# Linguistic Summarization and Outlier Detection of Blended Learning Data
## 3 Materials and Methods
### A Method of Extracting Linguistic Summaries Expressing Outliers
To specify the cardinality of objects satisfying summarizer \(S\) among the set of objects satisfying filter criterion \(F\), we propose to use the confidence measure \(\beta\) computed using Equation (6). \[\beta=\frac{\sum\limits_{i=1}^{n}\mu_{x}(\alpha_{i})\wedge\mu_{y}(\alpha_{i})}{ \sum\limits_{i=1}^{n}\mu_{x}(\alpha_{i})} \tag{6}\]  
The steps to detect outliers based on a set of linguistic summaries expressing common rules are as follows and visualized in Figure 3:  
_Step_ 1: With the summarizer \(S_{1}\) of the form \"_B is \(y\)_\" of each linguistic summary expressing common rules, if \(y\) is not \"_medium_\" (\"_medium_\" is the word that has neutral semantics), replace the word \(y\) with the word \(\overline{y}\) in the LFoC of attribute \(B\) having its semantic opposite to \(y\) to form a new summarizer \(S_{2}\). _Step_ 2: Compute the cardinality \(\beta\) of objects satisfying \"_B is \(\overline{y}\)_\" in the set of objects satisfying filter criterion \(F\) by Equation (6). _Step_ 3: If \(\beta>0\) (an outlier is detected), specify the new quantifier word \(Q_{2}\) and create a new linguistic summary expressing the outlier with new summarizer \(S_{2}\) and new quantifier word \(Q_{2}\). Figure 3: An algorithm model for outlier detection. _Step 4_: If all linguistic summaries expressing common rules are examined, go to Step 5; otherwise, go to Step 1. _Step 5_: Return a set of linguistic summaries expressing outliers. To perform the evaluation of the extracted outliers, we used the similarity function \(L(p1,p2)\) proposed in [9] to check the similarity between the normal linguistic summary \(p1\) and the outlier \(p2\). The similarity value is calculated as follows [9]:  
\[L(p1,\ p2)= \begin{cases}\gamma\text{\emph{es}}&if\ \sum_{k=0}^{m}H(p1_{k},p2_{k})<2\\ No&in\ \text{\emph{other case}}\end{cases} , \tag{7}\]  
where \(p1_{0}\) and \(p2_{0}\) are the indexes of the linguistic quantifier \(Q\) in \(Dom(Q)\) of \(p1\) and \(p2\), respectively; \(p1_{m}\) and \(p2_{m}\) are the indexes of the word summarizers of \(p1\) and \(p2\), respectively; \(p1_{k}\) and \(p2_{k}\), \(k=1,\ldots,\)\(m-1\), are the indexes of the words in the \(Dom(A_{k})\) of the qualifiers of \(p1\) and \(p2\), respectively. If the function \(L\) returns _Yes_, the two linguistic summaries are similar. The function \(H(p1_{k},p2_{k})\) determines the semantic distance between two words in the linguistic domain \(Dom(A_{k})\) of the linguistic variable \(A_{k}\) is calculated as follows [9]:  
\[H(p1_{k},\ p2_{k})= \begin{cases}1&if\quad|p1_{k}-p2_{k}|>round\ (20\%\times size \ (Dom(A_{k}))\ \text{or}\\ &if\quad\quad p1_{k}=0\ \text{\emph{and}}\ p2_{k}\ \
eq 0\qquad\qquad\text{or}\\ &if\quad\quad\ p1_{k}\
eq 0\ \text{\emph{and}}\ p2_{k}=0\\ 0&\qquad\qquad\text{in other case}\end{cases}  \tag{8}\]  
For example, if \(Dom(A_{k})=\) [\"_very few_\", \"_few_\", \"_little few_\", \"_medium_\", \"_little many_\", \"_many_\", \"_very many_\"], the word \"_few_\" at position 2 and the word \"_many_\" at position 6 have their distance \(|\,2-6\,|>\) round(20% \(\times\) 7) = 1. Therefore, the two words \"_few_\" and \"_many_\" are considered distinct.