Note
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Fit Multiple Data Sets¶
Fitting multiple (simulated) Gaussian data sets simultaneously.
All minimizers require the residual array to be one-dimensional. Therefore, in
the objective
function we need to flatten
the array before returning it.
TODO: this could/should be using the Model interface / built-in models!
import matplotlib.pyplot as plt
import numpy as np
from lmfit import Parameters, minimize, report_fit
def gauss(x, amp, cen, sigma):
"""Gaussian lineshape."""
return amp * np.exp(-(x-cen)**2 / (2.*sigma**2))
def gauss_dataset(params, i, x):
"""Calculate Gaussian lineshape from parameters for data set."""
amp = params[f'amp_{i+1}']
cen = params[f'cen_{i+1}']
sig = params[f'sig_{i+1}']
return gauss(x, amp, cen, sig)
def objective(params, x, data):
"""Calculate total residual for fits of Gaussians to several data sets."""
ndata, _ = data.shape
resid = 0.0*data[:]
# make residual per data set
for i in range(ndata):
resid[i, :] = data[i, :] - gauss_dataset(params, i, x)
# now flatten this to a 1D array, as minimize() needs
return resid.flatten()
Create five simulated Gaussian data sets
np.random.seed(2021)
x = np.linspace(-1, 2, 151)
data = []
for _ in np.arange(5):
amp = 0.60 + 9.50*np.random.rand()
cen = -0.20 + 1.20*np.random.rand()
sig = 0.25 + 0.03*np.random.rand()
dat = gauss(x, amp, cen, sig) + np.random.normal(size=x.size, scale=0.1)
data.append(dat)
data = np.array(data)
Create five sets of fitting parameters, one per data set
fit_params = Parameters()
for iy, y in enumerate(data):
fit_params.add(f'amp_{iy+1}', value=0.5, min=0.0, max=200)
fit_params.add(f'cen_{iy+1}', value=0.4, min=-2.0, max=2.0)
fit_params.add(f'sig_{iy+1}', value=0.3, min=0.01, max=3.0)
Constrain the values of sigma to be the same for all peaks by assigning sig_2, …, sig_5 to be equal to sig_1.
for iy in (2, 3, 4, 5):
fit_params[f'sig_{iy}'].expr = 'sig_1'
Run the global fit and show the fitting result
out = minimize(objective, fit_params, args=(x, data))
report_fit(out.params)
Out:
[[Variables]]
amp_1: 6.32742010 +/- 0.02279089 (0.36%) (init = 0.5)
cen_1: 0.68049261 +/- 0.00126458 (0.19%) (init = 0.4)
sig_1: 0.25755570 +/- 4.9426e-04 (0.19%) (init = 0.3)
amp_2: 6.98604753 +/- 0.02296733 (0.33%) (init = 0.5)
cen_2: 0.50433700 +/- 0.00114536 (0.23%) (init = 0.4)
sig_2: 0.25755570 +/- 4.9426e-04 (0.19%) == 'sig_1'
amp_3: 7.11643510 +/- 0.02300415 (0.32%) (init = 0.5)
cen_3: -0.08260274 +/- 0.00112437 (1.36%) (init = 0.4)
sig_3: 0.25755570 +/- 4.9426e-04 (0.19%) == 'sig_1'
amp_4: 6.10197422 +/- 0.02273421 (0.37%) (init = 0.5)
cen_4: 0.07386098 +/- 0.00131130 (1.78%) (init = 0.4)
sig_4: 0.25755570 +/- 4.9426e-04 (0.19%) == 'sig_1'
amp_5: 9.23910555 +/- 0.02368872 (0.26%) (init = 0.5)
cen_5: 0.34443083 +/- 8.6605e-04 (0.25%) (init = 0.4)
sig_5: 0.25755570 +/- 4.9426e-04 (0.19%) == 'sig_1'
[[Correlations]] (unreported correlations are < 0.100)
C(sig_1, amp_5) = -0.3742
C(sig_1, amp_3) = -0.2968
C(sig_1, amp_2) = -0.2919
C(amp_1, sig_1) = -0.2664
C(sig_1, amp_4) = -0.2575
C(amp_3, amp_5) = +0.1111
C(amp_2, amp_5) = +0.1092
Plot the data sets and fits
plt.figure()
for i in range(5):
y_fit = gauss_dataset(out.params, i, x)
plt.plot(x, data[i, :], 'o', x, y_fit, '-')

Total running time of the script: ( 0 minutes 0.294 seconds)