

![0_image_0.png](0_image_0.png)

As well as the situation described above  which results in the level   model M  the present paper also considers a further situation where the output from the level models is a set of class probabilities rather than a single class prediction  If model M i - 3 is used to classify an instance x in L j  let P ki x  denote the probability of the ith output class  and the vector Pkn Pk1 Xn ,...,Pki Xn ,...,Pk1 Xn gives the model s class probabilities for the nth instance assuming that there are I classes As the level    data  assemble together the class probability vector from all K models  along with the actual class L'CV
{ Yn; P1n; . . . ; Pkn; . . . ; PKn ; n n N }.

Denote the level model derived from this as M ′ to contrast it with M
The following two subsections describe the algorithms used as level and level gnnr alizers in the experiments reported in Section

## 2.1 Level-0 Generalizers

a decision tree learning Three learning algorithms are used as the level - generalizers— C
NB  a re implementation of a Naive Bayesian classifier  Cestnik algorithm Quinlan and IB a variant of a lazy learning algorithm Aha Kibler Albert w hich employs the p nearest neighbor method using a modified value difference metric for nominal For each of these learning algorithms we and binary attributes  Cost    Salzberg now show the formula that we use for the estimated output class probabilities P i x  for an instance x  where  in all cases  Ei Pi x C4.5: Consider the leaf of the decision tree at which the instance x falls  Let mi be the number of training instances with class i at this leaf  and suppose the majority class