Aggregative quantification for regression

Abstract

The problem of estimating the class distribution (or prevalence) for a new unlabelled dataset (from a possibly different distribution) is a very common problem which has been addressed in one way or another in the past decades. This problem has been recently reconsidered as a new task in data mining, renamed quantification when the estimation is performed as an aggregation (and possible adjustment) of a single-instance supervised model (e.g., a classifier). However, the study of quantification has been limited to classification, while it is clear that this problem also appears, perhaps even more frequently, with other predictive problems, such asregression. In this case, the goal is to determine a distribution or an aggregated indicator of the output variable for a new unlabelled dataset. In this paper, we introduce a comprehensive new taxonomy of quantification tasks, distinguishing between the estimation of the whole distribution and the estimation of some indicators (summary statistics), for both classification and regression. This distinction is especially useful for regression, since predictions are numerical values that can be aggregated in many different ways, as in multi-dimensional hierarchical data warehouses. We focus on aggregative quantification for regression and see that the approaches borrowed from classification do not work. We present several techniques based on segmentation which are able to produce accurate estimations of the expected value and the distribution of the output variable. We show experimentally that these methods especially excel for the relevant scenarios where training and test distributions dramatically differ.

A Review on Quantification Learning

“…The problem is defined by assigning an individual cost to each training example (its biomass) and the model must be able to predict the total biomass of each species. Bella et al [2014] focus on quantification for regression problems (see Section 11) and propose several novel techniques based on discretization. In this case, the examples in the training set D have attached a real value, y i ∈ Y = R. The authors distinguish between two types of regression quantification tasks: i) the estimation of the expected value for the given sample; and ii) the estimation of the whole distribution.…”

“…The main paper on quantification for regression problems, [Bella et al 2014], uses Relative Squared Error (RSE). RSE compares the squared error between the estimated mean and the actual one, normalizing such error with the variance observed in the test set…”

“…A more reliable option is to employ the average instead of the maximum. For that reason Bella et al [2014] prefer the Cramér-von Mises statistic for two samples. The U value is calculated from the empirical data sorted in increasing order …”

“…Most of the regression quantification methods are due to Bella et al [2014] using the learning setting described in Section 2.3. These methods adapt some Forman's methods (e.g.…”

“…This equation is applied for all the methods described here in the experiments reported in [Bella et al 2014]. Based on the RS method, Bella et al [2014] introduce the Adjusted Regress & Sum (ARS) method. Following a similar idea that AC and PAC methods, ARS adjusts the prediction of RS.…”