Not sure if someone had encountered this problem before: say given a time series, we need to determine the minmax. Usually we need to use some kernel smoother to extract second-derivative. It is easy to get minmax inside the series. But at the edge or end of the series, minmax is not deterministic because if newer abrupt sample come in to disrupt the trend then previous peak/trough would be changed. So I think this is a two-sided question:

  1. what smoother should be used, in particular smoother bandwidth, given the time series characteristics?
  2. How to determine the probability of the minmax from current available samples and one (or multiple) unknown incoming samples? Something like Prob(P_{minmax}|x_{t+1},x_{t},...x_{1}) and x_{t+1} is unknown incoming samples. Note that x_{t+1} could mean multiple new samples that disrupt previous minmax.

I don't think spline type of interpolation would help in the case. Not sure if I had described the issue clearly.

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    $\begingroup$ Could you define minmax? $\endgroup$ – Bob Jansen Dec 7 '17 at 20:10
  • $\begingroup$ minmax simply means local minimum and maximum after the smoother. $\endgroup$ – user30896 Dec 8 '17 at 0:38

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