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

[Simon Lewis]

Thank you Magnus, that was very helpful. I'm busy running through Bill
Riley's handbook, and trying to get a grip on this.
I'm not a stats expert at all, so the cogs are slowly turning!

Cheers,
Simon


On Fri, Feb 12, 2021 at 5:33 PM Magnus Danielson <magnus at rubidium.se> wrote:

> 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
>
>
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