Uncertainty analysis

# This file is part of ILTpy examples.
# Author : Dr. Davis Thomas Daniel
# Last updated : 25.08.2025

This example shows steps to generate uncertainty estimates for distributions. As an example, an EPR inversion recovery dataset is used. The same approach and functions can also be used for any multi-dimensional data using ILTpy.

The method iltstats of IltData class is used.

• Results after calling iltstats are stored in IltData.statistics

• IltData.statistics is a dictionary containing the following keys:

  • g_list : A list with indiviudal distributions from n inversions.

  • fit_list : A list with indiviudal fits from n inversions.

  • residuals_list : A list with indiviudal residuals from n inversions.

  • noise_samples : A list with noise samples used to generate n number of data samples.

  • data_samples : A list with n number of data samples.

  • g_mean : Mean of n distrbutions.

  • fit_mean : Mean of n fits.

  • residuals_mean : Mean of n residuals.

  • g_std : Standard deviation of n distributions.

  • fit_std : Standard deviation of n fits.

  • residuals_std : Standard deviation of n residuals.

  • g_conf_interval : A list containing g_mean-std_mean, g_mean+std_mean.

  • fit_conf_interval : A list containing fit_mean-fit_std, fit_mean+fit_std.

Imports

# import iltpy
import iltpy as ilt
print(f"ILTpy version: {ilt.__version__}")

# other libraries for handling data, plotting
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec
plt.style.use('../../../examples/latex_style.mplstyle')

ILTpy version: 1.0.0

Loading data

# Load data
data_EPR = np.loadtxt('../../../examples/EPR/inversion_recovery/data.txt')
t_EPR = np.loadtxt('../../../examples/EPR/inversion_recovery/dim1.txt')

coalIR = ilt.iltload(data=data_EPR,t=t_EPR)

Data preparation

# Set noise variance to 1.
## estimate noise level using some points at the end of inversion recovery trace
noise_lvl  = np.std(coalIR.data[900:])

# Scale the data with the noise level
coalIR.data = coalIR.data/noise_lvl

Initial inversion

# Initialize the IltData object
# Tau generated internally
coalIR.init(kernel=ilt.Exponential())

# Invert
coalIR.invert()

/Users/davisthomasdaniel/sciebo/JuGit/iltpy_iet1/iltpy/iltpy/input/parameters.py:697: UserWarning: tau[0] was not specified, proceeding with ILTpy-generated tau. Note that ILTpy-generated tau is only valid in case of NMR/EPR relaxation data.
  warnings.warn(tau_warn)

Starting iterations ...

  0%|          | 0/100 [00:00<?, ?it/s]100%|██████████| 100/100 [00:00<00:00, 686.74it/s]

Done.

Uncertainity analysis

• n controls the number of data samples which will be generated and n_jobs controls the number of parallel jobs to create.

coalIR.iltstats(n=200,n_jobs=-1)

[Parallel(n_jobs=-1)]: Using backend LokyBackend with 8 concurrent workers.
[Parallel(n_jobs=-1)]: Done  34 tasks      | elapsed:    2.1s
[Parallel(n_jobs=-1)]: Done 184 tasks      | elapsed:    6.2s
[Parallel(n_jobs=-1)]: Done 200 out of 200 | elapsed:    6.6s finished

Plot results

fig,ax = plt.subplots(figsize=(3,2.5))
ax.semilogx(coalIR.tau[0].flatten(),-coalIR.statistics['g_mean'],color='k',linewidth=0.5,label='Mean')
ax.fill_between(coalIR.tau[0].flatten(),*[-i for i in coalIR.statistics['g_conf_interval']],color='grey',alpha=0.4,label='Std. Dev.')
ax.legend()
plt.xlim(1e4,2e6)
_=plt.ylim(-1,22)