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.
- A Poisson process with varying intensity is spatially heterogeneous but has no spatial dependence.
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.
- A neighborhood mean, or convolution, of a “white noise” process is spatially homogeneous but has spatial dependence.
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.
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.
Spatial dependence is the causal spatial relationship of variable values (for themes defined over space, such as rainfall) or locations (for themes defined as objects, such as cities). Spatial dependence is measured as the existence of statistical dependence in a collection ofrandom variables or a collection of random variables, each of which is associated with a different geographical location. Spatial dependence is of importance in applications where it is reasonable to postulate the existence of corresponding set of random variables at locations that have not been included in a sample. Thus rainfall may be measured at a set of rain gauge locations, and such measurements can be considered as outcomes of random variables, but rainfall clearly occurs at other locations and would again be random. Since rainfall exhibits properties ofautocorrelation, spatial interpolation techniques can be used to estimate rainfall amounts at locations near measured locations.
As with other types of statistical dependence, the presence of spatial dependence generally leads to estimates of an average value from a sample being less accurate than had the samples been independent, although if negative dependence exists a sample average can be better than in the independent case. A different problem than that of estimating an overall average is that of spatial interpolation: here the problem is to estimate the unobserved random outcomes of variables at locations intermediate to places where measurements are made, on that there is spatial dependence between the observed and unobserved random variables.
Tools for exploring spatial dependence include: spatial correlation, spatial covariance functions and semivariograms.
Methods for spatial interpolation include Kriging, which is a type of best linear unbiased prediction.
The topic of spatial dependence is of importance to geostatistics and spatial analysis.