- hello,

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

lognormal.

Now,

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

lognormals.

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.

Best Regards

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

> lognormal.

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 the

If 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- 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

'relative semi-variogram'].

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

around 1.

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 ;-)

Isobel

http://geoecosse.bizland.com/whatsnew.htm