# Outlier Recognition via Linguistic Aggregation of Graph Databases
## 2 Fuzzy Sets and Linguistic Quantification of Statements  
In this section, we briefly review the basics of the linguistic quantification of statements in the sense of Zadeh [21]. A fuzzy set \(A\) in a finite non-empty \(\mathcal{X}=\{x_{1},\ldots,x_{N}\}\), \(N\in\mathbb{N}\), is denoted as \(A=\{(x,\mu_{A}(x))\colon x\in\mathcal{X}\}\), and \(\mu_{A}\colon\mathcal{X}\to[0,1]\) is its membership function. The intersection of fuzzy sets \(A\), \(B\) in \(\mathcal{X}\) is a fuzzy set \(A\cap B\) in \(\mathcal{X}\):  
\[\mu_{A\cap B}(x)=t(\mu_{A}(x),\mu_{B}(x)), \tag{1}\]  
where \(t\) is a triangular norm, e.g., min or product. The cardinality of \(A\), the so-called \(\Sigma\text{count}(A)\), _sigma-count_, is defined as [22]:  
\[\text{card}(A)=\Sigma\text{count}(A)=_{df}\sum_{x\in\mathcal{X}}\mu_{A}(x). \tag{2}\]A _relative cardinality_ of \(A\) with respect to a fuzzy set \(B\) is proposed:  
\[\text{card}(A\big{|}B)=_{df}\frac{\text{card}(A\cap B)}{\text{card}(B)}. \tag{3}\]  
For fuzzy set \(A\) in a continuous and uncountable universe of discourse \(\mathcal{Y}\), the following counterpart of (2) is proposed:  
\[\text{clm}(A)=_{df}\int_{y\in\mathcal{Y}}\mu_{A}(y)dy, \tag{4}\]  
where \"clm\" comes from \"cardinality-like measure\". Finally, the support of \(A\) in \(\mathcal{X}\) is a non-fuzzy set in \(\mathcal{X}\):  
\[\text{supp}(A)=\{x\colon\mu_{A}(x)>0\}. \tag{5}\]  
The generalized support of \(A\) is called _a-cut_, _alpha-cut_, and replaces the right side of inequality in Equation (5) with \(\alpha\in[0,1]\): \(A_{a}=\{x\colon\mu_{A}(x)>\alpha\}\). Alpha cuts are non-fuzzy sets and they are necessary to define the convexity of fuzzy set \(A\) in \(\mathcal{X}\): \(A\) is convex in \(\mathcal{X}\) iff each of its alpha-cuts is convex non-fuzzy set. \(A\) in \(\mathcal{X}\) is normal iff \(\sup_{x\in\mathcal{X}}\mu_{A}(x)=1\). Assume, \(S\), \(W\) are linguistically expressed characteristics of objects, represented by fuzzy sets in a finite \(\mathcal{D}=\{d_{1},d_{2},\ldots,d_{N}\}\), \(N\in\mathbb{N}\). \(Q\) is a relative fuzzy quantifier describing quantities of objects as ratios to a superset (usually, the universe of discourse of \(Q\)), e.g., \"many of\", \"about \(1/3\)\", \"less than half\", \"very few\", and \(Q\) is modeled by a fuzzy set that is convex and normal in \([0,1]\)[23] (see Equations (23)-(25) as examples of relative fuzzy quantifiers). The first and the second form of linguistically quantified statements by [PERSON] [21], are:  
\[Q\;d\text{'s are }S, \tag{6}\] \[Q\;d\text{'s being }W\text{ are }S, \tag{7}\]  
respectively. Their degrees of truth in terms of fuzzy logic are evaluated via Equation (3) as:  
\[T\big{(}Q\;d\text{'s are }S\big{)}=\mu_{Q}\big{(}\text{card}(S \big{|}\mathcal{D}\big{)}, \tag{8}\] \[T\big{(}Q\;d\text{'s being }W\text{ are }S\big{)}=\mu_{Q}\big{(}\text{card}(S \big{|}W\big{)}. \tag{9}\]  
It must be noticed that in the context of detecting outliers, only the so-called regular relative linguistic quantifier \(Q\) with the monotonically non-increasing \(\mu_{Q}\) is taken into account, i.e. :  
\[\mu_{Q}(0)=1,\;\;\mu_{Q}(1)=0, \tag{10}\] \[\text{card}(A)=\Sigma\text{count}(A)=_{df}\sum_{x\in\mathcal{X}}\mu_{A}(x). \tag{2}\]0  
Moreover, quality of the fuzzy quantified statements can be additionally evaluated with the given measures of quality of fuzzy quantifier \(Q\): \(T_{\text{supp}}\) and \(T_{\text{clm}}\)[23; 24]. They are based on support (5) and on the clm measure (4), respectively, of a fuzzy set that represents the fuzzy quantifier \(Q\):  
\[\text{card}(A)=\Sigma\text{count}(A)=_{df}\sum_{x\in\mathcal{X}}\mu_{A}(x). \tag{2}\]1 \[\text{card}(A)=\Sigma\text{count}(A)=_{df}\sum_{x\in\mathcal{X}}\mu_{A}(x). \tag{2}\]2  
Both measures depend on the presented characteristics of \(Q\) and their meaning is: the closer to \(1\), the more precise quantifier \(Q\) is.