| Edge-correction | ||||||
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By defining a point process in a bounded window, the distribution of the points is different on the edges of the window. How much, depends on the density. In the left picture below, a Strauss process with inhibition between the points is shown. Notice the high frequency of points on the edges of the window. This is of course due to the fact, that the density decreases drastically for each pair of points that are closer than R=0.01. Thus, by 'placing' points on the edges, the space is being 'used' most economically.
Observing a point pattern in a window, there are two possibilities: Either the point process also lives outside the window, or it doesn't.
If it doesn't, then a point process defined in the window might be applicable.
If the point pattern also lives outside the window, then a point process defined only in the window is not sufficient. However, for practical, and also often theoretically reasons, it is convinient to work with point processes defined in the window.
There are several ways to handle this problem, known as edge-correction. Se the excellent survay paper Baddeley (1999).
One way to handle the problem is torus edge-correction. It can be thought of as repeating the point pattern all over the plane. Another way to think of this method is to wrap the unit square as a torus (hence the name). Thus, the point relation is changed such that if a point is too close to the edges of the window, then the neighbourhood of the point is continued elsewhere in the window: Illustration of how the torus edge-correction works.
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Strauss point processes in the unit square (  ,  ,
R) = (500, 0.01, 0.1)
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| without edge-correction | with torus edge-correction |
In the paper Nielsen & Jensen (2001) we describe the torus edge-correction mathematically. Furthermore, we proove that the marginal distribution of a single point is uniform in the window.
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This page was last modified on September 28th 2001 |
