I have created a classifier which outputs behaviour of a curve (growth, decline, growth and decline etc), and have come up with an equation which tells me if there is enough variation in the oscillatory output to be growing or decreasing (not constant oscillation which would have a straight line through the median values).
However, although it works, and makes sense in my head.... I am unable to justify my reasoning, and my mathematical ability is not strong enough to try and derive it or explain it mathematically. Can anyone point me to a theory or something that agrees with this that perhaps I saw years ago and forgot? Or perhaps explain why it works so I can justify it?
abs(standard deviation – mean) > 1
If True, data is either Growing Oscillation or Declining Oscillation – enough variation for the data to be growing or decreasing – not Constant Oscillation.
This equation does not make sense. I can take any time series and add a constant to all the terms. That will not change the standard deviation, but it will add that constant to the mean. It has not changed anything in the oscillatory nature of the series. Similarly if we multiply by a constant the mean and standard deviation will scale by the constant, which can make it greater than or less than $1$ without changing the nature of the oscillation.
For example, if the series comes from $\sin t$ for some interval of time it will have a mean of $0$ and a standard deviation of less than $1$, so your test would say False. That seems to be correct because it is a constant oscillation. If it comes from $e^{-t/100}\sin t$ it will again come back False, but it is a declining oscillation. By contrast, if it comes from $100+\sin t$ or $100 + e^{-t/100}\sin t$ your formula will return True in both cases.