I couldn't agree more, that, once you add a correlated disturbance or 1/f^a power law noise, things get even messier. Gaussian is just the easiest to toss in.
I once herd a story from once upon a time that, if you bought a 10% resistor, what you ended up with is something like this in the figure attached. Of course 1% percent resistors (EIA96) are manufactured in high yield today, but I would guess some of this still applies to OCXOs, you aren't likely to find a gem in the D grade parts. After pre-aging for a couple of weeks they are either binned, labeled D, or the ones that show promise are left to age some more before being tested to C grade, etc, etc. On Wed, Nov 16, 2016 at 8:06 PM, Bob Camp <[email protected]> wrote: > Hi > > The issue in fitting over short time periods is that the noise is very much > *not* gaussian. You have effects from things like temperature and warmup > that *do* have trends to them. They will lead you off into all sorts of dark > holes fit wise. > > Bob > >> On Nov 16, 2016, at 6:48 PM, Scott Stobbe <[email protected]> wrote: >> >> A few different plots. I didn't have an intuitive feel for what the B >> coefficient in log term looks like on a plot, so that is the first >> plot. The same aging curve is plotted three times, with the exception >> of the B coefficient being scaled by 1/10, 1, 10 respectively. In hand >> waving terms, it does have an enormous impact during the first 30 days >> (or until Bt >>1), but from then on, it is just an additive offset. >> >> The next 4 plots are just sample fits with noise added. >> >> Finally the 6th plot is of just the first 30 days, the data would seem >> to be cleaner than what was shown as a sample in the paper, but the >> stability of the B coefficient in 10 monte-carlo runs is not great. >> But when plotted over a year the results are minimal. >> >> A1 A2 A3 >> 0.022914 6.8459 0.00016743 >> 0.022932 6.6702 0.00058768 >> 0.023206 5.7969 0.0026103 >> 0.023219 4.3127 0.0093793 >> 0.02374 2.8309 0.016838 >> 0.023119 5.0214 0.0061557 >> 0.023054 5.8399 0.0031886 >> 0.022782 9.8582 -0.0074089 >> 0.023279 3.7392 0.012161 >> 0.02345 4.1062 0.0095448 >> >> The only other thing to point out from this, is that the A2 and A3 >> coefficients are highly non-orthogonal, as A2 increases, A3 drops to >> make up the difference. >> >> On Wed, Nov 16, 2016 at 7:38 AM, Bob Camp <[email protected]> wrote: >>> Hi >>> >>> The original introduction of 55310 written by a couple of *very* good guys: >>> >>> http://tycho.usno.navy.mil/ptti/1987papers/Vol%2019_16.pdf >>> >>> A fairly current copy of 55310: >>> >>> https://nepp.nasa.gov/DocUploads/1F3275A6-9140-4C0C-864542DBF16EB1CC/MIL-PRF-55310.pdf >>> >>> The “right” equation is on page 47. It’s the “Bt+1” in the log that messes >>> up the fit. If you fit it without >>> the +1, the fit is *much* easier to do. The result isn’t quite right. >>> >>> Bob >>> >>> >>>> On Nov 15, 2016, at 11:58 PM, Scott Stobbe <[email protected]> >>>> wrote: >>>> >>>> Hi Bob, >>>> >>>> Do you recall if you fitted with true ordinary least squares, or fit with a >>>> recursive/iterative approach in a least squares sense. If the aging curve >>>> is linearizable, it isn't jumping out at me. >>>> >>>> If the model was hypothetically: >>>> F = A ln( B*t ) >>>> >>>> F = A ln(t) + Aln(B) >>>> >>>> which could easily be fit as >>>> F = A' X + B', where X = ln(t) >>>> >>>> It would appear stable32 uses an iterative approach for the non-linear >>>> problem >>>> >>>> "y(t) = a·ln(bt+1), where slope = y'(t) = ab/(bt+1) Determining the >>>> nonlinear log fit coefficients requires an iterative procedure. This >>>> involves setting b to an in initial value, linearizing the equation, >>>> solving for the other coefficients and the sum of the squared error, >>>> comparing that with an error criterion, and iterating until a satisfactory >>>> result is found. The key aspects to this numerical analysis process are >>>> establishing a satisfactory iteration factor and error criterion to assure >>>> both convergence and small residuals." >>>> >>>> http://www.stable32.com/Curve%20Fitting%20Features%20in%20Stable32.pdf >>>> >>>> Not sure what others do. >>>> >>>> >>>> On Mon, Nov 14, 2016 at 7:15 AM, Bob Camp <[email protected]> wrote: >>>> >>>>> Hi >>>>> >>>>> If you already *have* data over a year (or multiple years) the fit is >>>>> fairly easy. >>>>> If you try to do this with data from a few days or less, the whole fit >>>>> process is >>>>> a bit crazy. You also have *multiple* time