Should we model the change in mean explicitly, or rely on detrending?
One of the first things to deal with in the model setup is whether we ‘detrend’ the data first or not. As you know, in most saddle-node bifurcations the mean also moves since the stable node usually moves a bit to collide with the unstable node rather than just sitting there and waiting for the unstable node to hit it; which results in a gradual decrease of the mean prior to the actual catastrophe. It’s relatively typical to detrend any changes in the mean out (after all, sometimes this will also reflect things like seasonal variation etc., so it’s not a trivial thing to build into the model in a general way). Modeling the rate of change in mean makes it possible for a mean-only change to be sufficient to prefer the model of change over the OU model… However, detrending needs rigorous definition to justify the approach. Transforming the data can distort the signal (particularly obvious with the case of interpolation). I don’t really like detrending since it can be rather arbitrary in how you choose to go about it, but I suppose it’s preferable to specific modeling assumptions that try and capture the trend as well. There’s an interesting comparison of detrending with just a Gaussian filter vs using the exact model in this paper really nice paper on warning signals in epidemiology that I’ve only recently come across; which also does some elegant van Kampen expansions en route: https://doi.org/10.1007/s12080-013-0185-5.