# Benchmarking Change Detector Algorithms from Different Concept Drift Perspectives
## 2 Problem Statement
### Concept Drift  
Changes in data streams in non-stationary environments are reflected through alterations in their probability distributions, which are referred to as concept drifts. Formally, let \(C_{t}\) denote the true concept or data distribution at time \(t\). In a typical machine learning setting, we train a model \(M\) on a fixed dataset with the assumption that the distribution of the data will remain the same in the future. However, in a streaming scenario, the distribution of the data may shift or evolve over time, leading to a mismatch between the model and the true concept. This drift in the concept can lead to a decrease in the performance of the model over time. A concept drift can manifest in different forms, including abrupt, gradual, and incremental drift [21].  
Abrupt drift occurs when there is a sudden and significant change in the underlying data distribution. The new data distribution is often very different from the old one, and the model that was previously effective may become obsolete. The main challenge with abrupt drift is to quickly detect the change and adapt the model to the new data distribution. An example of abrupt drift could be a change in the way that a sensor is calibrated, resulting in a sudden shift in the readings.  
The second mentioned type is the gradual drift, which occurs when the underlying data distribution slowly changes over time. The change is often smooth, making it difficult to detect. The main challenge with gradual drift is to continuously monitor the data and update the model to reflect the new data distribution. A gradual drift could be exemplified as a change in consumer behavior over time, leading to a slow shift in product preferences.  
The last one, incremental drift, occurs when the data distribution changes over time, but the rate of change increases suddenly. The main challenge with incremental drift is that of detecting the change and adapting the model quickly enough to prevent performance degradation. An example of incremental drift could be a seasonal shift in the heating system, where the shift occurs gradually over several months but accelerates rapidly during the winter.  
According to [22; 23], the formal representation of a concept drift between two-time points (\(t\) and \(t+1\)) over a given period of time is:  
\[\exists X:P_{t}(X,y)\
eq P_{t+1}(X,y) \tag{1}\]  
In Equation (1), \(P_{t}\) represents the joint distribution between the set of input variables \(X\) and the target variable \(y\) at time \(t\). Similarly, \(P_{t+1}\) denotes the joint distribution at time \(t+1\). Concept drift is related to the covariance shift in the distribution, which means that a change in the joint probability of \(X\) and \(y\) indicates the occurrence of concept drift at time \(t\). Hence, \(P_{t}(X,y)\) can be defined as a composition of \(P_{t}(X)\times P_{t}(y|X)\). Change detection algorithms can detect concept drift by monitoring changes in the joint distribution \(P_{t}(X,y)\) over time. If the distribution of the input variables or the conditional distribution of the output variables change significantly over time, it can indicate the occurrence of concept drift. Change detection algorithms typically monitor the statistical properties of the data, such as the mean, variance, or distribution, to detect changes in the joint distribution. When a significant change is detected, the algorithm can trigger the retraining of the model or some other action to adapt to the new concept.