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Rule Generation

Once a decision tree has been constructed, it is a simple matter to convert it into an equivalent set of rules.

Converting a decision tree to rules before pruning has three main advantages:

  1. Converting to rules allows distinguishing among the different contexts in which a decision node is used.
  2. Converting to rules removes the distinction between attribute tests that occur near the root of the tree and those that occur near the leaves.
  3. Converting to rules improves readability.

To generate rules, trace each path in the decision tree, from root node to leaf node, recording the test outcomes as antecedents and the leaf-node classification as the consequent.

Rule Simplification Overview

Once a rule set has been devised:

  1. Eliminate unecessary rule antecedents to simplify the rules.
  2. Eliminate unecessary rules to simplify the rule set.

Contingency Tables

The following is a contingency table, a tabular representation of a rule.

  C1 C2 Marginal Sums
R1 x11 x12 R1T = x11 + x12
R2 x21 x22 R2T = x21 + x22
Marginal Sums CT1 = x11 + x21 CT2 = x12 + x22 T = x11 + x12 + x21 + x22

R1 and R2 represent the Boolean states of an antecedent for the conclusions C1 and C2
(C2 is the negation of C1).
x11, x12, x21 and x22 represent the frequencies of each antecedent-consequent pair.
R1T, R2T, CT1, CT2 are the marginal sums of the rows and columns, respectively.

The marginal sums and T, the total frequency of the table, are used to calculate expected cell values in step 3 of the test for independence.

Test for Independence

Given a contingency table of dimensions r by c (rows x columns):

  1. Calculate and fix the sizes of the marginal sums.

  2. Calculate the total frequency, T, using the marginal sums.

  3. Calculate the expected frequencies for each cell.

    The general formula for obtaining the expected frequency of any cell xij, 1ir, 1jc in a contingency table is given by:

    where RiT and CTj are the row total for ith row and the column total for jth column.

  4. Select the test to be used to calculate based on the highest expected frequency, m:

    if then use
    m 10 Chi-Square Test
    5 m 10 Yates' Correction for Continuity
    m 5 Fisher's Exact Test

  5. Calculate using the chosen test.

  6. Calculate the degrees of freedom.

    df = (r - 1)(c - 1)

  7. Use a chi-square table with and df to determine if the conclusions are independent from the antecedent at the selected level of significance, .

Chi-Square Formulae


Click here for an exercise in decision tree pruning.


Decision Lists

A decision list is a set of if-then statements.

It is searched sequentially for an appropriate if-then statement to be used as a rule.

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