Method for detecting significant increase in values?

Question

I have a list of values which start and continue close to 0 before increasing towards the end (see attached plot of these values). I am interested in the value at which this increase starts - I could do this by manually observing the plot and values, but I would like to explore a more robust method, and find the point at which a statistically significant increase in values starts. This seems like it would be a fairly simple procedure but I haven't found previous examples of this yet, and do not know the proper term for this sort of procedure.

Any suggestions as to statistical methods I could investigate would be much appreciated!

For reference, my values are derived from digitised coordinates on a curve, and each value is the distance from a coordinate to the circumference of a circle. So I am trying to find the point at which my coordinates deviate away from a circular progression.

Answer 1

This is a problem that falls under the category Change detection. For this particular problem, the change detection method called CUSUM seems very well suited because only too positive values are a problem.

There are more or less complicated ways of using CUSUM depending on what distribution to use. The simplest is to assume that both the steady states and the inflated states are normally distributed. In that case CUSUM becomes the cumulated sum and to detect when the cumulative sum is significantly different, one could use the V-shapes (see slides)