The concept of covariance of entropies can be understood as a way of quantifying how uncertainties in different signals vary together. Rather than monitoring each metric independently, the focus shifts to the relationships between sources of uncertainty. When signals that typically exhibit aligned behavior diverge, this can provide an early indicator of system anomalies.

While this terminology is uncommon, the underlying idea overlaps strongly with established constructs in information theory, particularly mutual information. Mutual information measures the reduction in uncertainty about one random variable given knowledge of another, and has been widely applied to anomaly detection and monitoring tasks.

Recent studies demonstrate several approaches that resonate with the notion of entropy covariance:

  • Mutual Information in Anomaly Detection
    Deep learning frameworks have been combined with mutual information matrices to improve anomaly detection in monitoring data, capturing nonlinear dependencies that traditional metrics miss (ScienceDirect, 2023).

  • Graph-Based Models
    Graph learning techniques have been employed where edges encode mutual information between time-series signals. These graph embeddings enable the detection of anomalies across multivariate systems (MDPI Electronics, 2024).

  • Mutual Information Matrices (PMIM)
    Projection methods based on Rényi entropy and mutual information matrices provide interpretable fault detection with low false alarm rates, highlighting the utility of entropy-based measures for system monitoring (arXiv, 2020).

  • Covariance-Based Approaches
    High-dimensional covariance estimation methods and Mahalanobis distance-based detectors continue to be effective in multivariate anomaly detection contexts, reflecting a parallel statistical tradition of modeling dependencies (Scholar’s Mine; ResearchGate, 2023).

Implications

Taken together, these results suggest that the idea of monitoring relationships between uncertainties is not only conceptually appealing but also supported by a substantial body of empirical work. While the term covariance of entropies is unconventional, its mathematical essence is closely related to mutual information and related dependency measures. The potential contribution lies in reframing these constructs for the governance of automated systems, where the operational role of relational monitoring may be as significant as the formal properties of the measures themselves.

Relation to Existing Principles

Although the terminology of covariance of entropies is uncommon, the underlying intent resonates with several established concepts:

  • Mutual Information (Shannon, 1948)
    Captures the reduction in uncertainty of one variable given knowledge of another. This is the closest mathematical construct, though typically applied to feature selection, channel capacity, or dependency estimation rather than governance.

  • Granger Causality and Transfer Entropy
    Widely used in time-series analysis to quantify directional influence between signals. These approaches emphasize prediction rather than governance.

  • Multivariate Anomaly Detection
    Contemporary research often employs mutual information matrices, covariance structures, or graph embeddings to identify system anomalies. The intent is usually fault detection, not higher-order self-regulation.

  • Cybernetics and Ashby’s Law of Requisite Variety (1950s)
    This principle states that effective regulation requires the regulator to have at least as much “variety” (possible states) as the system being regulated. Philosophically, this comes closest to the governance motivation implicit in entropy covariance.

Distinctive Intent

What remains relatively unexplored is the use of such measures not only for detection but also for governance: embedding the monitoring of relational uncertainty directly into the operational feedback loops of automated systems. In this sense, the intent behind covariance of entropies is not yet formalized as a principle or law in the literature, even though many of its mathematical components already exist.