Some Results on the Spatial Breakdown Point of Robust Point Estimates of the Variogram |
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| Authors: | R M Lark |
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| Institution: | (1) Rothamsted Research, Harpenden, Hertfordshire, AL5 2JQ, UK |
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| Abstract: | The effect of outliers on estimates of the variogram depends on how they are distributed in space. The ‘spatial breakdown
point’ is the largest proportion of observations which can be drawn from some arbitrary contaminating process without destroying
a robust variogram estimator, when they are arranged in the most damaging spatial pattern. A numerical method is presented
to find the spatial breakdown point for any sample array in two dimensions or more. It is shown by means of some examples
that such a numerical approach is needed to determine the spatial breakdown point for two or more dimensions, even on a regular
square sample grid, since previous conjectures about the spatial breakdown point in two dimensions do not hold. The ‘average
spatial breakdown point’ has been used as a basis for practical guidelines on the intensity of contaminating processes that
can be tolerated by robust variogram estimators. It is the largest proportion of contaminating observations in a data set
such that the breakdown point of the variance estimator used to obtain point estimates of the variogram is not exceeded by
the expected proportion of contaminated pairs of observations over any lag. In this paper the behaviour of the average spatial
breakdown point is investigated for cases where the contaminating process is spatially dependent. It is shown that in two
dimensions the average spatial breakdown point is 0.25. Finally, the ‘empirical spatial breakdown point’, a tool for the exploratory
analysis of spatial data thought to contain outliers, is introduced and demonstrated using data on metal content in the soils
of Sheffield, England. The empirical spatial breakdown point of a particular data set can be used to indicate whether the
distribution of possible contaminants is likely to undermine a robust variogram estimator. |
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| Keywords: | Variogram Robust Breakdown point Spatial breakdown point |
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