myencoding = GPVAL_ENCODING
set encoding utf8
set termoption enhanced
set style data points
set dummy x, y

print "Some examples how data fitting using nonlinear least squares fit can be done."
print ''

print "We fit a straight line to the data -- only as a demo without physical meaning."
l(x) = y0 + m*x
# Although gnuplot's internal fit backend can cope with it in this
# case, initializing variables to zero is a no, no, no!
#y0 = 0.0
#m = 0.0
y0 = 1.1
m = -0.1
print "  fit function:", GPFUN_l
print sprintf("  initial parameters: y0 = %g, m = %g", y0, m)
print "  fit command: fit l(x) 'lcdemo.dat' via y0, m\n"
set xlabel "Temperature T (°C)"
set ylabel "Density (g/cm^3)"
set key below
set title 'data set and initial parameters'
plot 'lcdemo.dat', l(x)

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fit l(x) 'lcdemo.dat' via y0, m
set title 'unweighted fit'
plot 'lcdemo.dat', l(x)

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# BM: The "errors" given in column 5 of lcdemo.dat are clearly not measurement errors
#     since the scatter in the data points is _much_ smaller.
set title 'data with experimental weigths'
plot 'lcdemo.dat' using 1:2:5 with errorbars
print ''
print "Now use the real single-measurement weights from column 5.  (Look at the file"
print "lcdemo.dat and compare the columns to see the difference.)"
print "Since these are weights we rescale the resulting parameter errors."
print "  fit settings: set fit errorscaling"
print "  fit command : fit l(x) 'lcdemo.dat' using 1:2:5 yerr via y0, m\n"

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set fit errorscaling
set fit covariancevariables
fit l(x) 'lcdemo.dat' using 1:2:5 yerr via y0, m
set title 'fit weighted by experimental weights'
plot 'lcdemo.dat' using 1:2:5 with errorbars, l(x)
set fit nocovariancevariables

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load 'density.fnc'
print ''
print "It's time now to try a more realistic model function:\n"
print "  ", GPFUN_density
print "  ", GPFUN_curve
print "  ", GPFUN_lowlin
print "  ", GPFUN_high
print "\ndensity(x) is a function which shall fit the whole temperature range using"
print "a ?: expression. It contains 6 model parameters which will all be varied. Now"
print "take the start parameters out of the file 'start.par' and plot the function."
print "  fit command: fit density(x) 'lcdemo.dat' using 1:2:5 yerror via 'start.par'\n"
load 'start.par'
set title 'initial parameters for realistic model function'
plot 'lcdemo.dat' using 1:2:5 with errorbars, density(x)

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fit density(x) 'lcdemo.dat' using 1:2:5 yerror via 'start.par'
set title 'fitted to realistic model function'
plot 'lcdemo.dat' using 1:2:5 with errorbars, density(x)

# FIXME: Nowadays this fit ends too quickly as to be useful any more...
#print ''
#print "Looks already rather nice?  We will now do the following:  Set the epsilon"
#print "fit3.dat lower so that we need more iteration steps to convergence."
#print "During fitting please hit Ctrl-C.  You will be asked:"
#print "  Stop, Continue, Execute"
#print "Try all options.  You may define a script using the FIT_SCRIPT environment"
#print "variable.  An example would be:"
#print "  FIT_SCRIPT=plot nonsense.dat"
#print "Normally you don't need to set FIT_SCRIPT since it defaults to 'replot'.  Please"
#print "note that FIT_SCRIPT cannot be set from inside gnuplot."
#print "  fit settings: set fit errorscaling limit 1e-10"
#print "  fit command : fit density(x) 'lcdemo.dat' using 1:2:5 via 'start.par'\n"
#pause -1 "Press enter to start the fit."

#set fit errorscaling limit 1e-10
#fit density(x) 'lcdemo.dat' using 1:2:5 yerror via 'start.par'
#set title 'fit with more iterations'
#plot 'lcdemo.dat' using 1:2:5 with errorbars, density(x)
#set fit limit 1e-5

