A New Approach of Exploiting Self-Adjoint Matrix Polynomials of Large Random Matrices for Anomaly Detection and Fault Location

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Abstract
Synchronized measurements of a large power grid enable an unprecedented opportunity to study the spatial-temporal correlations. Statistical analytics for those massive datasets start with high-dimensional data matrices. Uncertainty is ubiquitous in a future's power grid. These data matrices are recognized as random matrices. This new point of view is fundamental in our theoretical analysis since true covariance matrices cannot be estimated accurately in a high-dimensional regime. As an alternative, we consider large-dimensional sample covariance matrices in the asymptotic regime to replace the true covariance matrices. The self-adjoint polynomials of large-dimensional random matrices are studied as statistics for big data analytics. The calculation of the asymptotic spectrum distribution (ASD) for such a matrix polynomial is understandably challenging. This task is made possible by a recent breakthrough in free probability, an active research branch in random matrix theory. This is the very reason why the work of this paper is inspired initially. The new approach is interesting in many aspects. The mathematical reason may be most critical. The real-world problems can be solved using this approach, however.
Funding Information
  • National Natural Science Foundation of China (61571296)
  • National Science Foundation (CNS-1619250)

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