Fitting with time delay scan¶
Objective¶
Fitting with exponential decay model
Save and Load fitting result
Calculates species associated coefficent from fitting result
Evaluates F-test based confidence interval
In this example, we only deal with gaussian irf
# import needed module
import numpy as np
import matplotlib.pyplot as plt
import TRXASprefitpack
from TRXASprefitpack import solve_seq_model, rate_eq_conv
plt.rcParams["figure.figsize"] = (12,9)
Version information¶
print(TRXASprefitpack.__version__)
0.7.0
Fitting with exponential decay model¶
# Generates fake experiment data
# Model: 1 -> 2 -> GS
# lifetime tau1: 500 ps, tau2: 10 ns
# fwhm paramter of gaussian IRF: 100 ps
tau_1 = 500
tau_2 = 10000
fwhm = 100
# initial condition
y0 = np.array([1, 0, 0])
# set time range (mixed step)
t_seq1 = np.arange(-2500, -500, 100)
t_seq2 = np.arange(-500, 1500, 50)
t_seq3 = np.arange(1500, 5000, 250)
t_seq4 = np.arange(5000, 50000, 2500)
t_seq = np.hstack((t_seq1, t_seq2, t_seq3, t_seq4))
eigval_seq, V_seq, c_seq = solve_seq_model(np.array([tau_1, tau_2]), y0)
# Now generates measured transient signal
# Last element is ground state
abs_1 = [1, 1, 0]
abs_2 = [0.5, 0.8, 0]
abs_3 = [-0.5, 0.7, 0]
abs_4 = [0.6, 0.3, 0]
t0 = np.random.normal(0, fwhm, 4) # perturb time zero of each scan
# generate measured data
y_obs_1 = rate_eq_conv(t_seq-t0[0], fwhm, abs_1, eigval_seq, V_seq, c_seq, irf='g')
y_obs_2 = rate_eq_conv(t_seq-t0[1], fwhm, abs_2, eigval_seq, V_seq, c_seq, irf='g')
y_obs_3 = rate_eq_conv(t_seq-t0[2], fwhm, abs_3, eigval_seq, V_seq, c_seq, irf='g')
y_obs_4 = rate_eq_conv(t_seq-t0[3], fwhm, abs_4, eigval_seq, V_seq, c_seq, irf='g')
# generate random noise with (S/N = 20)
# Define noise level (S/N=20) w.r.t peak
eps_obs_1 = np.max(np.abs(y_obs_1))/20*np.ones_like(y_obs_1)
eps_obs_2 = np.max(np.abs(y_obs_2))/20*np.ones_like(y_obs_2)
eps_obs_3 = np.max(np.abs(y_obs_3))/20*np.ones_like(y_obs_3)
eps_obs_4 = np.max(np.abs(y_obs_4))/20*np.ones_like(y_obs_4)
# generate random noise
noise_1 = np.random.normal(0, eps_obs_1, t_seq.size)
noise_2 = np.random.normal(0, eps_obs_2, t_seq.size)
noise_3 = np.random.normal(0, eps_obs_3, t_seq.size)
noise_4 = np.random.normal(0, eps_obs_4, t_seq.size)
# generate measured intensity
i_obs_1 = y_obs_1 + noise_1
i_obs_2 = y_obs_2 + noise_2
i_obs_3 = y_obs_3 + noise_3
i_obs_4 = y_obs_4 + noise_4
# print real values
print('-'*24)
print(f'fwhm: {fwhm}')
print(f'tau_1: {tau_1}')
print(f'tau_2: {tau_2}')
for i in range(4):
print(f't_0_{i+1}: {t0[i]}')
print('-'*24)
print('Excited Species contribution')
print(f'scan 1: {abs_1[0]} \t {abs_1[1]}')
print(f'scan 2: {abs_2[0]} \t {abs_2[1]}')
print(f'scan 3: {abs_3[0]} \t {abs_3[1]}')
print(f'scan 4: {abs_4[0]} \t {abs_4[1]}')
param_exact = [fwhm, t0[0], t0[1], t0[2], t0[3], tau_1, tau_2]
------------------------
fwhm: 100
tau_1: 500
tau_2: 10000
t_0_1: -156.12041304890062
t_0_2: 38.61083766738686
t_0_3: -70.46010438461614
t_0_4: 96.11767660754525
------------------------
Excited Species contribution
scan 1: 1 1
scan 2: 0.5 0.8
