# Combining Advanced Feature-Selection Methods to Uncover Atypical Energy-Consumption Patterns
## 2 Methodology
### Fuzzy Rough Feature-Selection Method  
[PERSON]'s fuzzy set theory [32] extends classical set theory to handle uncertainty and vagueness. In classical set theory, an element either belongs to a set or does not. However, fuzzy set theory allows for partial membership, where elements can belong to a set with varying degrees between 0 and 1. This approach accommodates uncertainty about the boundaries of sets. Let \(V\) be the universe of discourse, which is the set of all possible elements under consideration in a given context. A fuzzy set \(A\) is defined as a set of ordered pairs \(A=\{(v,\mu_{A}(v)):v\in V,\mu_{A}(v)\in[0,1]\}\), where the membership function \(\mu_{A}:V\rightarrow[0,1]\) represents the degree of membership of element \(v\) in the fuzzy set \(A\). Rough set theory aims to handle incomplete or imprecise information by distinguishing between certain and uncertain knowledge [33]. Let \(U\) be the universe of objects, which includes all specific objects under analysis in the given context. While \(V\) represents all possible elements considered, \(U\) focuses on the particular set of objects being studied. Let \(R\subseteq U\times U\) be a relation representing the lack of knowledge about elements of \(U\). For a set of objects \(X\subseteq U\), \(R(x)\) denotes the equivalence class of \(R\) determined by element \(x\). The rough set \(X\) is composed of the tuple \(\langle X^{-},X^{+}\rangle\), where \(X^{-}\) is the lower approximation and \(X^{+}\) is the upper approximation, defined as  
\[X^{-}=\bigcup_{x\in U}\{R(x):R(x)\subseteq X\}\]  
\[X^{+}=\bigcup_{x\in U}\{R(x):R(x)\cap X\
eq\varnothing\}\]  
In the rough set, \(X^{-}\) represents the certain elements of \(X\) and \(X^{+}\) includes both certain and uncertain elements. The boundary region \(X^{0}\) delineates regions of complete information (positive region) and regions of uncertainty (boundary region): \(X^{0}=X^{+}-X^{-}\). The fuzzy rough sets [34] combine fuzzy set theory and rough set theory to handle uncertainty and imprecision in data. Let \(V\) be the universe of discourse, and \(A\subseteq V\) be a fuzzy set in \(V\). Let \(U\) be a non-empty universe, and \(R\) be a similarity relation on \(U\). For \(F\in f(U)\), the fuzzy rough set is the pair \(\langle R^{-}(F),R^{+}(F)\rangle\) of fuzzy sets on \(U\), such that for every \(u\in U\)  
\[R^{-}(F)(u)=\inf_{y\in U}\max\{1-R(u,y),F(y)\}\]  
\[R^{+}(F)(u)=\sup_{y\in U}\min\{R(u,y),F(y)\}\]  
The fuzzy rough sets feature-selection (FRFS) algorithm identifies relevant features that distinguish between different classes or concepts in a dataset. It uses rough set approximations to evaluate feature contributions and iteratively refines feature selection to balance relevance and redundancy in the selected feature subset. The algorithm considers the dataset's inherent properties to ensure optimal feature selection. The algorithm selects features that increase the positive region size until it matches the size of the positive region with all features or the required number of features. The concept of indiscernibility is central to locating a positive region. Suppose we have a non-empty set of objects \(U\) and a non-empty set of attributes \(A\). The indiscernibility of two objects \(u_{i}\) and \(u_{j}\) based on the sets of attributes in \(F\), where \(F\subseteq A\), is given by  
\[\text{IND}(F)=\{(u_{i},u_{j})\in U^{2}\mid\forall f\in F,f(u_{i})=f(u_{j})\}\]  
Using indiscernibility, we can identify the partition of \(U\) generated by IND(F). This partition is defined as  
\[U/\text{IND}(F)=\otimes\{U/\text{IND}(\{f\}):f\in F\},\]  
where \(\otimes\) for two sets \(A\) and \(B\) is represented by  
\[A\otimes B=\{X\cap Y:\forall X\in A,\forall Y\in B,X\cap Y\
eq\varnothing\}\]  
Let \(Q\) be an equivalence relation over \(U\).