# Analyze and generate “clumpy” distributions?

Are there standard ways of analyzing and generating "clumpy" distributions?

• analyze: how clumpy is a given point cloud (in 1d, 2d, nd), what are its clumpy coefficients?

• generate or synthesize a pseudo-random cloud with coefficients C

(These are the basics for any family of distributions, e.g. normal.)

There are many kinds of clumpiness in nature (traffic jams, clumpiness of climate changes), so it's a wide term with I imagine various attempts at description and various links to "classical" statistics.
I'm looking for an overview; pictures would be nice.

Added Friday 22 Oct: I had hoped to find methods for both analysis and synthesis,
abstract <-> real both ways; surely a wide-ranging abstraction must do both. Still looking ...

-
what type of data are you working with? From your examples I assume you are dealing with time series? –  nico Oct 21 '10 at 17:57

If assessing spatial auto-correlation is what your interested in, here is a paper that simulates data and evaluates different auto-regressive models in R.

Spatial autocorrelation and the selection of simultaneous autoregressive models by: W. D. Kissling, G. Carl Global Ecology and Biogeography, Vol. 17, No. 1. (January 2008), pp. 59-71. (PDF available here)

Unfortunately they do not have the code in R they used to generate the simulated data, but they do have the code available of how they fit each of the models in the supplementary material.

It would definately help though if you could be a little more clear about the nature of your data. Many of the techniques intended for spatial analysis will probably not be implemented in higher dimensional data, and I am sure there are other techniques that are more suitable. Some type of K-nearest neighbors technique might be useful, and make sure to change your search term from clumpy to cluster.

Some other references you may find helpful. I would imagine the best resources for simulating data in such a manner would be with packages in the R program.

Websites I suggest you check out the Spatstat R package page, and the R Cran Task View for spatial data. I would also suggest you check out the GeoDa center page, and you never know the OpenSpace Google group may have some helpful info. I also came across this R mailing list concerning geo data, but I have not combed the archive very much at this point (but I'm sure there is useful data in there).

Edit: For those interested in simulating a pre-specified amount of spatial auto-correlation in a distribution, I recently came across a paper that gives a quite simple recommended procedure (Dray, 2011, page 136);

I used the following steps to obtain a sample with a given autocorrelation level $\rho$: (1) generate a vector $y$ containing 100 iid normally distributed random values, (2) compute the inverse matrix $(I - \rho{W})^{-1}$ , and (3) premultiply the vector $y$ by the matrix obtained in (2) to obtain autocorrelated data in the vector $x$ (i.e., $x = (I - \rho{W})^{-1}y$ ).

The only thing not defined here is that $W$ is an A priori defined spatial weighting matrix. I'm not sure how this would translate to multivariate case, but hopefully it is helpful to someone!

Citation:

Dray, Stephane. 2011. A new perspective about Moran's coefficient: Spatial autocorrelation as a linear regression problem. Geographical Analysis 43(2):127-141. (unfortunately I did not come across a public pdf of the document)

-
Thanks @Andy W, looks relevant but Wiley want . "Nature of my data": messing about with Kdtree variants / KNN; as I said, looking for an overview. –  denis Oct 22 '10 at 13:45
@Denis, I have added in a link to the pdf. Often times if you search for the publication on Google Scholar a version is posted free to the public. The supplementary material I posted is free as well, and the R code would be insightful even without access to the paper. The description "messing about with Kdtree variants / KNN" is not specific enough to be helpful. Based on that I imagine what I quoted is not that useful unless you are explicitly interested in geographic data. –  Andy W Oct 22 '10 at 14:31
Thanks again @Andy W. Does my wish for both analysis and synthesis make sense ? –  denis Oct 22 '10 at 16:12
@Denis, yes your distinction between analysis and synthesis make sense given your description in the question (although I would use the word "simulate" instead of synthesize). I'm not sure if reducing clustering to a single coefficient for simulation will be all that insightful, and the geographic coefficients for clustering will likely be harder to implement in higher dimensional space and would probably take custom coding. The more refined a question the more applicable answers you will receive are. –  Andy W Oct 22 '10 at 16:33

I think suitable 'clumpy coefficients' are measures of spatial autocorrelation such as Moran's I and Geary's C. Spatial statistics is not my area and I don't know about simulation though.

-