# [SciPy-User] Nonlinear fit to multiple data sets with a shared parameter, and three variable parameters.

Troels Emtekær Linnet tlinnet@gmail....
Thu Apr 4 09:41:26 CDT 2013

```I got it to work perfectly with a test script, and I include here the
test-script to help others.

This one is for scipy.optimize.leastsq

#-------------------------------------------------------------------------------
# Name:        Test for global fitting with scipy.optimize.leastsq
# Purpose:      To understand how to do global fitting
# Thanks to:   Jonathan, Josef, Charles, Matt Newville and especially
Jonathan Helmus
# Reference:
http://mail.scipy.org/pipermail/scipy-user/2013-April/034401.html
# Author:      Troels Emtekaer Linnet
#
# Created:     04-04-2013
# Licence:     Free
#-------------------------------------------------------------------------------
#
import pylab as pl
import numpy as np
import scipy.optimize
#
############# Fitting functions ################
def sim(x, p):
b, a, c  = p # Unpacking of shared variables should come first, then
the vary parameters
return a*np.exp(-b * x) + c
#
def err(p, x, y):
return sim(x, p) - y
#
def err_global(P_arr, x_arr, y_arr):
toterr = np.array([])
s = nr_shared_par # Number of shared parameters. Getting from the set
global parameter
v = nr_vary_par # Number of parameters that vary. Getting from the set
global parameter
for i in range(len(x_arr)):
par = np.array(P_arr[:s])
par = np.concatenate((par,P_arr[s+i*v:s+i*v+v]))
#print p
x = x_arr[i]
y = y_arr[i]
erri = err(par, x, y)
toterr = np.concatenate((toterr, erri))
#print len(toterr), type(toterr)
#
def unpack_global(dic, p_list):
s = nr_shared_par # Number of shared parameters. Getting from the set
global parameter
v = nr_vary_par # Number of parameters that vary. Getting from the set
global parameter
for i in range(len(p_list)):
p = p_list[i]
par_shared = dic['gfit']['par'][:s]
par_vary = dic['gfit']['par'][s+i*v:s+i*v+v]
par_all = np.concatenate((par_shared, par_vary))
dic[str(p)]['gfit']['par'] = par_all # Store paramaters
# Calc other parameters for the fit
Yfit = sim(dic[str(p)]['X'], par_all)
dic[str(p)]['gfit']['Yfit'] = Yfit
residual = Yfit - dic[str(p)]['Yran']
dic[str(p)]['gfit']['residual'] = residual
chisq = sum(residual**2)
dic[str(p)]['gfit']['chisq'] = chisq
NDF = len(residual)-len(par_all)
dic[str(p)]['gfit']['NDF'] = NDF
dic[str(p)]['gfit']['what_is_this_called'] = np.sqrt(chisq/NDF)
dic[str(p)]['gfit']['redchisq'] = chisq/NDF
return()
################ Extract parameters from output of global fit
###########################
def getleastsstat(result):
# http://mail.scipy.org/pipermail/scipy-user/2009-March/020516.html
dic = {}
dic['par'], dic['cov_x'], dic['infodict'], dic['mesg'], dic['ier'] =
result
dic['residual'] = dic['infodict']['fvec']
dic['chisq']=sum(dic['residual']**2) # calculate final chi square
dic['NDF']=len(dic['residual'])-len(dic['par'])
dic['what_is_this_called'] = np.sqrt(dic['chisq']/dic['NDF'])
dic['redchisq'] = dic['chisq']/dic['NDF']
return(dic)
################ Random peak data generator ###########################
def gendat(nr):
pd = {}
for i in range(1,nr+1):
b = 0.15
a = np.random.random_integers(1, 80)/10.
c = np.random.random_integers(1, 80)/100.
pd[str(i)] = [b,a,c]
return(pd)
#############################################################################
## Start
#############################################################################
limit = 0.6   # Limit set for chisq test, to select peaks
# Global fitting
global nr_shared_par ; nr_shared_par = 1 # Number of shared parameters
global nr_vary_par ; nr_vary_par = 2 # Number of parameters that vary
#############################################################################
# set up the data
data_x = np.linspace(0, 20, 50)
pd = {} # Parameter dictionary, the "true" values of the data sets
pd['1'] = [0.15, 2.5, 0.5]       # parameters for the first trajectory
pd['2'] = [0.15, 4.2, 0.2]       # parameters for the second trajectory,
same b
pd['3'] = [0.15, 1.2, 0.3]       # parameters for the third trajectory,
same b
pd = gendat(9)  # You can generate a large number of peaks to test
#
#Start making a dictionary, which holds all data
dic = {}; dic['peaks']=range(1,len(pd)+1)
for p in dic['peaks']:
dic['%s'%p] = {}
dic[str(p)]['X'] = data_x
dic[str(p)]['Y'] = sim(data_x, pd[str(p)])
dic[str(p)]['Yran'] = dic[str(p)]['Y'] +
np.random.normal(size=len(dic[str(p)]['Y']), scale=0.12)
dic[str(p)]['fit'] = {}  # Make space for future fit results
dic[str(p)]['gfit'] = {}  # Male space for future global fit results
#print "keys for start dictionary:", dic.keys()
#
# independent fitting of the trajectories
for p in dic['peaks']:
pguess = [2.0, 2.0, 2.0]
res = scipy.optimize.leastsq(err, pguess, args=(dic[str(p)]['X'],
dic[str(p)]['Yran']), full_output=1)
res_dic = getleastsstat(res)
dic[str(p)]['fit'].update(res_dic)
Yfit = sim(dic[str(p)]['X'], dic[str(p)]['fit']['par'])
#Yfit2 = dic[str(p)]['Yran']+res_dic['residual']
#print sum(Yfit-Yfit2), "Test for difference in two ways to get the
fitted Y-values "
dic[str(p)]['fit']['Yfit'] = Yfit
print "Best fit parameter for peak %s"%p, dic[str(p)]['fit']['par'],
print "Compare to real paramaters", pd[str(p)]
#
# Make a selection flag, based on some test. Now a chisq value, but could
be a Ftest between a simple and advanced model fit.
