[time-nuts] exponential+linear fit

[Jim Lux]

On 10/4/13 1:18 PM, Joseph Gwinn wrote:
> On Fri, 04 Oct 2013 14:30:40 -0400, time-nuts-request at febo.com wrote:
>> ------------------------------
>>
>> Message: 4
>> Date: Fri, 04 Oct 2013 10:38:07 -0700
>> From: Jim Lux <jimlux at earthlink.net>
>> To: Discussion of precise time and frequency measurement
>> 	<time-nuts at febo.com>
>> Subject: [time-nuts] exponential+linear fit
>> Message-ID: <524EFCFF.8000802 at earthlink.net>
>> Content-Type: text/plain; charset=ISO-8859-1; format=flowed
>>
>> 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.
>>
>> I could do it in steps.. do a straight line to get k1 and k2, then fit
>> k3& k4 to the residual; or fit the exponential first, then do the
>> straight line., but I'm not sure that will minimize the error, or if it
>> matches the underlying model (a combination of a linear trend and
>> thermal effects) as well.
>>
>> I suppose I could do something like do the fit on the derivative, which
>> would be
>>
>> y'(t) = k2 + k3*k4*exp(k4*t)
>>
>> Then solve for the the k1.  In reality, I don't think I care as much
>> what the numbers are (particularly the k1 DC offset) so could probably
>> just integrate (numerically)
>>
>> y'()-k2-k3*k4*exp(k4*t) and get my sequence with the DC term, linear
>> drift, and exponential component removed.
>>
>>
>> The fear I have is that differentiating emphasizes noise.
>
> How many measured data points do you have?  If you have enough data,
> you can use the MatLab nlinfit() (nonlinear fit) function to fit the
> data directly to the y(t) equation.


I'm removing a slowly varying bias term from fairly noisy data.  Maybe 
several 10s of thousands of data points,
And I want to do it quickly on a slow processor.



>
> Because nlinfit uses a least-squares approach, and there are many
> coefficients to be found, a reasonable starting point is required.  The
> fit on the derivative, while probably too noisy to yield a useful final
> answer, would be one way to get some of the initial values.

I've done that and it works.. but I'm looking for a more "basic" sort of 
approach, given that I actually know something about the underlying model.

>
> I did just this recently, fitting a skew gaussian pdf to a histogram,
> using rough mean, standard deviation, and skewness computed from the
> histogram to seed nlinfit.  The mean, standard deviation, and skewness
> computed from the histogram are famously inaccurate if there is
> significant skew, but it was still good enough to keep nlinfit from
> getting lost.
>