Quadratic Discriminant Analysis

Quadratic Discriminant Analysis

1. Difference from LDA

QDA allows each class its own covariance $Σ_{c}$ → boundary becomes quadratic (curves, ellipses). More flexible, needs more data.

Here $det_{c}$ means determinant of $Σ_{c}$ (written in words to keep rendering safe).

2. LDA vs QDA

	LDA	QDA
Covariance	shared $Σ$	per-class $Σ_{c}$
Boundary	linear	quadratic
Params	fewer (stable)	many (needs big $n$)
Wins when	shapes similar	shapes differ

3. Practice

Regularize $Σ_{c}$ toward shared $Σ$ when $n$ small (shrinkage). Intermediate: regularized discriminant analysis blends LDA/QDA.

4. Worked Intuition

Class 0 tight blob, class 1 wide blob → QDA draws ellipse around class 0; LDA can only draw a line and misclassifies the ring.

5. Exam Points

• QDA vs LDA (own covariance → quadratic) — 5 marks, very frequent.
• Write discriminant with per-class term — 5 marks, frequent.
• Bias-variance reading (QDA lower bias, higher variance) — 3 marks.