[R-sig-Geo] 3D Point Pattern Summary Statistics

[Adrian.Baddeley at csiro.au]

Rajagopal Vijayaraghavan [vijayaraghavan at smart.mit.edu] wrote:

> I want to test the validity of a non-parametric model to simulate a 3D point pattern. [...]

> I use the F3est, K3est, pcf3Est and G3est functions to generate the envelopes from 
> this non-parametric model to then see if my observed data (that I used to generate the 
> stats for the energy function in the first place) fits into the envelope.

These functions are from the 'spatstat' package - but the question applies more generally.

> I find that the K, G and pcf of the observed data fits well within the envelopes of K, G and pcf
>  from the 99 simulated point patterns. However, I find that the F function of the observed is 
> slightly outside the envelope for the F from the simulations. My questions are:

> (i) If G, K and pcf (I acknowledge that K and pcf are related) fit into the envelope, then F should, should it not?

No, not necessarily. Different statistical procedures do not always agree when applied to the same data.

>  F(r) is the proportion of the nearest points in the point process that are less than r away from an
>  arbitrary point in the window. The K and G are related to some degree to this aren't they?

No, they come from the same circle of ideas but they are not directly related.
In some sense F is the 'opposite' of G. 

> (ii) How far out of the envelope is too far? I guess I am asking if the envelope test is a hard black/white 
> kind of rule or does one think of the context and justify how sincerely one must adhere to the envelope test 
> for these different summary statistics?

There are two basic kinds of envelopes: 'pointwise' and 'global', which have different statistical interpretations.

Assuming you used the spatstat function 'envelope' to generate the envelopes, the default is to produce a pointwise envelope.
The interpretation here is that, for a *fixed* value of distance r, the test rule which rejects the null hypothesis whenever the empirical curve
goes outside the critical boundary, has significance level alpha (usually alpha = 2/(nsim + 1)). 

If you use envelope() with global=TRUE, you get a global envelope. The interpretation is that the test rule which rejects the null
hypothesis if the empirical curve EVER goes outside the critical boundary for ANY value of distance r, has significance level alpha
(usually alpha = 1/(nsim+1)).

In your question, talking about the curve "fitting inside" the envelope, you effectively treated the envelope as if it was the 'global' kind.
This would be valid if you had set global=TRUE in the call to envelope.

This is explained in detail in the help file for 'envelope', or in the spatstat workshop notes (see www.spatstat.org)

Adrian Baddeley

 Prof Adrian Baddeley FAA
School of Earth & Environment                          |      CSIRO Mathematics, Informatics & Statistics
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