Variants of the Borda Count Method for Combining Ranked Classifier Hypotheses

COMBINING RANKED CLASSIFIER HYPOTHESES MERIJN VAN ERP AND LAMBERT SCHOMAKER NICI, P.O.Box 9104, 6500 HE Nijmegen, the Netherlands fM.vanErp, s homakerg ni i.kun.nl The Borda ount is a simple yet e e tive method of ombining rankings. In pattern re ognition, lassi ers are often able to return a ranked set of results. Several experiments have been ondu ted to test the ability of the Borda ount and two variant methods to ombine these ranked lassi er results. By using arti ial data, domain-spe i results were avoided. The results show the strength of the Borda ount when many errors o ur in the results, but also show its weakness in ase of a limited number of large ranking errors. 1 Introdu tion In all elds of pattern re ognition there exist multiple, di erent te hniques to lassify instan es of patterns, ea h approa h being hara terized by its own virtues and short omings. The idea of ombining the output of multiple lassi ers has been studied for several years 1;2;3;4 but it is still diÆ ult to hoose a suitable ombination algorithm. Using a ombination of lassi ers enables one to use all available knowledge and the extra omputing time be omes less of a problem with the urrent developments in omputer pro essing power. Instead of de ning the integration of lassi er opinions as a metalassi ation problem, we will fo us on less umbersome te hniques. This avoids the undesirable onsequen es of metalassi ation 5, i.e., (1) an extra, large amount of training data is needed and (2) for every lassi er that is added, the omplete metalassi er needs to be trained again. The most straightforward form of opinion integration is to let the lassiers ast a vote by forwarding the lass they prefer best. The lass with the most votes wins. This is alled plurality voting and while it is simple and quite e e tive, it la ks depth. With depth we mean that lassi ers often have a ranking of lasses to indi ate whi h are more likely andidates than others. Plurality voting only uses the absolute top of those rankings. In this arti le we will dis uss a method for ombining the rankings of di erent lassi ers, the Borda ount. The Borda ount is an easy, intuitively appealing, and powerful method of ombining di erent rankings. Moreover, it has some variants that may perform better on spe i lassi ation problems (see se tion 2). However, the theoreti al foundation of the approa h is less well developed then in the 443