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unset xlabel
unset ylabel


print "\n\nNow a brief demonstration of 3d fitting."
print "hemisphr.dat contains random points on a hemisphere of radius 1, but we let"
print "fit figure this out for us.  It takes many iterations, so we limit them to 50."
print "We also do not want intermediate results here."
print "  fit settings: set fit results maxiter 50"
#HBB: made this a lot harder: also fit the center of the sphere
#h(x,y) = sqrt(r*r - (x-x0)**2 - (y-y0)**2) + z0
#HBB 970522: distort the function, so it won't fit exactly:
h(x,y) = sqrt(r*r - (abs(x-x0))**2.2 - (abs(y-y0))**1.8) + z0
x0 = 0.1
y0 = 0.2
z0 = 0.3
r  = 0.5
set title 'the scattered points, and the initial parameter'
splot 'hemisphr.dat' using 1:2:3, h(x,y)
print "  fit function: " . GPFUN_h
print "  fit command : fit h(x,y) 'hemisphr.dat' using 1:2:3 via r, x0, y0, z0"

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set fit results maxiter 50 errorscaling
fit h(x,y) 'hemisphr.dat' using 1:2:3 via r, x0, y0, z0
set title 'the scattered points, fitted curve'
splot 'hemisphr.dat' using 1:2:3, h(x,y)

print ''
print "Notice, however, that this would converge much faster when fitted in a more"
print "appropriate co-ordinate system:"
print "  fit r 'hemisphr.dat' using 0:($1*$1+$2*$2+$3*$3) via r"
print "where we are fitting f(x)=r to the radii calculated as the data is read from"
print "the file.  No x value is required in this case."
print "(This is left as an exercise for the user)."
print ''

print "Another possibility is to prescale the variables (set fit prescale),"
print "which may improve convergence."
print "  fit settings: set fit maxiter 50 prescale"
print "  fit command : fit h(x,y) 'hemisphr.dat' using 1:2:3 via r, x0,y0,z0"
#pause -1 "Press enter to start the fit."

set fit maxiter 50 prescale
x0 = 0.1
y0 = 0.2
z0 = 0.3
r  = 0.5
#fit h(x,y) 'hemisphr.dat' using 1:2:3 via r, x0, y0, z0
#replot
set fit maxiter 0 noprescale brief

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print "\n\nNow an example on how to fit multi-branch functions."
print "The model consists of two branches, the first describing longitudinal sound"
print "velocity as function of propagation direction (upper data, from dataset 1),"
print "the second describing transverse sound velocity (lower data, from dataset 0)."
print ''
print "The model uses these data in order to fit elastic stiffnesses which occur"
print "differently in both branches."
load 'hexa.fnc'
load 'sound.par'
set title 'sound data, and model with initial parameters'
plot 'soundvel.dat', vlong(x), vtrans(x)
print "  fit function: " . GPFUN_f
print "                " . GPFUN_vlong
print "                " . GPFUN_vtrans
print "y will be the index of the dataset."
print "  fit command: fit f(x,y) 'soundvel.dat' using 1:-2:2 via 'sound.par'\n"

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fit f(x,y) 'soundvel.dat' using 1:-2:2 via 'sound.par'

set title 'pseudo-3d multi-branch fit to velocity data'
plot 'soundvel.dat', vlong(x), vtrans(x)
print ''
print "Look at the file 'hexa.fnc' to see how the branches are realized using the"
print "data index as input for a pseudo-3d fit."
print ''

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print "Next we only use every fifth data point for fitting by using the 'every'"
print "keyword. Note the faster fit and its result."
print "  fit command: fit f(x,y) 'soundvel.dat' every 5 using 1:-2:2 via 'sound.par'\n"
load 'sound.par'
plot 'soundvel.dat', vlong(x), vtrans(x)

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fit f(x,y) 'soundvel.dat' every 5 using 1:-2:2 via 'sound.par'
set title 'fitted only every 5th data point'
plot 'soundvel.dat', vlong(x), vtrans(x)
print ''
print "When you compare the results (see 'fit.log') you will note that the error of"
print "the fitted parameters have become larger, and the quality of the plot is only"
print "slightly affected."

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print ''
print "By marking some parameters as '# FIXED' in the parameter file, you fit only"
print "the others (c44 and c13 are fixed here)."
load 'sound2.par'
set title 'initial parameters'
plot 'soundvel.dat', vlong(x), vtrans(x)

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fit f(x,y) 'soundvel.dat' using 1:-2:2 via 'sound2.par'
set title 'fit with c44 and c13 fixed'
plot 'soundvel.dat', vlong(x), vtrans(x)
print ''
print "This has the same effect as specifying only the real free parameters by"
print "the 'via' syntax:"
print "  fit f(x) 'soundvel.dat' via c33, c11, phi0\n"

