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As from title I would like to know if exist a statistical test that can help me to identify a significant divergence between two similar time series. Specifically, looking the figure below, I would like to detect that the series start to diverge at time t1, ie when the difference between them starts to be significant. Moreover, I would also detect when the difference between the series returns to be not significant.

Is there any useful statistical test to do this?

enter image description here

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There are a few ways that come to mind. The first is to take the difference between the two series and create a "new series" . Analyze that series and empirically identify Pulses,Level Shifts/ Local Time Trends and a possible ARIMA component. The results will/might suggest any identifiable divergence. A second approach is to build a common ARIMA model for both time series and use the CHOW TEST to test for statistically significant parameters.

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Another approach that might work is to consider algorithms for change detection.

A first idea is to apply a change detection method like CUSUM on both series and compare the change points. In your example it is very likely that the red series will yield a change point at t1 whereas the yellow one would not. Interestingly, both red and yellow would probably both yield a change point at the first bump of the curve (depending on the sensibility of the CUSUM parameters) but you really don't mind as they behave similarly.

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Some options you may wish to consider:

  1. If you are looking for identifying a significant difference, a Statistical Process Control (SPC) chart using Western Electric rules might also help you identify that it is occurring. As @IrishStat has suggested, graphing the difference between the two time series is the best start. Then applying SPC rules based on analysis of a stable period of the two time series is good.

https://en.wikipedia.org/wiki/Western_Electric_rules

  1. A more detailed pragmatic approach is chronostatistics that is gathering wide acceptance in the mining industry for identifying change and the specific characteristics of noise in time series data. As you can imagine, in an environment where you are interested in 0.001% of the material, uncertainty in the sampling and variability of the process must be understood to know if you have a difference in two time series.

As a mine process engineer, I am used to dealing with time series data that is a lot more noisy than this and chronostatistics (proponents include Pierre Gy and Francis Pitard) allows identification of the errors introduced by the data sampling technique and other aspects of data gathering. More accessible papers (i.e. easier for non-professional statisticians) have been written by Tim Napier-Munn who has a very application-based approach to assessing time series data.

I am not aware of any open source papers but both of these authors have published through Elsevier.

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