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| Titel |
Distribution of kriging errors, the implications and how to communicate them |
| VerfasserIn |
Hongyi Li, Alice Milne, Richard Webster |
| Konferenz |
EGU General Assembly 2015
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 17 (2015) |
| Datensatznummer |
250106260
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| Publikation (Nr.) |
 EGU/EGU2015-5922.pdf |
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| Zusammenfassung |
| % Abstract for EGU meeting in 2015
% Begun by RW 5 January 2015
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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
($\sigma_{\rm K}^2$),
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
$\mbox{MSDR}=\mbox{MSE}/ \sigma_{\rm K}^2 \approx1$.
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 $\chi^2$ 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 (EC$_{\rm a}$) 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 EC$_{\rm a}$s 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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