# Prediction of High Capabilities in the Development of Kindergarten Children
## 3 Data Preprocessing
### Fundamentals of Rough Set Theory  
[PERSON] introduced Rough Set Theory in 1982 [14] to deal with vague and imprecise information. Since then, it have been successfully applied to data preprocessing in both cases and attributes selection [19]. Let \(A\) be a set of features and a non-empty set \(U\) (universe) of instances described by the features in \(A\); the pair (\(U\),\(A\)) is denoted as the information system. If every element of \(U\) has also an additional decision feature \(c\), then it is obtained a decision system, \(DS(U,A\cup\{c\})\), where \(c\
ot\in A\)[14].  
Classical (often called [PERSON]'s) RST considers that a feature \(A_{i}\in A\) distinguishes an instance \(x\) from another instance \(y\), and it is denoted by \(\mathit{Distinguishes}(A_{i},x,y)\), if and only if all their feature vales are different; that is, \(\mathit{Distinguishes}(A_{i},x,y)\leftrightarrow x(i)\
eq y(i),\ \forall A_{i}\in A\).  
Every subset of features \(B\) of \(A\) has associated a binary inseparability relation \(\mathit{IND}_{B}(U)\), which is formed by the set of pairs of instances indistinguishable by the relation; that is, the instances having the same feature values in the set of features B. Formally, \(\mathit{IND}_{B}(U)=\{(x,y)\in U\times U:\sim\mathit{Distinguishes}(B_{i},x,y), \ \forall B_{i}\in B\}\). An inseparability or indiscernibility relation defined by forming subsets of elements from \(U\) having the same feature values for a subset of features \(B\subseteq A\), is an equivalence relation.  
The indistinguishable instances form an equivalence class. The equivalence class of an instance \(x\) with respect to the indiscernibility relation induced by the features in \(B\), is denoted by \(\left[x\right]_{B}\).  
RST incorporates a very interesting concept, the reduct definition. A reduct is a set of features \(B\subseteq A\) such that \(\mathit{IND}_{B}(X)=\mathit{IND}_{A}(X)\); that is, both \(B\) and \(A\) generate the same partition of the universe \(U\). In [PERSON]'s words \"a reduct is the minimal set of attributes that enables the same classification of elements of the universe as the whole set of attributes. In other words, attributes that do not belong to a reduct are superfluous with regard to classification of elements of the universe\" [14].  
Following these considerations, the computation of the set of reducts in a dataset is a kind of feature selection (by deleting those features, which do not belong to the obtained reducts), and have been extensively used [20]. In this research, we include the computation of all reducts to perform feature selection the proposed algorithm.  
RST also considers that every concept can be roughly approximated. Let it be a decision system \(\mathsf{DS}=(\mathsf{U},\mathsf{A}\cup\{\mathsf{c}\})\) and let it be the sets \(B\) and \(X\) such that \(B\subseteq A\) and \(X\subseteq U\). The concept \(X\) can be roughly approximated using the information contained in \(B\) by constructing the \(B\)-inferior (\(B\)-lower) and \(B\)-superior (\(B\)-upper) approximations, denoted by \(\mathit{INF}_{B}(X)\) and \(\mathit{SUP}_{B}(X)\), respectively; and defined as follows: \(\mathit{INF}_{B}(X)=\left\{x\in U:\left[x\right]_{B}\subseteq X\right\}\) and \(\mathit{SUP}_{B}(X)=\left\{x\in U:\left[x\right]_{B}\cap X\
eq\varnothing\right\}\). The instances in \(\mathit{INF}_{B}(X)\) are with certainty members of \(X\) while the instances in \(\mathit{SUP}_{B}(X)\) are possible members of \(X\). The limit region for the concept \(X\) is computed as \(\mathit{LIM}_{B}(X)=\mathit{SUP}_{B}(X)-\mathit{INF}_{B}(X)\).  
The information in the lower and upper approximations of a rough set have been used for the task of selecting relevant instances [21]. In this paper, we also used that information. However, we use Minimum Neighborhood Rough Sets (MNRS) [21] instead of Pawlak's.