scan 3: -0.5 0.7
scan 4: 0.6 0.3
# plot model experimental data
plt.errorbar(t_seq, i_obs_1, eps_obs_1, label='1')
plt.errorbar(t_seq, i_obs_2, eps_obs_2, label='2')
plt.errorbar(t_seq, i_obs_3, eps_obs_3, label='3')
plt.errorbar(t_seq, i_obs_4, eps_obs_4, label='4')
plt.legend()
plt.show()
# import needed module for fitting
from TRXASprefitpack import fit_transient_exp
# time, intensity, eps should be sequence of numpy.ndarray
t = [t_seq]
intensity = [np.vstack((i_obs_1, i_obs_2, i_obs_3, i_obs_4)).T]
eps = [np.vstack((eps_obs_1, eps_obs_2, eps_obs_3, eps_obs_4)).T]
# set initial guess
irf = 'g' # shape of irf function
fwhm_init = 100
t0_init = np.array([0, 0, 0, 0])
# test with one decay module
tau_init = np.array([15000])
# use global optimization method: AMPGO
fit_result_decay_1 = fit_transient_exp(irf, fwhm_init, t0_init, tau_init, False,
method_glb='ampgo', t=t, intensity=intensity, eps=eps)
# print fitting result
print(fit_result_decay_1)
[Model information]
model : decay
irf: g
fwhm: 144.4151
eta: 0.0000
base: False
[Optimization Method]
global: ampgo
leastsq: trf
[Optimization Status]
nfev: 769
status: 0
global_opt msg: Requested Number of global iteration is finished.
leastsq_opt msg: `ftol` termination condition is satisfied.
[Optimization Results]
Total Data points: 368
Number of effective parameters: 10
Degree of Freedom: 358
Chi squared: 2180.6363
Reduced chi squared: 6.0912
AIC (Akaike Information Criterion statistic): 674.7784
BIC (Bayesian Information Criterion statistic): 713.8592
[Parameters]
fwhm_G: 144.41507069 +/- 23.65709134 ( 16.38%)
t_0_1_1: -152.23580141 +/- 12.88603562 ( 8.46%)
t_0_1_2: 76.21013523 +/- 12.57882417 ( 16.51%)
t_0_1_3: 200.00000000 +/- 14.64366187 ( 7.32%)
t_0_1_4: 62.15838170 +/- 18.81500401 ( 30.27%)
tau_1: 12175.91776145 +/- 742.50080195 ( 6.10%)
[Parameter Bound]
fwhm_G: 50 <= 144.41507069 <= 200
t_0_1_1: -200 <= -152.23580141 <= 200
t_0_1_2: -200 <= 76.21013523 <= 200
t_0_1_3: -200 <= 200.00000000 <= 200
t_0_1_4: -200 <= 62.15838170 <= 200
tau_1: 3200 <= 12175.91776145 <= 51200
[Component Contribution]
DataSet dataset_1:
#tscan tscan_1 tscan_2 tscan_3 tscan_4
decay 1 100.00% 100.00% 100.00% 100.00%
[Parameter Correlation]
Parameter Correlations > 0.1 are reported.
# plot fitting result and experimental data
color_lst = ['red', 'blue', 'green', 'black']
for i in range(4):
plt.errorbar(t[0], intensity[0][:, i], eps[0][:, i], label=f'expt {i+1}', color=color_lst[i])
plt.errorbar(t[0], fit_result_decay_1['fit'][0][:, i], label=f'fit {i+1}', color=color_lst[i])
plt.legend()
plt.show()
For scan 1 and 2, experimental data and fitting data match well. However for scan 3 and 4, they do not match at shorter time region (< 10000).
# plot with shorter time range
for i in range(4):
plt.errorbar(t[0], intensity[0][:, i], eps[0][:, i], label=f'expt {i+1}', color=color_lst[i])
plt.errorbar(t[0], fit_result_decay_1['fit'][0][:, i], label=f'fit {i+1}', color=color_lst[i])
plt.legend()
plt.xlim(-10*fwhm_init, 20*fwhm_init)
plt.show()
There may exists shorter lifetime component.
