Brownian Translational diffusion ∗ Resolution with bumps
Introduction
The objective of this notebook is to show how to use the Brownian Translational diffusion model to perform some fits using bumps .
The reference data were generated data corresponding to a Brownian Translational diffusion model with self-diffusion coefficient = 0.145 Å\(^2\times\)meV.
The model was convoluted with a Gaussian resolution function of FWHM = 0.1 meV, centered randomly in the range [-0.01, +0.01] meV.
Finally the data are sampled randomly from a Poisson distribution.
The data do not have a background.
Physical units
For information about unit conversion, please refer to the jupyter notebook called Convert_units.ipynb in the tools folder.
The dictionary of units defined in the cell below specify the units of the refined parameters adapted to the convention used in the experimental datafile.
[1]:
# Units of parameters for selected QENS model and experimental data
dict_physical_units = {'D': "meV.Angstrom^2",
'scale': "unit_of_signal.meV",
'center': "meV"}
Import libraries
[2]:
import ipywidgets
import numpy as np
import h5py
import QENSmodels
from scipy.integrate import simps
import bumps.names as bmp
from bumps import fitters
from bumps.formatnum import format_uncertainty_pm
import matplotlib.pyplot as plt
%matplotlib widget
Matplotlib is building the font cache; this may take a moment.
Setting of fitting
Load reference data
[3]:
path_to_data = './data/'
# Read the sample
with h5py.File(path_to_data + 'BrownianDiff_Sample.hdf', 'r') as f:
hw = f['entry1']['data1']['X'][:]
q = f['entry1']['data1']['Y'][:]
unit_w=f['entry1']['data1']['X'].attrs['long_name']
unit_q=f['entry1']['data1']['Y'].attrs['long_name']
sqw = np.transpose(f['entry1']['data1']['DATA'][:])
err = np.transpose(f['entry1']['data1']['errors'][:])
# Read resolution
with h5py.File(path_to_data + 'BrownianDiff_Resol.hdf', 'r') as f:
res = np.transpose(f['entry1']['data1']['DATA'][:])
# Force resolution function to have unit area
for i in range(len(q)):
area = simps(res[:,i], hw)
res[:,i] /= area
[4]:
fig, ax = plt.subplots(nrows=2, sharex=True)
for i in range(len(q)):
ax[0].semilogy(hw, sqw[:,i], label=f"q={q[i]:.1f}")
ax[1].semilogy(hw, res[:,i], label=f"q={q[i]:.1f}")
ax[1].legend(bbox_to_anchor=(1.1,1.3), loc='upper right', shadow=True)
ax[0].set_title('Signal')
ax[0].grid()
ax[1].set_title('Resolution')
ax[1].set_xlabel(f"$\hbar \omega ({dict_physical_units['center']})$")
ax[1].grid()
Display units of input data
Just for information in order to determine if a conversion of units is required before using the QENSmodels
[5]:
print(f"The names and units of `w` (`x`axis) and `q` are: {unit_w[0].decode()} and {unit_q[0].decode()}, respectively.")
The names and units of `w` (`x`axis) and `q` are: X and Y, respectively.
