Combining transition probabilities for stochastic simulation of categorical fields

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.

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:

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:

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)