I'm developing a code for fitting a data with a model which is convolution of two functions (Gaussian with multi exponential decay exp(Ax)+exp(Bx)+...). basically the fitting with only Gaussian and/or Gaussian modified https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution is working perfectly fine in Lmfit but using the builtin convolution (i.e if np.convolve of two functions is used Lmfit doesn't work.
I have tried many examples on internet, so far I realized that my functions returns inf or nan values and also data is not equally spaced for being used in convolution. I found a detour for the issue by using the mathematical expression of convolution and by using scipy.optimize.curve_fit .But it is a very clumsy and time consuming, I would like to find a way to making it more sophisticated and general by using a convolution of two functions and using lmfit where I can control the parameters a lot easier.
The data set is also included in comments as your reference.
w=0.1 # is constant
def CONVSum(x,w,*p):
n=np.int(len(p)/3)
A=p[:n]
B=p[n:2*n]
C=p[2*n:3*n]
# =======================================================================
# below formula is derived as mathematical expression of convoluted multi exponential components with a gaussian distribution based on the instruction given in http://www.np.ph.bham.ac.uk/research_resources/programs/halflife/gauss_exp_conv.pdf
# ======================================================================
fnct=sum(np.float64([A[i]*np.exp(-B[i]*((x-C[i])-(0.5*np.square(w)*B[i])))*(1+scipy.special.erf(((x-C[i])-(np.square(w)*B[i]))/(np.sqrt(2)*w))) for i in range(n)]))
fnct[np.isnan(fnct)]=0
fnct[fnct<1e-12]=0
return fnct
N=4 #number of exponential functions to be fitted
params = np.linspace(1, 0.0001, N*3); #parameters for a multiple exponential
popt,pcov = curve_fit(CONVSum,x,y,p0=params,
bounds=((0,0,0,0,-np.inf,-np.inf,-np.inf,-np.inf,-3,-3,-3,-3),
(1,1,1,1, np.inf, np.inf, np.inf, np.inf, 3, 3, 3, 3)),
maxfev = 1000000)
Any help or hint regarding the fitting with convolution of Gaussian and multiple exponential decay is highly appreciated, I prefer using lmfit since I can identify parameters very nicely and also to relate them to each other.
Ideally I want to fit my data with the parameters where some of them are shared among the data sets, some are delayed (+off_set).
Here is the equvalent of the cure fitting code given in question. I managed to creat this by using very great instruction and infromation in here and here. But still it needs to be developed.
# =============================================================================
# below formula is drived as mathematical expresion of convoluted multi exponential components with a gausian distribution based on the instruction given in http://www.np.ph.bham.ac.uk/research_resources/programs/halflife/gauss_exp_conv.pdf
# =============================================================================
def CONVSum(x,params):
fnct=sum(
np.float64([
(params['amp%s_%s'%(n,i)].value)*np.exp(-(params['dec%s_%s'%(n,i)].value)*((x-(params['cen%s_%s'%(n,i)].value))-
(0.5*np.square((params['sig%s_%s'%(n,i)].value))*(params['dec%s_%s'%(n,i)].value))))*
(1+scipy.special.erf(((x-(params['cen%s_%s'%(n,i)].value))-(np.square((params['sig%s_%s'%(n,i)].value))*
(params['dec%s_%s'%(n,i)].value)))/(np.sqrt(2)*(params['sig%s_%s'%(n,i)].value)))) for n in range(N) for i in wav
])
)
fnct=fnct/fnct.max()
return fnct
# =============================================================================
# this global fit were adapted from https://stackoverflow.com/questions/20339234/python-and-lmfit-how-to-fit-multiple-datasets-with-shared-parameters/20341726#20341726
# it is of very important thet we can identify the shared parameteres for datasets
# =============================================================================
def objective(params, x, data):
""" calculate total residual for fits to several data sets"""
ndata = data.shape[0]
resid = 0.0*data[:]
# make residual per data set
resid = data- CONVSum(x,params)
# now flatten this to a 1D array, as minimize() needs
return resid.flatten()
# selec datasets
x = df[949].index
data =df[949].values
# create required sets of parameters, one per data set
N=4 #number of exponential decays
wav=[949] #the desired data to be fitted
fit_params = Parameters()
for i in wav:
for n in range(N):
fit_params.add( 'amp%s_%s'%(n,i), value=1, min=0.0, max=1)
fit_params.add( 'dec%s_%s'%(n,i), value=0.5, min=-1e10, max=1e10)
fit_params.add( 'cen%s_%s'%(n,i), value=0.1, min=-3.0, max=1000)
fit_params.add( 'sig%s_%s'%(n,i), value=0.1, min=0.05, max=0.5)
# now we constrain some values to have the same value
# for example assigning sig_2, sig_3, .. sig_5 to be equal to sig_1
for i in wav:
for n in (1,2,3):
print(n,i)
fit_params['sig%s_%s'%(n,i)].expr='sig0_949'
fit_params['cen%s_%s'%(n,i)].expr='cen0_949'
# it will run the global fit to all the data sets
result = minimize(objective, fit_params, args=(x,data))
report_fit(result.params)
# plot the data sets and fits
plt.close('all')
plt.figure()
for i in wav:
y_fit = CONVSum(x,result.params)
plt.plot(x, data, 'o-', x, y_fit, '-')
plt.xscale('symlog')
plt.show()
fitted data with convolution of multi exponential and gausian
unfortunately the fitted results are not very satisfying, I am still looking for some advice to improve this.
