Combining transition probabilities for stochastic simulation of categorical fields

# Guofeng Cao and Phaedon C. Kyriakidis and Michael F. Goodchild

In the paper entitled "Combine Transition Probabilities for stochastic simulation of categorical fields", we propose a new geostatistical method for modeling categorical spatial data, such as land use classes and socio-economic statistics data.  In the proposed method, transition probabilities used as measures of spatial structure and the conditional or posterior (multi-point) probability is approximated by a weighted combination of pre-posterior (two-point) transition probabilities, while accounting for spatial interdependencies often ignored by existing approaches.

As supplementary material to this paper, this webpage lists the simulated results under different parameters settings for Truncated Gaussian based model (used as reference model), Spatial Markov Chain model, Tau model with weights equal one (i.e., permanence of ratios), and the our new proposed method. It is shown that the new proposed method achieves best results in every case. For more information, please check out our Matlab toolbox and tutorials at: http://www.geog.ucsb.edu/~cao/NGAtoolbox.rar.

# 1, Spherical model

Parameters configuration:

Three categories labeled as 1,2,3  and the class proportion are 0.35,0.4,0.25 respectively; simulation  is performed at the grid nodes of a 100*100 regular raster. The variogram model for the underlying Gaussian field is a nugget effect with sill 0.001 plus an isotropic spherical model with range 5 with sill 0.999. Sets of 50 simulations are generated with maximum of 15 neighbors used for each node.

A realization of unconditional simulation using Truncated Gaussian-based simulation (reference image)

### A realization of unconditional simulation using SMC model: A realization of unconditional simulation using Tau mode with weight = 1 A realization of unconditional simulation using Tau mode with OK weights Comparison of reproduced transiograms using different models: # 2, Exponential Model

## Parameters configuration:

Three categories labeled as 1,2,3 and the class proportion are 0.3,0.4,0.3 respectively; simulation  is performed at the grid nodes of a 100*100 regular raster. The variogram model for the underlying Gaussian field is a nugget effect with sill 0.001 plus an isotropic exponential model with range 10 with sill 0.999. Sets of 20 simulations are generated with maximum of 15 neighbors used for each node.

A realization of unconditional simulation using Truncated Gaussian-based simulation (reference image) A realization of unconditional simulation using SMC model A realization of unconditional simulation using Tau model with weight = 1 A realization of unconditional simulation using Tau model with OK weights Comparison of reproduced transiograms using different models: # 3, Gaussian Model

## Parameters configuration:

Three categories labeled as 1,2,3 and the class proportion are 0.3,0.4,0.3 respectively; simulation  is performed at the grid nodes of a 100*100 regular raster. The variogram model for the underlying Gaussian field is a nugget effect with sill 0.1 plus an isotropic Gaussian model with range 10 with sill 0.9. Sets of 20 simulations are generated with maximum of 15 neighbors used for each node.

A realization of unconditional simulation using Truncated Gaussian-based simulation (reference image) A realization of unconditional simulation using SMC model A realization of unconditional simulation using Tau model with weight = 1 A realization of unconditional simulation using Tau model with OK weights Comparison of reproduced transiograms using different models is as follows.  We get more fluctuations here than the previous examples due to the larger sill of nugget models (0.1 compared to 0.001) 