Let the mean feature vectors of three classes be...

Let the mean feature vectors of three classes be $\{22,-20\},\{20,20\}$, and $\{-18,-19\}$. Let the test vectors be $\{23,-20\},\{21,18\}$, and $\{-18,-19\}$. Find the three discriminant functions of the three classes. Classify the test vectors.

Key Concepts

Discriminant Functions

Discriminant functions are mathematical functions that assign a score to an input vector indicating how well it fits a particular class. In the context of pattern classification, they are used to decide between classes by comparing these scores, with the highest score (or, in distance-based methods, the lowest distance) indicating the selected class. When the classes are modeled by Gaussian distributions with equal covariance matrices, the discriminant functions often reduce to comparing distances from the mean vectors of the classes.

Mean Feature Vectors

Mean feature vectors represent the central or average value of the features for samples in each class, effectively summarizing the class characteristics. In many classification methods, particularly those assuming a Gaussian distribution, these means act as prototypes, and the distances between a test vector and these means are used for classification decisions.

Minimum Distance Classification

Minimum distance classification is a method where a test vector is assigned to the class whose mean feature vector is the closest in terms of Euclidean distance. This approach is effective when the classes are well represented by their means and when the covariance within each class is assumed to be the same, allowing the use of simplified discriminant functions that rely primarily on the distance from the class centroids.

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