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What is the difference between spatial dependence and spatial heterogeneity?

My question is motivated by readings in model specification problems in spatial econometrics, in particular Anselin (2010).

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    $\begingroup$ Reference would be useful. From my personal experience not all terminology is yet fixed in spatial econometrics, i.e. different authors might give different definitions. $\endgroup$ – mpiktas Nov 15 '11 at 13:22
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    $\begingroup$ I have a feeling Luc Anselin wrote more than one paper in 2010! A more specific citation (plus a link) would be useful (although he has used these terms since his Spatial Econometrics book printed in 1988). $\endgroup$ – Andy W Nov 15 '11 at 19:32
  • $\begingroup$ Thanks for the suggestion - I added a link to the paper. $\endgroup$ – mindless.panda Nov 15 '11 at 19:37
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These terms probably do not have a universally accepted technical definition, but their meanings are reasonably clear: they refer to second order and first order variation of a spatial process, respectively. Let's take them by order after first introducing some standard concepts.

A spatial process or spatial stochastic process can be thought of as a collection of random variables indexed by points in a space. (The variables have to satisfy some natural technical consistency conditions in order to qualify as a process: see the Kolmogorov Extension Theorem.)

Note that a spatial process is a model. It is valid to use multiple different (conflicting) models to analyze and describe the same data. For instance, models of naturally occurring concentrations of metals in soils may be purely stochastic for small regions (such as a hectare or less) whereas over large regions (extending many kilometers) it's usually important to describe underlying regional trends deterministically--that is, as a form of spatial heterogeneity.

Spatial heterogeneity is a property of a spatial process whose mean (or "intensity") varies from point to point.

The mean is a first order property of a random variable (that is, related to its first moment), whence spatial heterogeneity can be considered a first order property of a process.

Spatial dependence is a property of a spatial stochastic process in which the outcomes at different locations may be dependent.

Often we can measure dependence in terms of the covariance (second moment) or correlation of the random variables: in this sense, dependence can be thought of as a second-order property. (Sticklers will be quick to point out that correlation and independence are not the same, so equating dependence with second order properties, although intuitively helpful, is not generally valid.)

When you see patterns in spatial data, you can usually describe them either as heterogeneity or dependence (or both), depending on the purpose of the analysis, prior information, and the amount of data.

Some simple, well-studied examples illustrate these ideas.

Poisson process

In this figure, the square demarcates an area of higher spatial intensity. All point locations, however, are independent: the clustering and gaps in points are typical of independent randomly chosen locations.

Gaussian filter

The spatial dependence in this Gaussian process is apparent through the patterns of ridges and valleys. They are homogeneous, though: there is no trend overall. Note, however, that if we were to focus on a small part of this area, we might elect to treat it as an inhomogeneous process (that is, with a trend) instead. This illustrates how scale can influence the model we choose.

  • The previous process added to a deterministic function produces a process that is spatially dependent and heterogeneous.

Dependent heterogeneous process

This image shows a different realization of the random component of this process than used for the previous illustration, so the patterns of small undulations will not be exactly the same as before--but they will have the same statistical properties.

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    $\begingroup$ Amazing answer, as usual - very clear examples. $\endgroup$ – Matt Parker Nov 15 '11 at 16:22
  • $\begingroup$ Amazing answer, indeed. A small additional question/comment: If there exists a trend in the data (spatial heterogeneity) than there are areas where close observations are similar/have the same mean. Doesn't it follow that these observations are spatially dependent, at least in an informal sense? $\endgroup$ – Julian Dec 3 '15 at 15:19
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    $\begingroup$ @Julian Yes, that's quite right. This is why the underlying form of the process cannot uniquely be identified from an examination of the data alone. For further discussion see my answer at stats.stackexchange.com/a/35524 in which your conclusion is supported with a formal calculation. $\endgroup$ – whuber Dec 3 '15 at 15:23
  • $\begingroup$ I have another question, just to be clear: The second picture alone could be an example for spatial dependency as well as(!) spatial heterogeneity, right? As long as we do not know the process itself we cannot tell from the data, right? Thank you very much! $\endgroup$ – Julian Nov 20 '19 at 15:51
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    $\begingroup$ @Julian That's correct. It's partly a matter of scale: at a large scale (extending beyond the last image) one might elect to model all variation as random, with long-range correlations; but at the scale shown the better choice might be to model the longer-range "secular" variation as a deterministic trend. There isn't enough information at the scale of the image to decide which is the better model, but there's not really enough information to construct a fully random model. Other information (not present in the data) can often help in choosing the appropriate model. $\endgroup$ – whuber Nov 20 '19 at 16:00
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The notion of spatial heterogeneity in current spatial statistics is only used to characterize local variance of spatial dependence or regression. I suggested a broad perspective on spatial heterogeneity, which refers to the scaling pattern of far more small things than large ones. Importantly the scaling pattern recurs multiple times, measured by ht-index.

