Discriminant Analysis 2D: LDA vs QDA on Correlated Gaussian Clusters

Classification rule

LDA (shared Σ)

QDA (per-class Σ)

Class covariance

Correlation ρ_A (blue class) 0.60

Correlation ρ_B (orange class) -0.40

Mean separation |μ_A−μ_B| 4.0

Confidence ellipse (σ) 2.0

Points per class 130

Live readouts

Accuracy

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Misclassified

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Boundary shape

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Legend

Class A points (correct) Class B points (correct) Misclassified point Decision boundary curve

Drag the top panel to pan, scroll/pinch to zoom.

How it works

Each class k is modelled as a bivariate Gaussian N(μ_k, Σ_k) in the (x,y) plane. A Bayes classifier assigns a point x to the class with the largest discriminant score:

δ_k(x) = -½(x-μ_k)'Σ_k⁻¹(x-μ_k) - ½ln|Σ_k| + ln π_k

The decision boundary is the set where δ_A(x) = δ_B(x):

LDA — both classes are forced to share one pooled covariance Σ = (Σ_A+Σ_B)/2. The quadratic terms then cancel identically, leaving a linear equation → the boundary is a straight line.
QDA — each class keeps its own Σ_k. The quadratic terms no longer cancel, so the boundary is a genuine conic curve (ellipse/parabola/hyperbola depending on ρ_A, ρ_B).

The dashed ellipses are the Σ_k confidence ellipses: with unit variances and correlation ρ, the eigenvectors of Σ_k are always exactly (1,1)/√2 and (1,-1)/√2 (a ±45° tilt), with eigenvalues 1+ρ and 1-ρ — shown live in the eigenvalue bars below the heatmap.

Sample points are drawn once per class as raw standard-normal pairs (z₁,z₂) and mapped into correlated coordinates through the Cholesky factor L of Σ_k (x,y) = μ_k + L·(z₁,z₂) every time ρ changes, so the cloud continuously reshapes to match the ellipse instead of staying spherical.

The middle panel renders the raw score field g(x,y) = δ_A(x,y) - δ_B(x,y) as a blue↔orange heatmap; its zero-contour is exactly the boundary curve drawn above it. Drag ρ_A/ρ_B to reshape each class's spread, pull the separation slider, and toggle LDA/QDA to watch the boundary snap from a straight line to a curved conic — with live accuracy recomputed on every change.

Scatter · ellipses · boundary

Discriminant score field δ_A − δ_B

Covariance eigenvalues (1±ρ)