Four points form a square scatter whose mean is (1,1); centered vectors combine into outer products to build a scatter matrix of 4I, revealing the difference between maximum-likelihood (divide by n) and unbiased (divide by n-1) covariance estimates. The animation then shows how covariance matrices I, diag(4,1), and [[3,2],[2,3]] stretch and rotate Gaussian contour ellipses, and demonstrates Mahalanobis distance by comparing two equally Euclidean-distant points that differ sharply once covariance is accounted for. Useful for undergraduates studying probability, statistics, or machine learning foundations.
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# Covariance gives a Gaussian cloud its shape Four points at (0,0), (2,0), (0,2), and (2,2) move into view on fixed axes. Their mean is (1,1). Show centered vectors and accumulate their outer products to obtain a scatter matrix of 4 times the identity. Dividing by four gives the maximum-likelihood covariance I; dividing by three gives the unbiased sample estimate 4I/3. Next compare Gaussian contours with covariances I, diag(4,1), and [[3,2],[2,3]]. Their orientation and axis lengths must follow the matrices; the last ellipse has principal-axis ratio sqrt(5) to 1. Introduce Mahalanobis distance through two points, (2,0) and (0,2), under zero mean and covariance diag(4,1): their squared distances are 1 and 4 despite equal Euclidean distance. Label vertical density axes as density, not probability. Use black backgrounds, white axes, cyan contours, and clear English explanations. Audience: 二年级人工智能专业大学生