Geostatistical techniques focus on the spatial structure of a variable, i.e., the relationship between a variable as measured at a given point and that same variable measured at points at distances removed from the first point (see Isaaks and Srivastava, 1989).

A variable which takes on values according to its spatial location is known as a regionalized variable.Consider a variable z measured at location i, we can partition the total variability in z into three components:

z(i) = f(i) + s(i) + e

f(i) is some coarse scale forcing trend in the data

s(i) is local spatial dependency

e is error variance presumed to be normal
 

A variable is autocorrelated if it is possible to predict its value at a given point in space by knowing its value at other locations. Positive autocorrelation means that points at a certain distance away from each other have similar values. Negative autocorrelation means that they have unlike values. Autocorrelation is summarized by a structure function that captures the spatial structure of the variable. (see Griffith, 1987; Odland, 1988; and Legendre, 1993 for more information on spatial autocorrelation.)

The two most common structure functions are the semivariance (graphed as a semivariogram) and autocorrelation (as a correlogram). These depict spatial dependence on the ordinate versus the spatial separation or lag distance on the abscissa.

The most common measure of autocorrelation used by ecologists is called Moran's I Where wij is the weight at distance d, i.e., wij=1 if point j is within distance class d from point i, else wij=0. W is the sum of all the weights where i does not equal j.

The numerator is a cross-products covariance term. The denominator is a variance term, which makes I behave as a product moment correlation like Pearson's r. If points at a distance have similar measurements, then the numerator is either the product of two positive or two negative deviates (hence there is a + correlation). Reciprocally, if points at a distance have unlike values the numerator would be a product of deviates of opposite sign (yielding a negative correlation). I typically varies between 1 and -1.

 

Semivariance is used for two sorts of applications:

1) purely descriptive studies, in which a semivariogram is used to characterize the spatial structure of the data

2) and predictive applications used to interpolate between points.

A semivariogram is a plot of semivariance against distance

The sill is the value where the semivariance levels off, depicts the amount of variance

The range is the distance at which the leveling occurs, depicts where autocorrelation occurs The nugget is the semivariance at a distance of 0.

Semivariance Models used to fit semivariograms
 

The fractal dimension can be calculated from the slope of a double logarithmic plot of the semivariogram (Palmer, 1988).

D=(4-m)/2

If the values of two near values are no more or less different than two distant samples, the slope of the semivariogram will be 0 or D will be 2. Thus the fractal dimension is an index of the degree of spatial dependence of a variable. If slope is greater than 0 or if D<2 there is spatial dependence, i.e., autocorrelation.

References

Griffith, D.A. 1987. Spatial autocorrelation: a primer. Assoc. of American Geographers, Washington.

Isaaks, E.H., and R.M. Srivastava. 1989. An introduction to applied geostatistics. Oxford Univ. Press, New York.

Legendre, P. 1993. Spatial autocorrelation: trouble or new paradigm? Ecology 74:1659-1673.

Odland, J. 1988. Spatial autocorrelation. Sage Publications, Newbury Park, California.

Palmer, M.W. 1988. Fractal geometry: A tool for describing spatial pattern of plant communities. Vegetatio 75: 91-102.

Other Possible References

Burrough, P.A. 1987. Spatial aspects of ecological data. Pages 213-251 in R.H. G. Jongman, C.J.F. ter Braak, and O.F.R. van Tongeron (eds.), Data analysis in community and landscape ecology. PUDOC, Wageningen, the Netherlands.

Cliff, A.D., and J.K. Ord. 1981. Spatial processes: models and applications. Pion Ltd., London.

Czaplewski, R.L., R.M. Reich, and W.A. Bechtold. 1994. Spatial autocorrelation in growth of undisturbed natural pine stands across Georgia. Forest Science 40:314-328.

Cressie, N.A.C. 1993. Statistics for spatial data. Wiley, New York.

Legendre, P. and Fortin, M. 1989. Spatial pattern and ecological analysis. Vegetatio 80: 107-138.