If I have a time series of Xt observations. I convert them to returns by:
Rt = Ln(Pt) - Ln(Pt-1).
I then calculate the 20 period moving average and subtract from the Return to find the de-trended return DRt:
DRt = Rt- MA(Rt)
Lastly I calculate the 20 period Moving Average of the de-trended return and calculate the standard deviation SD using:
SDt = SQRT(SUM(MA(DRt)-DRt)^2/20)
So basically I am doing a differencing and de-trending on time series, and then finding the standard deviation.
My question is how do I convert back to the original scale the value MA(DRt) + SDt? I cant seem to get the transformations back correctly. I have done this:
SDt + MA(DRt) + MA(Rt) = Rt,
exp(Rt) = Yt - Xt,
where Yt is the value of SDt +MA(DRt) in terms of the original time series.
Yt = exp(Rt) + Xt
or in other words
Yt = exp(SDt + MA(DRt) + MA(Rt)) + Xt
Pretty sure this is not correct!
Also is there a better math function toolkit I can use to write out more clearly what I am doing on stackexchange?
Thanks for any help.