{"id":444,"date":"2014-01-21T23:09:12","date_gmt":"2014-01-22T04:09:12","guid":{"rendered":"http:\/\/homepages.uc.edu\/~yaozo\/wordpress\/?p=444"},"modified":"2014-01-21T23:09:12","modified_gmt":"2014-01-22T04:09:12","slug":"what-is-the-difference-between-spatial-dependence-and-spatial-heterogeneity-2","status":"publish","type":"post","link":"https:\/\/zhuoyao.net\/index.php\/2014\/01\/21\/what-is-the-difference-between-spatial-dependence-and-spatial-heterogeneity-2\/","title":{"rendered":"What is the difference between spatial dependence and spatial heterogeneity"},"content":{"rendered":"

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.<\/p>\n

A\u00a0spatial process<\/em>\u00a0or\u00a0spatial stochastic process<\/em>\u00a0can 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\u00a0Kolmogorov Extension Theorem<\/a>.)<\/p>\n

Note that a spatial process is a\u00a0model.<\/em>\u00a0It 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.<\/p>\n

Spatial heterogeneity<\/strong>\u00a0is a property of a spatial process whose mean (or “intensity”) varies from point to point.<\/p>\n

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.<\/p>\n

Spatial dependence<\/strong>\u00a0is a property of a spatial stochastic process in which the outcomes at different locations may be dependent.<\/p>\n

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.)<\/p>\n

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.<\/p>\n

Some simple, well-studied examples illustrate these ideas.<\/p>\n