fitting a non-linear trend on first example

I've fitted a non-linear trend on the data described in the first example, to give a better idea of what I am doing. I've used a robust estimation procedure and a 4-parameter non linear trend function (the green line). The good news is that the trend line tracks the data very well and yields a "credible" conclusion (the series is stable, i.e. not trending up or downwards. A shift occured around period 7 which propelled the trend to a higher plateau by period 22 -- the hit storm is entirely discounted because it lies so far out of the "usual" data range.)

The bad news is that the estimation procedure is sensitive to seed values given to parameters (i.e. the loss function is not well behaved). While the essential conclusion (that the series has reached a plateau at 172 or so) remains unchanged across different solutions, the S that you can see between periods 7-22 can be replaced by a concave curve (i.e. an extremely brief S, too short to be seen) starting at the 157 level.

To summarize -- a robust non-linear trend line appears to be a viable solution to provide a compact analysis of a time series such as the one described above, compared to OLS (linear) and Robust linear estimates.

I'll provide similar charts for a few other prototypical series over the next few days.

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