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Differentiate LDA and QDA; compute boundary

Quick Overview

This question evaluates understanding of Gaussian generative classification and discriminant analysis, specifically the effects of equal versus class-specific covariances on linear (LDA) and quadratic (QDA) decision boundaries, numerical discriminant comparison, and model selection with regularization and cross-validation.

Binary Gaussian Classification: LDA vs QDA

You are modeling a 2D binary Gaussian classifier with features (x, y):

Class 0: mean μ0 = (0, 0), covariance Σ0 = [[1, 0], [0, 1]].
Class 1: mean μ1 = (2, 0), covariance Σ1 = [[4, 0], [0, 1]].
Priors: π0 = 0.7, π1 = 0.3.

Assume natural logarithms throughout; additive constants common to both classes can be dropped in discriminants.

Tasks

Under the (incorrect) assumption Σ0 = Σ1, write the LDA discriminant and decision rule, and give the resulting linear decision boundary (explicit equation).
Using the true covariances, write the QDA discriminant and derive the quadratic decision boundary as an explicit scalar equation in x and y.
Classify the point x = (1, 1) under both LDA and QDA by numerically evaluating the discriminants.
With n = 60 observations per class, discuss when QDA is preferable vs when LDA is safer. Propose a regularized QDA that shrinks class-specific covariances toward a common Σ with a tunable shrinkage parameter, and explain how to select the parameter via cross-validation.

Quick Overview

Binary Gaussian Classification: LDA vs QDA

You are modeling a 2D binary Gaussian classifier with features (x, y):

Class 0: mean μ0 = (0, 0), covariance Σ0 = [[1, 0], [0, 1]].

Class 1: mean μ1 = (2, 0), covariance Σ1 = [[4, 0], [0, 1]].

Priors: π0 = 0.7, π1 = 0.3.

Assume natural logarithms throughout; additive constants common to both classes can be dropped in discriminants.

Tasks

Under the (incorrect) assumption Σ0 = Σ1, write the LDA discriminant and decision rule, and give the resulting linear decision boundary (explicit equation).

Using the true covariances, write the QDA discriminant and derive the quadratic decision boundary as an explicit scalar equation in x and y.

Classify the point x = (1, 1) under both LDA and QDA by numerically evaluating the discriminants.

With n = 60 observations per class, discuss when QDA is preferable vs when LDA is safer. Propose a regularized QDA that shrinks class-specific covariances toward a common Σ with a tunable shrinkage parameter, and explain how to select the parameter via cross-validation.

Differentiate LDA and QDA; compute boundary

Quick Overview

Binary Gaussian Classification: LDA vs QDA

Tasks

Solution

Comments (0)

Differentiate LDA and QDA; compute boundary

Quick Overview

Binary Gaussian Classification: LDA vs QDA

Tasks

Solution

Comments (0)