Create fitting model
[6]:
# Fitting model
def model_convol(x, q, scale=1, center=0, D=1, resolution=None):
model = QENSmodels.sqwBrownianTranslationalDiffusion(x, q, scale, center, D)
return np.convolve(model, resolution/resolution.sum(), mode='same')
# Fit
model_all_qs = []
for i in range(len(q)):
# Bumps fitting model
model_q = bmp.Curve(
model_convol,
hw, sqw[:,i], err[:,i],
name=f'q_{q[i]:.2f}',
q=q[i],
scale=1000,
center=0.0,
D=0.1,
resolution=res[:, i]
)
model_q.scale.range(0, 1e5)
model_q.center.range(-0.1, 0.1)
model_q.D.range(1e-12, 1)
# Q-independent parameters
if i == 0:
D_q = model_q.D
else:
model_q.D = D_q
model_all_qs.append(model_q)
problem = bmp.FitProblem(model_all_qs)
Display initial configuration: experimental data, fitting model with initial guesses
[7]:
slider = ipywidgets.IntSlider(value=0, min=0, max=len(q)-1, continuous_update=False)
output = ipywidgets.Output()
def fig_q(model, ax, q_index=0):
"""
Plot of experimental data, fitting model and residual for a selected q value
Parameters
----------
model: list of bumps.curve.Curves for all q
ax: matplotlib.axes to be updated when changing the ipywidgets
q_index: int
index of q to be plotted
"""
model = model[q_index]
ax[0].errorbar(model.x,
model.y,
yerr=model.dy,
label='experimental data',
color='C0')
ax[0].plot(model.x,
model.theory(),
label='theory (model)',
color='C1')
ax[0].set_title(f'Model {model.name} - $\chi^2$={problem.chisq_str()}')
ax[0].legend()
ax[1].plot(model.x, model.residuals(), marker='o', linewidth=0, markersize=3, color='C0')
with output:
fig, ax = plt.subplots(nrows=2, ncols=1, sharex=True)
ax[0].grid(); ax[1].grid()
ax[1].set_ylabel('Residual')
ax[1].set_xlabel(f"$\hbar \omega ({dict_physical_units['center']})$")
fig_q(model_all_qs, ax, 0)
def update_profile(change):
"""
Update plots for a new q-value
"""
with output:
ax[0].clear(); ax[1].lines.clear()
ax[0].grid()
fig_q(model_all_qs, ax, change['new'])
slider.observe(update_profile, names="value")
slider_label = ipywidgets.Label("q value to display")
slider_comp = ipywidgets.HBox([slider_label, slider])
ipywidgets.VBox([slider_comp, output])
[7]:
[8]:
problem.summarize().splitlines()
[8]:
[' D |......... 0.1 in (1e-12,1)',
' center ....|..... 0 in (-0.1,0.1)',
' scale |......... 1000 in (0,100000)',
' center ....|..... 0 in (-0.1,0.1)',
' scale |......... 1000 in (0,100000)',
' center ....|..... 0 in (-0.1,0.1)',
' scale |......... 1000 in (0,100000)',
' center ....|..... 0 in (-0.1,0.1)',
' scale |......... 1000 in (0,100000)',
' center ....|..... 0 in (-0.1,0.1)',
' scale |......... 1000 in (0,100000)',
' center ....|..... 0 in (-0.1,0.1)',
' scale |......... 1000 in (0,100000)',
' center ....|..... 0 in (-0.1,0.1)',
' scale |......... 1000 in (0,100000)',
' center ....|..... 0 in (-0.1,0.1)',
' scale |......... 1000 in (0,100000)',
' center ....|..... 0 in (-0.1,0.1)',
' scale |......... 1000 in (0,100000)',
' center ....|..... 0 in (-0.1,0.1)',
' scale |......... 1000 in (0,100000)']
Choice of minimizer for bumps
[9]:
options_dict = {}
for item in fitters.__dict__.keys():
if item.endswith('Fit') and fitters.__dict__[item].id in fitters.FIT_AVAILABLE_IDS:
options_dict[fitters.__dict__[item].name] = fitters.__dict__[item].id
w_choice_minimizer = ipywidgets.Dropdown(
options=list(options_dict.keys()),
value='Levenberg-Marquardt',
description='Minimizer:',
layout=ipywidgets.Layout(height='40px'))
w_choice_minimizer
[9]:
Setting for running bumps
[10]:
steps_fitting = ipywidgets.IntText(
value=100,
step=100,
description='Number of steps when fitting',
style={'description_width': 'initial'})
steps_fitting
[10]:
Running the fit
Run the fit using the minimizer defined above with a number of steps also specified above
[11]:
# Preview of the settings
print('Initial chisq', problem.chisq_str())
Initial chisq 429.0449(59)
[12]:
