I've been working with Allan variance for a while and I'm not really super excited about is purpose.

I understand that noise time recordings, if they include flicker noise, diverge in time. In other words the standard deviation increases logarithmically, but why is this a problem? Modern computers can calculate the st dev as a function of window size without any problems for millions of data points.

The plot of Allan dev against observation time multiples is nice, but your data has to be very clean of deterministic signals otherwise very ugly things appear. Also you can see different noise process in the std dev vs window size also or PSD.

It seems to me that this tool was mainly useful decades ago when computer were much more limited and other problems like dead time in measurements existed.

Am I missing something?


1 Answer 1


I currently work in the design of atomic clocks and precision frequency sources and pleased to report that the Allan Variance is still quite relevant and useful. In fact it's utility extends to convenient characterization of many non-stationary processes, well beyond its primary tool as a frequency stability assessment. (And as mentioned in its comments, it’s use to assess stability of accelerometers).

Traditional statistical approaches using the first and second order moments (mean and standard deviation, or equivalently squared as the variance) cannot be computed for non-stationary processes, since the variance will increase the longer you compute the parameter on the data set (diverging). The Allan Variance or Two-Sample Variance is a convenient statistical tool to provide a metric similar to the variance for non-stationary processes. It basically converts non-stationary data to stationary for a consistent metric that can be used to compare competing products or solutions. Applicable to other processes, I personally like how I can use the Allan Variance to quickly assess an optimum averaging time (what we would call the "Flicker Floor" if the independent variable is frequency), for which if we average up to that time we can get an improved estimate of the mean, but if we average any longer, the noise of that estimate only degrades. For example, this can then be applied to optimum time durations for channel equalization and estimation, or any process where we are computing a longer term average to estimate a parameter.

The following graphic copied from a NIST presentation shows how ADEV can reveal the different noise processes within a signal, and where there effects would be of concern (in terms of observation times, $\tau$). This was from a 2015 presentation by Marc Weiss and Kishan Shenoi.

Frequency Stability

I have further detailed the Allan Variance and its practical purpose at these other links starting with the link copied below (that includes links to other related posts):

Allan Variance vs Autocorrelation - Advantages

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    $\begingroup$ Very cool! Thanks for the nice answer. :-) $\endgroup$
    – Peter K.
    Nov 14, 2021 at 19:08
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    $\begingroup$ The place where I discovered Allan Variance was in characterizing IMU outputs -- gyro and accelerometer data, basically. It's useful for all the reasons given above. $\endgroup$
    – TimWescott
    Nov 14, 2021 at 20:35
  • $\begingroup$ Thanks for the answer Im glad also it is still used. About non stationary process, the standard deviation is sometimes well behaved in relation to the window size. Tipically in flicker noise or noises that have a "nice" PSD, the std dev increases smoothly with window size. I wonder however what happens when you have noise sources that are non stationary but also unruly, like for examples, people playing with the temperature of the room where the clock is but not at night nor weekends (or holidays). Certainly the Std dev will not be useful, but I wonder if the Allan Variance is more robuest in $\endgroup$
    – Ralph
    Nov 16, 2021 at 21:10
  • $\begingroup$ @Ralph Yes the Allan Deviation has nice characteristics for each type of noise in it's slope versus averaging time (tau) on a log scale. A white noise process will go down at 1/root-tau, flicker noise will be constant at all tau, random walk noise will go up as 1/root-tau, sinusoidal noise will have a clear pattern etc. I added a plot that helps demonstrate this. Further there are other variants (modified ADEV, etc for their ability to discern certain types of noise processes better). $\endgroup$ Nov 18, 2021 at 22:45

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