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Simple algorithm for online outlier detection of a generic time series

Observing the time series data I noticed there are some outstanding peaks (the picture below). I would like to automatically find out which on which consecutive days (x axis) these peaks appear and am thinking what would be the best way to do it.

During the different time periods the peak heights might be lower or higher so it is not appropriate just to set a global treshold so I guess some kind of sliding window for checking local irregularities would be the best choice.

Before creating own logic I would like to know whether there are any existing valuable methods for dealing with such problems. Otherwise I was thinking to create a time window, say 15 days, go through the observing period with time step 1 day and on each step calculate the average and the difference between the maximum value and average.

Is there any better (existing) algorithm which could be used to detect peaks?

Thank you!

enter image description here

niko
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