The Structural Representation of Proximity Matrices With MATLAB

1.1 The number.dat data file from Shepard, Kilpatric, and Cunningham (1975). 11 3.1 A proximity matrix, morse digits.dat, for the 10 Morse code symbols representing the first ten digits 9.1 Order-constrained least-squares approximations to the digit proximity data of Shepard et al. (1975); the upper-triangular portion is anti-Robinson and the lower-triangular portion is strongly-anti-9.2 The 45 subsets listed according to increasing diameter values that are contiguous in the object ordering used to display the upper-triangular portion of Table 9.1. The 22 subsets given in italics are redundant in the sense that they are proper subsets of another listed subset with the same diameter.. .. .. 164 9.3 The fourteen (nonredundant) subsets listed according to increasing diameter values are contiguous in the linear object ordering used to display the lower-10.1 The fifteen (nonredundant) subsets listed according to increasing diameter values are contiguous in the circular object ordering used to display the CSAR entries in Table 10. A circular strongly-anti-Robinson order-constrained least-squares approximations to the digit proximity data of Shepard et al. 9.1 Two 4 × 4 submatrices and the object subsets they induce, taken from the anti-Robinson matrix in the upper-triangular portion of Table 9.1. For (a), a graphical representation of the fitted values is possible; for (b), the anomaly indicated by the dashed lines prevents a consistent graphical representation 10.1 A graphical representation for the fitted values given by the circular strongly-anti-Robinson matrix in the lower-triangular portion of Table 10.2 (Vaf = 72.96%). Note that digit 3 is placed both in the first and the last positions in the ordering of the objects with the implication that the sequence continues in a circular manner. This circularity is indicated by the curved dashed line in the figure.. The task of linear unidimensional scaling (LUS) can be characterized as a specific data analysis problem: given a set of n objects, S = {O 1 , ..., O n }, and an n × n symmetric proximity matrix P = {p ij }, arrange the objects along a single dimension such that the induced n(n − 1)/2 interpoint distances between the objects reflect the proximities in P. The term " proximity " refers to any symmetric numerical measure of relationship between each object pair (p ij = p ji for 1 ≤ i, j ≤ n) and for which all self-proximities are considered irrelevant and set equal to zero (p ii = 0 …