Re: fitting parameters to a differential equation
- To: mathgroup at smc.vnet.net
- Subject: [mg108546] Re: fitting parameters to a differential equation
- From: dh <dh at metrohm.com>
- Date: Mon, 22 Mar 2010 06:41:37 -0500 (EST)
Hello,
here is some code to get GS filter coefficients.
l=Filterlength/2, der= derivative (0-> only smoothing), o: Order->
degree of ploy used.
GS[l_, der_: 0, o_: 2] :=
Module[{power, m}, power[x1_, x2_] := If[x1 == x2 == 0, 1, x1^x2];
m = Outer[power[#1, #2] &, Range[-l, l], Range[0, o]];
der! (LinearSolve[Transpose[m].m, Transpose[m]])[[der + 1]]
];
cheesr, Daniel
On 21.03.2010 10:33, Virgil Stokes wrote:
> On 17-Mar-2010 10:37, dh wrote:
>> Hi Eric,
>> I think your idea of usein the DE is sound. Nonlinear Fits are usually
>> nasty.
>> I assume that "I have a noise measurement" actually means: "I have a
>> noisy measurement". Otherwise we are out of luck.
>> To get the derivative I would use digital filters (assuming the data is
>> equidistant) that at the same time smooth and calculate the derivative.
>> Which filter you need depends on the amount of noise in your data. A
>> very simple to calculate type are called Golay Savitzky filters.
>>
>> Daniel
>>
>> On 16.03.2010 10:46, eric g wrote:
>>> Hello Group,
>>>
>>> does this make sense to you?:
>>>
>>> Suppose I have an second order differential equation on y[t]:
>>> y''+ay'+by=0, and I have a noise measurement of {y[t], @t1,t2,....tN}, I
>>> would like to fit 'a' and 'b' using the differential equation rather
>>> than using the solution.
>>>
>>> I will proceed like this:
>>>
>>> * take my y[t1],...,y[tN] measuremenst and do b-splines interpolation (I
>>> dont know what is the best way to do this), named yi[t], then find
>>> yi'[t], and yi''[t]
>>>
>>> * then I have an algebraic system on 'a,b' with N-equations (N is a big
>>> number) ayi'[t1]+by[t1]=-y''[t1],.....
>>>
>>> * how to use pseudoinverse to fit 'a' and 'b'? do you think this way may
>>> be better that a nonlinear fit (weighted nonlinear regression) using the
>>> solution of the equation? Do you think that this way may avoid the
>>> problem of finding the appropriate guess for the nonlinear fits
>>> algorithms with is ussually an issue?
>>>
>>> best regards,
>>> Eric
>>>
>>>
>>>
>>
> I wrote a Mathematica notebook about a year ago on LS smoothing for
> teaching purposes. This notebook gives some background on the SG
> smoother and contains code that can be used to estimate smoothed values
> (0th derivative) and derivatives of noisy TS data --- several examples
> of smoothing are included. If anyone is interested in this notebook,
> then just drop me an email with "SG Smoother" in the subject.
>
> --V
>
>
>
>
--
Daniel Huber
Metrohm AG
International Headquarters
Oberdorfstr. 68, CH-9101 Herisau / Switzerland
Phone +41 71 353 8606, Fax +41 71 353 89 01
Mail <mailto:dh at metrohm.com>
Web <http://www.metrohm.com