# Uncertainty Decomposition in velocity signal

I have a timeseries with repeated measured velocity signal (almost periodic signal). To find the mean time-varying velocity, I have decomposed the signal and computed the ensemble-average and the RMSE to estimate the error. The problem is that this estimated error contains : Error due to the decomposition method (the decomposed signals are not exaactly the same), error due to the measurement and turbulence. Is there any method to decompose those errors from the RMSE ? I've looked up the standard error but I'm not sure this method would be useful in my case.

The starting point in the approach is to use linear propagation. For a signal $$S$$ that is affected by parameters $$X_j$$, the uncertainty in measuring $$S$$ can be expressed as below.

$$\Delta^2 S = \sum \left(\frac{\partial S}{\partial X_j} \right)^2 \Delta^2 X_j$$

For this to work, you must have two things. First, you should have a theoretical expression of $$S(X_1, X_2, \ldots X_N)$$. You can then obtain the analytical expression for the partial terms (call them the sensitivity factors) in the expression for $$\Delta^2 S$$. Secondly, you should have independent measures or estimates of each of the $$\Delta X_j$$ uncertainty values.

You have lumped the uncertainty across the entire set of your data. Your data have three factors: time period, device uncertainty, and random uncertainty. You have to extract the time period out as a separate factor. You should analyze the data keeping time period constant. In essence, rather than viewing your results across one experiment with varying time periods, view your data as $$N$$ experiments of $$N$$ different time periods.

Finally, for any overall uncertainty $$\Delta X_j$$, the device uncertainty $$\delta_D X_j$$ and random (replicant measurement) uncertainty $$\delta_R X_j$$ can generally be expressed using a rule of quadrature.

$$\Delta^2 X_j = \delta^2_D X_j + \delta^2_R X_j$$

All of this presumes that uncertainties are uncorrelated. When the uncertainties have correlation, the approach still serves as a useful starting point.

• That gives me an idea about the theoretical approach. I actually thought about using plainly linear propagation but I was a bit confused. The idea of extracting the time period as separate factor is interesting. I'm left with the device uncertainty and random uncertainty (turbulence in my case, and that's what I'm looking for). Hence: for each time period t, I'd have: RMSE_t = device_uncertainty + turbulence_t. Then I extract turbulence_t. But to do this I need RMSE_t, I'm not sure I can estimate it from the total RMSE, unless I assume that RMSE_t = RMSE/N (or other hypothesis)? – Yassine May 19 '19 at 10:01
• @Yassine I do not understand exactly what you measure and exactly how your measured values are then to be used to extract physical insights about your system. Without this detailed information, I cannot make any other recommendations than what I have given as a general approach to uncertainty analysis. – Jeffrey J Weimer May 19 '19 at 19:22