# Linguistic Summarization and Outlier Detection of Blended Learning Data
## 2 Background Knowledge  
### Linguistic Summaries with Quantifier Word  
In 1982, [PERSON] proposed a method of extracting summary sentences in natural language from numeric data using fuzzy propositions in the form of structures with quantifier words [2]. In this subsection, some concepts and notations related to the problem of extracting a set of compact and optimal linguistic summaries with quantifier words are briefly presented.  
Let \(X=\{x_{1},x_{2},\ldots,x_{n}\}\) be data patterns representing data objects in a given dataset, and let \(A=\{A_{1},A_{2},\ldots,A_{m}\}\) be a set of considering attributes of \(X\). The linguistic summary sentence structure used in this paper is one of the following proposed structures:  
\[Q\text{ students are with }<\!\!S\!\!>\text{ score}, \tag{1}\]  
\[Q\text{ students with }<\!\!F\!\!>\text{ obtain }<\!\!S\!\!>\text{ score}, \tag{2}\]  
where \(S\) is the summarizer, which includes a linguistic word in the linguistic domain of the linguistic variable associated with the summarizer attribute representing the degree of the conclusion: for example, \"_extremely high_ mid-term score\", \"_very low_ final exam score\", etc.; \(Q\) is the quantifier word expressing the proportion of data patterns satisfying the summarizer \(S\) in relation to all patterns in the dataset as the sentence structure (1) or in relation to the number of patterns satisfying the filter criterion \(F\) as the sentence structure (2): for example, \"_few_\", \"_a half_\", \"_very many_\", etc.  
_Validity value_ or _Truth value_\(T\) has its numeric value in the normalized interval [0, 1] used to evaluate the truth degree of a linguistic summary, and it is considered the truth degree of a fuzzy proposition with a quantifier word.  
In the general form, the components \(F\) and \(S\) are the associations of linguistic predicates AND/OR, and each linguistic theme is determined by a pair of an attribute and a linguistic word: for example, \"_very high_ average quiz score and _very many_ slide views\", \"_few_ times doing quizzes and _long_ average practice time\", etc. In these filter criteria, the average quiz score, the number of slide views, the number of times doing quizzes, and the average practice time are the filter attributes. To compute the truth value of the fuzzy propositions of the structures (1) and (2), the fuzzy set-based computational semantics of linguistic words in those fuzzy propositions should be designed. \(\mu_{Q}\), \(\mu_{F}\), and \(\mu_{S}\) are denoted by the membership degrees of the computational fuzzy set-based semantics of words in \(Q\), \(F\), and \(S\), respectively. The truth value \(T\) is the fundamental measure that is applied to evaluate the quality of a linguistic summary and is computed using [PERSON]'s formula [30] for fuzzy predicates with quantifier words [9] as follows:  
\[T(Q\text{ }y\text{ are }S)=\mu_{Q}\left[\frac{1}{n}\sum_{i=1}^{n}\mu_{S}(x_{i})\right] \tag{3}\]  
\[T(QF\text{ }y\text{ are }S)=\mu_{Q}\left[\frac{\sum_{i=1}^{n}(\mu_{F}(x_{i}) \wedge\mu_{S}(x_{i}))}{\sum_{i=1}^{n}\mu_{F}(x_{i})}\right] \tag{4}\]  
The linguistic summaries with a truth value greater than a given threshold \(\delta\) are considered to represent the information and useful knowledge hidden in the given numeric dataset. Therefore, only such linguistic summaries are extracted. To ensure the high quality of the extracted linguistic summary set, the value \(\delta\) should be greater than or equal to 0.8. Despite setting a given value \(\delta\), the cardinality of summary sentences in the extracted set is still very large. Therefore, genetic algorithm models are applied to achieve the optimal set of extracted linguistic summaries [5,9,20,21]. Furthermore, some additional genetic operators, such as _imprecision_, _appropriateness_, _covering_, and _focus_, are also applied to evaluate and select the best summary set. They are also computed based on the membership degrees of the computational fuzzy set-based semantics of words in the summary sentences [19,31].