Hilbert space Gaussian process for multiple time series

[avehtari]

@Peter.Stewart

As a next step, I would like to extend the model to incorporate the individual ID of each animal - although individuals generally follow a similar migration pattern, there is variation in the exact timing and overall distance that they migrate. I am thinking that a way to model this would be some sort of hierarchical GP, a bit like the one in this tutorial. - each individual would have its own GP (planning to use Matérn 1/2 kernel for these) , but would contribute to an overall average migration trajectory (which would presumably be modelled by the slow trend + cyclic GP as in my current model).

However, I am struggling to code a Hilbert basis function version of this type of model. In particular, it’s unclear to me exactly how the indexing would work - presumably I’d need to have separate basis functions and spectral densities for each individual, and combine these using append_row in normal_id_glm? Am I thinking along the right lines here, or am I barking up the wrong tree?

If you assume that a prior smoothness of the ID specific functions is the same over IDs, then you can use the same length scale for all IDs, which then means that you can use the same basis functions and spectral densities for all IDs, and you only need ID specific coefficients. You might assume that the average function is smoother and you could have separate length scale and magnitude for the average, and then common length scale and magnitude for how the individuals are different from the average. Although you have a hierarchical model also for the length scales, it is likely not to help as the length scales are weakly identified from the data, and you would need very different behavior for different individuals to see the effect. Remember that the length scale parameter is part of the prior specification, and the posterior wiggliness of each individual GP depends on the coefficient values (in case HSGP).

@Peter.Stewart

To further complicate things, the GPS data for each individual have different start and end points (not all individuals were tagged at once, tags were collected at different times) and thus quite different lengths. Also, data from different individuals may be spaced differently (e.g. some GPS tags return daily fixes, while others only return weekly fixes).

This doesn’t matter. You are doing basis function regression and different individuals can have different covariate values.