[time-nuts] ADEV vs. OADEV

Thu Jan 22 11:40:15 UTC 2009

Ulrich Bangert wrote:
> Magnus,
>
>   
>> 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.
>>     
>
> I still do not understand the fine point! A estimator might have this
> property and that property and may perform this task good and another
> task bad, but at the basics we have a formula and if the formula is new
> or different from prior art then the thing needs an name of its own. In
> this sense the summation over square(y(i+1)-y(i)) is called the base of
> the "Allan variance/deviation" just for historical reasons. So the name
> is "Allen deviation" and it is defined by its formula.  
>
>   
>> 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 
>   
>> became the OADEV estimator and another estimator took the ADEV place. 
>> This is what I oppose without a more detailed look at things.
>>     
>
> The OAVAR/OADEV has this name of its own BECAUSE it includes the
> summands that are missed by the original AVAR/ADEV so its needs an name
> of its own.
>
>   
>> Somewhere along the line the established ADEV estimator became the
>>     
> OADEV estimator
>
> If you had said: "The currently established estimator for oscillator
> stability is the OADEV estimator" I would have perfectly agreed.
> However, ADEV does already point to a different thing, so to say "Today
> we call ADEV what was formerly called OADEV and what was formerly called
> ADEV now is also called different" is not excused with a certain
> sloppiness in language but simply wrong use of terms. Exactly this is
> the point why I said that the discussion is dangerous. This is not a
> change in paradigm this is a case of inaccurate use of scientifical
> terms.
>
> Best regards
> Ulrich
>
>
>
>   
>> -----Ursprungliche Nachricht-----
>> Von: time-nuts-bounces at febo.com 
>> [mailto:time-nuts-bounces at febo.com] Im Auftrag von Magnus Danielson
>> Gesendet: Donnerstag, 22. Januar 2009 10:57
>> An: Discussion of precise time and frequency measurement
>> Betreff: Re: [time-nuts] ADEV vs. OADEV
>>
>>
>> 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 
>> became the OADEV estimator and another estimator took the ADEV place. 
>> 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
>>
>>     
Ulrich. Magnus

Perhaps the situation is best summarised in /NIST special Publication
1065/ (can be downloaded as 2220.pdf
<http://tf.nist.gov/timefreq/general/pdf/2220.pdf> from NIST) wherein it
is stated that ADEV, AVAR are often taken to mean the overlapped form of
the Allan deviation at least in the US.

The attached figure graphically illustrates the difference between the
ordinary and the overlapped versions.

Bruce
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