## 5 The Determination of the Weights
The result of this optimization process is made of: (i) the usage categories \(C_{opt}\); (ii) the membership degrees of each segment usage to all categories, \(M_{opt}\). Let us assume that the lowest category is identified by \(h=1\). Hence, we take \(w_{k,h}=1-\mu_{j}\) in order to discover atypical, misused and outdated segments in the sentence. Table 2 summarizes the parameters of the system, together with their typical values. Such values have been derived by maximizing the performance of the system over a subset of the sample sentences used in the experimental results (Section 6). ## 6 Experimental Results

In order to test the effectiveness of the system, a collection of 80 sentences have been derived from the British National Corpus (BNC) [40]. More specifically, the extraction criterion was the following. First, the following list of the most frequent English word has been derived: _time_, _year_, _people_, _way_, _man_, _day_, _thing_, _child_[41]. Second, word pairs in the list have been used as a search criterion to find a collection of 30 sentences. Third, a new collection of 50 sentences has been produced by introducing mistakes in the first collection of sentences, and thus having 80 total sentences. In order to measure the system performance, let us consider the system as a classifier whose results (expectation) are compared under test with trusted external judgments (observation). A correct result (_true positive_) is then an atypical subsequence discovered in the sentence, whereas a correct absence of result (_true negative_) is a good sentence where no atypical subsequence has been discovered, _i.e._, the lowest usage category is empty. (Actually, the lowest usage category contains the zero usage value by default, and then this condition from a technical standpoint means that the category contains the zero usage value only) Hence, the terms _positive_ (the sentence is somewhere atypical) and _negative_ (the sentence is good) refer to the _expectation_, whereas the terms _true_ and _false_ refer to whether that expectation corresponds to the observation. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Description** & **Value** & **Reference** \\ \hline \(\Theta\) & Search engine & google, bing, yahoo, all, random & Section 4.2 \\ \(\pi\) & Number of snippet pages & from 2 to 5, to improve precision & Section 4.2 \\ \(o\) & Allowed overlapping \(n\)-grams & 1, 2 & Section 4.1 \\ \(l_{\text{MIN}}\), \(l_{\text{MAX}}\) & Allowed length of \(n\)-grams. & 3, 4 & Section 4.1 \\ \(f\) & Initial threshold of low usage & 3,000,00 & Section 5 \\ \(t_{\text{MIN}}\), \(t_{\text{MAX}}\) & Minimum and maximum font size & 1,030 & Section 5 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of the algorithm and their typical values. Figure 11 shows some examples of successful application of our system. Here, each black rectangle is a visual output of an input sentence. On the left side of the Figure, original BNC sentences are presented. All these sentences are correct from a grammatical standpoint, and then no atypical subsequences are available in the sentences. Hence, all cases on the left are true negatives. On the right side of the Figure, the same sentence of the left side is presented with some grammatical mistake, so as to have some atypical subsequence. In all cases, the system correctly identified the atypical segment. Hence, all cases on the right are true positive. It is worth noting, on the right of Figure 11k and Figure 11 (l), two examples of colored non-contiguous subsequences. Table 3 shows some values related to the sentences of Figure 11c. Here, it can be easily noticed that the atypical subsequence (represented in boldface) is characterized by a weight value \(w_{k,h}\) lower than the corresponding scaling factor \(f\). Figure 12 shows some peculiar examples of successful application of our system. Again, on the left side of the figure, original BNC sentences are presented.