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Imagine the following data set (from http://www.novaims.unl.pt/labnt/geosom/georepository/) which consists of three different regions:

Example from http://www.novaims.unl.pt/labnt/geosom/georepository/

Obviously, blue points are more like to be located close to other blue points. The same is true for red points. Therefore, high spatial dependency can be expected.

But what about spatial heterogeneity? Spatial heterogeneity states that the mean of a process varies with location. Going from left to right, the points (and the mean) shifts from blue to red to blue. So, this certainly is some variation. But on the other hand, comparing only the left and right side of the data, both regions, despite being distant and having different locations, have the same colour (and mean). This is certainly no variation. So, is this data set subject to spatial heterogeneity or not?

Funkwecker
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  • I would say, spatial heterogeneity describes varying means within each of the regions. Thus, is there a lot of variation in red, blue 1 and blue 2? – Iris Dec 02 '15 at 08:13
  • Maybe this helps – ymirsson Dec 02 '15 at 08:33
  • There is the landscape ecology concept of high localized variability, where higher variation in your example is germane, and then there is the geostatistics definition of the stationarity models where mean (type I), variance ( type II) or both (type III) are considered constant across the random field. The assumed model(s) of stationarity, along with isotropy, are assumptions in geostatistics (eg., kriging models). – Jeffrey Evans Dec 02 '15 at 18:08
  • You might find this is answered in the duplicate question on the [stats.SE] site: see http://stats.stackexchange.com/questions/18406. In the present case, do not confuse the process with its realization: whether the process is heterogeneous depends on your choice of how to model this realization. – whuber Dec 03 '15 at 00:32
  • Thanks @whuber. Your expertise is appreciated and very welcome. Just to verify that I understand you correctly: A (data-generating) process may be spatially heterogeneous, but it is not possible to say that a process is spatially heterogeneous just from the data alone. You need to know the process or at least have a suitable model of the process. – Funkwecker Dec 03 '15 at 07:09
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    Yes, I think that is basically correct--although I should hasten to add that a realization with a sufficient amount of data may be more consistent with some models than others. I recall that Andre Journel (Stanford U.) wrote extensively about this issue in the 1990's. Intuitively, in many situations what looks like spatial heterogeneity could plausibly be just correlated "noise." Unless you can obtain independent realizations of the same process (which is rare, except in space-time applications), you're stuck with the fact that any analysis of your data will rely on such modeling assumptions. – whuber Dec 03 '15 at 14:49

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