blocks along the diagonal for GM2, whereas clusiVAT shows three light blocks, including many tiny blocks (data points) along the diagonal. Although clusiVAT and FensiVAT both show three blocks for GM1 and GM2, FensiVAT provides a more convincing assessment because of the sharper contrast between diagonal blocks and the background. Moreover, FensiVAT takes only a small fraction of the time needed by clusiVAT for both data sets. The sizes of the diagonal blocks in all four images show the relative size of each cluster accurately, which supports the claim that nearMMRS sampling replicates (approximately) the same cluster distribution in the sample as the MMRS sampling used by clusiVAT. Finally, SL clustering of the samples and NPR extension to the rest of the data set is performed in the down space for different random projections, and majority-voting-based schemes are used to assign the final cluster labels. The pseudocode for FensiVAT is given in algorithm S22 in "Pseudocode for Various Algorithms Belonging to the Visual Assessment of Tendency Family." Coclustering The VAT family of algorithms tackles the problem of clustering tendency assessment and subsequent clustering for an n # n (square) dissimilarity (or, in more general terms, relational) matrix. An even more general form of relational data is rectangular. These data are represented by an m # n dissimilarity matrix D, where the entries are the pairwise dissimilarity values between m-row objects O r and n-column objects O c . An example comes from Web-document analysis, where the row objects are m webpages, the columns are n words, and the (dis) similarity entries are occurrence measures of words in the webpages [92]. Another important problem involving rectangular relational data is the analysis of gene-expression data, where the m rows correspond to genes and the n columns correspond to tissue samples or conditions [93]. In each case, the row and column objects may be nonintersecting sets, so structural relations between the row (or column) objects are unknown. Conventional relational clustering algorithms are ill equipped to deal with rectangular data. Additionally, the definition of a cluster as a group of similar objects takes on a new meaning. There can be groups of similar objects that are composed of only row objects, only column objects, only mixed objects (often called coclusters), and, finally, clusters in the union of all of the row and column objects. In other words, a rectangular dissimilarity matrix comprises four different clustering problems. Bezdek et al. [94] developed an approach for visually assessing cluster tendency for the objects represented by a rectangular relational data matrix D by assuming that D is an m # n (sub)matrix containing only m # n of the (m + n) # (m + n) possible pairwise dissimilarities 34 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE Apri l 2020 between objects in O = O r , O c . The full distance matrix D r , c of O was assumed to be of the form Dr,c = ; Dr D E, DT Dc where D T is the transpose of D. The coVAT coclustering VAT (coVAT) algorithm proposed in [94] first generates an estimate of D r and D c by interpreting the m rows and n columns of D as feature vectors representing m row objects and n column objects, respectively, and imputing the missing values using the (Euclidean) distance between them. The VAT algorithm is then applied to D r , c to generate the reordering indices of the objects in O = O r , O c . The coVAT algorithm then unshuffles the row objects from the column objects based on their indices to generate individual row and column reordering arrays: RP (for row permutation) and CP (for column permutation). The coVAT image is then produced by displaying a (scaled) u = [du i j] = [d RP(i),CP( j )], for version of the rectangular matrix D 1 # i # m and 1 # j # n , obtained by reordering the rows and columns of the original matrix D using the indices stored in RP and CP, respectively. Just as with VAT, dark u ) (not along any diagonal, and not necessarily blocks in I (D square) suggest the existence of coclusters. The pseudocode for coVAT is given in algorithm S23 in "Pseudocode for Various Algorithms Belonging to the Visual Assessment of Tendency Family." The basic idea of coVAT is embodied in Figure 15, which is excerpted from [94] (Figures 1 and 3). Figure 15(a) depicts a set of n = 20 points labeled as row objects (the circles) and m = 40 points labeled as column objects (the squares). It may be helpful to imagine the circles as women (5) and the squares as men (4) who have congregated at the locations shown in five small groups. Three of the groups are "pure," or unmixed: the two sets of squares at the top and the centrally located set of circles at the bottom. The lower left and lower right clusters are mixed groups of circles and squares, that is, coclusters. The spatial coordinates of these points are used only to compute Euclidean distances between the circles and squares, yielding a rectangular dissimilarity matrix for input to coVAT. Evidently, there are three clusters (O r) in the row objects (ignoring the column objects), four clusters (O c) in the column objects (ignoring the row objects), five clusters (O r , c) in the union of the row and column objects, and two mixed coclusters in O r , c . Figure 15(b)-(e) shows the coVAT images built from the rectangular input data for each of these four cases. The numbers of clusters for each of the four clustering problems are seen in the images as dark blocks: diagonal for the three square subproblems and nondiagonal for the coclustering problem. In this simple example, coVAT images provide a good visual estimate for possible cluster structure in all four problems. The coVAT algorithm was extended to the coiVAT algorithm in [95] by applying a path-based distance transform used by the iVAT algorithm.

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