The NBI method is especially efficient in processing datasets that do not display any visible, obvious, self-evident correlations. A discovery of hidden, non-obvious information contained in original similarity matrices is the key feature of the NBI-algorithm. Fig. 6 is the illustration of a system of objects in the form of 300 scattered points which, based on a common logic, visually represent 42 clusters. The clustering produced by NBI-algorithm demonstrates the same logic (see Fig. 7 and 8 depicting the dendrogram and tree obtained by processing of the said set of points). Figure 9 shows sequences of the points where each point is numbered in accordance with the clusters they belong to (as per logical visual distribution shown on Fig. 6). Red color is used when all members of a cluster come out together, and black color is used when at least one point of a cluster is not in place. Sections A, B, and C in Fig. 9 correspond to: (A) the results obtained by sorting performed based on similarity scores as per the similarity matrix; (B) results of one-step ultimate transformation of the same matrix; and (C) completed unsupervised clustering by Meaning Finder. As is seen, upon simple sorting, only 6.7% of points fall into their own clusters, while a one-step transformation results in a three-fold increase in the number of the points correctly attributed to respective clusters. Upon completion of the data processing, all points are located within respective clusters as outlined in Fig. 6.

The above example demonstrates how certain links that exist but are not seen in an original similarity matrix are discovered by the NBI-algorithm in the process of matrix transformation. The core procedure is not the sorting of objects with the purpose of arranging them in a peculiar order, but a multi-stage process of data benefication, based on complex non-linear logic, unlike most of statistical methods that are based on linearity. Output results produced by the NBI are, as a rule, non-additive and oftentimes represent unanticipated, heuristic decisions. This is especially well seen in Fig. 10 illustrating the process of transformation of the similarity matrix for the 300 points displayed in Fig. 6. Logarithms of coefficients of similarities of each point to point 1-1 are plotted on X-axis, and same to point 33-1, on Y-axis. This is an example that perfectly demonstrates non-linearity and interactivity of the operations taking place during the NBI-processing of similarity matrices. Each of the nine consecutive transformations produces results that are seemingly unexpected and illogical, whereas the final computation output is strictly logical and fully in line with human logic. We would like to reiterate that all operations, including metric selection and clustering, are performed automatically and autonomously.

A further illustration to the above is presented in the animated plot in Fig. 11 (to view Fig. 11, click on: http://www.matrixreasoning.com/animation/anim.html). There are 40 points scattered in a plane, of which point 1 is changing its position, moving from the bottom left corner of the plane to the upper right corner. As one can see, a position of one point determines the way all the points are aggregated in clusters. The displayed interactive, holistic and nonlinear attributions of the NBI techniques make the respective reasoning mechanism close to human reasoning.

Another demonstration of the NBI ability to view analyzed data as holistic and nonlinear systems and to take into account even minor quantitative changes is presented and discussed in the following section in the example of partitioning of a graph of size 10.