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There are more sophisticated methods for getting the best possible curve fi t for a number of parameters when the curve position is a non-linear funct ion of each of the parameters.

I got exposed to the Marquardt algorithm when I was doing my Ph.D. For my p roblem the Fletcher-Powell approach worked better, and I ended up coding it for myself - in Fortran 4.

I had the advantage that I could work out (and sum) the derivatives of the least squares sum against each parameters at the same time as I was numeric ally integrating the function I was trying to fit to the data, so the proce ss of numerically estimating the second derivatives was a bit more accurat e than it might have been.

You will have a lot more parameters than I had (three - with a brief and un successful excursion into four) but most of your won't make any difference over most of the curve, so you shouldn't have lot of trouble with parameter interaction - where the best fit for one depends heavily on all the others . Marquardt seemed to work better for lots of parameters - my tutor from T heory of Computation 1 had written his M.Sc. on the subject, and that was w hat I got from it when I got to read it back then.

Well... in the real analog hardware world... there are too many other limitation, blemisihes and long term drift issues that negate the utility of any higher accuracy. 10 bits to set the parameter values are also a major hardware hit is a blown fuse memory. In this case the parameters, by specific design, are constant, that is independent of the variables being modelled.

The initial core xtal temperature variation of around +/-20 ppm is compensated by chebys to around 50 ppb. The xtal wiggles on top of that at around 200 ppb or so. The goal is get the wiggles down by a factor of 10. This means an overall cancelation of the error across temperature of 1000:1. The next level to the 2 ppb is not really realistic for a non ovenised oscillator. (1-0.9999) don't like to stay that way over time. Ovenised oscillators would take a raw 50 ppb down to < 0.5 ppb much more reliably. However, they take a lot of power.

In Excel one can use any error function. I tried a few, such as minimising the absolute error and so on. Don't make any relevant difference though.

-- Kevin Aylward

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It's not so much a question of higher accuracy as having a procedure that d oes the job properly.

The Excel Solver function seems to employ a "Generalized Reduced Gradient ( GRG) Nonlinear" approach, whatever that means. Multi-parameter nonlinear re gression mostly seems to involves setting up a multidimensional goodness-of

-fit surface and sliding down it to a local minimum, for which you do need gradients - first derivative against each parameter - and second derivativ es against each pair of parameters.

This evil test problem generates a "banana valley" where there are lot of p airs of parameters that give you a respectable fit, and a good minimiser wi ll track straight along the bottom of the valley to find the pair of parame ter values that give you the best fit. Bad optimisers zig-zag across the va lley and get very confused when they get to the point where valley bends, w hich means they need a lot more iterations.

Clearly Excel works well enough for your purposes, but it doesn't look as i f you know how it works. This is risky.

g the absolute error and so on. Don't make any relevant difference though.

Minimising the root mean square deviation is widely popular, not least beca use it avoids nasty discontinuities. The fact that you could get away with minimising the absolute error does suggests that whatever Excel Solver is d oing is fairly crude and unsophisticated.

Physical beauty is grossly unfair, but its appreciation is immensely powerful and hardwired into us. Fortunately tastes vary, so most of us are able to find someone who finds us attractive.

I've always wondered why humans have such radical extremes of beauty. Are there superstar-level beautiful individuals among cows or birds or worms? And do they have wild swings in what is considered beautiful and stylish?

Cryptomnesia

RL

Oh, in birds, certainly; birds-of-paradise have some amazing plumage (so attractive, that it's illegal to hunt them, or keep the feathers, because the New Guinea natives want to conserve the critters).

Why the BIRDS find them attractive, I don't know. Peacocks, too, have a lot of feathers that aren't good for anything except display. Peahens presumably care...

Because human beings, made in the image of God, can appreciate the Good, the True, and the Beautiful. (None of which are really in the eye, mind, or heart of the beholder.) Ask Aristotle next time he stops by for a brewski. ;)

Cheers

Phil Hobbs

Here's one for you:

CH

but that is the human "valuation" do the sheeps agree?

I wondered that at first, but ewes don't get consulted in the matter. Rams pretty-much have a policy of "if it moves, fight it or f*ck it".

There are more recent, and more useful, opinions on the subject, none of which depend on Platonic ideals.

There are wild swings in what is considered stylish - style is just a device that lets you show that you care about how people perceive you.

Historical art shows that being plump was considered more attractive in a woman back when it was harder to get enough to eat.

Scrawny women still don't ovulate - and still aren't seen as attractive - but it's lot less important today to have fat reserves against a hard winter.

The patents were issued in September 2020

US10790831B2 US10790831B2

Basically the same patent, but one used for temperature compensation one for linearizing a varacter, i'm not complaining.. :-)

-- Kevin Aylward