Discriminant Analysis: Classifying Correlated Gaussian Clusters

Classification rule

LDA (shared Σ)

QDA (per-class Σ)

Class covariance

Correlation ρ_A (blue class) 0.00

Correlation ρ_B (orange class) 0.00

Mean separation |μ_A−μ_B| 4.0

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 surface

How it works

Each class k is modelled as a multivariate Gaussian N(μ_k, Σ_k). 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). Here Σ_x,y carries the ρ-controlled correlation while the z-variance is identical and independent for both classes, so the z-terms cancel and the boundary is a vertical surface over a curve in the (x,y) plane:

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 flat plane.
QDA — each class keeps its own Σ_k. The quadratic terms no longer cancel, so the boundary is a genuine conic curve extruded into a curved 3D sheet (ellipse/parabola/hyperbola depending on ρ_A, ρ_B).

The wireframe ellipsoids are the Σ_k covariance ellipsoids: since Σ decouples into a 2×2 (x,y) block plus a scalar z-variance, its eigenvectors are exactly (1,1,0)/√2 and (1,-1,0)/√2 with eigenvalues 1±ρ, plus (0,0,1) with eigenvalue 1 — which is why every ellipsoid here is tilted at exactly ±45° in the xy-plane.

Drag ρ_A/ρ_B to reshape each class's spread, drag the separation slider to pull the class means apart, and toggle LDA/QDA to watch the boundary snap from a flat plane to a curved quadric — with live accuracy recomputed on every change.

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