# try with double exponential decay
# set initial guess
from tabnanny import verbose
irf = 'g' # shape of irf function
fwhm_init = 100
t0_init = np.array([0, 0, 0, 0])
# test with two decay module
tau_init = np.array([300, 15000])
fit_result_decay_2 = fit_transient_exp(irf, fwhm_init, t0_init, tau_init, False,
method_glb='ampgo', kwargs_lsq={'verbose' : 2}, t=t, intensity=intensity, eps=eps)
Iteration Total nfev Cost Cost reduction Step norm Optimality
0 1 1.8288e+02 1.13e-03
1 2 1.8288e+02 7.13e-11 2.25e-03 3.89e-05
`ftol` termination condition is satisfied.
Function evaluations 2, initial cost 1.8288e+02, final cost 1.8288e+02, first-order optimality 3.89e-05.
# print fitting result
print(fit_result_decay_2)
[Model information]
model : decay
irf: g
fwhm: 101.5631
eta: 0.0000
base: False
[Optimization Method]
global: ampgo
leastsq: trf
[Optimization Status]
nfev: 965
status: 0
global_opt msg: Requested Number of global iteration is finished.
leastsq_opt msg: `ftol` termination condition is satisfied.
[Optimization Results]
Total Data points: 368
Number of effective parameters: 15
Degree of Freedom: 353
Chi squared: 365.7519
Reduced chi squared: 1.0361
AIC (Akaike Information Criterion statistic): 27.745
BIC (Bayesian Information Criterion statistic): 86.3663
[Parameters]
fwhm_G: 101.56312034 +/- 8.65279856 ( 8.52%)
t_0_1_1: -153.31713009 +/- 4.86051817 ( 3.17%)
t_0_1_2: 43.99363705 +/- 6.94025884 ( 15.78%)
t_0_1_3: -67.50417257 +/- 6.36057697 ( 9.42%)
t_0_1_4: 101.54236607 +/- 4.39923343 ( 4.33%)
tau_1: 497.19877854 +/- 19.08450237 ( 3.84%)
tau_2: 9901.14794048 +/- 284.67177431 ( 2.88%)
[Parameter Bound]
fwhm_G: 50 <= 101.56312034 <= 200
t_0_1_1: -200 <= -153.31713009 <= 200
t_0_1_2: -200 <= 43.99363705 <= 200
t_0_1_3: -200 <= -67.50417257 <= 200
t_0_1_4: -200 <= 101.54236607 <= 200
tau_1: 50 <= 497.19877854 <= 800
tau_2: 3200 <= 9901.14794048 <= 51200
[Component Contribution]
DataSet dataset_1:
#tscan tscan_1 tscan_2 tscan_3 tscan_4
decay 1 -4.41% -28.71% -62.25% 48.48%
decay 2 95.59% 71.29% 37.75% 51.52%
[Parameter Correlation]
Parameter Correlations > 0.1 are reported.
(tau_1, fwhm_G) = -0.17
(tau_1, t_0_1_3) = -0.345
(tau_1, t_0_1_4) = -0.126
(tau_2, tau_1) = -0.366
# plot fitting result and experimental data
color_lst = ['red', 'blue', 'green', 'black']
for i in range(4):
plt.errorbar(t[0], intensity[0][:, i], eps[0][:, i], label=f'expt {i+1}', color=color_lst[i])
plt.errorbar(t[0], fit_result_decay_2['fit'][0][:, i], label=f'fit {i+1}', color=color_lst[i])
plt.legend()
plt.show()
# plot with shorter time range
for i in range(4):
plt.errorbar(t[0], intensity[0][:, i], eps[0][:, i], label=f'expt {i+1}', color=color_lst[i])
plt.errorbar(t[0], fit_result_decay_2['fit'][0][:, i], label=f'fit {i+1}', color=color_lst[i])
plt.legend()
plt.xlim(-10*fwhm_init, 20*fwhm_init)
plt.show()
Two decay model fits well
# Compare fitting value and exact value
for i in range(len(fit_result_decay_2['x'])):
print(f"{fit_result_decay_2['param_name'][i]}: {fit_result_decay_2['x'][i]} (fit) \t {param_exact[i]} (exact)")
fwhm_G: 101.563120337142 (fit) 100 (exact)
t_0_1_1: -153.31713008689923 (fit) -156.12041304890062 (exact)
t_0_1_2: 43.9936370546416 (fit) 38.61083766738686 (exact)
t_0_1_3: -67.50417256681469 (fit) -70.46010438461614 (exact)
t_0_1_4: 101.54236606639758 (fit) 96.11767660754525 (exact)
tau_1: 497.1987785448846 (fit) 500 (exact)
tau_2: 9901.14794048022 (fit) 10000 (exact)
Fitting result and exact result are match well. For future use or transfer your fitting result to your collaborator or superviser, you want to save or load fitting result from file.
