[time-nuts] Allan Deviation 68-95-99 rule

[Magnus Danielson]

Hi,

On 2021-02-12 12:25, Simon Lewis wrote:
> Hi everyone,
>
> Novice question, but does the 1-sigma, 2-sigma, 3-sigma (68-95-99) rule
> apply to Allan and modified deviations? That is, can I say that that if my
> MDEV is 1e-11 for 1s, 99% of samples fall within 3 MDEVs? I know that the
> standard variance is the same as the ADEV for white FM, but are the
> coloured components an issue in doing this?

I think this is a very good question! Quite insightful actually.

To put it bluntly, no, it's not valid.

The Allan Deviation just as Standard Deviation is an average of squared
noise, and not average of noise. This changes the distribution from
normal distribution to Chi-square distribution. So, you can not use the
same basic rules.

To complicate the matter, the Chi-squared distribution depends on the
degrees of freedom you have in the measure. The degrees of freedom
depends on the number of samples you use, but also on other details in
the filtering mechanism, and how that affects the noise, and it turns
out the noise type as in power law slope. So, you can now find
estimators of degrees of freedom for the number of samples and then
different for noise-type.

So, the confidence interval is set from the type of noise, number of
samples, processing type and the Chi-square scale properties, which will
be asymmetric around the average value compared to the classical normal
distribution. Similar to the classical normal distribution you have the
confidence value for the range, such as 95%.

The comes the question, how close to the Chi-square does your estimation
of say ADEV or MDEV turn out? Also, be aware, if you have systematic
noise in there, it will not be valid estimation.

Cheers,
Magnus