constants involved on most >>>>> OCXO’s. >>>>> The result is that an earlier fit will have a shorter time constant (and >>>>> will ultimately >>>>> die out). You may not be able to separate the 25 year curve from the 3 >>>>> month >>>>> curve with only 3 months of data. >>>>> >>>>> Bob >>>>> >>>>>> On Nov 13, 2016, at 10:59 PM, Scott Stobbe <[email protected]> >>>>> wrote: >>>>>> >>>>>> On Mon, Nov 7, 2016 at 10:34 AM, Scott Stobbe <[email protected]> >>>>>> wrote: >>>>>> >>>>>>> Here is a sample data point taken from http://tycho.usno.navy.mil/ptt >>>>>>> i/1987papers/Vol%2019_16.pdf; the first that showed up on a google >>>>> search. >>>>>>> >>>>>>> Year Aging [PPB] dF/dt [PPT/Day] >>>>>>> 1 180.51 63.884 >>>>>>> 2 196.65 31.93 >>>>>>> 5 218 12.769 >>>>>>> 9 231.69 7.0934 >>>>>>> 10 234.15 6.384 >>>>>>> 25 255.5 2.5535 >>>>>>> >>>>>>> If you have a set of coefficients you believe to be representative of >>>>> your >>>>>>> OCXO, we can give those a go. >>>>>>> >>>>>>> >>>>>> I thought I would come back to this sample data point and see what the >>>>>> impact of using a 1st order estimate for the log function would entail. >>>>>> >>>>>> The coefficients supplied in the paper are the following: >>>>>> A1 = 0.0233; >>>>>> A2 = 4.4583; >>>>>> A3 = 0.0082; >>>>>> >>>>>> F = A1*ln( A2*x +1 ) + A3; where x is time in days >>>>>> >>>>>> Fdot = (A1*A2)/(A2*x +1) >>>>>> >>>>>> Fdotdot = -(A1*A2^2)/(A2*x +1)^2 >>>>>> >>>>>> When x is large, the derivatives are approximately: >>>>>> >>>>>> Fdot ~= A1/x >>>>>> >>>>>> Fdotdot ~= -A1/x^2 >>>>>> >>>>>> It's worth noting that, just as it is visually apparent from the graph, >>>>> the >>>>>> aging becomes more linear as time progresses, the second, third, ..., >>>>>> derivatives drop off faster than the first. >>>>>> >>>>>> A first order taylor series of the aging would be, >>>>>> >>>>>> T1(x, xo) = A3 + A1*ln(A2*xo + 1) + (A1*A2)(x - xo)/(A2*xo +1) + O( >>>>>> (x-xo)^2 ) >>>>>> >>>>>> The remainder (error) term for a 1st order taylor series of F would be: >>>>>> R(x) = Fdotdot(c) * ((x-xo)^2)/(2!); where c is some value between >>>>> x >>>>>> and xo. >>>>>> >>>>>> So, take for example, forward projecting the drift one day after the >>>>> 365th >>>>>> day using a first order model, >>>>>> xo = 365 >>>>>> >>>>>> Fdot(365) = 63.796 PPT/day, alternatively the approximate derivative >>>>>> is: 63.836 PPT/day >>>>>> >>>>>> |R(366)| = 0.087339 PPT (more than likely, this is no where near 1 >>>>>> DAC LSB on the EFC line) >>>>>> >>>>>> More than likely you wouldn't try to project 7 days out, but considering >>>>>> only the generalized effects of aging, the error would be: >>>>>> >>>>>> |R(372)| = 4.282 PPT (So on the 7th day, a 1st order model starts to >>>>>> degrade into a few DAC LSB) >>>>>> >>>>>> In the case of forward projecting aging for one day, using a 1st order >>>>>> model versus the full logarithmic model, would likely be a discrepancy of >>>>>> less than one dac LSB. >>>>>> _______________________________________________ >>>>>> time-nuts mailing list -- [email protected] >>>>>> To unsubscribe, go to https://www.febo.com/cgi-bin/ >>>>> mailman/listinfo/time-nuts >>>>>> and follow the instructions there. >>>>> >>>>> _______________________________________________ >>>>> time-nuts mailing list -- [email protected] >>>>> To unsubscribe, go to https://www.febo.com/cgi-bin/ >>>>> mailman/listinfo/time-nuts >>>>> and follow the instructions there. >>>>> >>>> _______________________________________________ >>>> time-nuts mailing list -- [email protected] >>>> To unsubscribe, go to >>>> https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts >>>> and follow the instructions there. >>> >>> _______________________________________________ >>> time-nuts mailing list -- [email protected] >>> To unsubscribe, go to >>> https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts >>> and follow the instructions there. >> <AGING_30DAYS_0p5ppb.png><AGING_30DAYS_0p5ppb_simple.png><AGING_30DAYS_0p5ppb_zoomin.png><AGING_30DAYS_5ppb.png><AGING_30DAYS_5ppb_simple.png><AGING_SCALE_A2.png>_______________________________________________ >> time-nuts mailing list -- [email protected] >> To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts >> and follow the instructions there. > > _______________________________________________ > time-nuts mailing list -- [email protected] > To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts > and follow the instructions there.
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