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.
