fitting a sloped gaussian

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Scarlet Passer 2021-2-17

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评论： dpb 2021-2-18

Hello!

So I have some data from a compton scattering experiment and I want to fit my peak with a gaussian and find the standard deviation, the amplitute, and the area under the curve. I've tried the 'gauss3' fit and it works pretty well but I'm not sure where to go from there. Because the background is sloped I don't know how to tackle the area without getting unwanted space from beneath the sloped background. All I want is the area of the curve itself. This is what it looks like:

Thanks in advance!

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dpb 2021-2-17

You didn't fit a background term and fitted three peaks over a section of the overall spectrum that doesn't include most of the LH side of the one peak around 450. Why did you do that?

Your resultant fit is

>> f50
f50 = 
     General model Gauss3:
     f50(x) = 
              a1*exp(-((x-b1)/c1)^2) + a2*exp(-((x-b2)/c2)^2) + 
              a3*exp(-((x-b3)/c3)^2)
     Coefficients (with 95% confidence bounds):
       a1 =       55.65  (53.18, 58.13)
       b1 =       456.5  (454.6, 458.4)
       c1 =       24.17  (22.01, 26.33)
       a2 =       2.944  (-2.806, 8.694)
       b2 =       424.3  (403.4, 445.2)
       c2 =       15.13  (-15.05, 45.31)
       a3 =   3.156e+14  (-3.132e+17, 3.138e+17)
       b3 =  -1.296e+04  (-4.457e+05, 4.198e+05)
       c3 =        2430  (-3.635e+04, 4.121e+04)
>> 

NB: that the interval for a2 encompases zero and is ~3+/6--the range is 2X the coefficient so is not significant (as would not be unexpected since there's no sign of any second peak there at all in that region). Then note that the estimates for the third peak are just totally nonsensical.

You need a fit for one peak over the range of the peak plus a background term; you could choose an exponential decay as a reasonable model.

I don't have the time to actually work on this at the moment; just thought would point out these observations quickly now; maybe this evening I can get back. The example in gauss3.m does show fitting a model with an exponential peak; it does also have three gaussian terms; you need to eliminate two of those if only going to fit the one peak and not the full spectrum.

Alex Sha 2021-2-18

If use the combination of 5 Gauss function, the result will be much better:

y = a1*exp(-((x-b1)/c1)^2)+a2*exp(-((x-b2)/c2)^2)+a3*exp(-((x-b3)/c3)^2)+a4*exp(-((x-b4)/c4)^2)+a5*exp(-((x-b5)/c5)^2)

Root of Mean Square Error (RMSE): 10.2702586328186
Sum of Squared Residual: 108115.167694609
Correlation Coef. (R): 0.995045924385973
R-Square: 0.990116391637135
Parameter	Best Estimate
----------	-------------
a1	92.7498096695778
a2	-1769.94724857124
a3	-145.468199223536
a4	60.6523446676163
a5	151714694.184991
b1	199.060149101054
b2	-61.3707462724171
b3	46.4480539913997
b4	456.40444211691
b5	-2441.31620849057
c1	17.1540546396503
c2	95.5557434478509
c3	26.7272034725448
c4	-31.7643462462293
c5	-711.910806560653

dpb 2021-2-18

But there are "issues" here...

a2, a3 < 0

b2, b5 < 0

c4, c5 < 0

and no background term estimated at all, either. I didn't have time to do anything more last night, sorry...

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Answer 1

dpb 2021-2-18

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编辑：dpb 2021-2-18

First pass...overall not so great; need to do more to figure out how to model background. My spectroscopy background has dealt only with gamma-spec so not familiar with what the accepted methods are for Compton scattering data; the decaying exponential can produce a nice fit overall, but the baseline estimate is going to be very variable from run to run I suspect. I tried a couple ranges; your limited LH on the peak and another wider with similar results. But, for starters,

N1=find(energy>400,1);              % this returns the 464 you used, btw...by happenstance
E=energy(N1:end); S=int50(N1:end);  % shorthand variables for the chosen range
% define fitting function including exponential decay background 
fnPk=@(a,b,a1,b1,c1,x) a*exp(-b*(x-E(1)))+a1*exp(-((x-b1)/c1).^2);
fnBG=@(a,b,x) a*exp(-b*(x-E(1)));   % shorthand for the background term for later...
f50=fit(E,S,fnPk,'Lower',[25 0 0 0 0],'StartPoint',[50 0.001 60 450 20]);

This returns

>> f50
f50 = 
     General model:
     f50(x) = a*exp(-b*(x-E(1)))+a1*exp(-((x-b1)/c1).^2)
     Coefficients (with 95% confidence bounds):
       a =          25  (23.31, 26.69)
       b =    0.004602  (0.004254, 0.00495)
       a1 =        54.7  (52.99, 56.41)
       b1 =       456.1  (455.5, 456.7)
       c1 =       24.55  (23.51, 25.59)
>> figure
>> plot(E,S), hold on, plot(f50)
>> hBG=plot(E,fnBG(fitPK.a,fitPK.b,E),'k-');
>> hLG=legend; hLG.String={'Fitted model','Background'};

results in

A wider swath using N1=find(E>350,1) resulted in very similar fitting; a little higher background level but still significantly below the visual location when estimated in the combined model.

Had a similar issue years and years ago in another world with a NaI scintillation setup where were trying to use a quadratic for background; we ended up fitting the two independently with OLS for the background quadratic and the gaussian separately with a combined Marquardt routine. It improved things some, but the background estimation was always problematical for reproducibility from repitition to repition of measurements of the same sample.

That was for an online industrial instrument; we ended up finally having to average successive measurments as simply didn't have sufficient computing power at the time to do anything more sophisticated. This, of course, was 30 years ago by now...

Good luck, I'll do a very quick search for Compton spectra and see if anything shows up if get a chance...