Let's say we had a time series data that has a periodic component but also has some low and high frequency components. Let us now say we band pass this data in a certain frequency band (using an appropriate filter) and then take the rms value of the resulting time series.
My question is, what is the statistical error in the rms value that I have obtained, given that I have used a filter on it? How do I quantify the effect that the filter has on the final rms value?
Thanks very much!
The two questions are actually quite independent of each other, depending on the type of noise in the system. For example, if it has colored high-frequency noise, and your band pass filters out a low frequency periodic component, the rms value of this component will (theoretically) have no statistical error. The effect of the filter in this case will be large because it filters out the high-frequency noise. If, however, there is white noise or colored noise in the frequency band of your filter, there will be a statistical error depending on the signal-to-noise ratio in this frequency band. The effect of the filter can be low or even zero in this case, depending on the frequency distribution of the noise.
The effect of the filter can even be large without reducing the statistical error at all, e.g. if your time series is completely noise-free, and the band pass selects one of several periodic components of the signal.
So it's difficult to give a general answer without knowing the structure of your time series in detail.
maybe you can use the ideal filter for compare how well your filter technique filter low, middle and high frequency. Try in stata with the gain filter. If you need moore info you can write me
Thank you very much for all of your suggestions! Thank you William Lizarazo, I will try that out. Regards, Ric
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