Real-case example -> Reaction-Diffusion model

[lacerbi]

• Let's think of something.

[GClarte]

I think the best example we can have are those similar to lotka-volterra, we can imagine some variations (for example with more than a single species, or with 'unknown' species). For a real life dataset, there is the CIGALE (python, https://cigale.lam.fr/) simulator that produces simulations of galaxy spectrum (do not ask for details). We can change one of the main parameters (SFH) and produce several datasets. In this, X and Y corresponds to the profile of the galaxy, and we split the profile to get context/test. This has the advantage of having a real dataset available as well, but it might be a little complex to make work.

Another possibility, very similar, is to work on meta review: several regression problems supposed to be on the same model, that produce different results. I found an article on the subject (https://www.sciencedirect.com/science/article/abs/pii/S0167947309001765), but the data is not available simple.

Finally, it is possible to imagine a 3-level hierarchical model: \alpha -> \beta -> \mu -> x, where each point of the regression would be on [\mu -> x] (some x serve as input, the other as output), the context are some \mu's (which is why we need an additional level). This is very artificial (especially the assumption that everything is iid) but it's the only case where the dimensions of the input are exchangeable.

[GClarte]

• [ ] find an example with larger dimension of input (~3,4), for example several similar regressions.
• [ ] find an example (the same ?) where the input dimension present translational invariance (or other invariance to rigid transformation)

[lacerbi]

Just to be clear, the ideal example is a regression problem that has:

• Large dimensional input (more than $D=2$); we don't care about the output being more than $1D$.
• At least some of the input dimensions exhibit translational invariance, or invariance to rigid transformations (i.e., equivalent to stationarity and isotropy properties for kernels).

[GClarte]

The asteroid example is not feasible, the code is in Julia and would require to link both (I don't think it's possible as it requires a lot of things on the cluster). I propose to work on SPDE (https://www.scitepress.org/papers/2007/20647/20647.pdf) to model blurring of images. There's a lot of possibilities in particular with higher dimension, small datasets (we generate images from a similar process, eg markov field) and we can have invariances

[GClarte]

TODO:

• [ ] run example of latent inference
• [x] add details on simulations, parameters, etc.

[GClarte]

• [x] the results for RD have all been rerun during the submission of the supplement, so I won't relaunch it again.
• [x] for the rebuttal, I contacted a computer scientist working with medicine researchers, to know if the RD model is used and if there were some dataset existing
• [ ] For the rebuttal, I will change the parameters of the simulation to get something more interesting and relaunch everything.