1. EPA Presentation March 13,2003 G. P. Patil

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

3. 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.

4. 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.

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

6. 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).

7. 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}.

8. 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.

9. 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.

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

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

13. 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.

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

15. 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.

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

17. 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.

18. 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.

19. 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.

20. 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.

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

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

23. 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.

24. Logo for Statistics, Environment, Health, Ecology, and Society

25. National Mortality Maps and Health Statistics
Health Service Areas, Counties, Zip Codes, …
Geographical Patterns for Health Resource Allocation
Study Areas for Putative Sources of Health Hazard
Balance between dilution effect and edge effect
Case Event Analysis and Ecological Analysis
Thresholds, contours, corresponding data
Regional Comparisons and Rankings with Multiple Indicators/Criteria
Choices of Reference/Control Areas

26. Statistical Ecology, Environmental Statistics, Health Statistics—3 Multiscale Advanced Raster Map Analysis System Initiative—2
Partially Ordered Sets and Hasse Diagrams
Multiple indicators, comparisons, fuzzy rankings
Intrinsic hierarchical groups, reference areas
Performance measures, composite indices
System Design and Development
BAT, BPT, and synergistic collaboration
Bilateral and multilateral partnerships

27. Spatial Complexity with Multiple Indicators: Echelons Approach
Compare echelon features among indicators for consistency/inconsistency:
Order
Number of ancestors (distance from root of tree)
Compression by treating features as pseudo-bands

28. Multiple Criteria Analysis Multiple Indicators Partial Ordering Procedures
Cells are objects of primary interest, such as countries, states, watersheds, counties, etc.
Cell comparisons and rankings are the goals
Suite of indicators are available on each cell
Different indicators have different comparative messages, i.e., partial instead of linear ordering
Hasse diagrams for visualization of partial orders. Multi-level diagram whose top level of nodes consists of all maximal elements in the partially ordered set of objects. Next level consists of all maximal elements when top level is removed from the partially ordered set, etc. Nodes are joined by segments when they are immediately comparable.

29. Multiple Criteria Analysis Multiple Indicators Partial Ordering Procedures
Issues to be addressed:
Crisp rankings, interval rankings, fuzzy rankings
Fuzzy comparisons
Echelon analysis of partially ordered sets with ordinal response levels determined by successive levels in the Hasse diagram
Hasse diagram metrics: height, width, dimension, ambiguity (departure from linear order), etc.
Hasse diagram stochastics (random structure on the indicators or random structure on Hasse diagram)
Hasse diagram comparisons, e.g., compare Hasse diagrams for different regions

30. Health, Environment,and Raster Map Analysis System Multiple Indicators
Air Quality, Heat, and Pediatric Asthma
Urban Heat Island, Urban Sprawl
Infectious and Vector-borne Diseases
Lyme Disease, West Nile Virus, Scistosomaisis, Malaria
UV Radiation
Landscape Health
Ecosystem Health
Human Health
CDC, EPA, DOT, NASA, USBS

31. Second stage screening
Multiple Criteria Analysis Multiple Indicators and Choices Health Statistics Disease Etiology, Health Policy, Resource Allocation
First stage screening
Significant clusters by SaTScan and/or
upper surface level echelon sets
Second stage screening
Multicriteria noteworthy clusters by partially ordered sets and Hass diagrams
Final stage screening
Follow up clusters for etiology, intervention
based on multiple criteria using Hass diagrams

32. Case Study –UNEP - PSU Nationwide Human Environment Index worldwide
Construction and Evaluation of HEI
Multiple Indicators and Comparisons without Integration of Indicators
Hasse Diagrams, fuzzy rankings, and visualizations
Handbook
Interactive Queries