Department of Image Processing

One of the basic motivations for using statistics is to decide whether your observed data corresponds well to a particular model. This is can be considered as hypothesis testing - the null hypothesis is that the data follows the model, the alternative hypothesis is that it does not. If a complicated model is assumed for the null hypothesis, it is reasonable to use simulations from the model for the testing rather than its theoretical characteristics (which may not be available at all).

Sometimes it is appropriate to describe your data with a functional characteristic, say F(r), for a range of values of the argument r. An example would be a binary image showing some (random) set and F(r) would be the area of the set dilated by r. If a model is assumed for the random set, we can calculate values of F(r) from simulated images (which follow the model) in order to see the typical behaviour of F(r) under the model. Then we compare the values of F(r) calculated from the original image to this typical behaviour to see if the data follows the model or not.

The problematic part is hidden behind the word "compare". For many years inappropriate methods were used simply because there was no better choice. However, recent development in spatial statistics has solved the problem and the results are easily applicable in many other fields. We will discuss both the inappropriate methods to see the problems and finaly the new approach that remedies them.

[1] Loosmore, N. B. & Ford, E. D. (2006). Statistical inference using the G or K point pattern spatial statistics. Ecology 87, 1925–1931.
[2] Grabarnik, P., Myllymäki, M. & Stoyan, D. (2011). Correct testing of mark independence for marked point patterns. Ecological Modelling 222, 3888–3894.
[3] Myllymäki, P., Mrkvička, T., Seijo, H., Grabarnik, P. (2014+). Global envelope tests for spatial processes. arXiv:1307.0239.