[time-nuts] σ vs s in ADEV
attila at kinali.ch
Mon Jan 9 13:18:04 EST 2017
God kväll Magnus,
On Wed, 4 Jan 2017 22:13:04 +0100
Magnus Danielson <magnus at rubidium.dyndns.org> wrote:
> > My question is two-fold: Why is (n) being used even though it's known
> > to be an biased estimator? And why do people not use s when using (n-1)?
> First off all, you need keep number of phase samples (N) or number
> frequency samples (M) separate.
> As you derivate the phase samples, you loose the phase bias from the
> samples, so the remaining degree of freedom becomes one less. This is
> the same as number of frequency samples, so any average will be (N-1)
> which is the number of frequency samples M, so M=N-1 is motivated both ways.
> Now, as you do an Allan Deviation/Variance estimator, you do second
> derivation, so they the also the frequency bias gets derivated out, and
> another degree of freedom is lost, so as you average you have only M-1
> drift estimates which is what you average over, or N-2.
My statistics is still pretty weak, but I think that the degree of freedom,
as you use it here, does not matter.
The sums of the formulas in  and  are over (M-1) and (N-2) elements,
respectively. The sums are then divided by (M-1) and (N-2) as well.
Which means we are in the case of σ, ie division by (n) and not (n-1) as it
would be the case for s.
> The ADEV core function is just the square of second derivate of phase,
> and then you do an ensemble average over those squares.
> A hint for the use of s, consider the frequency stability. See Allan 1966.
I guess you are refering to . Yes Allan does give tables on the expected
difference of variance for some types of noise, but not explicitly on why
σ and not s is being used.
 "Handbook of Frequency Stability Analysis" NIST Special Pub 1065,
by W.J. Riley, 2008
 "Statistics of Atomic Frequency Standards", by David Allan, 1966
Any simple idea will be worded in the most complicated way.
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