Calculation of patch size?

[cdjemiel]

Hello RGeostats Forum,

I am a beginner in the field of geostatistics so sorry for my semantics which may be approximate.
I'm using your great package to make maps in the field of biogeography and I have some questions especially about the size of the patches. I use the Range to get an idea but I have a doubt, is it the radius?

To create a model I can nest several structures as for example here:

model.OK = model.auto(vario.data.OK,
struct = melem.name(c(1,2),
flag.norm.sill = FALSE,
wmode = 2, draw=FALSE)

and I get this:

Covariance Share
---------------
Nugget Effect
- Sill = 0.804
exponential
- Sill = 0.311
- Range = 593950.900
- Theo. Range = 198265.699
Total Sill = 1.115

Could you tell me what is the difference between Range and Theo. Range. In my case, what should I anticipate?

I can also have several Range / Theo. Range :

Covariance Share
---------------
Nugget Effect
- Sill = 0.621
exponential
- Sill = 0.219
- Range = 73162.818
- Theo. Range = 24422.351
Spherical
- Sill = 0.177
- Range = 185649.957
Total Sill = 1.016

Should I take the smaller Range? make the sum ? or is there a calculation to get the "true Range"?

Congratulations again and thank you for this package.

Best regards

Christophe

[Didier Renard]

Hello and thanks for using RGeostats.

Here are some information that you need to better understand the output of the package.
You have provided an experimental variogram (square of differences between values at samples separated by a distance) and let the package suggest the corresponding Model (obtained by applying an automatic fitting procedure [model.auto]). You are interested in the case of a single variable (so I will not introduce the multivariate case).
A Model (let us call it "a Variogram Model" to avoid ambiguity) in general is the combination of several basic structures. A basic structure is one of the specific variogram functions (let us name some of them for example: nugget effect, spherical, exponential, cubic, gaussian, ...)
Each basic structure is very specific as it has to ensure some mathematical properties (conditional definite positiveness) in order to lead to positive variances in the Kriging equations. It is essentially a function of the distance. It is characterized by its formula.
The common feature of these basic structures g(h) is their generic formula:

g(h) = C * f(h / a)
where C stands for the "sill" and a stands for the "range". Obviously the function "f" varies per basic structure. Some (few) basic structures use a third parameter. One basic structure (nugget effect) has no range (or say its range is equal to an infinitely small distance).

When several basic structures are involved, the final (nested) Model simply corresponds to the following sum:

g(h) = C1 * f1(h / a1) + C2 * f2(h / a2) + ...
Where C1 and C2 are the sills of the difference basic structures, and a1 and a2 their ranges.
Note that the automatic procedure is a nice piece of software as, given ONE experimental variogram, it finds the optimal combination of basic structures (can be one or several) and the optimal set of parameters (sill and range for each basic structure). A high-dimensional problem!!!
This is the case of your example where the Model is composed of the sum of a Nugget Effect and Exponential in one case or sum of Nugget Effect, Exponential and Spherical in the other case.

Coming back to an individual basic structure... Most of them present a shape which starts from 0 (at zero distance), increase with distance until it reach a plateau at a given specific distance: this plateau is the "sill" and the specific distance the "range". For some basic structures, the curve does not reach the plateau exactly (asymptotic behavior). In this case, we distinguish between the (theoretical) range [which corresponds to the value of "a" in the formula] and the "practical" range [which conventionally corresponds to the distance at which the curve reaches 95% of the sill].
This is the case for the exponential basic structure for example where, when a=198000, the corresponding practical range is roughly three times larger, i.e. 594000. The ratio between practical and theoretical ranges differ per basic structure. In the case of the spherical (which reaches the sill with no asymptotic behavior, both are equal.

Given this introductory information, the size of your patch is not an easy question. It is probably easier to discuss in terms of covariance than variogram (same curve mirrored vertically): when the variogram reaches the sill, the covariance falls to zero.
Then we can say than:
- when the model is reduced to a single basic structure, the patch size can be compared to the practical range dimension: two samples whose distance is larger than this distance are not correlated anymore
- when the model is nested, the patch size could be represented by the largest practical range.

In the latter case, it is interesting to pay attention to the sills of the different basic structures. For example in your second fit (nugget effect + exponential + spherical), the total variance of your variable is 1.016 (total sill). The model has three components:
- a large nugget effect 0.621 (out of 1.106) which corresponds to the noise or very short scale variability
- a small spherical component with a sill of 0.177 (out of 1.016) whose range (extension reaches a distance of 186000.
- a more significant exponential structure with sill of 0.219 (out of 1.016) whose practical range extends up to 732000.
We could phrase this situation by saying that any correlation vanishes at 732000 distance units; that 0.219/1.106 percent of the variation fades away at 186000 distance units, and that a 0.621/1.016 percent of the variability remains whatever the smallest distance that you can consider between samples.

Hoper these information will help you.