Trivariate Normal 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 a bivariate model simultaneously evaluates two related sensors, such as temperature and humidity, the trivariate normal distribution enables the inclusion of a third critical variable, such as air pressure, to establish a more 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 trivariate scaling factor and inverse covariance matrix, the model can distinguish between benign stochastic fluctuations and genuine early-stage hardware malfunctions with high granularity. The study demonstrates that by understanding how three sensors dance together through covariance, systems can identify contextual anomalies that individually appear normal but statistically violate the learned operational relationship.