Covariance and Correlation: From Factorization to Fitness Games
Covariance and correlation are foundational statistical tools that reveal how variables interact within data systems. At their core, covariance measures the degree to which two variables change together—positively, negatively, or not at all—across observations. Correlation extends this idea by normalizing covariance, producing a dimensionless value between -1 and 1 that quantifies both strength and direction of linear dependence. Together, they expose dependence structures invisible in raw data, guiding decisions in science, finance, and interactive design.
Mathematical Foundations: Eigenvalues, Matrices, and Probabilistic FoundationsIn linear algebra, eigenvalues derived from the characteristic equation det(A − λI) = 0 determine system stability and behavior in multivariate models. These λ values indicate how much a transformation stretches space along principal directions—critical in understanding variance distribution in datasets.