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What are the main theorical differences between the homogeneous and inhomogeneous Poisson point process? What are the aspects and condition of my data that I can determine which point process best describe my data?

Your help is greatly appreciated!

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A homogeneous Poisson point process is also called complete spatial randomness described by a single parameter called the intensity (number of points per unit area). It distributes a random number of points completely randomly and uniformly in any given set. The number of points falling in two disjoint sets are independent random variables.

An inhomogeneous Poisson point process also has independence between disjoint sets but the points are not uniformly distributed. Rather the points are unevenly distributed according to the intensity function of the process.

If you only are deciding between these two models you basically need to decide whether there are signs of inhomogeneous intensity in the data. There are many tests for this. A simple one is quadrat counting where you divide your study region into disjoint subsets of equal area and use a chi-square test statistic to judge whether the count distribution is significantly non-homogeneous. This requires you to choose the size of the subsets which says something about the spatial scale you are looking for inhomogeneity at.

For point patterns in 2D the R package spatstat has facilities to do this.

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