Classifying Observations into Two Groups

Procedure

For classifying into two groups ($G_1$, $G_2$), Fisher’s method compares $\mathbf{y}$ to group means, assuming equal covariance matrices.

1: Compute the Discriminant Function

Projects $\mathbf{y}$ onto a line maximizing group separation.

Equation (9.1)

$$z={\mathbf{a}}^{\prime }\mathbf{y}=({\overline{\mathbf{y}}}_1-{\overline{\mathbf{y}}}_2{)}^{\prime }{\mathbf{S}}_{\text{pl}}^{-1}\mathbf{y}$$

where ${\mathbf{S}}_{\text{pl}}$ is the pooled covariance matrix TODO how to compute spl.

See Two-Group Discriminant Analysis or Multiple Group Discriminant Analysis to read further.

2: Determine the Cutoff

Equation (9.5)

Assign to $G_1$ (group 1) if the following inequality applies: $z>\frac{1}{2}({\bar{z}}_1+{\bar{z}}_2)$ where:

• ${\bar{z}}_1={\mathbf{a}}^{\prime }{\overline{\mathbf{y}}}_1$
• ${\bar{z}}_2={\mathbf{a}}^{\prime }{\overline{\mathbf{y}}}_2$
• $z$: Discriminant score of new observation

Otherwise, assign to $G_2$.

3: Classify the observation

Compare $z$ to the cutoff and assign accordingly.

Here’s a new section for “Classification into Several Groups” to add to your current notes on Chapter 9. It follows your style, focusing on intuition, key formulas, and structure, while covering the multi-group case from Section 9.3 of the PDF.