Coverage is measured by the number of examples that satisfy the rule. Notice that we start this specialization process with a rule and we are trying to compare it with its specializations. For each candidate specialization we calculate the gain in fitting error and the loss in coverage compared to this original rule. These factors are calculated by :

(2)

where

The quality of the candidate specialization is calculated as a weighted average of these two values^[5]. It remains an open question how to set the weight of each of the factors. These weights represent a trade-off between generality and correctness of the learned rules. The bigger the weight on GainMAD the more specific the rules get. As a result the final theory will probably have too many rules. On the contrary, if we favor coverage we will get fewer rules but probably less accurate. This can be interesting in noisy domains where we don't want to overfit the noise by producing too specific rules. As this is highly domain dependent we let the user tune these weights^[6]. R² introduces a further degree of flexibility by allowing some limited variation on these weights, as we will explain below. The formula for calculating the quality of a specialization is as follows :

(3)

R² lets the user specify an interval for the value of wgain. This interval states the minimum and maximum value for the weight of GainMAD in the quality of a rule. The actual value of the parameter is calculated as a function of the value of GainMAD. The idea is that if the value of GainMAD is very high then we put more weight on this factor, and vice-versa. In summary, given an interval [w1 ... w2] the actual value of wgain is calculated as :

(4)

This formulation of a flexible weighted average is very similar to the one used in system Yails [17]. We intend to compare this simple approach to one using the Minimum Description Length (MDL) theory [14]. This methodology would give an elegant framework for integrating these properties with the issue of rule complexity. This could be important if we want to stress the advantages of ML-based systems in terms of comprehensibility.

We will now illustrate these ideas with the example we have been using. Imagine that after the step of building a model for an uncovered region we have as result the rule R3 previously shown. Suppose that this rule covers 15 examples and has a MAD of 0.45. One possible specialization of this rule could be :

R31 IF X3 5 X2 < 2 THEN Y = -4.2 + 0.6 X4 + 0.99 X2

Notice that this rule has as conclusion a refined model of the one in the original rule R3. This is a consequence of the change of the set of examples that the rule is covering. If this new rule covered only 12 of the previous examples but had a MAD of 0.24 then we would get :

If the interval for the weight on GainMAD was [0.6..0.85] then wgain would be given by 0.6 + (0.85-0.6) 0.467 = 0.717, and finally the quality of this specialization would be:

Q = 0.467 0.717 + (1-0.2) (1-0.717) =0.561

R² would now compare this value to the quality of the original rule and decide whether this specialization is better.

We still did not refer how the quality of the original rule is assessed^[7]. We decided to use the difference 1-MAD has the quality of these original models^[8]. In this case rule R3 would have a value of 0.55. This makes the specialization more attractive and the next step followed by R² would be to try to specialize this new rule and so on until no better specialization of the current best rule exists.