# Fast Reaction to Sudden Concept Drift in the Absence of Class Labels
## 3 The Proposed Approach
### Pairwise Diversity Measures  
Diversity is an important characteristic of ensembles in the standard, static data context. Measuring diversity can be useful to analyse the effectiveness of a diversity-inducing method. Therefore, many researchers consider diversity to prune a number of component classifiers [24, 25], measure the diversity in decision forests [26] or characterize the diversity between classifiers [27]. In addition, there have been scant attempts at promoting diversity. For example, the authors of [28] discuss the impact of diversity on online ensemble learning and reactions to drift by modifying the Poisson distribution used in online bagging. However, by doing so, they only measure the accuracy of the modified ensemble, not its diversity. Recently, another work considers diversity [29] in an experiment using sliding windows. Therefore, to the best of our knowledge, our approach is the first semisupervised drift detection method to measure the diversity of component classifiers directly and use it as a base for drift detection. In contrast, the previous approaches used classification accuracy to detect drifts. Informally, for a pair of classifiers in a binary classification problem, each component gives one of two possible predictions for each example, let us say 0 (negative class) or 1 (positive class). If we take the predictions of each component, we can calculate the disagreement between these predictions. Therefore, it is about the disagreement between the predictions of pairs of component classifiers, regardless of the true labels. Formally, the diversity of component classifiers (e.g., disagreement) can be calculated in pairs, for example, let \(X=x_{1},\ldots,x_{n}\) be a labelled data set and \(y^{\prime}_{v}=[y^{\prime}_{v}\ (x_{1}),\ \ldots,y^{\prime}_{v}\ (x_{n})]\) an n-dimensional binary vector that represents the output of a classifier \(h_{v}\), such that \(y^{\prime}_{v}\ (x_{j})=1\), if \(h_{v}\) correctly predicts the class label, and 0 otherwise. Table 1 presents all the possible outcomes for a pair of classifiers \(h_{u}\) and \(h_{v}\), such that \(h_{u}=h_{v}\), where \(N^{ab}\) is the number of instances \(x_{j}\in X\) for which \(y^{\prime}_{u}\ (x_{j})=a\) and \(y^{\prime}_{v}\ (x_{j})=b\). Therefore, all the probabilities of \(N^{ab}\) are as follows. * \(N^{10}\) number of examples where \(h_{u}\) predicts class 1 and \(h_{v}\) predicts class 0. * \(N^{01}\) number of examples where \(h_{v}\) predicts class 1 and \(h_{u}\) predicts class 0. * \(N^{11}\) number of examples where \(h_{u}\) predicts class 1 and \(h_{v}\) predicts class 1. * \(N^{00}\) number of examples where \(h_{u}\) predicts class 0 and \(h_{v}\) predicts class 0. Formally speaking, the disagreement measure is defined as the ratio of the number of inconsistent decisions over the total number of observations. Also, we can say it is the ratio between the number of observations for which one classifier is correct and the other is incorrect. Consequently, the diversity measure reflects the variety of responses of classifiers to changes of incoming data. The disagreement measure is probably the most intuitive measure of diversity between a pair of classifiers [30]. Thus, calculating the diversity between two base classifiers (\(h_{u}\) and \(h_{v}\)) using the disagreement measure is measured by Equation (1):  
\[D_{u:v}=N^{10}+N^{01} \tag{1}\]  
On the other hand, the PH test considers a variable \(m_{T}\) which measures the accumulated differences between observed values \(e\) (error estimates). Two basic approaches to calculate these values prequantitially are _fading factors_ and _sliding windows_. Over time, the fading factors deduct outdated information via multiplying the former summary by a factor and then adding a new value calculated using the incoming data. Other approaches use the sliding windows to hold at each time