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MUV1 PULSE ANALYSIS 14/4/2025 ---------Two-piece (split) normal fits-------------------------------------
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INTRODUCTION
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As shown below, a split-normal function gives a good description of MUV1 data.
The chi**2/ndf  of the fit of the function to the data is ~1; high chi**2 fits are probably
due to an underestimate of the errors for high-amplitude pulses.

The split normal PDF is defined in terms of the mode, amplitude, sigma1 and sigma2
of the time distribution of the number of counts in a MUV1 signal. From these variables, the
mean and area of the pulse can be calculated.

 split-normal reference.



ONE-PEAK FITS
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  muvfit1.pdf  One peak fits.   two-piece normal (2PN) HIGH + LOW  AMP. 11/4/25

Comment:  The pulse data is well fitted by a two-piece normal.  Error set = 4 . 5-constraint fits.
Fitted variables:  baseline + amplitude, time, sigma1,sigma2 of single peak. 
Fits: sigma1 (lhs) ~ 30 ns,  sigma2 (rhs) ~ 43 ns;   mean = mode + 10 ns, error of mode ~ 4 ns.
Overall RMS  ~ 36 ns. Mean baseline ~ 1 count.

Further work is required to remove  early out-of-time data from the fits and to study high chi**2 and high sigma fits.
Also, it would be useful to have muon and pion MC data over a range of momenta to check these results. 



  muvfit1devb.pdf  One peak fits. dev. no baseline fit. 
Fitted variables:   amplitude, time, sigma1,sigma2 of single peak. 
The baseline is set at the minimum count of the pulse data.
This 6-constraint fit fits the pulse data well. The results of the fit agree within errors with those obtained with a 5-constraint
(baseline-incuded) fit.  
Note the profile plot on page 55  that demonstrates that the error on the number of counts is
amplitude dependent.


muvfit1.f muvfit1.kumac xfun used.
Fits made using HBOOK routine HFITH.


TWO-PEAK FITS
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  muvfit2.pdf  Two-peak fits. two-piece normal (2PN) HIGH + LOW  AMP. 11/4/25

Comment: Two-piece normal fits data well. Error set = 4. 2-constraint fits.
Fitted variables: amplitude,time,sigma1,sigma2 for peak1; amplitude,time,sigma1,sigma2 for peak2. Baseline = 0.
Fits: Mean sigmas fitted 31 ns ,35 ns  apparently inconsistent with single peak fits (see muvfi1.pdf).

Work is needed to determine the point error as a function of amplitude and to study high chi**2 fits.

muvfit2.f muvfit2.kumac fun6 used.
Fits made using HBOOK routine HFITH.

NOTE: pulses selected with max - min greater than 25 counts;  peak - min greater than 15 counts.
      Plots are in units  (counts - minimum counts).

 split-normal reference.



General remark
--------------

A model of the MUV pulse distribution is required to determine:

1) the mode  of the distribution;
2) the mean for a asymmetric distribution;
3) the amplitude at the mode;
4) the area of the distribution;
5) the size of the tails;
6) the level of background;
7) the error-matrix for the above variables.

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cd MUVdev
./clg753 x x   
./x  to produce x.dat
paw  exec x     for x.dat to x.ps
./trans x       for x.ps tc  public_html/newplots/x.pdf


public_html/MUV1.html


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Development and old fits.

  muvfit1dev.pdf  One peak fits. dev. residuals added 24/04/25 
  muvfit1deva.pdf  One peak fits. dev. baseline fit iplots 141 601 added. errors fixed 4 28/4/25
  muvfit1devb.pdf  One peak fits. dev. no baseline fit. 

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  muvfit1devc.pdf  One peak fits. dev. no baseline fit. error studies.
  errors set in muvfit1devc.f  010525




      hh = cont(kk)  ! mod 010425
      if(hh.gt.0.)errors(kk) = sqrt(hh/2.)
      if(hh.lt.200.)errors(kk) = 2.
      if((hh.ge.100.).and.(hh.lt.400.))errors(kk) = 8. 

Structure  in chi**2 / amplitude  ??

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  muvfit2dev.pdf  Two-peak fits. dev.

  muvplots5askew2.pdf   dev. version,  two-piece normal (2PN) HIGH +LOW  AMP. 26/3/25
  muvplotskew41.pdf   dev. version, 2-peak fits, two-piece normal (2PN) HIGH AMP. 10/4/25