Generalized Multivariate Gaussian Distribution in Machine Learning for Industrial Anomaly Detection

In the field of Machine Learning (ML), transitioning from univariate monitoring to a multivariate framework allows for a sophisticated analysis of complex industrial systems. While bivariate models evaluate two related sensors, the generalized multivariate Gaussian distribution (d x d) enables the inclusion of multiple critical variables—such as air pressure, vibration, temperature, and humidity—to establish a robust dynamic mathematical baseline for operational normality. This research presents a paradigm shift in industrial monitoring by transitioning from rigid, static threshold systems to a framework of probabilistic intelligence. By deriving the scaling factor and inverse covariance matrix for d-dimensions, the model can distinguish between benign stochastic fluctuations and genuine early-stage hardware malfunctions with high granularity. The study demonstrates that by understanding how multiple sensors dance together through covariance, systems can identify contextual anomalies that individually appear normal but statistically violate the learned operational relationship across the entire sensor network.