Algorithm S24. coVAT2 [95], [96]. Input : D - m # n rectangular dissimilarity matrix Output: D )), Dt r , and Dt c - Reordered dissimilarity matrices 1 Build estimates of D r and D c by interpreting the m rows and n columns of D as the feature vector representing m row objects and n column objects respectively 2 Apply VAT to D r generating permutation array RP = " RP ^1h, P ^2h f, P ^m h, 3 Apply VAT to D c generating permutation array CP = " CP ^1h, C ^2h f, C ^nh, 4 for p ! 1 to m do 5 for q ! 1 to n do 6 D ))p,q = D RP ,CP 7 end 8 end 9 Create I ^D )) h p q Visual Assessment of Clustering Tendency The VAT algorithm, introduced in [11], is a method for visually assessing the clustering tendency in a set of objects O = " o 1, o 2, f, o n ,, whether they are represented by object feature vectors or numerical pairwise dissimilarity values. The input matrix D, generally any dissimilarity matrix, but usually a pairwise distance matrix built from vector data inputs, is reordered to obtain D) using a modified Prim's algorithm. The image I (D) ), when displayed as a grayscale image, shows possible clusters as dark blocks along the diagonal. The pseudocode for VAT is given in algorithm S1 in "Pseudocode for Various Algorithms Belonging to the Visual Assessment of Tendency Family." Our first example replicates part of [13, Fig. 2]. Figure 1(a) is a scatterplot of three subsets, X 1, X 2, and X 3, drawn from Gaussian distributions centered at (0, 0), (3, 4), and (6, 0), with cardinalities of ; X 1 ; = 750, ; X 2 ; = 1,750, ; X 3 ; = 2,500, respectively. All three distributions had the same covariance matrix: 0.1 0 E. Ri = ; 0 0.1 Algorithm S25. Scalable coVAT (scoVAT) [98]. Input : D - m # n rectangular dissimilarity matrix, M and N are large c l - An overestimate of the true but unknown number of clusters c m - Row sample size n - column sample size Output: D )m # n - scoVAT reordered dissimilarity matrix 1 Build estimates of [D r ]M # M and [D c]N # N by interpreting the M rows and N columns of D as the feature vector representing M row objects and N column objects respectively 2 Apply siVAT on [D r ]M # M returning m sampled rows 3 Apply siVAT on [D c]N # N returning n sampled rows 4 Build D m # n by extracting m sampled rows and n sampled columns from D M # N 5 Apply coVAT on D m # n image in Figure 1(b). However, we will usually use Euclidean distance, so we suppress the set X and subscript E unless clarity demands it. Improvements in the VAT RDI for Complex Cluster Structure and Noisy Data Although VAT can often provide a useful estimate of the number of clusters in a data set that have globular compact separated clusters, the VAT image can often be inconclusive, especially if the cluster structure in the data set is complex. To illustrate this point, Figure 2(a) and (b) shows two data 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -0.5 Figure 1(b) is the VAT image of the Euclidean distance matrix D E, whose ijth entry is the Euclidean distance d E,ij = < x i - x j <. The structure of the data in Figure 1(a) is represented in Figure 1(b) by the three dark diagonal blocks. For the three compact, well-separated clusters in X, the sizes of the three blocks in the VAT image correspond exactly to the cardinalities of the three subsets, beginning with the largest block for X 3 at the top, X 2 in the middle, and X 1 at the bottom. This shows how the VAT image can suggest both the number and sizes of clusters in the data. In addition, the VAT image in Figure 1(b) depends on the choice of distance used to build the input dissimilarity matrix. A change in the metric might result in a different VAT image for the same data set, which explains the explicit notation I (D)E (X )) for the 0 (a) (c) 0.5 1 -1 -1 -0.5 0 (b) 0.5 1 (d) Figure 2. The data scatterplots and VAT images for the two data sets: (a) a three-line data set, (b) a three-ring data set, (c) I (D *) for the three-line data set, and (d) I (D *) for the three-ring data set. Ap ri l 2020 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 19

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