# Outlier Recognition via Linguistic Aggregation of Graph Databases
## 5 Application Example  
The proposed algorithms for outlier recognition via linguistic information are now implemented on the sequence obtained (see Table 1). Because of specificity of data and possible connections/relations between them, a graph representation has been chosen for the set of complaints submitted. The general schema of the experiment is to filter (select) the vertices representing complaints themselves and represent them as the sequence of objects, the parameters of which are inputs for algorithms detecting outliers (see Section 3). Next, properties of interest of these objects are fuzzified, which means their crisp values are assigned to labels and corresponding fuzzy sets: \"date received\" to {early spring, middle spring, summer, autumn, early winter, winter} in \(\mathcal{X}_{1}\), \"county per capita income\" to {poor, middle, rich} in \(\mathcal{X}_{2}\), and \"time of sending complaint\" to {short, average, long} in \(\mathcal{X}_{3}\), see sample linguistic values of chosen properties of objects in \(\mathcal{D}\) (23)-(25). \(S_{1}\) represents label \"early spring\" in \(\mathcal{X}_{1}=\{1,2,\ldots,366\}\)--days in a year with \(\mu_{S_{1}}\) given  
\[\mu_{S_{1}}(x)= \begin{cases}\frac{x-79}{14},&79\leq x\leq 93\\ \frac{-x+107}{14},&93\leq x\leq 107\end{cases}  \tag{23}\]  
and 0 otherwise. \(S_{2}\) represents label \"rich country\" in \(\mathcal{X}_{2}=\{0,1,\ldots,70\}\)--per capita income (in USD thousands) in the county the submission comes from, with \(\mu_{S_{2}}(x)\):  
\[\mu_{S_{2}}(x)= \begin{cases}\frac{x-52}{6},&52\leq x\leq 58\\ \frac{-x+64}{58},&58\leq x\leq 64\end{cases}  \tag{24}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**ID**} & \multirow{2}{*}{**Company**} & \multirow{2}{*}{\
\begin{tabular}{c} **Date** \\ **Received** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Sent by** \\ **CFPB** \\ \end{tabular} } & \multicolumn{2}{c}{**Per**} & \multirow{2}{*}{**Product**} & \multirow{2}{*}{**Zipcode**} \\  & & & & & **Capita** & & \\
801691 & \
\begin{tabular}{c} STONELEIGH \\ RECOVERY \\ ASSOCIATES, \\ LLC \\ NATIONSTAR \\ mortgage \\ WELLS \\ \end{tabular} & 11.04.2014 & 09.04.2014 & \$56,791 &
\begin{tabular}{c} debt \\ collection \\ \end{tabular} & 94930 \\
305167 & \begin{tabular}{c} SATONELEIGH \\ RECOVERY \\ ASSOCIATES, \\ LLC \\ NATIONSTAR \\ mortgage \\ WELLS \\ FARGO \& \\ COMPANY \\ WELLS \\ PARGO \& \\ COMPANY \\ PARGO \& \\ COMPANY \\ BARCLAYS \\ 809768 & \
\begin{tabular}{c} BANK \\ DELAWARE \\ \end{tabular} & 15.04.2014 & 15.04.2014 & \$26,924 &
\begin{tabular}{c} credit \\ card \\ bank \\ \end{tabular} & 87124 \\
941562 & \
\begin{tabular}{c} COMERICA \\ \end{tabular} & 17.07.2014 & 17.07.2014 & \$23,023 &
\begin{tabular}{c} account \\ or \\ service \\ \end{tabular} & 48653 \\
720703 & \
\begin{tabular}{c} Vision \\ Financial \\ Corp. \\ \end{tabular} & 21.05.2012 & 21.05.2012 & \$60,929 &
\begin{tabular}{c} debt \\ collection \\ \end{tabular} & 75089 \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sample records of dataset \(D\) obtained as a sequence from the graph database processed, see Section 4 and Figure 1.  
and 0 otherwise. \(S_{3}\) represents label \"average time\" in \(\mathcal{X}_{3}=\{0,1,\ldots,30\}\)--numbers of days between receiving and sending the complaint to a company by CFPB (Consumer Financial Protection Bureau), with \(\mu_{S_{3}}(x)\):  
\[\mu_{S_{3}}(x)= \begin{cases}\frac{x-2}{4},&2\leq x\leq 6\\ \frac{-x+10}{4},&6\leq x\in 10\end{cases}  \tag{25}\]  
and 0 otherwise. \(S_{4}\) is a non-fuzzy set representing one of the labels: {Older American, Servicemember, Older American and Servicemember, none}. The relative linguistic quantifiers proposed to be applied in Algorithm 2, according to Definition 2, are \(Q_{1}=\) \"very few\", \(Q_{2}=\) \"close to 0\", \(Q_{3}=\) \"almost none\". They are illustrated in Figure 2 and their membership functions for \(r\in[0,1]\subset\mathbb{R}\) are:  
\[\mu_{Q_{1}}(r)=