Representing Uncertainty of Qualitative Thematic Maps with an Inter-map Cell Swapping Heuristic

EHLSCHLAEGER, Charles R. (chuck@everest.hunter.cuny.edu), Hunter College, Department of Geography, New York, NY 10021

Key Words: uncertainty, spatial data, thematic maps, area class maps, categorical coverage maps, data fusion, conflation, conditional simulation, unconditional simulation, geostatistics, statistical modelling, Monte Carlo simulation

Monte Carlo simulation, a technique that generates many versions of possible application results, is a popular method for representing application uncertainty. Monte Carlo simulation can be used to generate many sets of application input maps with each input map consisting of a possible version of reality. Representing the total distribution of potential application results during the Monte Carlo process requires two critical components: a thematic class probability model and a model of spatial autocorrelation for these same classes. This paper introduces a combination of techniques including an inter-map cell-swapping algorithm, a class probability model, and a new spatial statistic. Together, these techniques allow for the generation of spatially autocorrelated random qualitative thematic maps for the purpose of representing spatial application uncertainty.

This paper assumes a generalized thematic map of the study area is available. A generalized map, for the purpose of this paper, is a map containing qualitative thematic information at a resolution too coarse and/or attribute information too inaccurate to be useful for a particular application. Qualitative thematic maps also are known as "area class maps" or "categorical coverage" maps. Samples of application quality data must also be available. For the purpose of this paper, application quality data are qualitative thematic information accurate enough and with a resolution fine enough to achieve useful application results. Class information from the application quality data is used to determine the spatial statistics of all generated maps. Samples of application quality data can either be in the study area, or outside the study area if the application quality data intersects generalized thematic data. If the application quality data are inside the study area, the Monte Carlo process will be a conditional simulation, otherwise, the Monte Carlo process will be an unconditional simulation. Intersecting application quality data and the generalized thematic map allows a conflation technique defining an algorithm that will generate the probability vector for any generalized map cell. A probability vector is a set of values representing the likelihood each class will be located in that particular cell. This probability model analyzes each application quality data cell intersecting the generalized map by looking at the distance within a generalized map class polygon, as well as the distance away from the closest cell of other classes in the generalized map. During these analyses, the generalized map will be re-sampled to the resolution of the application quality data.

Spatial autocorrelation is measured with a new spatial statistic named densogram. A densogram measures the proportion of each application quality cell's class (MG1) at various distances from cells of the same class. A densogram and a variogram of a binary map provides the same measure while the densogram is computationally quicker when calculating statistic changes during the cell-swapping heuristic. The quality of a randomly generated Monte Carlo input map will be based on how closely its densogram matches the densogram of application quality data using a weighed least squares analysis.

The inter-map cell swapping heuristic begins with a set of random thematic maps. Each cell within each random thematic map is given a class value based on the probability vector defined by the location of class polygons in the surrounding generalized map. Cells are swapped at the same location from different maps if the resulting maps' densograms provide a better overall fit to the application quality data's densogram. This heuristic is continued until no more possible cell swaps are possible. The inter-map swapping technique described in this paper works for both conditional and unconditional simulation of qualitative thematic data because the proportion of class values within a cell will never change, preserving the probability model. The inter-map swapping technique is flexible, allowing for any spatial statistic to be used, whether global or regional, alone or in combination, with other spatial statistics.

Source code and sample data demonstrating the inter-map cell-swapping heuristic is included in this paper's appendix. The software is written in Java, and works for versions 1.1.7a and later. The software is part of the Research Geographic Information System (RGIS). RGIS is a public domain GISystem designed by the author as a research and educational tool. RGIS is used in Hunter College's geographic programming class to demonstrate object-oriented programming and will run on all commonly used workstations. RGIS allows GISystem raster data from Arc/Info and ArcView as inputs.