Re: AI-GEOSTATS: average semi-variogram

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• Dear Koen, ... Thats my view of the forest: When you directly measure the mean value of blocks, with the blocks all of equal shape B, that is nothing else than
Message 1 of 9 , Mar 2, 2004
Dear Koen,

> I did measure the point values in a grid in a plot/block. I don't have to
> estimate them, I just need a spacially depended estimate of the whole
> plot/block variance.
>
> The problem is that I would like to automate the procedure of calculating
> that within block variance because variogram models need to be fit
> visually most of the time, wich is time consuming. When repeating
> calculations on randomized data this isn't an option. So if gamma(V,V)
> really is model property rather then a data property I have a problem...
>
> If you can see the trees for the forest, because I don't.
> Koen.

Thats my view of the forest:

When you directly measure the mean value of blocks, with the blocks all of
equal shape B, that is nothing else than measuring points x in another random
process Y , which is related to original process Z by the simple relation
Y(x) = \int_{B} Z(x+w) dw / Area(B). (1)
But anyway that is a random process in its own right and can be handled by
ordinary pointwise geostatistics. That is in the same way true, when we have
a block approximated by datapoints:

Y(x) = 1/|B| \sum_{w in B} Z(x+w) (2)

That the variograms of Z and Y are related in the way

gamma_Y(x,x') = 1/|B|^2 \int \int gamma_Z(x+w,x'+w') dw dw'

or resp.

gamma_Y(x,x') = 1/|B|^2 \sum_{w in B} \sum_{w' in B} gamma_Z(x+w,x'+w')

stays a biproduct, which would more or less (or more less) allow to infer
gamma_Z from gamma_Y. However you don't seem to need relation. You can go
along with gamma_Y only.

The first problem is that you measure according to equation (2) but you want
to estimate according to equation (1). Anyway (2) should well approximate
(1).

The second problem is: The variogram of Y should be very smooth in the
origing, but it is most sure not Gaussian.

Gerald

-------------------------------------------------
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Professor als Juniorprofessor für Statistik
http://www.math-inf.uni-greifswald.de/statistik/

office: Franz-Mehring-Str. 48, 1.Etage rechts
e-mail: Gerald.Boogaart@...
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paper-mail:
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Institut für Mathematik und Informatik
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• If you have gridded observed data at close and regular spacing for the block, then why bother with a Gamma(V,V)? Just calculate the variance for the block
Message 2 of 9 , Mar 2, 2004
If you have gridded observed data at close and regular spacing for the block, then why bother with a Gamma(V,V)? Just calculate the variance for the block directly.

On the one hand you need "a spatially dependent estimate of the whole plot/block variance" but on the other hand you want to dispense with modelling any correlation (whether variogram, or correlogram, etc) because it's "time consuming." Not sure how to tackle this one, something has to give. How about implementing an inverse distance weighting scheme to derive the within block variance instead of working with a variogram measure?

Cheers,

Syed

>> Note that gamma(V,V) is a variogram model property, not a data
>> property: in the figure you mention, the big dots are data points,
>> the small dots discretize the block but do not indicate observed
>> data.
>
>I know that you can discretize block estimates with a pretty good accuracy
>with a limited amount of points. But, I was convinced that I could use
>grided observed data and gamma(V,V) as an estimate for the variance within
>my grided block.
>
>To clarify,
>
>I did measure the point values in a grid in a plot/block. I don't have to
>estimate them, I just need a spacially depended estimate of the whole
>plot/block variance.
>
>The problem is that I would like to automate the procedure of calculating
>that within block variance because variogram models need to be fit
>visually most of the time, wich is time consuming. When repeating
>calculations on randomized data this isn't an option. So if gamma(V,V)
>really is model property rather then a data property I have a problem...
>
>If you can see the trees for the forest, because I don't.
>Koen.
>
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>
>

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• ... This is indeed an option and a simple one. The original sampling design was roughly tested with a RMSE/MSE analysis. Or, taking the statistics of all the
Message 3 of 9 , Mar 2, 2004
> If you have gridded observed data at close and regular spacing for the
> block, then why bother with a Gamma(V,V)? Just calculate the variance
> for the block directly.

This is indeed an option and a simple one. The original sampling design
was roughly tested with a RMSE/MSE analysis. Or, taking the statistics of
all the grid data as a reference and then sampling smaller and smaller
portions of this data to get an idea of the block variance and how many
samples could suffice.

> On the one hand you need "a spatially dependent estimate of the whole
> plot/block variance" but on the other hand you want to dispense with
> modelling any correlation (whether variogram, or correlogram, etc)
> because it's "time consuming."

Time consuming isn't exactly bad but I think that if I have to repeat
fitting curves a lot of times because of the randomizations etc. Result, I
would be fitting curves more then I would like and probably is useful.

> Not sure how to tackle this one, something has to give.

Indeed, that's what I'm finding out right now. Implementing a technique in
my work that's as efficient as possible without losing to much accuracy.

> How about implementing an inverse distance weighting scheme to derive
> the within block variance instead of working with a variogram measure?

I'll look into that...
Thanks,
Koen.

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• ... Hard one without the mathematical writing, I looked it up and you refer to following calculations I presume (end of the page):
Message 4 of 9 , Mar 2, 2004
> ...
> gamma_Y(x,x') = 1/|B|^2 \sum_{w in B} \sum_{w' in B} gamma_Z(x+w,x'+w')
>
> stays a biproduct, which would more or less (or more less) allow to infer
> gamma_Z from gamma_Y. However you don't seem to need relation. You can go
> along with gamma_Y only.

Hard one without the mathematical writing, I looked it up and you refer to
following calculations I presume (end of the page):

http://uk.geocities.com/drisobelclark/PG1979/Chapter_3/Part2.htm

> The first problem is that you measure according to equation (2) but you
> want
> to estimate according to equation (1). Anyway (2) should well approximate
> (1).

It could do the trick, but then again... as someone asked before is it
worth the effort, considering the pretty dense sampling grid in the
plot/block maybe this isn't the best way to approach the problem!?

> The second problem is: The variogram of Y should be very smooth in the
> origing, but it is most sure not Gaussian.

Correct on that one.
Or, I've got data that is highly skewed (LAI, leaf area indexes) and other
data that isn't (that much, reflectance/albedo measurements). So getting a
variogram in the first place has horror qualities (transforming,
backtransforming and that mess) even if I do the actual fitting on all the
data from multiple plots/blocks.

Thanks for all the input,
Koen.

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