. Compute Empirical Correlation Matrix<br>2. Apply Hierarchical (Turbulence) Model<br>3. Fit Extended MP Distribution"]
    D --> E["Key Finding 1<br>Turbulent Market Hypothesis<br>Markets behave like complex turbulent systems<br>creating characteristic scales"]
    D --> F["Key Finding 2<br>Effective Noise Reduction<br>Method captures more variance<br>improves true correlation inference"]
    E --> G["Outcome<br>Enhanced understanding of financial market structure<br>and improved correlation matrix estimation"]
    F --> G