EPA Presentation March 13,2003 G. P. Patil

This report is very disappointing. What kind of software are you using?

Figure 1. With two indicators, each object a divides indicator space into four quadrants. Objects in the second and fourth quadrants are ambiguous in making comparisons with a.

Figure 2. Contour of index H passing through object a Figure 2. Contour of index H passing through object a. A linear index is shown on the left and a non-linear index on the right.

Figure 3. The top two diagrams depict valid contours while the bottom two diagrams depict invalid contours.

Figure 4. The tradeoff or substitutability between height and weight in assessing the size of a person. The tradeoff is constant with a linear index (left) but varies across indicator space with a nonlinear index (right).

Figure 5. Hasse diagrams for four different posets Figure 5. Hasse diagrams for four different posets. Poset D has a disconnected Hasse diagram with two connected components {a, c, e} and {b, d}.

Figure 6. Bottom-up Hasse diagrams for the posets of Figure 5 Figure 6. Bottom-up Hasse diagrams for the posets of Figure 5. Hasse diagrams for Posets A and B are unchanged.

Figure 7. Hasse diagram for the four countries of Table 1 Figure 7. Hasse diagram for the four countries of Table 1. Note that it has the same structure as Poset A in Figure 5.

Figure 8. Hasse diagram for all 106 countries. Labels are the HEI ranks. The diagram is connected.

Figure 9. Hasse diagram for the countries of Western Europe Figure 9. Hasse diagram for the countries of Western Europe. The diagram is connected.

Figure 10. Hasse diagram for Latin America Figure 10. Hasse diagram for Latin America. There are four connected components. Three of these components are isolates; the remaining component contains 13 countries.

Figure 11. Hasse diagram for the 52 watersheds in the primary component. Labels are (arbitrary) row numbers in the data matrix.

Figure 12. Map of the Mid-Atlantic region showing the primary Hasse component (shaded). Geographically, there are three connected components of which two are small and located near the periphery of the region.

Figure 13: Hasse diagrams (right) of the two possible rankings for the poset on the left.

Figure 14. Rank-intervals for all 106 countries Figure 14. Rank-intervals for all 106 countries. The intervals (countries) are labeled by their midpoints as shown along the horizontal axis. For each interval, the lower endpoint and the upper endpoint are shown vertically. The length of each interval corresponds to the ambiguity inherent in attempting to rank that country among all 106 countries.

Figure 15. Rank-intervals for all 106 countries, plotted against their HEI rank. The HEI rank appears as the 45-degree line. The HEI tends to be optimistic (closer to the lower endpoint) for better-ranked countries and pessimistic (closer to the upper endpoint) for poorer-ranked countries.

Figure 16. A ranking of a poset determines a linear Hasse diagram Figure 16. A ranking of a poset determines a linear Hasse diagram. The numerical rank assigned to each element is that element’s depth in the Hasse diagram.

Figure 17. Hasse diagram of Poset B (left) and a decision tree enumerating all possible linear extensions of the poset (right). Every downward path through the decision tree determines a linear extension. Dashed links in the decision tree are not implied by the partial order and are called jumps. If one tried to trace the linear extension in the original Hasse diagram, a “jump” would be required at each dashed link. Note that there is a pure-jump linear extension (path a, b, c, d, e, f) in which every link is a jump.

Figure 18. Histograms of the rank-frequency distributions for Poset B.

Figure 19. Cumulative rank-frequency distributions for Poset B.

Figure 20. (Left) Two iterations of the CRF operator are required to transform this poset into a linear ordering. (Right) A poset for which the CRF operator produces ties in the final linear ordering.

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