- MLR is the best of the four learning algorithms to use as the level   generalizer for obtaining the model M'
- When obtained using MLR M' has lower predictive error rate than the best model selected by J fold cross validation  for almost all datasets used in the experiments
- Another advantage of MLR over the other three level - generalizers is its interpretability The weights α k t indicate the different contributions that each level   model k makes to the prediction classes
- Model M' can be derived by MLR with or without non negativity constraints   Such constraints make little difference to the model s predictive accuracy
- The use of non negativity constraints in MLR has the advantage of interpretability   Non negative weights akt support easier interpretation of the extent to which each model contributes to each prediction class
- When derived using MLR  model M ′ compares favorably with majority vote
- MLR provides a method of combining the confidence generated by the level —models into a final decision For various reasons NB C and IB are not suitable for this task

## 4. Comparison With Arcing And Bagging

This section compares the results of stacking C     NB and IB  with the results of arcing called boosting by its originator  Schapire             and bagging that are reported by Breiman c Both arcing and bagging employ sampling techniques to modify the data b distribution in order to produce multiple models from a single learning algorithm  To combine the decisions of the individual models  arcing uses a weighted majority vote and bagging uses an unweighted majority vote  Breiman reports that both arcing and bagging can substantially improve the predictive accuracy of a single model derived using a base learning algorithm

## 4.1  Experimental Results

First we describe the differences between the experimental procedures   Our results for staing are averaged over ten fold cross validation for all datasets except Waveform  whi
is averaged over ten repeated trials Standard errors are also shown Results for arcing and ben bagging are those obtained by Breiman c which are averaged over b trials In Breiman s experiments  each trial uses a random     split to form the training and test sets for all datasets except Waveform Also note that the Waveform dataset we used has irrelevant attributes  but Breiman used a version without irrelevant attributes   which would be expected to degrade the performance of level   generalizers in our experiments   In both training instances were used for this dataset but we used test instances cases whereas Breiman used       Arcing and bagging are done with    decision tree models derived from CART  Breiman et al.

in each trial