Talk:Allan variance

message: The current version of IEEE Std 1139 is 1139-2008 (http://dx.doi.org/10.1109/IEEESTD.2009.6581834) and supersedes the one references here (1139-1999)  — Preceding unsigned comment added by 88.208.89.131 (talk) 18:18, 2 November 2016 (UTC)
thread_title:  References 

message: there is missing a concise explanation of variable names:
with the first equation should be explained what y, T and tau are. x appears without any explanation ... later appears the nebulous remark that y is the derivative of x - but what is x ?
the oscillator model has a variable name for the amplitude (big) V (13th letter of alpabet). 
The name of frequency is (small) v (same letter).
This is even for a quality C article not sufficient.
--Janschween (talk) 14:52, 31 October 2010 (UTC)
thread_title:  variable names 

message: Could someone add an explanation for high-school-level people as well?  —Preceding unsigned comment added by 82.136.198.205 (talk) 13:15, 19 January 2008 (UTC)
replies: text: I took a stab at it.--Srleffler 05:58, 29 November 2005 (UTC)
thread_title: explanation for high-school-level

message: This article has been revised as part of a large-scale clean-up project of multiple article copyright infringement.  Earlier text must not be restored, unless it can be verified to be free of infringement. For legal reasons, Wikipedia cannot accept copyrighted text or images borrowed from other web sites or printed material; such additions must be deleted. Contributors may use sources as a source of information, but not as a source of sentences or phrases. Accordingly, the material may be rewritten, but only if it does not infringe on the copyright of the original or plagiarize from that source. Please see our guideline on non-free text for how to properly implement limited quotations of copyrighted text. Wikipedia takes copyright violations very seriously. --r.e.b. (talk) 03:05, 24 December 2009 (UTC)
replies: text: Considering that the article now has undergone a major overhaul, which has essentially removed the old structure by freshly written material, I think that this copyright notice may be dropped as there will be less incentive to revert the removal. I will have a look at the old text and see if there is any meaningful contents not already covered. Cfmd (talk) 01:58, 12 April 2010 (UTC)
thread_title:  Copyright problem 

message: ===Sample indexing convention===
The casual reader will notice that the convention being used here for numbering and indexing phase and average frequency samples differs from most other texts. The traditional convention numbers time error samples x_i with i from 1 to N and average fractional frequency samples y_i from 1 to M. The mapping functions lets
:x_i = x(i*T)
which looks OK at first. But then considering that the average fractional frequency integration goes from 0 to tau will let the first sample cover the time from x(T) to x(T+tau) rather than starting at x(0) up to x(tau) which one would believe it to be and in fact what most implementations would do. IEEE 1139 has the constant x_0 added, but to no help since you expect the first average to cover x(x_0) to x(x_0 + tau). To circumvent this issue four different approaches can be used
1) Offset the mapping function with T such that it becomes
:x_i = x((i-1)T)
which will solve the problem, but leaving the explanation of the somewhat unexplained -1 term.
2) Offset the difference function in the Allan variance such that it uses the backward frequency derivate.
This would break the relationship to the M-variance function which is not a derivate but becomes that in the special case of doing a 2-sample variance.
3) Offset the average fractional frequency function to use back-ward integration.
:\bar{y}_i = \frac{1}{\tau}\int\limits_{-T}^{-T+\tau}y(t+t_v) \, dt_v = \frac{x(iT-T+\tau) - x(iT-T)}{\tau}
which again is inelegant. Notice how the continous x(t) function is needed as the T vs tau subtle detail can't be expressed in the indexed sample variant, another inelegance in presentation.
4) Use a straight mapping and offset the predictor to run from 0 rather than 1.
From these choices solution 4 has been selected as M-sample variance with dead-time can be made to colapase nicely into the Allan variance when M=2 and T=tau.
This convention has proven useful when implementing in base-0 indexed languages such as C.
One should also consider what happens for different taus, either by evaluating for different taus or by concatenating samples to create an extended measurement. In all these case we want the starting point to remain the same, 0 (or some offset x_0) for all these cases. Using an offset like T or tau would create a shift which needs to be cancelled correctly.
The used convention and traditional convention does not deliver different results, but the used convention is hopefully more consistent and useful over the range of variances that is included in this series of articles. I believe that the convention chosen is for best consistency and presentation solution available, at the cost of not being exactly transcribed from the referenced papers or IEEE 1139. Cfmd (talk) 22:56, 27 April 2010 (UTC)
thread_title:  Conventions 

