Conditioning a point process is the concept being investigated in this post. The marked point processes are of interest. Figure A demonstrates a phenomenon being investigated for modelling. We recognize a feature such can be modeled by a point as the center and an angle (Figures B and C). Therefore we construct a point process based on the central points (Figure C). Then we develop our model by adding a mark as representative of angle (Figures C and D).

Figure (i) demonstrates a field being studied in which only some samples in some regions are available. Some regions therefore are unsampled and unknown. As described above we generate a marked point process to get a model describing the phenomenon in entire space. Figure (ii) is the result. As it can be seen the left hand side of (ii) does not match the existing samples. There are indeed generated stochastically. figure (iii) demonstrates the final goal in which a full map of the phenomenon has been modeled in entire space conditioned to the existing samples at the left hand side.
Question: How can we do it?
Note: The conditional modelling referred to conditional simulations those are common in geostatistics cannot be applied for this case at least directly. For example, a grade estimation made by Kriging or IDW is already conditioned to the existing samples. This is because an interpolation (weighted averaging) is being used for estimation which makes sense for grade. However for the case being studied here, there is no such understanding for variation of angle (as one of variables of interest). Furthermore, the output of the estimations/simulations in geostatistics etc is often a full grid which means you have a point very close to the sampling point. Indeed the location itself is not considered as variable while for PP stochastic generated locations are the base. The marks also are generated by means of stochastic models where again cannot be adjusted regarding to nearby samples.

Updates: of course an answer is of interest if you don't mind!

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  • $\begingroup$ It depends on the nature of the process. If you have a particular application in mind, perhaps you could share that information. Otherwise, you're really asking generically for a list of techniques people have used to perform conditional simulations (of which there are very many) and that might not be useful to you. $\endgroup$ – whuber Oct 21 '11 at 15:31
  • $\begingroup$ More explanation was added. Just as a quick note, none of conditional methods used in geostatistics etc could be applied satisfactorily for this case. The point relies on the stochastic nature of point processes. How can it be preserved to be stochastic while being conditioned to sampling data? If you could list the methods that could be used for the case being discussed here I will share my points and experiences regarding difficulties in the application of those techniques for the problem being addressed in this post. For the best of my knowledge this is very challenging question. $\endgroup$ – Developer Oct 22 '11 at 1:28
  • $\begingroup$ As an example, you could create a sample of points consistent with the observed ones and then simulate a random field of marks at those points conditioned on the observed values. Variations of angle can be handled in various ways, such as by means of a bivariate distribution of (cos, sin) values. If there is a dependency between the point locations and the marked values, a more creative solution will be called for. But are you really in such a situation? An example would help clarify this. $\endgroup$ – whuber Oct 24 '11 at 13:12
  • $\begingroup$ @whuber 1- ...consistent...how to generate samples that are consistent? There are samples (hard data=conditioning) and stochastic generated points. 2- for the rest of your comment I'm still thinking how to implement. Why not giving me more explanation with examples as answer! I really appreciate it in advance. $\endgroup$ – Developer Oct 24 '11 at 13:59

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