Spatial dependence

In mathematical statistics, spatial dependence is a measure for the degree of associative dependence between independently measured values in a temporally or in situ ordered set, determined in samples selected at positions with different coordinates in a sample space, or taken from a sampling unit. In this context, independently measured implies that neither the primary sample selection stage, nor the sample preparation stage or the analytical stage in the measurement hierarchy add uncertainties to any contiguous subset of the complete set of measured values.

Spatial dependence is verified by applying analysis of variance to the variance of a set and the first variance term of the ordered set (closest/shortest lag) and comparing the observed F-value between these variances with values of F-distributions at 5% or 1% probability with applicable degrees of freedom. Deriving additional variance terms makes it possible to chart a sampling variogram. All measured values in ordered sets are used to derive first variance terms but higher variance terms have fewer degrees of freedom. For example, the second variance term of the ordered set does not take into account the last but one measured value. This is why each higher term has two fewer degrees of freedom than the previous term. Loss of degrees of freedom during reiteration impacts large sets of measured values most of all. Those who study our own sample space of time should correct for the loss of degrees of freedom when deriving sampling variograms.

A significant degree of spatial dependence gives a higher degree of precision for the central value of a set of measured values. Therefore, testing for spatial dependence makes scientific sense in many applications in mineral exploration, mining, mineral processing, smelting and refining, and in a broad range of engineering and scientific disciplines. In mathematical statistics, a set of n measured values with equal weights gives df = n − 1 degrees of freedom and the ordered set gives df(o) = 2(n − j) degrees of freedom for the jth variance term. In contrast, a set of n calculated values such as functionally dependent kriged estimates gives exactly zero degrees of freedom. Degrees of freedom are positive integers for measured values with equal weights but become positive irrationals for measured values with variable weights.

Controversy

The assumption of spatial dependency in geostatistics is controversial. In his letter of October 15, 1992, to the Editor of the Journal for Mathematical Geology, Dr A G Journel, the lead author of Mining Geostatistics and a Stanford professor, postulates, “The very reason for geostatistics or spatial statistics in general is the acceptance (a decision rather) that spatially distributed data should be considered a priori as dependent one to another, unless proven otherwise”. However, Journel was troubled when analysis of variance proved a significant degree of spatial dependence between gold grades of ordered rounds in a decline. Journel's hypothesis, “ In presence of spatial dependence the classical notion of degrees of freedom vanished: n spatially dependent data do not provide n degrees of freedom” is puzzling. If his hypothesis were indeed true, then analysis of variance cannot possibly be applied to verify spatial dependence and chart a sampling variogram.

References

1. Hald, A, 1952, Statistical Theory with Engineering Applications, John Wiley & Sons, New York
2. ISO/DIS 11648, Statistical aspects of sampling from bulk materials, Part 1: General principles
3. Journel, A G and Huijbregts, C J, Mining Geostatistics, Academic Press Inc, London
4. Lipschutz, S, Theory and Problems of Probability, McCraw Hill Book Company, 1968
5. Volk, W, Applied Statistics for Engineers, Krieger Publishing Company, Huntington, New York, 1980
6. Youden, W J, 1951, Statistical Methods for Chemists: John Wiley & Sons, New York

See also

• Spatial analysis
• Geostatistics