God kväll Magnus, On Wed, 4 Jan 2017 22:13:04 +0100 Magnus Danielson <mag...@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 [1] and [2] 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. Yes. > A hint for the use of s, consider the frequency stability. See Allan 1966. I guess you are refering to [3]. 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. Attila Kinali [1] https://en.wikipedia.org/wiki/Allan_variance#Fixed_.CF.84_estimators [2] "Handbook of Frequency Stability Analysis" NIST Special Pub 1065, by W.J. Riley, 2008 http://tf.nist.gov/timefreq/general/pdf/2220.pdf [3] "Statistics of Atomic Frequency Standards", by David Allan, 1966 -- Malek's Law: Any simple idea will be worded in the most complicated way. _______________________________________________ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.