So it sounds like from your October 25th comment that you are interested in algorithmically finding and characterizing two main features: the initial response decay followed by a cycle of increased response and subsequent decay. I assume that the data are observed at discrete time intervals.
Here is what I would try:
- Use a routine like numpy.ma.polyfit to fit, say, a 4th degree polynomial through your data. This should account for the initial drop followed by the rise/drop, but smooth out the numerous but minor fluctuations. Hopefully this degree of polynomial would be flexible enough to fit other, similar series well. The main goal I think would be to get a function that accounts for the major pattern you are looking for.
- Use Python routines for computing the derivative of the polynomial function fit to the data. Example routines are scipy.misc.derivative and numpy.diff. You are looking for the time values where the 1st derivative is zero, indicating a possible local min or max of the function. A second derivative test could be used to confirm which point corresponds to a min or max. Presumably you will have three such points if the graph you showed is representative. Note that the sage project could be very valuable here.
At this point you'll have the time values associated with
a. the start of the initial decay
b. the start of the upswing
c. the start of the second decay
You can then do what you want analytically to assess the changes.
It may be best to let the data speak for itself: across multiple series, when you apply this method, what is the typical size change at the upswing, when does it typically occur into the decay period, and how long does it last? And what does the distribution of this upswing look like in terms of where, how big, and how long? Knowing these statistics, you can better characterize a particular upswing as being within tolerance, with respect to where in time it occurs as well as it size and duration. The key from my understanding would be to easily identify where these changes are occurring. The rest of what I have described is straight-forward to calculate.