# Fast Reaction to Sudden Concept Drift in the Absence of Class Labels
## 2 Problem Definition and Related Work  
According to the Bayesian decision theory [9], a classification model can be described by the prior probabilities of classes \(p(y)\) and class conditional probabilities \(p(x|y)\) for all classes \(y\in\{K_{1},\ldots,K_{c}\}\), where \(c\) is the number of predefined classes. A data point \(x_{i}\in X\) has \(y_{j}\) for all classes \(y\in\{\mathcal{O},1,\ldots,\mathbb{C}\}\). If a data point \(x_{j}\in X\) has \(y_{j}=\emptyset\), then it is unlabelled. Therefore, the classification decision for instance \(X\) at equal costs of mistake is made based on maximal a posteriori probability, which for class \(y\) can be represented as \(p(y|X)=p(y)p(X|y)/p(X)\) where \(p(X)=\sum_{y=1}^{c}p(y)p(X|y)\).  
In nonstationary environments, changes in a data stream are reflected by the probability distributions in an event called concept drift. Concept drift means that the concept about which the data are being collected may shift after some minimal stable period [8]. Formally, concept drift between time point \(t^{0}\) and time point \(t^{1}\) can be defined as follows.  
For a given data stream \(S\), we say that concept drift occurs between two distinct points in time, \(t\) and \(t+\Delta\), \(iff\)\(\exists\)\(x\): \(p^{t}(x,y)\
eq p^{t+\Delta}(x,y)\) where \(p^{t}\) denotes the joint distribution at time \(t\) between the set of input attributes and the class label.  
By considering this, any changes in incoming data can be characterized by changes in components of the Bayesian decision theory [10]:  
* Prior probabilities \(p(y)\) are prone to changes.
* Probabilities \(p(X|y)\) of class conditional are also prone to changes.
* Consequently, posterior probabilities \(p(y|X)\) may/may not change.  
Based on the cause and effect of these changes, as shown in Figure 1, two types of drift are identified: real drift and virtual drift [8; 11; 12]. Real drift is defined as changes in \(p(y|X)\). Note that such changes can occur with or without changes in \(p(x)\); thus, they may or may not be visible from the data distribution without knowing the true class labels. Virtual drift is defined as changes in the \(p(x)\) or class \(p(y)\) distributions that do not affect \(p(y|X)\). As we are mostly interested in the effect of concept drift on classification, we focus on methods that use true class labels to detect drift. We, therefore, concentrate mainly on real drift, regardless of whether they are visible from the input data distribution \(p(x)\).  
Furthermore, in terms of class distribution, researchers differentiate how these changes happen, as shown in Figure 2. A sudden/abrupt drift happens when the source distribution in \(S^{t}\) at a moment in time \(t\) is suddenly substituted by another distribution \(S^{t+1}\). Gradual drift is connected with a slower rate of change and it refers to a transition stage where examples of two different distributions \(P^{j}\) and \(P^{j+1}\) are mixed, whereas in recurring drift, after a period of time, previous concepts may reappear again. We emphasise that the proposed drift detector has been designed mainly for sudden/abrupt drift.  
Figure 1: Types of drifts: circles represent instances, different colors represent different classes.