The interactive film is in the book's Companion

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Mahalanobis Distance: The Ellipsoids of a Gaussian

Chapter 13 · Mathematics for AI and Machine Learning · Advanced · 1:34

An interactive live film: the level sets of a 3D Gaussian's quadratic form are ellipsoids with semi-axes √λᵢ along the eigenvectors; on the floor, a Euclidean circle and a Mahalanobis ellipse pass through the reader's point. Find a point at Euclidean distance 1.5 and Mahalanobis distance 1.2.

Narration

A Gaussian in three dimensions. Its density depends on x only through one quadratic form: x minus mu, transposed, times the inverse covariance, times x minus mu.

Surfaces where that quadratic form is constant are ellipsoids around the mean. This one is where it equals one.

The rest of the narration is in the book's Companion.

More from Chapter 13

Interactive1:35Normalizing Flows: Chaining Invertible MapsChapter 13Preview1:38Independence as an Outer ProductChapter 13Preview1:40Conditional Gaussian: A Slice of the BellChapter 13Preview1:22Random Fourier FeaturesChapter 13Preview1:46Change of Variables: Same Probability, Different AreaChapter 13Preview1:35The von Mises-Fisher Distribution: A Gaussian for DirectionsChapter 13Preview1:42The Dirichlet Distribution: A Density on the SimplexChapter 13Preview1:14Gaussian Mixture Model: Multimodal TerrainChapter 13Preview1:41Whitening: PCA and ZCAChapter 13Preview1:36Beta and Bernoulli: Updating with Pseudo-CountsChapter 13Preview1:43ROC Curve and AUC: the Mass Above the DiagonalChapter 13Preview1:40Hoeffding's Inequality: How Fast a Sample Mean SettlesChapter 13Preview1:25Temperature Scaling: Calibrating a SoftmaxChapter 13Preview2:09Random Directions Are Nearly OrthogonalChapter 13