juliohm
commented
3 years ago Thanks for raising this issue. I like the idea of providing better support for LMC. Currently, we can combine the different variograms with scalars and sums, but perhaps we need to introduce an algebraic structure with array coefficients, similar to what you suggested.
My proposal is to have something like:
g = 0.2*NuggetEffect() + 0.8*SphericalVariogram(range=10.0) + 0.5*ExponentialVariogram(range=100.0)
We can currently write this code just fine, but I think you meant to say that it is hard to extract the coefficients with an API? Notice that 0.5*ExponentialVariogram() will return a ScaledVariogram which stores the coefficient c=0.5 as a scalar. Currently, we have no API to query this scalar, or more generally to decompose a compositive additive model into its constituent parts.
Perhaps we can address this issue in two steps. First, we need to figure out how we can generalize the scalar variogram models to LMC models. We need to make sure that we have a good design here to avoid over-complicating user code downstream. Second, we need to have methods that can extract the structures from an additive model with the corresponding coefficients. This should be easy after we have a good design for the first part.
struct NestedVariogram{T,D} <: Variogram{T,D} scales :: T[] structures :: VariogramStrucutres[] end
This seems like a good alternative approach to our current combination of SumVariogram and ScaledVariogram. Instead of modeling the specific sums and scalings, this type you wrote above would contain a list of the coefficients and a list of simple variogram models. I will digest the idea with more time. I cannot find a use case right now where it would be important to know that we have a SumVariogram or ScaledVariogram specifically. It seems that a simple NestedVariogram as you suggested would be fine. @rmcaixeta what do you think about this?
Excited to discuss this further :+1:
exepulveda
rmcaixeta
I'm raising this issue for discussion on Nested Variogram.
The motivation is for having a better design of variograms that helps us to incorporate turning bands (co-) simulations and to implement a linear model of coregionalization (LMC)
Currently we have unary structures that have a property
nugget, although nugget effect is a valid structure. We already have a PR for [Nugget effect] (https://github.com/JuliaEarth/Variography.jl/pull/31) which is the first step. Also there are two mechanisms to make nested structures:g = SphericalVariogram() + ExponentialVariogram()which is a SumVariogram. This is OK and it can be composed as follow to include more than two structures:g = NuggetEffect() + SphericalVariogram() + ExponentialVariogram()which is a SumVariogram that contains NuggetEffect and a nested SumVariogram.My proposal is to have something like:
g = 0.2*NuggetEffect() + 0.8*SphericalVariogram(range=10.0) + 0.5*ExponentialVariogram(range=100.0)The scale factors could be stored separately to facilitate internal calculations and be ready for a LMC, for example for two variables:
lcm = [0.2 0.1; 0.1 0.2] *NuggetEffect() + [0.8 0.4; 0.4 0.8]*SphericalVariogram(range=10.0) + [0.5 0.3; 0.3 05]*ExponentialVariogram(range=100.0)NestedVariogram would be just a Array of SimpleStructures.
For a smooth transition we can introduce SphericalStructure, ExponentialStructure, etc. and redefine SphericalVariogram (as an example) as follows:
SphericalVariogram(nugget=n, sill=s, range=r) = NestedVariogram(scales=[n, (s-n)], structures=[NuggetEffect(), SphericalStructure(range=r)])Internally NestedVariogram would look like:
I cannot wait for your comments :-)