[ai-geostats] question about kriging with skewed distribution
I have a question about what is/should be typically done when kriging is
used for spatial interpolation of a process X(z) where z gives spatial
location (e.g. z=(x,y) with cartesian coordinates x,y) and X(z) has a
skewed continuous distribution with nonnegative support. For instance
if all data are in the form of point samples, X(z)'s can obviously be
transformed by taking logs to Y(z)=log(X(z)) which are exactly (with
lognormal X's) or approximately Gaussian, so that kriging can be done
comfortably (and the result backtransformed with easy correction for the
fact that E f(X) is generally not equal to f(E X), based on the formula
for lognormal expected value or Taylor expansion).
If at least some data are not point samples, but correspond to the
regional averages, then problem occurs due to the facts that: i) sum
of lognormals is not lognormal, ii) the log of the sum (or average)
of lognormals is not normal.
Obviously, one can do:
i) the kriging on logs anyway with some hand-waving (effectively
replacing sums by products based on delta method),
ii) or one can (quite inefficiently) work with original data without log
transformation and argue that at least method of moments estimators
are invoked (with proper weighting),
iii)or one can use some kind of Monte Carlo computationally-intensive
approach to compute likelihood (or posterior) based on sums of
At this point, I am not interested in either of the three. My question
is whether people used some other parametric family (it cannot be
lognormal) of marginal distributions with positive support, positive
skew, that is closed under convolution (or under taking weighted
averages, to be more general) - so that the regional averages and point
values will have distribution of the same type, differing only in
parameters (just like in normal case and real support case). One
possibility would be gamma, what about others?
Thanks in advance for any suggestions.
Ing. Marek Brabec, PhD
> hello,As far as i know, traditional geostatistics as originated in Matheron is
> I have a question about what is/should be typically done when kriging is
> used for spatial interpolation of a process X(z) where z gives spatial
> location (e.g. z=(x,y) with cartesian coordinates x,y) and X(z) has a
> skewed continuous distribution with nonnegative support. For instance
distribution-free. The analysis does not require a pre-experimental
probability model for the data, thus it does not rely on any
post-experimental likelihood function. So it does not matter, for kriging,
if the data are skewed or has any shape whatsoever. That is the theory at
least. People may still want to work with symmetrical distributions
because they may not be entirely confortable with the theory?
> Now,Yes, though the data may only be a little lognormal. If it is exactly
> if all data are in the form of point samples, X(z)'s can obviously be
> transformed by taking logs to Y(z)=log(X(z)) which are exactly (with
> lognormal X's) or approximately Gaussian, so that kriging can be done
> comfortably (and the result backtransformed with easy correction for the
> fact that E f(X) is generally not equal to f(E X), based on the formula
> for lognormal expected value or Taylor expansion).
lognormal then the parameter of the Box-Cox transformation is 0, but
values like -0.1 or +0.1 can produce more symetrical distributions . This
parameter can be estimated along with spatial correlation function
parameters to let the data decide what precise transformation makes it
look more Gaussian. For this you would need to set up a formal statistical
model for the data instead of following the traditional distribution-free
methodology. Check the info on geoR, a contributred package to R.
> If at least some data are not point samples, but correspond to theIf you don't have raw data but averages then within the likelihood-based
> regional averages, then problem occurs due to the facts that: i) sum
> of lognormals is not lognormal, ii) the log of the sum (or average)
> of lognormals is not normal.
approach you may want to think of a marginal likelihood model to carry
over the uncertainty associated with the averaging into the final
analysis. I think this is rather complicated. On the other hand, maybe
there is no such problem within the traditional distribution-free school
because the uncertainty associated to the fitting of the spatial model
ususally is ignored.
[snip the rest for brevity]
- Ruben (et al)
It is true that Matheron's theory is based on no
distributional assumptions. In fact, there is no
requirement for the distribution to be the same at
every location in the study area.
The necessity for using traditional geostatistical
theory is that the 'difference between two values'
should have a common distribution for a specified
distance (and possibly direction). The form of this
distribution is irrelevant but it needs to possess a
mean and variance.
The problem lies not with the theory but with the
practice. If you have the whole 'realisation' you can
calculate the true average and variance and the shape
of each distribution is irrelevant. If you have only a
few samples, then you can only find estimates for the
means and variances at each distance.
If the underlying distribution is highly skewed then,
unless you have ideal conditions (large number of
samples, regular sampling locations), your estimate of
the variance will be unstable -- influenced by the
average of the samples included in the particular
estimate. There was a huge amount of debate about this
"proportional effect" back in the 70s [search for
So, you have two potential problems:
(1) you may not get any true picture of the
semi-variogram due to the uncertainty associated with
each point exacerbated by the proportional effect;
(2) you may not wish to use an averaging technique
such as kriging on skewed samples. All of Sichel's
(mining) and much of Krige's work was motivated by the
fact that local averaging is not sensible when your
data has a coefficient of variation greater than
The theory is terrific, witness its survival for over
40 years and its proliferation over many fields of
application. However, real life isn't so tidy at the
sharp end ;-)