A Review on Quantification Learning <s>  <s> Many business applications track changes over time, for example, measuring the monthly prevalence of influenza incidents. In situations where a classifier is needed to identify the relevant incidents, imperfect classification accuracy can cause substantial bias in estimating class prevalence. The paper defines two research challenges for machine learning. The `quantification' task is to accurately estimate the number of positive cases (or class distribution) in a test set, using a training set that may have a substantially different distribution. The `cost quantification' variant estimates the total cost associated with the positive class, where each case is tagged with a cost attribute, such as the expense to resolve the case. Quantification has a very different utility model from traditional classification research. For both forms of quantification, the paper describes a variety of methods and evaluates them with a suitable methodology, revealing which methods give reliable estimates when training data is scarce, the testing class distribution differs widely from training, and the positive class is rare, e.g., 1% positives. These strengths can make quantification practical for business use, even where classification accuracy is poor. <s> BIB001 </s> A Review on Quantification Learning <s>  <s> Multi-instance (MI) learning is a variant of inductive machine learning, where each learning example contains a bag of instances instead of a single feature vector. The term commonly refers to the supervised setting, where each bag is associated with a label. This type of representation is a natural fit for a number of real-world learning scenarios, including drug activity prediction and image classification, hence many MI learning algorithms have been proposed. An yM I learning method must relate instances to bag-level class labels, but many types of relationships between instances and class labels are possible. Although all early work in MI learning assumes a specific MI concept class known to be appropriate for a drug activity prediction domain; this ‘standard MI assumption’ is not guaranteed to hold in other domains. Much of the recent work in MI learning has concentrated on a relaxed view of the MI problem, where the standard MI assumption is dropped, and alternative assumptions are considered instead. However, often it is not clearly stated what particular assumption is used and how it relates to other assumptions that have been proposed. In this paper, we aim to clarify the use of alternative MI assumptions by reviewing the work done in this area. <s> BIB002
A Review on Quantification Learning 74:3 are usually denoted as {+1, −1}, representing the positive and the negative classes, respectively. Whether a binary-or multi-class classification problem, the learning task at hand is to induce h from corresponding D to predict classes on the unseen data. The main difference between quantification and classification is that a quantification model, or quantifier, does not make predictions for individual instances but for samples, that is, groups of instances. This also occurs in multi-instance learning BIB002 in which the model is designed to assign a class label for a given bag 1 of instances, representing usually that at least one of the individual examples of the bag belongs to the assigned class. The goal of quantification learning is to predict an aggregate magnitude for the whole bag or sample. For instance, in multiclass quantification, the model predicts the prevalence of each class in the sample. Formally, N X denotes a multi-set of examples from X. A multi-set is represented by the number of times that each x ∈ X appears in that set or sample. Each element of the predicted vector,p j , represents the probability of the class j in the sample. Thus, their sum is 1, l j=1p j = 1, and the quantifier returns the probability distribution of the classes. The goal is to predict a class probability distribution [p 1 , . . . ,p l ] that is as close as possible to the true one [p 1 , . . . ,p l ]. As it occurs in supervised classification, multi-class quantification (l > 2) is a more complex learning task than binary quantification (l = 2), because it requires a multivariate predictive model. In binary quantification, the quantifier produces univariate predictions, typically the estimated prevalence of the positive class, denoted asp,h : because the prevalence of the negative class can be trivially obtained usingp:n = 1 −p. It is worth mentioning that some loss functions compare the whole predicted distribution [p,n] and the true one [p, n] , but other loss functions only compare the predicted prevalence for the positive class,p, with the actual one, p (see Section 4.1). Most of the quantification methods are designed for binary quantification. BIB001 proposes to use the one-versus-all approach to adapt binary quantifiers to multi-class quantification. The implementation of one-versus-all for multi-class quantification is quite straightforward: (1) Training: Given a multi-class training set with l classes, l binary quantifiers, {h 1 , . . . ,h l }, must be learned. In the training set for binary quantifierh j , D j , all the examples that belongs to class j are labelled as positives, with the rest being negatives. (2) Testing: Given a multi-class test set T , each binary quantifier,h j is applied over T to obtain p 0 j , an initial estimation of the prevalence for class j. (3) Final Prediction: The initial prevalences computed,p 0 j , are normalized so they sum to 1: Forman (2008) argues that the normalization process may compensate for imperfect quantification due to class imbalance.