The quantification objective is to predict the class distribution of a data sample. Therefore, this task intrinsically involves a drift in the class distribution that causes a mismatch between the training and test sets. However, existing quantification approaches assume that the feature distribution is stationary. We analyse for the first time how score-based quantifiers are affected by concept drifts and propose a novel drift-resilient quantifier for binary classes. Our proposal does not model the different types of concept drifts. Instead, we model the changes that such changes cause in the classification scores. This observation simplifies our analysis since distribution changes can only increase, decrease or maintain the overlap of the positive and negative classes in a rank induced by the scores. Our paper has two main contributions. The first one is MoSS, a model for synthetic scores. We use this model to show that state-of-the-art quantifiers underperform in the occurrence of any concept drift that changes the score distribution. Our second contribution is a quantifier, DySyn, that uses MoSS to estimate the class distribution. We show that DySyn statistically outperforms state-of-the-art quantifiers in a comprehensive comparison with real-world and benchmark datasets in the presence of concept drifts.

Quantification is a thriving research area that develops methods to estimate the class prior probabilities in an unlabelled set of observations. Quantification and classification share several similarities. For instance, the most straightforward quantification method, Classify & Count (CC), directly counts the output of a classifier. However, CC has a systematic bias that makes it increasingly misestimate the counts as the class distribution drifts away from a distribution it perfectly quantifies. This issue has motivated the development of more reliable quantification methods. Such newer methods can consistently outperform CC at the cost of a significant increase in processing requirements. Yet, for a large number of applications, quantification speed is an additional criterion that must be considered. Frequently, quantification methods need to deal with large amounts of data or fast-paced streams, as it is the case of news feeding, tweets and sensor data. In this paper, we propose Sample Mean Matching (SMM), a highly efficient algorithm able to quantify billions of data instances per second. We compare SMM to a set of 14 established and state-of-the-art quantifiers in an empirical analysis comprising 25 benchmark and real-world datasets. We show that SMM is competitive with state-of-the-art methods with no statistical difference in counting accuracy, and it is orders of magnitude faster than the vast majority of the algorithms.

Quantification is a task similar to classification in the sense that it learns from a labeled training set. However, quantification is not interested in predicting the class of each observation, but rather measure the class distribution in the test set. The community has developed performance measures and experimental setups tailored to quantification tasks. Nonetheless, we argue that a critical variable, the size of the test sets, remains ignored. Such disregard has three main detrimental effects. First, it implicitly assumes that quantifiers will perform equally well for different test set sizes. Second, it increases the risk of cherry-picking by selecting a test set size for which a particular proposal performs best. Finally, it disregards the importance of designing methods that are suitable for different test set sizes. We discuss these issues with the support of one of the broadest experimental evaluations ever performed, with three main outcomes. (i) We empirically demonstrate the importance of the test set size to assess quantifiers. (ii) We show that current quantifiers generally have a mediocre performance on the smallest test sets. (iii) We propose a metalearning scheme to select the best quantifier based on the test size that can outperform the best single quantification method.

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.

Many real-world applications in the batch and data stream settings with data shift pose restrictions to the access to class labels after the deployment of a classification or quantification model. However, a significant portion of the data stream literature assumes that actual labels are instantaneously available after issuing their corresponding classifications. In this paper, we explore a different set of assumptions without relying on the availability of class labels. We assume that, although the distribution of the data may change over time, it will switch between one of a handful of well-known distributions. Still, we allow the proportions of the classes to vary. In these conditions, we propose the first method that can accurately identify the correct context of data samples and simultaneously estimate the proportion of the positive class. This estimate can be further used to adjust a classification decision threshold and improve classification accuracy. Finally, the method is very efficient regarding time and memory requirements, fitting data stream applications.