Permutation invariant Gaussian matrix models for financial correlation matrices

ArXiv ID: 2306.04569 “View on arXiv”

Authors: Unknown

Abstract

We construct an ensemble of correlation matrices from high-frequency foreign exchange market data, with one matrix for every day for 446 days. The matrices are symmetric and have vanishing diagonal elements after subtracting the identity matrix. For this case, we construct the general permutation invariant Gaussian matrix model, which has 4 parameters characterised using the representation theory of symmetric groups. The permutation invariant polynomial functions of the symmetric, diagonally vanishing matrices have a basis labelled by undirected loop-less graphs. Using the expectation values of the general linear and quadratic permutation invariant functions of the matrices in the dataset, the 4 parameters of the matrix model are determined. The model then predicts the expectation values of the cubic and quartic polynomials. These predictions are compared to the data to give strong evidence for a good overall fit of the permutation invariant Gaussian matrix model. The linear, quadratic, cubic and quartic polynomial functions are then used to define low-dimensional feature vectors for the days associated to the matrices. These vectors, with choices informed by the refined structure of small non-Gaussianities, are found to be effective as a tool for anomaly detection in market states: statistically significant correlations are established between atypical days as defined using these feature vectors, and days with significant economic events as recognized in standard foreign exchange economic calendars. They are also shown to be useful as a tool for ranking pairs of days in terms of their similarity, yielding a strongly statistically significant correlation with a ranking based on a higher dimensional proxy for visual similarity.

Keywords: Gaussian Matrix Model, Representation Theory, Correlation Matrices, Anomaly Detection, High-Frequency Data, Foreign Exchange

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 6.8/10
• Quadrant: Holy Grail
• Why: The paper employs advanced representation theory of symmetric groups and graph-based invariants to construct a 4-parameter matrix model, resulting in very high mathematical density. Empirically, it tests predictions against 446 days of high-frequency FX data, performs anomaly detection, and correlates with economic events, though the dataset size is limited and the scope is more theoretical than fully backtest-ready.

  flowchart TD
    A["Research Goal:<br>Model & Analyze FX<br>Correlation Matrices"] --> B["Data: High-Frequency<br>FX Data (446 Days)"]
    B --> C["Construct<br>Permutation Invariant<br>Gaussian Matrix Model"]
    C --> D["Fit 4 Parameters<br>using Linear/Quadratic<br>Moments"]
    D --> E["Predict<br>Cubic/Quartic Moments"]
    E --> F{"Compare<br>Predictions vs Data"}
    F -- Good Fit --> G["Key Findings:<br>Anomaly Detection<br>& Day Similarity Ranking"]
    F -- Poor Fit --> H["Reject Model"]