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print "\n\nHere comes an example of a rather complex function.\n"
set xlabel "Delta (degrees)"
set ylabel "Reflectivity"
load 'reflect.fnc'
#HBB 970522: Changed initial values to something sensible, i.e. 
#  something an experienced user of fit would actually use.
#  The fit limit is also reduced, to ensure a better fit.
eta = 1.2e-4
tc  = 1.8e-3

print "First a plot with all parameters set to initial values."
set title 'data and initial parameters'
plot 'moli3.dat' w e, R(x)
print "Now fit the model function to the data."
print "  fit settings: set fit limit 1e-10"
print "  fit function: " . GPFUN_R
print "                " . GPFUN_a
print "                " . GPFUN_W
print "  initial parameters:"
print "                " . sprintf("eta = %g", eta)
print "                " . sprintf("tc  = %g", tc)
print "  fit command : fit R(x) 'moli3.dat' u 1:2:3 zerror via eta, tc\n"

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set fit limit 1e-10 noerrorscaling
fit R(x) 'moli3.dat' u 1:2:3 zerror via eta, tc
set title 'fitted parameters'
replot

#HBB 970522: added comment on result of last fit.
print ''
print "Looking at the plot of the resulting fit curve, you can see that this function"
print "doesn't really fit this set of data points.  This would normally be a reason to"
print "to check for measurement problems not yet accounted for, and maybe even re-think"
print "the theoretic prediction in use."
print ''

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set xlabel 'x'
set ylabel 'y'
set zlabel 'z'
set ticslevel .2
set zrange [-3:3]
print ''
print 'Next we show a fit with three independent variables.'
print 'The file fit3.dat has four columns, with values of the three independent'
print 'variables x, y and t, and the''resulting value z.  The data lines are in four'
print 'sections, with t being constant within each section.  The sections are separated'
print 'by two blank lines, so we can select sections with "index" modifiers.  Here are'
print 'the data in the first section, where t = -3.'
print ''
print 'We will fit the function a0/(1 + a1*x**2 + a2*y**2) to these data.  Since at'
print 'this point we have two independent variables, our "using" spec has four entries,'
print 'representing x:y:z:s (where s is the estimated error in the z value).'

a0=1; a1=.1; a2=.1
f1(x,y)=a0/(1+a1*x**2+a2*y**2)
print "  fit function: ", GPFUN_f1
print "  fit command : fit f1(x,y) 'fit3.dat' index 0 using 1:2:4 via a0,a1,a2\n"
set title "data and initial parameters"
splot f1(x,y), 'fit3.dat' in 0 u 1:2:4

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set fit errorscaling
fit f1(x,y) 'fit3.dat' index 0 using 1:2:4 via a0,a1,a2
set title "fit to data with t = -3"
splot f1(x,y), 'fit3.dat' in 0 u 1:2:4

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print ''
print "Here is the last set of data where t = 3."
print "We fit the same function to this set."
print "  fit function: ", GPFUN_f1
print "  fit command : fit f1(x,y) 'fit3.dat' index 3 using 1:2:4 via a0,a1,a2\n"
set title "fit to data with t = +3, initial parameters"
splot f1(x,y), 'fit3.dat' in 3 u 1:2:4

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fit f1(x,y) 'fit3.dat' in 3 u 1:2:4 via a0,a1,a2
set title "fit to data with t = +3"
splot f1(x,y), 'fit3.dat' in 3 u 1:2:4

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print ''
print "We also have data for several intermediate values of t.  We will fit the"
print "function f(x,y,t)=a0*t/(1+a1*x**2+a2*y**2) to all the data."
f(x,y,t)=a0*t/(1+a1*x**2+a2*y**2)
print "  fit function: ", GPFUN_f
print "  fit command : fit f(x,y,t) 'fit3.dat' u 1:2:3:4 via a0,a1,a2\n"
set title "data for all indices t, initial parameters"
splot f(x,y,1), 'fit3.dat' u 1:2:4

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fit f(x,y,t) 'fit3.dat' u 1:2:3:4 via a0,a1,a2
print ''
print 'Here are all the data together.'
set title "fit to all data"
splot f(x,y,3),  'fit3.dat' in 3 u 1:2:4, \
      f(x,y,1),  'fit3.dat' in 2 u 1:2:4, \
      f(x,y,-1), 'fit3.dat' in 1 u 1:2:4, \
      f(x,y,-3), 'fit3.dat' in 0 u 1:2:4

print ''
print 'You can use ranges to rename variables and/or limit the data included in the'
print 'fit.  The first range corresponds to the first "using" entry, etc. For example,'
print 'we could have gotten the same fit like this:'
print '   fit [lon=*:*][lat=*:*][time=*:*]        \'
print '     a0*time/(1 + a1*lon**2 + a2*lat**2)   \'
print '     "fit3.dat" u 1:2:3:4 via a0,a1,a2'

