I am working on time-series signals for which I need to fit a model (like MVAR) that could describe the process satisfactorily. However, my signal has a unit root and is highly non-stationary because of which it doesn't meet the required stationarity criteria for MVAR. With highly non-stationary, I mean it is of differencing order of 7 which makes the interpretation difficult. So, I was wondering if there is any way to make the signals stationary by adding some noise (preferably white) of some specified frequency range. Thank you!
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$\begingroup$ um, "white" and "of some specified frequency range" are literally opposites! Can't be both at the same time. $\endgroup$– Marcus MüllerCommented Jan 27, 2021 at 19:20
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2$\begingroup$ well, so here's the problem I have with your question: you're trying to fit a model to your signal that doesn't apply – as you notice yourself. Now you want to make your signal worse by adding noise – this literally reduces the information you can get from your signal through observation. That in itself is a bit of a counter-intuitive approach! Now, I'm all for interesting approaches, but it's not clear to me what properties you need for that noise to have. For example,if you allow the noise to be correlated to the signal,you can simply add the inverse signal as noise and set your signal=0, $\endgroup$– Marcus MüllerCommented Jan 27, 2021 at 19:27
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$\begingroup$ but I don't think this is at all what you intended to do! However, you don't say what you actually want to do! Why is MVAR a model you consider if your signal is strongly non-stationary? There's certainly a reason behind that! There's methods of making a signal that has non-stationary properties "look" stationary if you don't look too closely (we do that intentionally to decorrelate noise in radio receivers, for example!), but I can't tell whether they will help you at all, since we really don't know what kind of analysis you're going to be doing afterwards. $\endgroup$– Marcus MüllerCommented Jan 27, 2021 at 19:30
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1 Answer
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One processing approach is to use the Two-Sample Variance (also called the Allan Variance), which takes the differencing over intervals of increasing duration from which it can be directly observed over what interval the original signal (unprocessed) can be assumed to be stationary, and for other durations this processing can convert non-stationary signals to be stationary for further statistical evaluation. This is used widely in the Clock world due to the non-stationary characteristics of oscillator phase noise, but is applicable to other signals as well that have similar non-stationary noise processes (1/f noise, drift, etc).
More information on the Allan Variance (or it’s square root the Allan Deviation, ADEV) is here at Wikipedia:
https://en.wikipedia.org/wiki/Allan_variance
Differencing is a high pass comb filtering response, and works in the case of converting non-stationary clock frequency noise since the non-stationary aspects of the noise are more pronounced as you get closer and closer to the carrier. You can also see this with daily stock market prices where the cost vs time is a random walk process (non-stationary) but the daily difference would be a stationary white Gaussian noise process (assuming no memory in the market which is typically valid for prices from one day to the next- even from one minute to the next which is why traders are concerned with ns time intervals with regards to trading information / communications)
And I have other posts here on DSP.SE that demonstrate the application and utility of the Allan Variance and Allan Deviation beyond the evaluation of Clocks:
see
How to interpret Allan Deviation plot for gyroscope?
and
What determines the accuracy of the phase result in a DFT bin?
answered Jan 27, 2021 at 21:09
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$\begingroup$ Ohhh! I never made that mental transfer from clock metric to generally applicable metric! $\endgroup$ Commented Jan 27, 2021 at 21:47
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1$\begingroup$ I felt exactly the same excitement when THAT occurred to me; once I had to really look under the hood as to what was going on- plus further insight for me (and not explained anywhere else that I am aware of): i developed a frequency domain equivalent signal processing view of ADEV; but that may be because where my thinking is coming from but perhaps not generally helpful. $\endgroup$ Commented Jan 27, 2021 at 21:54
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$\begingroup$ @MarcusMüller so I use it in all statistical tests where I am trying to extract a mean in the presence of noise- I take the ADEV or whatever the signal is and then I know the optimum averaging interval to get the best estimate: if I average longer the noise gets worst if I am beyond the “flicker floor”. Now extend that to the optimum depth for an equalizing FIR filter .... cool huh? $\endgroup$ Commented Jan 27, 2021 at 21:57
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1$\begingroup$ Clever! I see how that can be useful for time-variant system identification! Also, it reminds me of the DAB+ (OFDM audio broadcasting) receiver's lagged-correlation based cyclic-prefix-detector that a student of mine and I came up with to improve performance of a reference implementation. (Basically: if your ADEV's nice and low, you're comparing a CP with the end of the same OFDM symbol, that's a pretty robust timing estimator) $\endgroup$ Commented Jan 27, 2021 at 22:01
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1$\begingroup$ I'll have to mull this over slightly, might be interesting for characterizing the dispersive effects of nonlinear media (optical fiber) along a distance, rather than a time axis. $\endgroup$ Commented Jan 27, 2021 at 22:01