sets consisting of data points along three parallel lines and three concentric circles. Most human observers would agree that the three lines and rings in Figure 2(a) and (b) constitute three clusters. Their VAT images are shown in Figure 2(c) and (d), respectively, and neither image gives any indication about the presence of three clusters in these two data sets. The problem for VAT images of these two data sets is that the clusters are not "clouds" of points; rather, they are "stringy." To alleviate this problem, a variety of improvements have been proposed in the literature. These fall into two major categories: one uses graph-based distances, and the other involves spectral graph theory. These two approaches are discussed next. Wang et al. [14] proposed an improved VAT (iVAT) method to enhance the RDI generated by the VAT algorithm by transforming the input dissimilarity matrix with a pathbased distance measure. The path-based dissimilarity measure is based on the idea that, if two objects oi and o j are very far from each other (reflected by a large distance value d i j), but there is a path connecting them through a sequence of other objects, such that the distances between any two successive objects are small, then d i j should be adjusted to a smaller value to reflect this connection. This adjustment reflects the idea that, no matter how far the distance between two objects may be, they should be considered as coming from one cluster if they are connected by a set of successive objects forming dense regions (reflecting the characteristic of elongated clusters). An efficient formulation of the iVAT algorithm (based on recursion), which significantly reduces its computational complexity from O(n 3 ) (for the iVAT implementation presented in [14]) to O(n 2 ) was proposed by Havens and Bezdek [15]. Their iVAT implementation begins by first finding the VAT reordered dissimilarity matrix D) and then transforming the input distances in the distance matrix D ) = [d )ij] by path-based minimax distances Dl ) = [dl ij) ], given by dl)ij = min max D)p[h]p[h + 1], (1) p ! P 1 # h #; p ; ij (a) (b) Figure 3. The iVAT images for the two data sets in Figure 2(a) and (b): (a) I (D l*) for the three-line data set and (b) I (D l*) for the three-ring data set. 20 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE Apri l 2020 where P i j is the set of all paths from object i(oi) to object j (o j) in the VAT-generated MST of O. This formulation is not only computationally efficient, but it also retains a direct relationship between the VAT and iVAT images, thus making it feasible to directly extract single-linkage (SL) clusters from the iVAT image. This improved version of VAT now appears in almost all of the literature based on algorithms in the VAT family. The pseudocode for iVAT is given in algorithm S2 of "Pseudocode for Various Algorithms Belonging to the Visual Assessment of Tendency Family." Figure 3 shows the iVAT images of the two data sets shown in Figure 2(a) and (b). Both iVAT images give a clear and accurate portrayal of the structure of the input data and suggest the presence of three clusters by three dark blocks along the diagonal. Again, the sizes of the three dark blocks indicate the relative sizes of the three clusters in the data. As opposed to iVAT, which first uses the VAT algorithm on the raw distance matrix and then uses a graph-based distance to improve the RDI quality, an alternate approach was employed in [16] and [17] that first transforms the raw distance matrix before feeding it to the VAT algorithm. Markov random-field VAT (MrfVAT), proposed in [16], modifies the input-distance matrix using Markov random fields, which updates each object with its local information dynamically and maximizes a global probability measure. Similarly, the approach presented in [17] takes a refined co-association matrix, which was originally used in ensemble clustering, as an initial similarity matrix and transforms it by a path-based measure before applying it to VAT. These methods can deal with data sets that have complex cluster structures (where VAT is likely to fail) and can reveal the relationship of clusters hierarchically. The -MrfVAT images of the two complex data sets shown in Figure 2(a) and (b) are similar to the iVAT images shown in Figure 3. Since the differences between the iVAT and M - rfVAT images for the threeline and three-ring data sets are indistinguishable to the human eye, they are not included in this article. Another set of algorithms that improve VAT-generated RDIs are based on spectral graph theory. The spectral VAT (SpecVAT) algorithm [18] addresses the limitation of VAT in highlighting the complex cluster structure present in a data set by first mapping the raw distance matrix D to a graph embedding space Dl before reordering it by using the VAT algorithm. SpecVAT first converts D to a weighted affinity matrix and then performs spectral decomposition of the normalized Laplacian of the weighted affinity matrix. It then transforms the original feature vector representation of data points using the k largest eigenvectors. The VAT algorithm applied to the transformed representation of the data points produces RDIs that have much sharper contrast between dark blocks along the diagonal and the remaining pixels in the image. The output of the SpecVAT algorithm is a set of images (corresponding to different values of k) from which we can visually choose a "best" SpecVAT image in terms of clarity and block structure.

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