Modern Multidimensional Scaling: Theory and Applications by Ingwer Borg;Patrick Groenen

By Ingwer Borg;Patrick Groenen

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A Ruler-and-Compass Approach to Ratio MDS For convenience in laying out the map, we first identify those cities that are farthest from one another. 1 shows that these are the cities 2 and 3, whose distance is d 23 = 1212 units. We then want to place two points on a piece of paper such that their distance is proportional to d 23 = 1212 units. To do this, we choose a scale factor, s, so that the reconstructed map has a convenient overall size. 0004125 so that s · 1212 = 5. 1 are then multiplied by s.

These definitions constitute the Hefner ( 1958) generalization of the Thurstone ( 1927) Case-V model. Wagenaar and Padmos (1971) and Spence and Graef (1974) report simulation studies based on the Hefner model. 7. 0 to CXJ, when represented in 1D through 5D (Spence & Graef, 1974). ponents sampled from N(O, a 2 ) to each of its coordinates. This leads to error-affected distances that are subsequently taken as proximities. In contrast to the study by Spence and Ogilvie (1973) described above, this simulation allows one to specify the true (underlying) MDS space as the point configuration used in computing the proximities.

The curves indicate again that Stress depends on n and m. One also notes that each additional dimension reduces Stress increasingly less. 6 show. 03 should result in reliable cutoff values for testing this Ho. 5 and curves interpolated therein for n = 12, 13, ... 12) 40 3. 35. 12) can be used to estimate expected "random" Stress for the range n = 10, ... , 60 and m = 1, ... , 5. We will see that the Stress values in all real-data MDS applications discussed in this book lie definitely under the values expected for Ha.

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