message: The casual reader will notice that the convention being used here for numbering and indexing phase and average frequency samples differs from most other texts. The traditional convention numbers time error samples x_i with i from 1 to N and average fractional frequency samples y_i from 1 to M. The mapping functions lets
:x_i = x(i*T)
which looks OK at first. But then considering that the average fractional frequency integration goes from 0 to tau will let the first sample cover the time from x(T) to x(T+tau) rather than starting at x(0) up to x(tau) which one would believe it to be and in fact what most implementations would do. IEEE 1139 has the constant x_0 added, but to no help since you expect the first average to cover x(x_0) to x(x_0 + tau). To circumvent this issue four different approaches can be used
1) Offset the mapping function with T such that it becomes
:x_i = x((i-1)T)
which will solve the problem, but leaving the explanation of the somewhat unexplained -1 term.
2) Offset the difference function in the Allan variance such that it uses the backward frequency derivate.
This would break the relationship to the M-variance function which is not a derivate but becomes that in the special case of doing a 2-sample variance.
3) Offset the average fractional frequency function to use back-ward integration.
:\bar{y}_i = \frac{1}{\tau}\int\limits_{-T}^{-T+\tau}y(t+t_v) \, dt_v = \frac{x(iT-T+\tau) - x(iT-T)}{\tau}
which again is inelegant. Notice how the continous x(t) function is needed as the T vs tau subtle detail can't be expressed in the indexed sample variant, another inelegance in presentation.
4) Use a straight mapping and offset the predictor to run from 0 rather than 1.
From these choices solution 4 has been selected as M-sample variance with dead-time can be made to colapase nicely into the Allan variance when M=2 and T=tau.
This convention has proven useful when implementing in base-0 indexed languages such as C.
One should also consider what happens for different taus, either by evaluating for different taus or by concatenating samples to create an extended measurement. In all these case we want the starting point to remain the same, 0 (or some offset x_0) for all these cases. Using an offset like T or tau would create a shift which needs to be cancelled correctly.
The used convention and traditional convention does not deliver different results, but the used convention is hopefully more consistent and useful over the range of variances that is included in this series of articles. I believe that the convention chosen is for best consistency and presentation solution available, at the cost of not being exactly transcribed from the referenced papers or IEEE 1139. Cfmd (talk) 22:56, 27 April 2010 (UTC)
thread_title: Sample indexing convention

message: === - vs. – ===
User:Michael Hardy and User:Materialscientist makes changes to or from - and –. This location is here to have them agree on which is the correct variant and then we will use only that format convention. My goal is to see the oscillation go away. Cfmd (talk) 17:57, 8 May 2010 (UTC)
replies: text: Upon cursory look, I haven't seen oscillations, but Michael rather fixed what I missed. Hyphens are not to be used for page ranges, minus signs, separation of sentence parts, etc. Materialscientist (talk) 00:17, 9 May 2010 (UTC)

text: en-dashes are used in ranges of pages, ranges of years, months, etc., things named after two people (e.g. the Cauchy–Riemann equations). Michael Hardy (talk) 01:16, 9 May 2010 (UTC)
thread_title:  Formatting 

message: User:Michael Hardy and User:Materialscientist makes changes to or from - and –. This location is here to have them agree on which is the correct variant and then we will use only that format convention. My goal is to see the oscillation go away. Cfmd (talk) 17:57, 8 May 2010 (UTC)
replies: text: Upon cursory look, I haven't seen oscillations, but Michael rather fixed what I missed. Hyphens are not to be used for page ranges, minus signs, separation of sentence parts, etc. Materialscientist (talk) 00:17, 9 May 2010 (UTC)

text: en-dashes are used in ranges of pages, ranges of years, months, etc., things named after two people (e.g. the Cauchy–Riemann equations). Michael Hardy (talk) 01:16, 9 May 2010 (UTC)
thread_title:  - vs. &ndash; 

