Quantification is an expanding research topic in Machine Learning literature. While in classification we are interested in obtaining the class of individual observations, in quantification we want to estimate the total number of instances that belong to each class. This subtle difference allows the development of several algorithms that incur smaller and more consistent errors than counting the classes issued by a classifier. Among such new quantification methods, one particular family stands out due to its accuracy, simplicity, and ability to operate with imbalanced training samples: Mixture Models (MM). Despite these desirable traits, MM, as a class of algorithms, lacks a more in-depth understanding concerning the influence of internal parameters on its performance. In this paper, we generalize MM with a base framework called DyS: Distribution y-Similarity. With this framework, we perform a thorough evaluation of the most critical design decisions of MM models. For instance, we assess 15 dissimilarity functions to compare histograms with varying numbers of bins from 2 to 110 and, for the first time, make a connection between quantification accuracy and test sample size, with experiments covering 24 public benchmark datasets. We conclude that, when tuned, Topsøe is the histogram distance function that consistently leads to smaller quantification errors and, therefore, is recommended to general use, notwithstanding Hellinger Distance’s popularity. To rid MM models of the dependency on a choice for the number of histogram bins, we introduce two dissimilarity functions that can operate directly on observations. We show that SORD, one of such measures, presents performance that is slightly inferior to the tuned Topsøe, while not requiring the sensible parameterization of the number of bins.

Most supervised learning methods consider that each dataset instance is associated with a unique label. However, there are several domains in which the instances are associated with a set of labels (a multi-label). An alternative to investigate properties of multi-label data and their relationship with the learning performance consists in exploratory data analysis. This approach aims to obtain a better understanding of the data by using different techniques, most of them related to graphic representations. This work proposes ML-EDA, a framework for multi-label exploratory data analysis, which is publicly available in the Internet. The framework has been designed considering extensibility and maintainability as its main goals. Moreover, ML-EDA can directly process, among others, the information provided by MULAN, a framework for multi-label learning frequently used by the community. Some of the ML-EDA facilities are illustrated using benchmark multi-label datasets, highlighting its use as an additional resource to investigate multi-label data.

Defining the attributes in terms of fuzzy sets is an essential part in designing a fuzzy system. The main tasks involved in defining the fuzzy data base include deciding the type of fuzzy set (triangular, trapezoidal, etc), the number of fuzzy sets for each attribute, and their distribution in each attribute domain. In the absence of an expert, these definitions can be done empirically or by using automatic methods. In this paper, we present four different methods to estimate the number of fuzzy sets for a dataset. The first defines the same number of fuzzy sets for all attributes, while the other three flexibly estimate different numbers of fuzzy sets for each attribute of a given dataset. The aim of this paper is to provide fast and practicable methods to define fuzzy data bases, previously to the generation of the fuzzy rule base by more costly approaches, such as genetic fuzzy systems. These methods are evaluated using the FuzzyDT method, which generates a fuzzy decision tree based on the C4.5 classic method, on 11 datasets. The results are compared in terms of accuracy and number of generated rules. The results showed that the flexible estimation of the number of fuzzy sets obtained better error rates for the datasets used in the experiments.