In a two-class classification problem using discriminant functions $\delta_k$ for class $k$, the location of the separating hyperplane can be determined based on the values of the discriminant functions. Here’s the explanation for each scenario:

1. Where $\delta_1 > \delta_2$:

   • This indicates that the point in question is classified as belonging to class 1. The region where $\delta_1 > \delta_2$ is considered the decision region for class 1.

2. Where $\delta_1 < \delta_2$:

   • This indicates that the point in question is classified as belonging to class 2. The region where $\delta_1 < \delta_2$ is considered the decision region for class 2.

3. Where $\delta_1 = \delta_2$:

   • The separating hyperplane is located where the discriminant functions for the two classes are equal. This is the decision boundary that separates the two classes. The equation of the separating hyperplane can be written as: $\delta_1(\mathbf{x}) = \delta_2(\mathbf{x})$

4. Where $\delta_1 + \delta_2 = 1$:

   • This condition does not typically describe the location of the separating hyperplane in standard discriminant analysis. Instead, it might represent some normalization condition or constraint specific to a particular problem setup. Generally, the sum of discriminant functions being equal to 1 does not directly relate to the decision boundary or separating hyperplane.

Additional Questions

1. How are discriminant functions $\delta_1$ and $\delta_2$ defined in a linear discriminant analysis?
2. What is the geometric interpretation of a separating hyperplane in a two-dimensional space?
3. How can the discriminant functions be derived from the probabilistic models of each class?
4. What role does the covariance matrix play in the formation of the discriminant functions?
5. How does the Mahalanobis distance relate to discriminant functions?
6. What are the assumptions underlying linear discriminant analysis?
7. How does quadratic discriminant analysis differ from linear discriminant analysis?
8. What is the impact of different prior probabilities on the location of the separating hyperplane?

Tip

Understanding the role of the covariance matrix in discriminant analysis can greatly enhance your ability to interpret the results and assumptions of the model.