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I'm looking for an effective way to measure changes in spatial location in a time series data set (whether an observation repeats in the same location between successive time-steps, or if it shifts and by how much).

What I have is a point file, showing the presence/absence of a species in four stages, in study region A. I have a similar dataset for four other regions (B-D). I am looking for an approach to measure how much movement there is in region A over the four time-steps, and if this is greater, lesser, or equal to comparable movements in regions B-D. Each region can be run as a separate analysis.

I am using ArcGIS (10.1 and 10.2), as well as GME. I don't find any out of the box solution to what I'm trying to show. The closest is to use a summary measure of point-distance. The closest theoretical approach I have found is Colwell's (1974) concept of temporal/spatial constancy, used in Ecology.

If anyone has suggestions or knows of a tool in ecological modeling for measuring constancy, this would be very helpful.

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You could try a matrix regression to derive the temporal correlations (ie., Partial Mantel Test on two pairwise-distance matrices or Mantel Test on single, cross-distance matrix) or you could apply Dutilleul's (1993) modified t-test.

Sorry, I just can't think of an "out-of-box" solution available in ArcGIS or GME. You may want to take a look at the "Spatial Analysis in Macroecology" or "PASSaGE" GUI software which are both designed for pattern and spatial statistical analysis. There may be a suitable tool available in one of them. I know that Michael Rosenberg is consistently adding new functionality to PASSaGE and am fairly sure that there is a Mantel test available.

If you are expecting large deviations, you could also address change in autocorrelation structure using the family of point pattern analysis (PPA) statistics. In this case a appraoch may be the Ripley's-K (Ripley 1976), standardized as Besag's-L (Besag & Diggle 1977), where you can look at spatial structure by distance lag. Whereas, this is available in ArcGIS, there is only a univariate implementation and you are needing a bivariate cross K. Lynch & Moorcroft (2008) did propose a spatiotemporal extension of the Ripley's-K.

I would note that it would be critical to perform a Monte Carlo simulation thus, producing a simulation envelope, to ensure that the results are significant.

References

Besag, J. & Diggle, P.J. (1977). Simple Monte Carlo tests for spatial pattern, Applied Statistics, 26:327–333

Dutilleul, P. (1993). Modifying the t-test for assessing the correlation between two spatial processes. Biometrics 49:305-314.

Lynch, H.J., & P.R. Moorcroft (2008) A spatiotemporal Ripley’s K-function to analyze interactions between spruce budworm and fire in British Columbia, Canada. Canadian Journal of Forest Research, 38:3112–3119

Ripley, B.D. (1976). The second-order analysis of stationary point processes, Journal of Applied Probability, 13:255–266

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