Distance Based Generalisation

Many distance-based methods in machine learning are able to identify similar cases or prototypes from which decisions can be made. The explanation given is usually based on expressions such as ”because case a is similar to case b”. However, a more general or meaningful pattern, such as ”because case a has properties x and y (as b has)” is usually more difficult to find. Even in this case, the connection of this pattern with the original distance-based method is generally unclear, or even inconsistent. In this paper, we study the connection between the concept of distance (or similarity) and the concept of generalisation. More precisely, we define several conditions a sensible distance-based generalisation must have. From here, we are able to tell whether a generalisation operator for a pattern representation language is consistent with the metric space defined by the underlying distance. We show sensible pattern languages and generalisation operators for typical data types: nominal, numerical, sets and lists. We also explore a possible relationship between the wellknown concepts of lgg and distances between terms, and the definition of generalisation presented in this paper. keywords: distance-based methods, generalisation operators, lgg, metric space.