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Could someone explain to me: the concept of "conditioning" in spatial statistics in a fairly advanced context?
Here is an example to clarify the question:
Step 1) generate a 2D point process, here 6 realization are shown:

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Step 2) choose a region from one realization

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Step 3) try to fill the rest of same size region, here I used the same PP:

enter image description here

Step 4) remove the overlapped region and merge with given part:

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Step 5) assume you did right!

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Step 6) ask question now: Is the result is a conditioned PP on a set of given points?

Step 7) wait keeping hope that somebody will answer your question thoroughly:)

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Although the context may be "advanced," the sense of "conditioning" is elementary. To see this, simplify: focus on two locations $1$ and $2$ in the domain of a spatial process. Let the value at $1$ be $X$ and the value at $2$ be $Y$. Then the multiple realizations of step 1 are sampling from the bivariate distribution $(X,Y)$ and the steps 2-5 are observing one value $x$ of $X$ and then sampling from the conditional distribution $(X,Y) | X=x$. – whuber♦ Feb 6 '12 at 15:53
@whuber Could you demonstrate what you said and provide a complete answer? That can be useful for learning to me and future readers. Furthermore, it can be possible to me to narrow my question according to your answer. – Developer Feb 7 '12 at 2:37

1 Answer

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The answer is clearly "yes". Your resulting pattern at the end of step 5 is conditioned on the points in the top corner. Imagine doing steps 3,4 and 5 again. You'll get the same points in the top left corner, and different points elsewhere.

There's also the element of working out how you've generated the new points given the points in the corner. Did you use the number of points in the small square to estimate the density, and then generate the new points conditional on that density? There's another conditioning.

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I got your second point. For the problem above I just used the same PP setting that I had. So both blue and red generations are from the same PP. There are however something makes me not satisfied with the resulting realization. 1) It could be from a region that locally is dense so not matching well with superimposed one. 2) original PP was a PPP, however after merging only one region not sure the realization stays Poisson. Hope you got the point, if I am not talking precisely, please keep kindness and explain more. – Developer Feb 6 '12 at 14:10

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