Do trend detection algorithms use differential equation? If not then why? if they do what are some of them?

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All the trend detection algorithms(for social networks) that I have seen use some kind of an average based on the previous data and for every new recording they check how far it is from the average. The problem I see with this approach is:

  1. how can we remove noises from the past data?

  2. We would need something that works on a sequence of data and not on a particular point, it is not about the peaks but whether a particular sequence of data will lead to a one.

  3. Another problem with average based methods are lack of past data