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.