def settings_selected_optimizer(chosen_minimizer):
"""
List the settings available for the selected optimizer
This list can be used as arguments for the `fit` function
"""
assert type(chosen_minimizer) == ipywidgets.widgets.widget_selection.Dropdown
for item in fitters.__dict__.keys():
if item.endswith('Fit') and \
fitters.__dict__[item].id == options_dict[chosen_minimizer.value]:
return [elt[0] for elt in fitters.__dict__[item].settings]
[13]:
print((f"With {w_choice_minimizer.value} optimizer, "
f"you can use {settings_selected_optimizer(w_choice_minimizer)} as arguments of `fit`"))
With Levenberg-Marquardt optimizer, you can use ['steps', 'ftol', 'xtol'] as arguments of `fit`
[14]:
result = fitters.fit(
problem,
starts=10,
keep_best=True,
method=options_dict[w_choice_minimizer.value],
steps=int(steps_fitting.value)
)
Showing the results
[15]:
problem.summarize().splitlines()
[15]:
[' D .|........ 0.144129 in (1e-12,1)',
' center ....|..... -0.000474452 in (-0.1,0.1)',
' scale |......... 116.475 in (0,100000)',
' center .....|.... 0.000197478 in (-0.1,0.1)',
' scale |......... 155.008 in (0,100000)',
' center ....|..... -0.000933528 in (-0.1,0.1)',
' scale |......... 229.015 in (0,100000)',
' center ....|..... -0.000301288 in (-0.1,0.1)',
' scale |......... 332.64 in (0,100000)',
' center .....|.... 0.00017421 in (-0.1,0.1)',
' scale |......... 484.245 in (0,100000)',
' center .....|.... 0.000605385 in (-0.1,0.1)',
' scale |......... 676.618 in (0,100000)',
' center ....|..... -0.00211537 in (-0.1,0.1)',
' scale |......... 904.544 in (0,100000)',
' center .....|.... 0.000697686 in (-0.1,0.1)',
' scale |......... 1176.02 in (0,100000)',
' center .....|.... 0.000651925 in (-0.1,0.1)',
' scale |......... 1475.37 in (0,100000)',
' center ....|..... -0.000256062 in (-0.1,0.1)',
' scale |......... 1823.9 in (0,100000)']
[16]:
# Other method to display the results of the fit (chi**2 and parameters' values)
print("final chisq", problem.chisq_str())
for k, v, dv in zip(problem.labels(), result.x, result.dx):
if k in dict_physical_units.keys():
print(k, ":", format_uncertainty_pm(v, dv), dict_physical_units[k])
else:
print(k, ":", format_uncertainty_pm(v, dv))
final chisq 0.9995(59)
D : 0.14413 +/- 0.00029 meV.Angstrom^2
center : -0.47e-3 +/- 0.56e-3 meV
scale : 116.5 +/- 1.1 unit_of_signal.meV
center : 0.20e-3 +/- 0.52e-3 meV
scale : 155.0 +/- 1.3 unit_of_signal.meV
center : -0.93e-3 +/- 0.66e-3 meV
scale : 229.0 +/- 1.5 unit_of_signal.meV
center : -0.30e-3 +/- 0.82e-3 meV
scale : 332.6 +/- 1.9 unit_of_signal.meV
center : 0.2e-3 +/- 1.2e-3 meV
scale : 484.2 +/- 2.3 unit_of_signal.meV
center : 0.6e-3 +/- 1.2e-3 meV
scale : 676.6 +/- 2.7 unit_of_signal.meV
center : -2.1e-3 +/- 1.3e-3 meV
scale : 904.5 +/- 3.2 unit_of_signal.meV
center : 0.7e-3 +/- 1.5e-3 meV
scale : 1176.0 +/- 3.7 unit_of_signal.meV
center : 0.7e-3 +/- 1.3e-3 meV
scale : 1475.4 +/- 4.2 unit_of_signal.meV
center : -0.3e-3 +/- 2.2e-3 meV
scale : 1823.9 +/- 4.8 unit_of_signal.meV
Display final configuration: experimental data, fitting model with output of fitting for the refined parameters
[17]:
slider1 = ipywidgets.IntSlider(value=0, min=0, max=len(q)-1, continuous_update=False)
output1 = ipywidgets.Output()
with output1:
fig1, ax1 = plt.subplots(nrows=2, ncols=1, sharex=True)
ax1[0].grid(); ax1[1].grid()
ax1[1].set_ylabel('Residual')
ax1[1].set_xlabel(f"$\hbar \omega ({dict_physical_units['center']})$")
fig_q(model_all_qs, ax1, 0)
def update_profile1(change):
"""
Update plots for a new q-value
"""
with output1:
ax1[0].clear(); ax1[1].lines.clear()
ax1[0].grid()
fig_q(model_all_qs, ax1, change['new'])
slider1.observe(update_profile1, names="value")
slider1_label = ipywidgets.Label("q value to display")
slider1_comp = ipywidgets.HBox([slider1_label, slider1])
ipywidgets.VBox([slider1_comp, output1])
[17]:
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