Magnus Danielson magnus at rubidium.dyndns.org
Thu Jan 22 04:56:49 EST 2009

```Ulrich,

Ulrich Bangert skrev:
> Magnus,
>
> I am aware that you know a lot about these things. Nevertheless I
> believe you are starting a most dangerous discussion in the sense that
> you put some terms into question of which I believed that they have well
> been established.

I have only recently seen the OADEV being used where as I have seen
countless articles on calculations of these without encountering them,
so from my standpoint OADEV is not well established, which is why I
raised the question in order to "shake the tree" to see what fruits that
I have missed.

> For that reason let me test where we agree and where
> not:
>
> Mr. Allan decided that for his new statistical measure the summation
> shall run over
>
> square(y(i+1)-y(i))
>
> for frequency data and over
>
> square(x(i+2)-2*x(i+1)+(xi))
>
> for phase data. Both in contrast to the standard deviation where the
> summation runs over squares of distances from the mean. This new
> variance was called "Allan variance" and its square root "Allan
> deviation" to honor Mr. Allan for his work. This variance/deviation has
> a certain "overlapping aspect" since a single y(i) or x(i) appears in
> multiple terms of the summation. Agreed?

Yes, yes....

Actually, what you describe is the estimator formulas rather than
definition. This is also targeting the fine point that I am trying to
make. It's not about the basic definition, but accepted convention to
denote the estimators.

> Both terms require that the elements with subsequent indices are spaced
> apart at the "Tau" for wich the computation shall be done. Considered a
> number of phase measurements spaced 1 s apart then the computation will
> run over
>
> square(x(i+2)-2*x(i+1)+(xi))
>
> for Tau = 1 s. If you are going to compute for Tau = 2 s from the SAME
> data set you will have to use the "original" samples
>
> square(x(5)-2*x(3)+x(1))
>
> for the first summand and
>
> square(x(7)-2*x(5)+x(3))
>
> for the second summand and
>
> square(x(9)-2*x(7)+x(5))
>
> for the third summand and so on. All indices are incremented by two
> between neighbour summands because the next summand is 2 s (or two
> original samples) apart from the current summand. Agreed?

Yes, yes...

> As we notice the summation leaves out a number of summands where the
> elements are also spaced 2 s apart, for example
>
> square(x(6)-2*x(4)+x(2))
>
> or
>
> square(x(8)-2*x(6)+x(4))
>
> If we use these additional terms in the summation the number of summands
> increases a lot and improves the confidence interval of the estimation,
> even though the added summands are NOT completely statistical
> independend from the original ones and therefore this measure shall be
> clearly distincted from the original Allan variance/deviation. The
> summation over the original terms plus the added terms delivers the
> "Overlapping Allan variance/deviation" in conjunction with a suitable
> normation factor. Agreed?

Disagree. The estimator formulation that is classically used includes
these "missed" tau0 steps that you claim that OAVAR/OADEV includes. This
is my point. Somewhere along the line the established ADEV estimator
This is what I oppose without a more detailed look at things.

I agree that it changes the statistical properties in terms of
confidence interval, but it also change the frequency dependence. The
analysis on frequency dependency needs to be redone as I suspect they do
not always agree.

Cheers,
Magnus

```