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Titel |
Distribution of kriging errors, the implications and how to communicate them |
VerfasserIn |
Hong Yi Li, Alice Milne, Richard Webster |
Konferenz |
EGU General Assembly 2016
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Medientyp |
Artikel
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Sprache |
en
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Digitales Dokument |
PDF |
Erschienen |
In: GRA - Volume 18 (2016) |
Datensatznummer |
250124817
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Publikation (Nr.) |
EGU/EGU2016-4307.pdf |
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Zusammenfassung |
Kriging in one form or another has become perhaps the most popular method for spatial
prediction in environmental science. Each prediction is unbiased and of minimum variance,
which itself is estimated. The kriging variances depend on the mathematical model chosen
to describe the spatial variation; different models, however plausible, give rise to
different minimized variances. Practitioners often compare models by so-called
cross-validation before finally choosing the most appropriate for their kriging. One proceeds
as follows. One removes a unit (a sampling point) from the whole set, kriges the
value there and compares the kriged value with the value observed to obtain the
deviation or error. One repeats the process for each and every point in turn and for all
plausible models. One then computes the mean errors (MEs) and the mean of the
squared errors (MSEs). Ideally a squared error should equal the corresponding
kriging variance (σK2), and so one is advised to choose the model for which on
average the squared errors most nearly equal the kriging variances, i.e. the ratio
MSDR = MSE∕σK2 ≈ 1.
Maximum likelihood estimation of models almost guarantees that the MSDR equals 1,
and so the kriging variances are unbiased predictors of the squared error across the region.
The method is based on the assumption that the errors have a normal distribution. The
squared deviation ratio (SDR) should therefore be distributed as χ2 with one degree of
freedom with a median of 0.455. We have found that often the median of the SDR
(MedSDR) is less, in some instances much less, than 0.455 even though the mean of the
SDR is close to 1. It seems that in these cases the distributions of the errors are
leptokurtic, i.e. they have an excess of predictions close to the true values, excesses
near the extremes and a dearth of predictions in between. In these cases the kriging
variances are poor measures of the uncertainty at individual sites. The uncertainty is
typically under-estimated for the extreme observations and compensated for by over
estimating for other observations. Statisticians must tell users when they present maps of
predictions.
We illustrate the situation with results from mapping salinity in land reclaimed from the
Yangtze delta in the Gulf of Hangzhou, China. There the apparent electrical conductivity
(ECa) of the topsoil was measured at 525 points in a field of 2.3 ha. The marginal
distribution of the observations was strongly positively skewed, and so the observed ECas
were transformed to their logarithms to give an approximately symmetric distribution. That
distribution was strongly platykurtic with short tails and no evident outliers. The
logarithms were analysed as a mixed model of quadratic drift plus correlated random
residuals with a spherical variogram. The kriged predictions that deviated from their
true values with an MSDR of 0.993, but with a medSDR=0.324. The coefficient of
kurtosis of the deviations was 1.45, i.e. substantially larger than 0 for a normal
distribution.
The reasons for this behaviour are being sought. The most likely explanation is that there
are spatial outliers, i.e. points at which the observed values that differ markedly from those at
their their closest neighbours. |
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