Spatial Autocorrelation

A time series is said to be autocorrelated if it is possible to predict the value of the series at some time from recent values of the series. For example, yesterday's temperature at noon often is a good predictor of today's temperature at noon, and the value of stock market indexes similarly bears stronger resemblance to very recent values than to historic values. Underlying these observations is the notion that some phenomena vary relatively slowly through time. Spatial autocorrelation refers to similar behavior in space; however, unlike the temporal case, space may be two- or even three-dimensional. A general statement by Waldo Tobler, often termed Tobler's first law of geography, asserts that spatial autocorrelation is positive for nearly all geographic phenomena.

Numerous indexes of spatial autocorrelation are in common use. Many are based on a simple extension of the Pearson coefficient of bivariate correlation, which is defined as the covariance between the two variables divided by the product of the standard deviations. In the case of autocorrelation, there is only one variable, so the denominator is the variable's variance and the covariance is the mean product of each value with neighboring values rather than the mean product of each value with the corresponding value of the other variable.

The definition of neighboring depends on the nature of the sampling scheme. When the variable is sampled over a raster, two cells can be regarded as neighbors if they share a common edge or if they share either an edge or a corner. When the variable is sampled over an irregular tessellation, as with summary statistics from the census, it is common to define two cases as neighbors if they share a common edge. More generally, when wij is defined as the weight used in comparing the value cases i and j of the variable, these schemes can be seen as providing ways of defining w as a binary indicator of adjacency. Continuous scaled definitions of w are available based on decreasing functions of distance, for example, negative exponential functions.

Spatial autocorrelation is of interest in numerous disciplines, and the precise ways in which it is commonly measured vary substantially. In human geography, where data often are encountered in the form of summary statistics for irregularly shaped reporting zones, the common measures are the indexes defined by Moran and Geary, notated I and c, respectively. I is essentially the Pearson correlation coefficient defined as in the previous paragraph using a user-defined matrix of weights. Thus, its fixed points are 0 when there is no tendency for neighboring values to be more similar than distant values (the precise expected value of the index is –1 / [n – 1], where n is the number of observations), positive when neighboring values tend to be more similar than distant values, and negative when neighboring values tend to be less similar than distant values. The Geary index's numerator is the mean weighted sum of differences between values and has a confusingly different set of fixed points: between 0 and 1 when spatial autocorrelation is positive, 1 when it is absent, and greater than 1 when it is negative.

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