Spatial Dependence

Spatial dependence is a characteristic of the distribution of geographic data, that is, data where the location of observations is explicitly taken into account. It combines the notion of attribute similarity with that of locational similarity. Not only is there dependence (correlation) between observations for a given variable, but also this dependence shows a spatial structure such as closer locations being more similar than locations that are farther apart. This is a formal expression of Tobler's first law of geography. Spatial dependence stands in contrast to spatial randomness, that is, the absence of any spatial structure in the data. The concept of spatial randomness is the reference or null hypothesis against which potential patterns are compared.

In practice, spatial dependence is quantified by means of a measure of spatial autocorrelation, that is, a summary of the data that formalizes closeness (as distance or contiguity) and similarity (as crossproduct correlation or squared difference). The specific measures used depend on the nature of the spatial observations, that is, whether they are considered to be locations of events (e.g., the locations of accidents), discrete observations (e.g., variables observed[Page 452] in counties), or samples from a continuous surface (e.g., measurements obtained at weather stations). Each of these three settings requires a different statistical framework, referred to as point pattern analysis, lattice data analysis, or geostatistics, respectively. For each of these frameworks, a number of test statistics have been developed to assess whether the null hypothesis of spatial randomness can be rejected. Detecting spatial autocorrelation is important because the standard assumptions of statistical inference (independent random sample) no longer hold. In addition, formal models of spatial dependence are required for the estimation and prediction of theories of spatial interaction and other spatially explicit phenomena.

Regression analysis applied to observations that are cross sections in space can be extended to incorporate the notion of spatial correlation (i.e., in spatial regression analysis) or spatial econometrics. Not only does this allow for the modeling of spatial interaction, but also the correction for the presence of spatial correlation may result in better parameter estimates. In addition, explicitly accounting for spatial autocorrelation yields improved spatial prediction in geostatistical models (i.e., kriging).

Spatial autocorrelation statistics can be classified either as indicators of global spatial autocorrelation (clustering) or as indicators of local spatial autocorrelation (clusters). Global spatial autocorrelation is a characteristic of the spatial distribution of the data as a whole and does not suggest where the patterns are located. There are two basic results: positive spatial autocorrelation and negative spatial autocorrelation. Positive spatial autocorrelation indicates a clustering of like observations, although this may be high values, low values, or a combination of both. Negative spatial autocorrelation suggests a checkerboard pattern, where high values tend to be surrounded by low values and vice versa.

Local indicators of spatial autocorrelation (LISA) are specific to each observation. They suggest whether locations are part of a local cluster or are local spatial outliers. Local clusters are collections of observations that are similar, showing either high values or low values. They often are characterized as “hot spots.” Local spatial outliers are locations that have values that are much higher or much lower than neighboring locations.

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