Albany New York (1) G. P. Patil

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

Space Age and Stone Age Syndrome Data: Space Age/Stone Age Analysis: Space Age/Stone Age Data Space Age Stone Age Analysis Space Age + + Stone Age +

Geospatial Cell-based Data Kinds of Data Cell as a Unit (Regular grid layout) Categorical Ordinal Numerical Multivariate Numerical Cell as an Object (Irregular cell sizes and shapes) Partially Ordered

Consider a 21st century digital government scenario of the following nature: What message does a remote sensing-derived land cover land use map have about the large landscape it represents? And at what scale and at what level of detail? Does the spatial pattern of the map reveal any societal, ecological, environmental condition of the landscape? And therefore can it be an indicator of change?

Consider a 21st century digital government scenario of the following nature: How do you automate the assessment of the spatial structure and behavior of change to discover critical areas, hot spots, and their corridors? Is the map accurate? How accurate is it? How do you assess the accuracy of the map? Of the change map over time for change detection?

Consider a 21st century digital government scenario of the following nature What are the implications of the kind and amount of change and accuracy on what matters, whether climate change, carbon emission, water resources, urban sprawl, biodiversity, indicator species, or early warning, or others. And with what confidence, even with a single map/change-map?

The needed partnership research is expected to find answers to these questions and a few more that involve multicategorial raster maps based on remote sensing and other geospatial data. It is also expected to design a prototype and user-friendly advanced raster map analysis system for digital governance.

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

National Mortality Maps and Statistics Geographic Patterns—1 Mortality rate due to a specific cause of death Elevated rates areas, patterns Ordinal thematic maps Transition pattern, transitionogram Transition matrices; spatial association with varying distance Comparatives with different causes of death

National Mortality Maps and Statistics Geographic Patterns—2 Surface topology and spatial structure High mortality area delineation Hotspots, clusters, outbreaks, corridors Surface smoothing Masking of true geographic patterns? Echelon analysis, original surface, smoothed surface

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.

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

Figure 1. Example of perfect positive and perfect negative correlation between two coordinates (variables).

Hasse Diagram (all countries)

Ranking Partially Ordered Sets – 1 S = partially ordered set (poset) with elements a, b, c, …. How can we rank the elements of S consistent with the partial order? Such rankings are called linear extensions of the partial order. Different people with different perceptions and priorities may choose different rankings. How many rankings assign rank 1 to element a ? Rank 2 ? Rank 3 ? , etc. If rankings are chosen randomly (equal probability), what is the likelihood that element x receives rank i ?

Ranking Partially Ordered Sets – 2 An Example Poset (Hasse Diagram) Some linear extensions a a a b b a b c c b a a c d b e c c c e b d d e e f d d e e d f f f f f Jump Size: 3 1 5 4 2 Jump or Imputed Link (-------) is a link in the ranking that is not implied by the partial order

Ranking Partially Ordered Sets – 5 Poset (Hasse Diagram) Linear extension decision tree a b a b c d c b a d e b c d c d a e f b e d e d c e d c c d d e f d e f e d e f e f e f f f f f e f f e f e f f e f e f e Jump Size: 1 3 3 2 3 5 4 3 3 2 4 3 4 4 2 2

Ranking Partially Ordered Sets – 3 In the example from the preceding slide, there are a total of 16 linear extensions, giving the following frequency table. Rank Element 1 2 3 4 5 6 Totals a 9 16 b 7 c d e f 10 Each (normalized) row gives the rank-frequency distribution for that element Each (normalized) column gives a rank-assignment distribution across the poset

Ranking Partially Ordered Sets – 3a Rank-Frequency Distributions Element a Element b Element c Element d Element e Element f Rank Rank

Cumulative Rank Frequency Operator – 5 An Example of the Procedure In the example from the preceding slide, there are a total of 16 linear extensions, giving the following cumulative frequency table. Rank Element 1 2 3 4 5 6 a 9 14 16 b 7 12 15 c 10 d e f Each entry gives the number of linear extensions in which the element (row label) receives a rank equal to or better that the column heading

Cumulative Rank Frequency Operator – 6 An Example of the Procedure 16 The curves are stacked one above the other and the result is a linear ordering of the elements: a > b > c > d > e > f

Original Poset (Hasse Diagram) Cumulative Rank Frequency Operator – 8 An example where F results in ties Original Poset (Hasse Diagram) a c b d a b, c (tied) d F Ties reflect symmetries among incomparable elements in the original Hasse diagram Elements that are comparable in the original Hasse diagram will not become tied after applying F operator

Original Poset (Hasse Diagram) Cumulative Rank Frequency Operator – 7 An example where F must be iterated a f e b d c h g F F 2 Original Poset (Hasse Diagram) a f e b d c h g a f e b c g d h

Ranking Possible Disease Clusters in the State of New York Data Matrix

Composite Indexes -- 1 I1, I2, . . ., Ip indicators for ranking elements of some set G(I1, I2, . . ., Ip) = composite index Many possible choices for G: Only general requirement is that G must be increasing in each indicator separately

Composite Indexes -- 2 Each choice of G determines a set of G-contours in indicator space and thereby determines a set of substitution or trade-off rules among the indicators Contour of constant G x substitutes for y  y Indicator 2 (y)  x  y  x Indicator 1 (x)

Comparison of Ranking Methods Composite Indicator Approach Requires choice of composite index G Implicitly or explicitly, requires choice of substitution rules among different indicators (this is often like comparing apples and oranges) Difficult to achieve a consensus on choice of G. Final decision is often made on basis of mathematical simplicity instead of scientific substance Once G is chosen, future elements are easily incorporated into the ranking without changing relative ranks of earlier elements

Comparison of Ranking Methods Poset Cumulative Rank Frequency Approach Entirely objective---no arbitrary choices involved Computationally challenging (typically requires combinatorial MCMC) Final ranking applies only to the given set of elements and reflects overall structure of entire Hasse diagram If new elements are added to the collection to be ranked, all computations must be redone and relative rankings of earlier elements may change

Incorporating Judgment Poset Cumulative Rank Frequency Approach Certain of the indicators may be deemed more important than the others Such differential importance can be accommodated by the poset cumulative rank frequency approach Instead of the uniform distribution on the set of linear extensions, we may use an appropriately weighted probability distribution  , e.g.,

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