https://www.researchgate.net/publication/236627484_Ht-Index_for_Quantifying_the_Fractal_or_Scaling_Structure_of_Geographic_Features

Under the new definition, spatial heterogeneity should be formulated as a scaling law. Thus the heterogeneity is power law like rather than Gaussian distribution like.

With this broad perspective, both spatial dependence and heterogeneity depict the true picture of the Earth's surface. There are far more small things than large ones across all scales or globally, but things are more or less similar at one scale or locally; see this paper for more details.

https://www.researchgate.net/publication/282310447_A_Fractal_Perspective_on_Scale_in_Geography

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    $\begingroup$ I think this post would benefit from making a more explicit comparison (particularly noting any distinction) between heterogeneity and dependence. The question asked what the difference between the two is. I can see "both spatial dependence and heterogeneity depict a true picture of the Earth's surface" notes a similarity between the concepts, but what is the distinction between them? Do they depict this picture in different ways? $\endgroup$ – Silverfish Feb 26 '16 at 7:04
  • $\begingroup$ There is a big difference between the two under the new definition of heterogeneity, yet little difference between the two under the old definition of heterogeneity. Under the old definition, spatial heterogeneity refers to how spatial dependence or regression varies from one local place to another. Under the new definition of heterogeneity (which is essentially the same definition as that in other sciences such as biology and physics), spatial heterogeneity is formulated as a scaling law being universal and general. I think the distinction is not just technical, but at the level of paradigm. $\endgroup$ – Bin Jiang Feb 26 '16 at 7:13
  • $\begingroup$ Thanks. I think the answer would benefit from including some of that discussion (there is an edit button at the bottom of it). I appreciate this may be treated in the linked articles, but we like our answers to be self-contained rather than relying on external links. $\endgroup$ – Silverfish Feb 26 '16 at 7:57
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The question is dependent upon the mathematical definition of the two concepts. There are already several definitions of spatial autocorrelation like Moran's I, but few of spatial heterogeneity, probably because the latter is scale dependent and would be different in distinct scales. I defined the spatial stratified heterogeneity (the full paper is expected online on 12 March 2016 in journal Ecological Indicators):

A measure of spatial stratified heterogeneity

Jin-Feng Wang1*, Tong-Lin Zhang2, Bo-Jie Fu3

ABSTRACT

Spatial stratified heterogeneity, referring to the within-strata variance less than the between strata-variance, is ubiquitous in ecological phenomena, such as ecological zones and many ecological variables. Spatial stratified heterogeneity reflects the essence of nature, implies potential distinct mechanisms by strata, suggests possible determinants of the observed process, allows the representativeness of observations of the earth, and enforces the applicability of statistical inferences. In this paper, we propose a q-statistic method to measure the degree of spatial stratified heterogeneity and to test its significance. The q value is within [0, 1] (0 if a spatial stratification of heterogeneity is not significant, and 1 if there is a perfect spatial stratification of heterogeneity). The exact probability density function is derived. The q-statistic is illustrated by two examples, wherein we assess the spatial stratified heterogeneities of a hand map and the distribution of the annual NDVI in China. --Jinfeng Wang 2016-3-8

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