Help L Function

Mon, Jan 13, 2014 5:23 PM
Firstly some comments to the original questioner.

      1. If you really need to do a formal test and obtain a p-value,
      then before doing this, it is important to verify that the point pattern is homogeneous
      (at least that the density of points per unit area is not spatially-varying)
      because this is an underlying assumption of the 'L' function. 
      In spatstat you can use quadrat.test, kstest or bermantest for example.

      2. The result of *any* test based on L will depend on the value or values of distance 'r'
      that are used. If you choose to use mad.test() or dclf.test() then it is best
      to choose the interval of 'r' values to be slightly larger than the range over which
      you suspect a spatial interaction is likely.

      3. Assuming you want to test CSR (complete spatial randomness) against the
      alternative of clustering or repulsion, it would be best to use the argument
      simulate=expression(runifpoint(n, W)) in the call to 'envelope',
      where n is the number of points in your dataset, and W is the window.
      Then the simulated patterns have the same number of points as the dataset.

      4. An alternative to a hypothesis test would be a confidence interval for the
       true L function. You can get this using the spatstat function 'lohboot' for example
       (or using the asymptotic variance estimates provided in the Lest/Kest functions).

And some comments to the experts:
 
      5. Establishing that the point process is stationary, is crucial to the validity of these methods.

      6. There is no uniformly most powerful test.
        If the alternative hypothesis is a point process with finite range R, 
        and we use mad.test() or dclf.test() over the interval of r values [0, s], 
        then the power of both tests is maximised when s is slightly greater than R, 
        and dclf.test() achieves a higher maximum power than mad.test(), 
        but the power of dclf.test() often drops off dramatically as s increases beyond R, 
        while the power of mad.test() never does. See Baddeley et al (2014)

      7. Therefore the optimal choice of test depends on how much we know
       about the (maximum) range of interaction R. If we know R = 4 metres, 
       then we should probably choose dclf.test() with s = 4.5. If we don't know
       anything about the spatial interactions, then we should use mad.test().

      8. Monte Carlo tests based on fitted models are typically conservative
       (i.e. true probability of type I error is smaller than nominal probability alpha;
       true p-value is smaller than calculated p-value).
       In this context the conservatism can be extreme.
       This can be avoided in the case where CSR is the null hypothesis,
       by conditioning on n, as advised in #3 above. Otherwise it is a problem.

Reference: 

   @Article{bdhlmn14,
  author =	 {A. Baddeley and P.J. Diggle and A. Hardegen and
                  T. Lawrence and R. Milne and G.M. Nair},
  title =	 {On tests of spatial pattern based on simulation
                  envelopes},
  journal =	 {Ecological Monographs},
  year =	 2014,
  note =	 {In press}
}
      


Prof Adrian Baddeley FAA
University of Western Australia
Help L Function

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