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

[Magnus Danielson]

Hi Simon,

Yeah, I remember those days too. I learned this from the NBS/NIST line
of educational papers. The NIST TN 1337 is recommended reading. The Bill
Riley handbook is for sure a good read, even if it may not dwell very in
deeply into why things are the way they are.

A main issue I failed to cover good enough is that if you have one or
more sinusoidal disturbances, they survive through the derivations and
take over as being dominant, and then the assumptions for confidence
values is not valid.

Keep asking questions as you stumble onto mysteries. I'm sure we can
help answer them.

Cheers,
Magnus

On 2021-02-13 17:09, Simon Lewis wrote:
> 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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