Computers & Geosciences, Vol. 2, pp. 437--438. Pergamon Press, 1976. Printed in Great Britain AUTOCORRELATION COEFFICIENTS FROM IRREGULARLY SPACED AREAL DATA S. HENLEY Instituteof Geological Sciences, MurchisonHouse, Edinburgh,Scotland (Received 5 May 1976) A~tract--A FORTRAN subroutine is presented for computation of the generalized Moran autocorrelation coefficient and Geary coefficient from irregularlyspaced data points. The algorithmused assumes isotropy of the data, so no directional informationis obtained. Key Words: FORTRAN. Autocorrelation. INTRODUCTION Existing methods of investigating spatial autocorrelation of geological data concentrate on regularly gddded data because of the simplicity of the serial-correlation al- gorithm. In a regular grid, interpoint spacings are simply multiples of the cell dimensions, and thus it is easy to obtain a correlogram indicating the variation of autocorre- lation with distance between data points (Parsley, 1971). If the data are distributed irregularly, a grid interpolation algorithm first must be applied. Cliff and Ord (1973) concentrate on autocorrelation between adjacent regions rather than data points, and are not concerned with the relationship of correlation and distance, but they suggest a number of autocorrelation coefficients which might be used with data that are not gridded. THE kL6ORITI-Ibl The serial-correlation coefficient used by Parsley, for separation of k cells between points is: r(k)=n~ zlzi+~ n-k z•, where zi is the deviation from the mean of the ith point in any row or column of the grid. Two coefficients used by Cliff and Ord seem suitable for adaption to irregularly distributed point data: these are the Moran and Geary statistics. Moran's coefficient is: I = n ~. .8~jz~z 2A ~=~ z~ 2, where z~ is the deviation from the mean, n is the number of regions (or data points), 8,j is set at 1 for adiacent regions (or points) or zero otherwise, and A is defined as: I n A = 2 ~ L~, i=l where L, is the number of regions (points) adjacent to the ith region (point). Geary's coefficient, using the same notation, is: 2 ~ c = (n - 1) 8~(z,- z~)/4A ~ z~-. Cliff and Ord have generalized these coefficients by replacing the binary weights 8~j by coefficients w~j of a generalized weighting matrix W. The generalized Moran autocorrelation coefficient is: I = n w~jz~z i W z~ ~ i#i. and the generalized Geary coefficient is: " /2W~ z2, c = (. - I) ~ w,~(z, - z f i~j '=" where W in each situation is the sum of all coefficients w 0. These generalized coefficients now may be used in a flexible manner and can be applied directly to data points rather than a pattern of contiguous regions. The serial-correlation coefficient used by Parsley is a form of the Moran coefficient, for example. For irregularly spaced data points, we may take the Euclidean distance matrix and build a set of weighting matrices W such that each matrix contains weightings of 1 for point separations within a given distance range and 0 for all other distances (longer or shorter). These matrices W may be used to compute a set of Moran or Geary type coefficients for autocorrelation at different distances allowing us to plot a correiogram. We are partitioning effectively the set of all distances into a number of sets of distances within given ranges. For instance, given data over an area 100 km x 100 kin, we may examine autoeor- relation within the interpoint distance ranges 0-10kin, 10-20 km ..... An assumption is made that the data are isotropic, thus no information on directional properties is obtained. SIJBROIYrINE The subroutine ARAUT, to compute a generalized Moran coefficient through a range of distance classes, is abstracted from a program GARAUT in the G-EXEC data-handling system. It is possible to convert this subroutine to compute a Geary coefficient by replacing line 38 by: SZZ(K) = SZZ(K) + (Z(I) - Z(J)) ** 2 437