Institute of Stochastics

Functionality

The GeoStoch library offers a huge variety of functionality for various areas. It has been developed and is extended by an interdisciplinary group of mathematicians and computer scientists which ensures that it is up to date regarding the state of the art. Because of these permanent extensions, the following presentation only reflects the status quo.

Simulation

The GeoStoch library currently provides simulation programs for the following models:

  • 1D point processes such as the Wiener point process, the Cox point process, the Markov point process, and so on.
  • Point processes such as (homogeneous and inhomogeneous) Poisson point processes, cluster point processes, hard core point processes and Gibbs point processes (e.g. the Strauss point process and Geyer's saturation point process).
  • Germ-grain processes for an arbitrary point process and primary grain.
  • Tessellation models such as line tessellation, Voronoi tessellation, Delaunay tessellation for an arbitrary underlying point process, nestings and superpositions thereof.
  • Typical cells of a Poisson line tessellation, Poisson Voronoi tessellation, Poisson Voronoi tessellation and Cox Voronoi tessellation induced by lines.
  • Random fields such as Gaussian fields and Poisson shot noise processes.

Statistical analysis of point patterns

The GeoStoch library provides estimators for the following point process characteristics:

  • Intensity
  • (Bivariate) Pair correlation function
  • (Bivariate) Ripley's K function
  • (Bivariate) L function
  • (Bivariate) Nearest neighbour distance distribution function
  • J function
  • Mark correlation function
  • (Distance-dependent) Simpson indices

Note that the sampling window does not have to be rectangular and most of the estimators are implemented for the 2D and 3D case. Besides the purely descriptive statistical analysis GeoStoch also provides methods to fit the following point process models to a given point pattern:

  • (Homogeneous and inhomogeneous) Poisson point process
  • Matern cluster point process
  • Gibbs point processes (e.g. Strauss point process, inhomogeneous marked saturation point process)

Furthermore, several statistical tests are implemented concerning point pattern analysis:

  • Quadrat count method and Monte Carlo rank tests to test the null hypothesis of a Poisson point process
  • A Monte Carlo rank test to test the null hypothesis of independent labelling

Statistical analysis of network structures

The GeoStoch library provides several statistical tools to analyze a given network:

  • Estimation of line segment length distribution, orientation distribution, meshwidth distribution and vector of parameters (mean number of vertices, mean number of edges, mean number of cells, total length of edges - always per unit area)
  • Efficient fitting of non-iterated and iterated (nested as well as superposed tessellation models) using the estimated parameter vector
  • Validation of model choice
  • Estimation of shortest path lengths and subscriber line lengths

Image processing

In the GeoStoch library there is an abstraction for (digital) images that is independent of the dimension. Furthermore, pixels may have arbitrary values, depending on the concrete type of the image. Possible types are

  • Binary images
  • Grayscale images
  • Complex-valued images

The followinge image operations from the field of mathematical morphology are implemented:

  • Erosion and dilation (distance transformation algorithm)
  • (Adaptive) Opening and closing 
  • Median filter
  • Threshold operator
  • Boolean operations: union, intersection and complement of binary images
  • Arithmetic operations: addition, subtraction and multiplication of images
  • Convolution of two images (FFT algorithm)

Image analysis (Minkowski functionals)

The Minkowski functionals are in the 2D case proportional to the area, perimeter, and Euler number and in the 3D case proportional to the volume, surface, mean curvature, and Euler number. For both cases, different estimators have been implemented:

  • Steiner's formula
  • Principal kinematic formula

Furthermore, the covariance can be estimated from a binary image using methods of the library. Statistical area fraction tests on an hypothetical area fraction of a single binary image are possible as well as tests on the equality of the area fractions of two binary images.

Extrapolation methods (Kriging)

For the extrapolation of data the following methods are implemented:

  • Inverse distance
  • Ordinary kriging
  • Simple kriging

The GeoStoch library contains modules for the estimation and adaption of variograms, which play an important role in this context. Currently, there are three different variogram estimators. Several isotropic and non-isotropic models can be used as theoretical variograms:

  • Gauss
  • Exponential
  • Spherical
  • Nugget effect

Import and export

The GeoStoch library offers the possibility to import and export (digital) images in PBM, PGM and PPM format. Using command line tools, these image formats can be converted into all common image formats. The results of simulations, which are continuous (as opposed to discrete), can be exported into Encapsulated PostScript (EPS) format or into digital images.