# Real-Time Anomaly Detection with Subspace Periodic Clustering Approach
## 3 Problem Definitions  
In below, we present some important terms and definitions used in this paper. **Definition 1**. : _Let U be a non-empty finite set of objects, \(A\), a finite set of attributes, and \(V=\cup\ V_{a}\), where \(V_{a}\) is the domain of the attribute \(a\in A\), then quadruple \(S=(U,\ A,\ V,f)\) defines a set-valued information system [60]. A function f:U \(\times\)\(A\)\(\rightarrow\)V is defined as \(1\leq f(x,a)\in V_{a}\), \(\forall\ x\in U\), \(a\in A\). Also, we take the attribute set \(A=\{C\cup[d];\ C\cap[d]=\phi;C\), the set of conditional and [\(d\)] the decision attributes]._  
**Definition 2**. : _If the domain of a conditional attribute in \(C\) can be arranged in ascending or descending order of preference, then such attribute is called as criterion [36]. If every conditional attribute of \(C\) is a criterion, then the information system is known as the set-valued ordered information system [60]._  
**Definition 3**. : _The attribute is an inclusion criterion if the values of some objects in U under a conditional attribute of \(C\) can be sorted according to inclusion increasing or decreasing preferences [60]._  
**Definition 4**. : _Let us define a relation \(R^{\geq}_{A}\) on a set-valued ordered information system [60] with inclusion increasing preference as, [see e,g., [46]]_  
\[R^{\geq}_{A}=\{(y,x)\in U\times U:f(y,a)\geq f(x,a)\forall a\in A\} \tag{1}\]  
_then \(R^{\geq}_{A}\) is said to be the dominance relation on U. When \((y,x)\in R^{\geq}_{A}\) then \(y\geq_{A}x\), that means \(y\) is at least as good as \(x\) with reference to \(A\)._  
**Property 1**. : _The inclusion dominance relation \(R^{\geq}_{A}\)[60] is (i) reflexive, (ii) unsymmetric, and (iii) transitive._  
**Definition 5**. : _For \(x\in U\), the dominance class [36; 37] of \(x\) is defined as_  
\[[x]^{\geq}_{A}=\Big{\{}yeU:(y,x)\epsilon R^{\geq}_{A}\Big{\}}=\{yeU:f(y,a) \geq f(x,a),\forall aeU\} \tag{2}\]_where \(U_{A}^{\geq}=\left\{[x]_{A}^{\geq}:xeU\right\}\) is the family of dominance classes._  
**Remark 1. **\(\mathbf{U}_{A}^{\geq}\) _is not a partition of U, rather it creates a covering of U, that is \(U=\cup\left[x\right]_{A}^{\geq}.\)_  
**Definition 6. **_For any set-valued ordered information system \(S=\{U,\,A,\,V,\,f\}\) and for a given \(X\subseteq U\), the upper approximation, lower approximation and boundary regions of \(X\) are respectively expressed as [36; 37]_  
\[\mathbf{U}_{A}^{\geq}(X)=\left\{\mathbf{x}\mathbf{e}\mathbf{U}:[x]_{A}^{\geq}\cap X\
eq\mathbf{\phi }\right\} \tag{3}\]  
\[\mathbf{L}_{A}^{\geq}(X)=\left\{\mathbf{x}\mathbf{e}\mathbf{U}:\left[x]_{A}^{\geq}\subseteq X\right\} \tag{4}\]  
\[B_{A}^{\geq}(X)=\mathbf{U}_{A}^{\geq}(X)-\mathbf{L}_{A}^{\geq}(X) \tag{5}\]  
**Definition 7. **_For a set-valued ordered information system, \(S=\{U,\,A,\,V,\,f\}\), \(B\) (\(\subseteq A\)) is termed as criterion reduction of \(S\) if \(\mathbf{R}_{A}^{\geq}=\mathbf{R}_{B}^{\geq}\) and \(\mathbf{R}_{A}^{\geq}\
eq\mathbf{R}_{A}^{\geq}\) for any \(M\subseteq A\)._ On the other way, _a minimal attribute set \(B\) is a criterion reduction of \(S\) if \(\mathbf{R}_{A}^{\geq}=\mathbf{R}_{B}^{\geq}\)[36; 37]._  
**Definition 8. **_A CORE is a minimal subset of attributes such that none of its elements can be removed without violating the nano topology generated by the lower and upper approximations. The CORE(\(A\)) is given by CORE(\(A\)) = \(\left\{\mathbf{a}\mathbf{e}\mathbf{A}:\mathbf{R}_{A}^{\geq}\
eq\mathbf{R}_{A-\{\mathbf{a}\}}^{\geq}\right\}\) [see e.g., [36; 37]]._  
**Definition 9. **_Let \(\mathbf{R}_{C}^{\geq}\) be a dominance relation on U, then \(\mathbf{\tau}_{C}^{\geq}(X)=\left\{\mathbf{U},\mathbf{\phi},\mathbf{U}_{C}^{\geq}(X),\mathbf{L}_{ C}^{\geq}(X),\mathbf{B}_{C}^{\geq}(X)\right\}\) forms a nano topology [36; 37] on U with respect to \(X\). As well, \(\mathbf{\beta}_{C}^{\geq}(X)=\left\{\mathbf{U},\mathbf{U}_{C}^{\geq}(X),\mathbf{L}_{C}^{\geq}( X)\right\}\) is the basis for \(\mathbf{\tau}_{C}^{\geq}(X)\).