Rule models

We can best understand rule models using the principles of discrete mathematics. Let's review some of these principles.

Let X be a set of features, the feature space, and C be a set of classes. We can define the ideal classifier for X as follows:

c: X → C

A set of examples in the feature space with class c is defined as follows:

D = {(x₁, c( x₁)), ... , (x_n, c( x_n)) ⊆ X × C

A splitting of X is partitioning X into a set of mutually exclusive subsets X₁....X_s, so we can say the following:

X = X1 ∪ .. ∪ Xs

This induces a splitting of D into D₁,...D_s. We define Dj where j = 1,...,s and is {(x,c(x) ∈ D | x ∈ Xj)}.

This is just defining a subset in X called Xj where all the members of Xj are perfectly classified.

In the following table we define a number of measurements using sums of indicator functions. An indicator function uses the notation where I[...] is equal to one if the statement between the square brackets is true and zero...