ADR suggestion: More advanced models

[henrikjacobsenfys]

General

This is work in progress, and not the highest priority, but I need to get the thoughts written down.

Right now, the model for each data set (intensity as function of energy at a particular coordinate) is a ComponentCollection, which is a sum of components. These can be standard components or custom components through the ExpressionComponent. However, some users have more advanced needs.

A simple example is the recently implemented DeltaLorentz model, which includes a delta function and a Lorentzian, both multiplied by the Debye-Waller factor:

$I= K \exp \left( \frac{-\langle u^2 \rangle Q^2}{3} \right)[A_0 \delta(E) + A_1 L(E, \Gamma)]$,

where $K$ is the scale factor, $\langle u^2 \rangle$ is the mean square displacement, $Q$ is
the scattering vector, $A_0$ and $A_1$ are the relative amplitudes of the delta function and
Lorentzian, respectively, with the constraint that $A_0+A_1=1$, and $L(E, \Gamma)$ is the
Lorentzian function with width $\Gamma$. $A_0$, $A_1$ and the width of the Lorentzian can be
the same at all $Q$ or be allowed to vary with $Q$.

A more advanced example that has been requested is this beast:

Clearly, I will not be able to implement all models that user needs one by one. A more general approach is needed.

Abandon or extend ComponentCollection?

The ComponentCollection as the model for each coordinate may be too simple a concept, and maybe something more advanced is needed. I'm not sure yet. Perhaps the concept of a component can be expanded, or a concept of multiplying components can be introduced?

A possible solution part one: ExpressionComponent

ExpressionComponent can be extended to allow standard components already in the library. So a user may write

expression = "A*Lorentzian(width=Gamma)*Lorentzian(width=Gamma2)",

and A, Gamma, Gamma2 would be created as Parameters, and the area and center of the Lorentzian would be fixed at their default values

A possible solution part two: ExpressionModel

This would be a model that depends e.g. on Q in addition to energy. Right now I imagine the user writes the expression they want, and are allowed to make Parameters dependent on other Parameters, e.g. for Brownian diffusion:

model = ExpresionModel(
expression = "A*Lorentzian(width=Gamma)",
dependent_parameters = {'Gamma': "D*Q**2"}, 
independent_parameters="D",

It could also be possible to specify which coordinates there are, e.g. "energy" and "Q", but I'm unsure yet.. this is just a quick draft.

There should also be a way to indicate which parameters are global and which should be allowed to be independently fitted at each Q. Something like

model = ExpresionModel(
expression = "scale*exp(-u_squared*Q**2/3)*(A_0*DeltaFunction(energy)+A_1*Lorentzian(energy, width=Gamma)",
local_parameters=["A_0", "A_1"]

I'll keep thinking about this, but your thoughts are also very welcome :)