# Explainability and Interpretability in Concept and Data Drift: A Systematic Literature Review
## 2 Background: Drift Phenomena and Explainable/Interpretable AI
### Definitions and Notations  
**Definition 1**. : _(Data Stream): It is an ordered sequence of data points, typically indexed by time and potentially unbounded. Formally, a data stream can be represented as \(S=(z_{1},z_{2},z_{3},\dots)\), where each element \((z_{i}\in\mathcal{Z})\) for some domain \((\mathcal{Z})\) arrives at time \(t_{i}\). We assume that each \(z_{i}\) is generated by an underlying probability distribution \((\mathcal{D}_{t_{i}})\) that may vary with time. In other words, a data stream is a sequence \(((z_{i},t_{i})_{i\geq 1})\) with \(t_{1}<t_{2}<\cdots\), and it allows only single-pass, incremental processing of its elements due to its continuous, potentially infinite nature [9,10]._  
**Definition 2**. : _(Drift): It refers to any systematic change in the data stream's statistical properties over time. Given a data stream \((S=(z_{i})_{i\geq 1})\) and letting \(\mathcal{D}_{t}\) denote the probability distribution of data at time \(t\), a drift is said to occur if there exist two time points \((t_{1}<t_{2})\) such that \(\mathcal{D}_{t_{1}}\
eq\mathcal{D}_{t_{2}}\). In other words, the data-generating process is nonstationary, and the probability law governing observations in the stream changes at some point [11]._  
**Definition 3**. : _(Data Drift): It is a specific type of drift concerning changes in the distribution of the input data over time, without necessarily altering the underlying concept that maps inputs to outputs. It is also known as covariate shift or virtual drift. Formally, consider a supervised data stream of input-output pairs \((\{(x_{i},y_{i})\})\), where \(x_{i}\in\mathcal{X}\) represents a feature vector and \(y_{i}\in\mathcal{Y}\) represents labels [11]. Data drift occurs when the marginal feature distribution changes; i.e., there exist times \((t_{1},t_{2})\) such that_  
\[P_{t_{1}}(X)\
eq P_{t_{2}}(X)\]  
**Definition 4**. : _(Concept Drift): It is a specific type of drift that affects the conditional distribution of the output given the input and thus alters the model's true mapping from \(\mathcal{X}\) to \(\mathcal{Y}\) over time [11]._Concept drift can manifest as a shift in class boundaries, changing correlations between features and the target, or a complete change of the task's true target function [12]. Formally, in a supervised stream with a joint distribution \(\left(P_{t}(X,Y)\right)\) at time \(t\), concept drift means there exist times \(\left(t_{1},t_{2}\right)\) such that_  
\[P_{t_{1}}(Y\mid X)\
eq P_{t_{2}}(Y\mid X)\]  
**Definition 5**: _(Explainability): It is the ability of a predictive model or system to provide intelligible explanations for its behavior or decisions [13]. An explanation is typically an interpretable description of the internal reasoning or the key factors that led to a particular outcome. Formally, consider a model \(\left(f:\mathcal{X}\rightarrow\mathcal{Y}\right)\). We say that \(f\) is explainable if there exists an explanation function \(\left(\xi\right)\) mapping the model's input-output pairs to a human-interpretable domain \(\left(\mathcal{H}\right)\) (such as natural language statements, logical rules, or visual indicators),_  
\[\xi:\mathcal{X}\times\mathcal{Y}\rightarrow\mathcal{H},\]  
_such that for any input \(\left(x\in\mathcal{X}\right)\) with a prediction \(\left(y=f(x)\right)\), the output \(\left(\xi(x,y)\in\mathcal{H}\right)\) articulates why \(f\) arrived at that decision. In simpler terms, explainability means the model can communicate the reasons for its predictions in understandable terms [14]._  
**Definition 6**: _(Interpretability): It is the degree to which a human can understand the cause of a decision or the workings of a model [15]. It is an intrinsic property of the model that describes how transparently its internal mechanics can be understood in human terms [16]. A model is interpretable if its structure and parameters are such that a user can directly follow the logic of how inputs are transformed into outputs [17].