# Fast Reaction to Sudden Concept Drift in the Absence of Class Labels
## 2 Problem Definition and Related Work
### Existing Concept Drift Detection Methods  
This section describes some well-known drift detectors that are related to our work:  
DDM: Drift Detection Method. This method, developed in [13], is the most recognized work in this field. The method uses a classifier error that is normally reduced as more training examples are used. When an increase in the number of training examples also increases the classifier error, the model must be rebuilt. This indicates that sudden drift might have occurred. This model introduces a warning level when the error rates have increased to a certain level. Upon reaching the warning level, incoming new examples will be treated in a special window. If within this special window the error keeps increasing to a drift level, then the current classifier is rebuilt by learning from examples within the special window. EDDM: This method, proposed by the authors of [14], is an adjustment of DDM. The same warning alarm mechanism that is used by DDM is used in EDDM, however; instead of monitoring error rate, EDDM monitors the distance between two sequential errors. Consequently, when the concepts of incoming data are stable, the distance between the two sequential errors decrease; otherwise, warnings and drifts are signalled. FHDDM [15] is a method for detecting drifts using Hoeffding's inequality through a window of size \(n\). This window detects a drift if a considerable variation is observed between the current probabilities and the maximum of correct predictions. By using the error's probability (\(\delta\), default 10-7), FHDDM finds the variations among the probabilities (\(\Delta P\)) and the threshold (\(\in\)). Consequently, FHDDM will signal a drift if \(\Delta P\geq\in\). PH Test: This method introduced in [16] is typically used for detecting drifts in applications of signal processing. In this work, between the observed values and their mean till current time \(T\), a cumulative difference is represented as a variable \(mT\). The variable of \(mT\) is calculated via \(m_{T}=\sum_{t=1}^{T}(x_{t}-x_{T}^{\prime}-\delta)\), where \(x_{T}=\frac{1}{T}\sum_{t=1}^{T}x_{t}\) and \(\delta\) matches to the changes in magnitude that are allowed. The minimum \(m_{T}\) which is denoted as \(M_{T}\), is also updated through \(M_{T}=min(m_{T},t=1\dots T)\). Subsequently, a concept drift is signalled when there is a considerable variation between \(m_{T}\) and \(M_{T}\) i.e., \(PH_{T}\): \(m_{T}-M_{T}>\lambda\) where \(\lambda\) is a user predefined threshold. However, increasing the value of \(\lambda\) causes fewer false alarms, but it increases the false negative rate. SEED: This method, proposed in [17], within a window \(W\), compares two sub-windows. The older part of this window is dropped when there is a distinct average exhibited by the two sub-windows. Hoeffding Inequality with Bonferroni correction is used by SEED to calculate its test statistic, performing block compression in order to remove unnecessary cut points and then blocks that are homogeneous in nature are merged. STEPD: STEPD is a method proposed by [18] over two windows, namely, recent and older, which calculates statistical tests with continuity correction. Within a concept, the accuracy of the learner using the two windows is expected to be stable. When there is a significant difference in the accuracy of the recent window, warning and drift are signalled. Figure 2: Concept Drift Patterns. RDDM: This method, developed in [19], solves the problem of a performance losing of DDM, which is caused by decreasing sensitivity which requires many examples for drift detection. This method uses a mechanism to abandon older examples, regularly recalculating the statistics of RDDM which are responsible for detecting drifts. The authors found that RDDM delivers higher accuracy than DDM in most cases, by detecting drifts earlier, despite an increase in false positives and memory consumption. DetectA: This method proposed in [20] is a proactive approach called Detect Abrupt Drift (DetectA) for sudden drift detection.