sel_p = []
for p in dic['peaks']:
chisq = dic[str(p)]['fit']['chisq']
if chisq < limit:
dic[str(p)]['Pval'] = 1.0
#print "Peak %s passed test"%p
sel_p.append(p)
else:
dic[str(p)]['Pval'] = False
#print sel_p
#
# Global fitting
# Pick up x,y-values and parameters that passed the test
X_arr = []
Y_arr = []
P_arr = [1.0] # Pack guess for shared values in first.
dic['gfit'] = {} # Make room for globat fit result
for p in sel_p:
par = dic[str(p)]['fit']['par']
X_arr.append(dic[str(p)]['X'])
Y_arr.append(dic[str(p)]['Yran'])
P_arr.append(par[1])
P_arr.append(par[2])
#print P_arr
res = scipy.optimize.leastsq(err_global, P_arr, args=(X_arr, Y_arr),
full_output=1)  # Do the fitting
res_dic = getleastsstat(res) # Extract parameters from result
dic['gfit'].update(res_dic) # Update the data dictionary from the returned
parameter
unpack_global(dic, sel_p) # Unpack the paramerts into the selected peaks
#
# Check result
for p in sel_p:
print p, "Single fit%s"%dic[str(p)]['fit']['par'], "Global
fit%s"%dic[str(p)]['gfit']['par'] , "Real par%s"%pd[str(p)]
#print p, "Single fit%s"%(dic[str(p)]['fit']['par']-pd[str(p)]),
"Global fit%s"%(dic[str(p)]['gfit']['par']-pd[str(p)])
#
# Start plotting
fig = pl.figure('Peak')
for i in range(len(sel_p)):
p = sel_p[i]
# Create figure
X = dic[str(p)]['X']
Y = dic[str(p)]['Y']
Ymeas = dic[str(p)]['Yran']
Yfit = dic[str(p)]['fit']['Yfit']
Yfit_global = dic[str(p)]['gfit']['Yfit']
rpar = pd[str(p)]
fpar = dic[str(p)]['fit']['par']
gpar = dic[str(p)]['gfit']['par']
fchisq = dic[str(p)]['fit']['chisq']
gchisq = dic[str(p)]['gfit']['chisq']
# plot
ax.plot(X,Y,".-",label='real. Peak: %s'%p)
ax.plot(X,Ymeas,'o',label='Measured (real+noise)')
ax.plot(X,Yfit,'.-',label='leastsq fit. chisq:%3.3f'%fchisq)
ax.plot(X,Yfit_global,'.-',label='global fit. chisq:%3.3f'%gchisq)
# annotate
ax.annotate('p%s. real    par: %1.3f %1.3f %1.3f'%(p,
rpar[0],rpar[1],rpar[2]), xy=(1,1), xycoords='data', xytext=(0.4, 0.8),
textcoords='axes fraction')
ax.annotate('p%s. single  par: %1.3f %1.3f %1.3f'%(p,
fpar[0],fpar[1],fpar[2]), xy=(1,1), xycoords='data', xytext=(0.4, 0.6),
textcoords='axes fraction')
ax.annotate('p%s. global  par: %1.3f %1.3f %1.3f'%(p,
gpar[0],gpar[1],gpar[2]), xy=(1,1), xycoords='data', xytext=(0.4, 0.4),
textcoords='axes fraction')
# set title and axis name
#ax.set_title('Fitting for peak %s'%p)
ax.set_ylabel('Decay')
# Put legend to the right
box = ax.get_position()
ax.set_position([box.x0, box.y0, box.width * 0.8, box.height]) # Shink
current axis by 20%
ax.legend(loc='center left', bbox_to_anchor=(1, 0.5),prop={'size':8}) #
Put a legend to the right of the current axis
ax.grid('on')
ax.set_xlabel('Time')
#
pl.show()
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