[time-nuts] exponential+linear fit

[Marek Peca]

Hello, Jim,

On Fri, 4 Oct 2013, Jim Lux wrote:
>
> I'm trying to find a good way to do a combination exponential/linear fit (for 
> baseline removal).  It's modeling phase for a moving source plus a thermal 
> transient, so the underlying physics is the linear term (the phase varies 
> linearly with time, since the velocity is constant) plus the temperature 
> effect.
>
> the general equation is y(t) = k1 + k2*t + k3*exp(k4*t)
>
> Working in matlab/octave, but that's just the tool, I'm looking for some 
> numerical analysis insight.
> (...etc)

The control & estimation practice is to model such a disturbance as a 
response of LTI system to, in your particular case, (almost) 
deterministic excitation.

Specifically, the exp() term is a response of 1st-order lowpass 
(1/(1+s*T). The k1+k2*t is a response of double cascaded integrator 
(1/s^2). If you need to fit the model to data, the inputs to the 
subsystems should be found (inverse filtration for simple systems, 
spectral factorization and inverse filtration for the more complicated 
ones). If you don't know the T, a greybox identification should be done in 
addition.

What is interesting: the model and often also the fit methods are the same 
either for the deterministic signal (such as your y(t)), as well as for 
the stochastic signals, like colored noise. Following statements from 
optimal estimation theory, both white noise as well as Dirac (delta) 
function share some common properties w.r.t. to LTI system description, 
and it is enoiugh to describe (and fit, estimate, etc.) then with the same 
tools. (Of course, this is limited to the linear system domain, L2 
criterions etc.)

Similar matter of clock drifts have been briefly discussed in our recent 
paper, dealing with two-clock ensembling:
Clock Composition by Wiener Filtering Illustrated on Two Atomic Clocks 
(M.Peca, V.Michalek, M.Vacek) http://rtime.felk.cvut.cz/~pecam1/eftf/

In case of interest, feel free to contact me directly.

Best regards,
Marek