[time-nuts] σ vs s in ADEV
Tom Van Baak
tvb at LeapSecond.com
Wed Jan 4 19:26:22 EST 2017
The plain ADEV calculation is essentially a measure of unexpected or unwanted drift in frequency; which is the 1st difference of frequency error; the 2nd difference of phase error; the 3rd difference in clock time itself.
When measuring the quality of a clock, the key idea is that initial phase doesn't matter (you can always manually set the time), and even initial frequency doesn't matter (you can often adjust the rate: whether pendulum, quartz or atomic clock), and so a more honest measure of intrinsic timekeeper stability is its ability to maintain frequency; that is, statistically speaking, the lower the change in frequency, tau to tau, the better. Change in frequency is frequency drift.
If you have N phase samples, you get N-1 frequency samples and N-2 drift samples. The standard ADEV calculation is simply based on the mean of those drift samples. (and you know Hadamard takes this one step deeper).
If you look a the code at http://leapsecond.com/tools/adev_lib.c you'll see I avoid the confusing issue of N-1, N, N+1 and simply count the number of terms in the rms sum. Not only does that give the correct result but IMHO it make it clear what is being averaged. The code passes the official NBS ADEV sample suite, agrees with Bill's Stable32, is used in John's TimeLab, and also Mark's Lady Heather.
I've never quite understood the pedantic separation of "sample" and "population" mean that statistic textbooks and academics love to discuss. They clearly have never measured oscillators. In my experience if you think there's an important difference between N and N-1, then that's nature's way of telling you to go back to sleep and wait until tomorrow when you have more data. If your N is too small your ADEV wanders all over the place (TimeLab is good at displaying this in real-time) -- meaning that the distinction between sample (n-1) and population (n) mean is beyond ridiculous; even if there's a "correct" textbook answer.
----- Original Message -----
From: "Attila Kinali" <attila at kinali.ch>
To: "Discussion of precise time and frequency measurement" <time-nuts at febo.com>
Sent: Wednesday, January 04, 2017 12:12 PM
Subject: [time-nuts] σ vs s in ADEV
A small detail caught my eye, when reading a paper that informally
introduced ADEV. In statistics, when calculating a variance over
a sample of a population the square-sum is divided by (n-1)(denoted by s in
statistics) instead of (n) (denoted by σ) in order to account for a small bias
the "standard" variance introduces
(c.f. https://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation )
In almost all literature I have seen, ADEV is defined using an average,
i.e. dividing by (n) and very few use (n-1).
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)?
It is upon moral qualities that a society is ultimately founded. All
the prosperity and technological sophistication in the world is of no
use without that foundation.
-- Miss Matheson, The Diamond Age, Neil Stephenson
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