Spatial Autocorrelation

A time series is said to be autocorrelated if it is possible to predict the value of the series at a given time from recent measured values of the series. For example, yesterday's temperature at noon is often a good predictor of today's temperature at noon; and the value of stock market indices similarly bears stronger resemblance to immediately previous 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, though unlike the temporal case, space may be twoor even three-dimensional. A general statement by Tobler, often termed Tobler's first law of geography, asserts that spatial autocorrelation is positive for almost all geographic phenomena.

Numerous indices 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 (values are first adjusted by subtracting the mean).

The definition of neighboring depends on the nature of the sampling scheme. If the variable is sampled over a raster, then two cells can be regarded as neighbors if they share a common edge (“rook's case”), or if they share either an edge or a corner (“queen's case”). If the variable is sampled over an irregular tesselation, as with summary statistics from the census, then it is common to define two cases as neighbors if they share a common edge. More generally, define wij as the weight used in comparing the value cases i and j of the variable. Then, these schemes can be seen as providing ways of defining weights w as binary indicators of adjacency. Other, continuousscaled definitions of weights w are available based on length of common boundary or decreasing functions of distance (for example, negative exponential functions). Such definitions may capture the effects of spatial separation better than simple indicators of adjacency, which give the same weight to short as to long common boundaries and no weight to pairs of areas that may be close in space but not adjacent.