# Linguistic Summarization and Outlier Detection of Blended Learning Data
## 2 Background Knowledge
### Generating Interpretable Multi-Level Semantic Structures Based on Enlarged Hedge Algebras
In the EHA approach [10; 24], an interval quantifying mapping (SQM) value of a word is a mapping \(f\): \(Dom(A)\to P[0,1]\) (\(P[0,1]\)--all sub-intervals of \([0,1]\)) that specifies an interval semantic core of the word, and it is considered the numerical semantics of that word. Therefore, the interval semantic core can be used to design the small base of the trapezoidal computational fuzzy set-based semantics of the word. In [23], based on the interval semantic cores (interval SQL values), a procedure to automatically produce such computational semantics of words from a set of given fuzziness parameter values of EHA was proposed. As stated in [23], the fuzziness parameter values of EHA are comprised of the fuzziness measure values of one of two primary words \(c^{-}\) (e.g., \"_low_\", \"_few_\", \"_short_\", etc.) and \(c^{+}\) (\"_high_\", \"_many_\", \"_long_\", etc. ), i.e., \(m(c^{-})\) or \(m(c^{+})\); the fuzziness measure values of the least word 0, neutral word \(W\), and greatest word 1, i.e., \(m(1)\), \(m(W)\), and \(m(1)\), respectively; and the fuzziness measure values of the linguistic hedges, denoted by \(\mu(h_{i})\), \(j=0\), \(\ldots\), \(n\), where \(h_{j}\) is a linguistic hedge including an artificial hedge \(h_{0}\), and \(n\) is the number of non-artificial linguistic hedges. For example, when \(k\) is set to 3, two primary words are \(c^{-}\) (e.g., \"_low_\") and \(c^{+}\) (e.g., \"_high_\"), the linguistic hedge set is {\(L\) (_Little_}, \(V\) (_Very_)), the steps for producing linguistic words are as follows: First, five words {0, \(c^{-}\), W, \(c^{+}\), 1} at level 1 are specified. Second, the hedges \(L\), \(h_{0}\), and \(V\) linearly act on two primary words, \(c^{-}\) and \(c^{+}\), to produce new words at level 2, including [\(Vc^{-}\), \(h_{0}c^{-}\), \(Lc^{-}\), \(Lc^{+}\), \(h_{0}c^{+}\), \(Vc^{+}\)]. Two words, \(h_{0}c^{-}\) and \(h_{0}c^{+}\), become the semantic cores of \(c^{-}\) and \(c^{+}\), respectively. Therefore, all words of levels 1 and 2 are {0, \(Vc^{-},c^{-},Lc^{-}\), \(W\), \(Lc^{+}\), \(c^{+}\), \(Vc^{+}\), 1}. Last, the hedges \(L\), \(h_{0}\), and \(V\) linearly act on the words at level 2 including [\(Vc^{-}\), \(Lc^{-}\), \(Lc^{+}\), \(Vc^{+}\)] to produce new words at level 3 including [\(VVc^{-}\), \(h_{0}Vc^{-}\), \(LVc^{-}\), \(LLc^{-}\), \(h_{0}Lc^{-}\), \(VLC^{-}\), \(VLC^{+}\), \(h_{0}Lc^{+}\), \(ILc^{+}\), \(LVc^{+}\), \(h_{0}Vc^{+}\), \(VVc^{+}\)]. Four words \(h_{0}Vc^{-}\), \(h_{0}Lc^{-}\), \(h_{0}Lc^{+}\), and \(h_{0}Vc^{+}\) become semantic cores of \(Vc^{-}\), \(Lc^{-}\), \(Lc^{+}\), and \(Vc^{+}\), respectively. Consequently, all linguistic words at levels 1, 2, and 3 in the LFoC generated by acting two linguistic hedges on two primary words are {0, \(VVc^{-}\), \(Vc^{-}\), \(Vc^{-}\), \(Llc^{-}\), \(Lc^{-}\), \(Vlc^{-}\), \(W\), \(Vlc^{+}\), \(LLc^{+}\), \(+Lc^{+}\), \(LVc^{+}\), \(Vc^{+}\), 1} and their semantic cores. These words are arranged by their semantic order. In [10; 35], a method of producing the trapezoidal fuzzy sets that constitute an interpretable multi-level semantic structure of a given LFoC of \(A\) satisfying the concept of interpretability of [PERSON] [36] was also proposed. Therefore, when the fuzziness parameter values are given, the fuzziness measure values and interval SQM values of those words are computed. Consequently, an interpretable fuzzy set-based multi-level semantic structure is automatically constructed as in Figure 1[10; 35]. As can be seen in Figure 1, for example, the word \(Vc^{-}\) is produced from the word \(c^{-}\) by acting the hedge \(V\) on \(c^{-}\); the semantics of \(Vc^{-}\) is more specific than \(c^{-}\), but it still conveys the original semantics of \(c^{-}\). Therefore, the support of the fuzzy set of \(Vc^{-}\) must be within the support of \(c^{-}\). It is the same for all other words at levels greater than 1.