## 3 Models of Vector Presentation of Text Documents Using Lexical and Semantic Approaches
The results in Figure 5a were derived using the occurrence frequency of in the document (2) and the smoothed inverse document frequency (5), whereas the results in Figure 5b were derived using the double normalized occurrence frequency of in the document (3) and the inverse document frequency (4). As you can see in Figure 5, in the second case, we have a better separation of respective clusters (the cluster colors correspond to the same topics that are presented in Figure 4). The comparison of Figures 4b and 5b shows that, in some cases, the semantic approach can provide a clearer separation of clusters (in Figure 5b, five distinct clusters can be isolated, and they are circled with ovals, whereas in Figure 4b, there are only four of them, and they are also circled with ovals). It is also worth noting that in Figures 4 and 5 we can see large overlap areas of some clusters. This is due to specifics of text documents, since a document can belong to several topics at once (e.g., \"industry\" and \"economics\"; \"economics\" and \"agriculture\"; etc.). This creates a fundamentally unsolvable problem of achieving uniqueness, high accuracy, and completeness of clustering, especially for texts of general, i.e., non-special, topics. However, we can draw the general conclusion that the choice of a TF-IDF model for the subsequent calculation of metrics between vectors can have a significant impact on the accuracy and completeness of the further clustering of text documents. Figure 2:2 Graphical representation of the cosine metric values for the marked corpus of documents with two different TF-IDF presentations using the term occurrence frequency in documents: **(a)** the double normalized frequency of an object in the document (3) and the inverse document frequency (4) (here 1, 2, 3 are the numbers of clusters); **(b)** the occurrence frequency of an object in the document (2) and the probabilistic inverse document frequency (6) (here 1, 2, 3, 4, 5 are the numbers of clusters). The comparison of Figures 4b, 5b and 6 shows that, when using the average value of cosine metrics, we can generally see a clearer division of documents into clusters than when using the lexical and semantic approaches only: in Figure 6, there are six distinct clusters (these clusters are numbered 1, 2, 3, 4, 6, and 8), whereas in Figure 5b, there are five of them (these clusters are numbered 1, 2, 4, 5, and 6), and in Figure 4b, there are only four of them (these clusters are numbered 1, 2, 3, and 5). Therefore, we can conclude that the use of averaged values of cosine metrics that take both the lexical and semantic features of documents into account can improve the clustering quality and provide clearer division into distinct clusters. The approach described above allows us to consider new clustering algorithms based on bonds between the lexical and semantic properties of documents and, e.g., the statistical properties of the similarity metric distributions of their vectors. Algorithm for Clustering Text Documents Based on Their Lexico-Semantic Bond and Statistical Properties of the Similarity Metric Distributions of Their Vectors  
Step 1. Take the matrix \(\mathcal{M}\) of the lexico-semantic bond between text document vectors (it has a square shape of size M \(\times\) M, where M is the number of documents in the corpus) and replace the elements along the main diagonal with zeros in order to exclude the bond of a vector with itself. Step 2. For each row of the lexico-semantic bond matrix \(\mathcal{M}\), find the sum of the values of its elements and select the row with the maximum sum value. This row corresponds to the document that best matches the meaning of the other documents in this text corpus. The vector corresponding to this row will form the first cluster (let this row have the number \(i\)). Step 3.