definition of alpha in GP spectral densities

[sbfnk]

The vectorised formulation of the Matern 3/2 kernel spectral density in #742 has come with a slight change in definition of alpha - what is now called alpha is actually sqrt(alpha), i.e. the square root of the magnitude.

This has a few consequences:

• the description is not correct
• the prior specification is tighter than before, leading to some unexpected changes in the fits, see e.g. evaluate output difference between package versions after updating EpiNow2 epiverse-trace/howto#81

To fix this we could

1. update the descriptions and the prior for alpha
2. change alpha to sqrt(alpha) in the spectral density of the Matern 3/2 kernel

Option (1) would be more computationally efficient at least the Matern 3/2 kernels. The downside is that we've changed a parameter in the model so will have to update the model description (esp. https://github.com/epiforecasts/EpiNow2/blob/main/vignettes/gaussian_process_implementation_details.Rmd) and perhaps issue a note to the user (although I suspect this is not necessarily a parameter many users have been tweaking).

[seabbs]

I think everywhere it has gone from the variance to the standard deviation (i.e in all kernels)? I agree this needs to be better doc'd if true and if it is a special case in the matern than this needs to be doc'd as well

[sbfnk]

Looking at this in the context of #831 I realised that we used to model alpha^2 (the square of the magnitude), then briefly switched to alpha (as in the paper), now back to alpha^2, i.e. alpha is indeed on the standard deviation scale.

[sbfnk]

Discussion of prior value continued in #834