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Say I have a e-commerce that operates in multiples countries. I can check if my sales are decreasing in a particular country by looking at the plot of the sales for that country over time.

Is there a way to identify a change of trend in various sub-series at a time?

Besides a change of trend, can I identify other abrupt changes such as in the picture below?

enter image description here

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I submitted your data to my visual automatic intervention detection procedure ( my trained eye ) , it identified a possible downwards trend for periods 1-100 and then upwards trend from 100-250 and three level shifts (intercept shift ) at 260 , 340 and 420. If I had your data I would submit it to AUTOBOX a commercially available package that I have helped to develop to confirm or correct my (probably flawed) first impression.

In all seriousness what you are after is an approach ( developed software ) that simultaneously deals with detecting and incorporating arima , level shifts , local time trends , seasonal pulses and pulses while adjusting for time varying parameters and tine varying error variance.

The problem/opportunity is to simultaneously identify BOTH arima structure & the Interventions.

Here is an interesting example enter image description here

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