Density-Based Fuzzy Clustering as a First Step to Learning the Rules: Challenges and Possible Solutions

In many practical situations, it is necessary to cluster given situations, i.e., to divide them into groups so that situations within each group are similar to each other. This is how we humans usually make decisions: instead of taking into account all the tiny details of a situation, we classify the situation into one of the few groups, and then make a decision depending on the group containing a given situation. When we have many situations, we can describe the probability density of different situations. In terms of this density, clusters are connected sets with higher density separated by sets of smaller density. It is therefore reasonable to define clusters as connected components of the set of all the situations in which the density exceeds a certain threshold t. This idea indeed leads to reasonable clustering. It turns out that the resulting clustering works best if we use a Gaussian function for smoothing when estimating the density, and we select a threshold in a certain way. In this paper, we provide a theoretical explanation for this empirical optimality. We also show how the above clustering algorithm can be modified so that it takes into account that we are not absolutely sure whether each observed situation is of the type in which we are interested, and takes into account that some situations almost belong to a cluster