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print "\n"
print "The fit command can handle errors in the independent variable, too."
print "The problem shown here is Pearson's data with York's weights.\n"
# Pearson's data and York's weights
$PearsonYork << EOD
#i x_i  W_xi  y_i  W_yi
 1 0.0 1000.0 5.9   1.0
 2 0.9 1000.0 5.4   1.8
 3 1.8  500.0 4.4   4.0
 4 2.6  800.0 4.6   8.0
 5 3.3  200.0 3.5  20.0
 6 4.4   80.0 3.7  20.0
 7 5.2   60.0 2.8  70.0
 8 6.1   20.0 2.8  70.0
 9 6.5    1.8 2.4 100.0
10 7.4    1.0 1.5 500.0
EOD

print "First draw the data with uncertainties and the initial function."
set xlabel "x"
set ylabel "y"
set key right
set yrange [0:8]
set xrange [-1:9]

# fit settings
set fit limit 1.0e-8 start_lambda 1 prescale noerrorscaling errorvariables

# fit function and initial values
f(x)  = a1 + a2*x
a1 = 5.
a2 = -0.5
fy(x) = a1y + a2y*x
a1y = 5.
a2y = -0.5

set title "Pearson's data and York's weights\noriginal data and the initial function"
plot \
  $PearsonYork using 2:4:(sqrt(1./$3)):(sqrt(1./$5)) with xyerrorbars lt -1 title 'data', \
  f(x) lw 2 lt 1 title 'initial function'
  
msg = "Press enter to fit the data using no error values"
print "\n". msg . "\n"

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# do the fit
fit f(x) $PearsonYork using 2:4 via a1, a2

set title "Pearson's data and York's weights\nfunction fit with no error terms"
plot \
  $PearsonYork using 2:4:(sqrt(1./$3)):(sqrt(1./$5)) lt -1 with xyerrorbars title 'data', \
  f(x) lw 2 lt 1 title 'fit using no error terms'

msg = "Press enter to fit the data using only the uncertainties of the y-values."
print "\n". msg . "\n"

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# do the fit
fit fy(x) $PearsonYork using 2:4:(sqrt(1./$5)) yerrors via a1y, a2y

set title "Pearson's data and York's weights\nfunction fit with yerror keyword"
plot \
  $PearsonYork using 2:4:(sqrt(1./$3)):(sqrt(1./$5)) lt -1 with xyerrorbars title 'data', \
  fy(x) lw 2 lt 2 title 'fit using only dy'

msg = "Press enter to fit the data using the uncertainties of the x and y values."
print "\n". msg . "\n"

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# fit function and initial values
fxy(x) = a1xy + a2xy*x
a1xy = 5
a2xy = -0.5
#a1xy = 5.48
#a2xy = -0.481

# do the fit
fit fxy(x) $PearsonYork using 2:4:(sqrt(1./$3)):(sqrt(1./$5)) xyerrors via a1xy, a2xy

set title "Pearson's data and York's weights\nfunction fit with xyerror keyword"
plot \
  $PearsonYork using 2:4:(sqrt(1./$3)):(sqrt(1./$5)) with xyerrorbars lt -1 title 'data', \
  f(x) lw 2 lt 1 title 'fit with no errors', \
  fy(x) lw 2 lt 2 title 'fit using only dy', \
  fxy(x) lw 2 lt 3 title 'fit using dx and dy'

print "\n\nSummary of the fit results:"
print         "                        a1           a2           a1_err       a2_err"
print         "------------------------------------------------------------------------"
print sprintf("initial values          %.3e  %.2e", 5, -0.5)
print sprintf("our result              %.3e  %.2e   %.3e  %.2e", a1xy, a2xy, a1xy_err, a2xy_err)
print sprintf("ROOT Minuit             %.3e  %.2e   %.3e  %.2e", 5.47990e+000, -4.80531e-001, 2.92561e-001, 5.76089e-002)
print         "------------------------------------------------------------------------"


print ''
print "You can have a look at all previous fit results by looking into the file"
print "'fit.log' (or whatever you defined the environment variable 'FIT_LOG' to)."
print "Remember that this file will always be appended to, so remove it from time"
print "to time."

print ''

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set encoding myencoding
# release datablock
undefine $PearsonYork