Moran's I: I (d) = —h = 1 i = 1- for h * i (13.1)

The yhs and y/s are the values of the observed variable at sites h and i. Before computing spatial autocorrelation coefficients, a matrix of geographic distances D = Dhi] among observation sites must be calculated. In the construction of a correlogram, spatial autocorrelation coefficients are computed, in turn, for the various distance classes d. The weights Whi are Kronecker deltas (as in eq. 7.20); the weights take the value Whi = 1 when sites h and i are at distance d and wu = 0 otherwise. In this way, only the pairs of sites (h, i) within the stated distance class (d) are taken into account in the calculation of any given coefficient. This approach is illustrated in Fig. 13.3. W is the sum of the weights Whi for the given distance class, i.e. the number of pairs used to calculate the coefficient. For a given distance class, the weights Wj are written in a (n x n) matrix W. Jumars et al. (1977) present ecological examples where the distance-1 or distance-2 among adjacent sites is used for weight instead of 1's.

The numerators of eqs. 13.1 and 13.2 are written with summations involving each pair of objects twice; in eq. 13.2 for example, the terms y - yp2 and y - yh)2 are both used in the summation. This allows for cases where the distance matrix D or the weight matrix W is asymmetric. In studies of the dispersion of pollutants in soil, for instance, drainage may make it more difficult to go from A to B than from B to A; this may be recorded as a larger distance from A to B than from B to A. In spatio-temporal analyses, an observed value may influence a later value at the same or a different site, but not the reverse. An impossible connection may be coded by a very large value of distance. In most applications, however, the geographic distance matrix among sites is symmetric and the coefficients may be computed from the half-matrix of distances; the formulae remain the same, in that case, because W, as well as the sum in the numerator, are half the values computed over the whole distance matrix D (except h = i).

One may use distances along a network of connections (Subsection 13.3.1) instead of straight-line geographic distances; this includes the "chess moves" for regularly-spaced points as obtained from systematic sampling designs: rook's, bishop's, or king's connections (see Fig. 13.19). For very broad-scale studies, involving a whole ocean for instance, "great-circle distances", i.e. distances along earth's curved surface, should be used instead of straight-line distances through the earth crust.

Figure 13.3 Construction of correlograms. Left: data series observed along a single geographic axis (10 equispaced observations). Moran's I and Geary's c statistics are computed from pairs of observations found at preselected distances (d = 1, d = 2, d = 3, etc.). Right: correlograms are graphs of the autocorrelation statistics plotted against distance. Dark squares: significant autocorrelation statistics (p < 0.05). Lower right: histogram showing the number of pairs in each distance class. Coefficients for the larger distance values (grey zones in correlograms) should not be considered in correlograms, nor interpreted, because they are based on a small number of pairs (test with low power) and only include the pairs of points bordering the series or surface.

Figure 13.3 Construction of correlograms. Left: data series observed along a single geographic axis (10 equispaced observations). Moran's I and Geary's c statistics are computed from pairs of observations found at preselected distances (d = 1, d = 2, d = 3, etc.). Right: correlograms are graphs of the autocorrelation statistics plotted against distance. Dark squares: significant autocorrelation statistics (p < 0.05). Lower right: histogram showing the number of pairs in each distance class. Coefficients for the larger distance values (grey zones in correlograms) should not be considered in correlograms, nor interpreted, because they are based on a small number of pairs (test with low power) and only include the pairs of points bordering the series or surface.

Moran's I formula is related to Pearson's correlation coefficient; its numerator is a covariance, comparing the values found at all pairs of points in turn, while its denominator is the maximum-likelihood estimator of the variance (i.e. division by n instead of n - 1); in Pearson's r, the denominator is the product of the standard deviations of the two variables (eq. 4.7), whereas in Moran's I there is only one variable involved. Moran's I mainly differs from Pearson's r in that the sums in the numerator and denominator of eq. 13.1 do not involve the same number of terms; only the terms corresponding to distances within the given class are considered in the numerator whereas all pairs are taken into account in the denominator. Moran's I usually takes values in the interval [-1, +1] although values lower than -1 or higher than +1 may occasionally be obtained. Positive autocorrelation in the data translates into positive values of I; negative autocorrelation produces negative values.

Readers who are familiar with correlograms in time series analysis will be reassured to know that, when a problem involves equispaced observations along a single physical dimension, as in Fig. 13.3, calculating Moran's I for the different distance classes is nearly the same as computing the autocorrelation coefficient of time series analysis (Fig. 12.5, eq. 12.6); a small numeric difference results from the divisions by (n - k - 1) and (n - 1), respectively, in the numerator and denominator of eq. 12.6, whereas division is by (n - k ) and (n ), respectively, in the numerator and denominator of Moran's I formula (eq. 13.1).

Geary's c coefficient is a distance-type function; it varies from 0 to some unspecified value larger than 1. Its numerator sums the squared differences between values found at the various pairs of sites being compared. A Geary's c correlogram varies as the reverse of a Moran's I correlogram; strong autocorrelation produces high values of I and low values of c (Fig. 13.3). Positive autocorrelation translates in values of c between 0 and 1 whereas negative autocorrelation produces values larger than 1. Hence, the reference 'no correlation' value is c = 1 in Geary's correlograms.

For sites lying on a surface or in a volume, geographic distances do not naturally fall into a small number of values; this is true for regular grids as well as random or other forms of irregular sampling designs. Distance values must be grouped into distance classes; in this way, each spatial autocorrelation coefficient can be computed using several comparisons of sampling sites.

Numerical example. In Fig. 13.4 (artificial data), 10 sites have been located at random into a 1-km2 sampling area. Euclidean (geographic) distances were computed among sites. The number of classes is arbitrary and left to the user's decision. A compromise has to be made between resolution of the correlogram (more resolution when there are more, narrower classes) and power of the test (more power when there are more pairs in a distance class). Sturge's rule is often used to decide about the number of classes in histograms; it was used here and gave:

Number of classes = 1 + 3.3log10(m) = 1 + 3.3log10(45) = 6.46 (13.3)

where m is, in the present case, the number of distances in the upper (or lower) triangular matrix; the number was rounded to the nearest integer (i.e. 6). The distance matrix was thus recoded into 6 classes, ascribing the class number (1 to 6) to all distances within a class of the histogram.

An alternative to distance classes with equal widths would be to create distance classes containing the same number of pairs (notwithstanding tied values); distance classes formed in this way are of unequal widths. The advantage is that the tests of significance have the same power across all distance classes because they are based upon the same number of pairs of observations. The disadvantages are that limits of the distance classes are more difficult to find and correlograms are harder to draw.

Spatial autocorrelation coefficients can be tested for significance and confidence intervals can be computed. With proper correction for multiple testing, one can determine whether a significant spatial structure is present in the data and what are the distance classes showing significant positive or negative autocorrelation. Tests of significance require, however, that certain conditions specified below be fulfilled.

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