message: In lack of finding another way to get assessment, I simply try to set the class and importance of the article. I beleive it has matured significantly, and to the best of my knowledge it should meet the C grade at least, possibly B grade. If not, I want to learn what is missing and correct it.
As for importance, within the measurement field of time and frequency, Allan variance is a key concept. For that field, tagging the importance as mid is an understatement, where as for others outside of that field it may seem specialized. However, considering that more units directly or indirectly reverts back to the SI second, the tools to assess the stability of time becomes necessary for those fields as well, and indeed we have seen the use of Allan variance for other measurements than pure time, such as in accelerometers. Cfmd (talk) 01:01, 16 May 2010 (UTC)
thread_title:  Measurement class and importance 

message: I reverted the removal of the section on modified Allan estimators. Please do not remove it without discussion here. There is a line of estimators achieving the modified variant of the Allan variance, but they line up to the same scale. The section is intended to explain the benefits, and is the part which goes into most detail where as the hints about it exists in more places. Cfmd (talk) 12:14, 7 June 2012 (UTC)
thread_title:  Modified Allan estimators section 

message: There seems to be a rather opaque normalisation in the table that gives equations for Allan variance/deviation in terms of Sf.  
That is to say: I would expect Allan deviation to be inversely proportional to the carrier frequency, but the carrier frequency does not appear anywhere in the table.  It could of course be hidden in the parameter h (h-1, h-2, etc.) but this is not locally apparent.
  
Please clarify or correct.
ThanksPhysicistQuery (talk) 18:20, 30 November 2015 (UTC)
thread_title:  Allan Variance Power Law Response (table) 

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thread_title:  External links modified 

message: There appear to be some typographic errors in these equations in the original source, which includes a PDF annotation referring to an erratum that I could not track down.  Google scanned a series of books containing a reproduction of the original article, but that scan seems to cut off right before the page where the appropriately-numbered erratum would appear.
As a general principle, we ought to expect df to scale linearly with N when n is held constant.  My intuition for offering this guideline is as follows.  The Allan variance estimator takes (in effect) a second-order finite difference of the phase time-series.  Among the five noise models, the one with the strongest long-time correlation behavior, the frequency random walk, is the double-integral of white noise.  All of the other noise models have less spectral constant at low frequencies, and hence weaker long-time correlations.  Taking a second-order finite difference of a double integral of white noise gives you back more-or-less white noise.  So Allan variance samples taken far apart (compared to n) under any of the five noise models will be more-or-less independent.  Thus, when N is much greater than n, the number of effective degrees of freedom had better scale linearly with N.
This general principle is violated by the expressions given in the article (as of Jan. 23, 2020) for FPM and FFM in the n=1 case.
In the FPM expression, the original source omits the inverse-square-root operation entirely.  They write exp(ln() * ln()).  That is clearly wrong; df would then scale like exp(ln(N)^2) for large N.  The inverse square root does not make sense, either, scaling like exp(1/ln(N)).  I have not found any source that corroborates the inverse-square-root version.  The correction that makes sense is a non-inverted square root.  Perhaps the original authors took the geometric mean of two fits on two different log-log plots?  In any event, with the exponent changed from (-1/2) to (+1/2), the expression gains sensible asymptotic behavior for large N: unless I am mistaken, it runs like (N-1) √((2n+1)/(8n)).
As for the n=1 FFM case:  The original authors cite, in an earlier paragraph, the work of a number of other authors (Lesage, Audoin, and Yoshimura).  Those equations are numbered (6.5).  The authors then cast some doubt upon them.  Nevertheless, it appears that they intended for the n=1 FFM expression in their results, (6.6), to be a copy of that in (6.5); however, the copy-editor seems to have missed that the (N-2) expression in the numerator must be squared.  Not only does this resolve the conflict between (6.5) and (6.6), it also recovers sensible asymptotic behavior for large N.
Mumble mumble original research-- I can't make the edit without locating a better source, but I definitely want to leave this breadcrumb here for the next person who gets confused.  Good luck!  — Preceding unsigned comment added by Piannucci (talk • contribs) 10:30, 23 January 2020 (UTC)
thread_title:  Effective degrees of freedom