# save fitting result to file
from TRXASprefitpack import save_TransientResult, load_TransientResult
save_TransientResult(fit_result_decay_2, 'example_decay_2') # save fitting result to example_decay_2.h5
loaded_result = load_TransientResult('example_decay_2') # load fitting result from example_decay_2.h5
Now deduce species associated difference coefficient from sequential decay model
y0 = np.array([1, 0, 0]) # initial cond
eigval, V, c = solve_seq_model(loaded_result['x'][5:], y0)
# compute scaled V matrix
V_scale = np.einsum('j,ij->ij', c, V)
diff_abs_fit = np.linalg.solve(V_scale[:-1, :-1].T, loaded_result['c'][0]) # slice last column and row corresponding to ground state
# compare with exact result
print('-'*24)
print('[Species Associated Difference Coefficent]')
print('scan # \t ex 1 (fit) \t ex 1 (exact) \t ex 2 (fit) \t ex 2 (exact)')
print(f'1 \t {diff_abs_fit[0,0]} \t {abs_1[0]} \t {diff_abs_fit[1,0]} \t {abs_1[1]}')
print(f'2 \t {diff_abs_fit[0,1]} \t {abs_2[0]} \t {diff_abs_fit[1,1]} \t {abs_2[1]}')
print(f'3 \t {diff_abs_fit[0,2]} \t {abs_3[0]} \t {diff_abs_fit[1,2]} \t {abs_3[1]}')
print(f'4 \t {diff_abs_fit[0,3]} \t {abs_4[0]} \t {diff_abs_fit[1,3]} \t {abs_4[1]}')
------------------------
[Species Associated Difference Coefficent]
scan # ex 1 (fit) ex 1 (exact) ex 2 (fit) ex 2 (exact)
1 1.0037847366325658 1 0.9995440788449074 1
2 0.5064294402029949 0.5 0.805300497592594 0.8
3 -0.4772831402102139 -0.5 0.6988293554639244 0.7
4 0.6038784961845008 0.6 0.29551241800998834 0.3
It also matches well, as expected.
The error of paramter reported from Transient Driver is based on Asymptotic Standard Error.
However, strictly, ASE cannot be used in non-linear regression.
TRXASprefitpack provides alternative error estimation based on F-test.
from TRXASprefitpack import confidence_interval
ci_result = confidence_interval(loaded_result, 0.05) # set significant level: 0.05 -> 95% confidence level
print(ci_result) # report confidence interval
[Report for Confidence Interval]
Method: f
Significance level: 5.000000e-02
[Confidence interval]
101.56312034 - 17.18281259 <= fwhm_G <= 101.56312034 + 18.07428539
-153.31713009 - 9.49174207 <= t_0_1_1 <= -153.31713009 + 9.44520352
43.99363705 - 12.62990633 <= t_0_1_2 <= 43.99363705 + 12.24103969
-67.50417257 - 12.37327962 <= t_0_1_3 <= -67.50417257 + 12.61249769
101.54236607 - 9.33060371 <= t_0_1_4 <= 101.54236607 + 9.37813454
497.19877854 - 37.12480863 <= tau_1 <= 497.19877854 + 39.7229684
9901.14794048 - 551.12555403 <= tau_2 <= 9901.14794048 + 573.87599666
# compare with ase
from scipy.stats import norm
factor = norm.ppf(1-0.05/2)
print('[Confidence interval (from ASE)]')
for i in range(loaded_result['param_name'].size):
print(f"{loaded_result['x'][i] :.8f} - {factor*loaded_result['x_eps'][i] :.8f}",
f"<= {loaded_result['param_name'][i]} <=", f"{loaded_result['x'][i] :.8f} + {factor*loaded_result['x_eps'][i] :.8f}")
[Confidence interval (from ASE)]
101.56312034 - 16.95917355 <= fwhm_G <= 101.56312034 + 16.95917355
-153.31713009 - 9.52644056 <= t_0_1_1 <= -153.31713009 + 9.52644056
43.99363705 - 13.60265737 <= t_0_1_2 <= 43.99363705 + 13.60265737
-67.50417257 - 12.46650178 <= t_0_1_3 <= -67.50417257 + 12.46650178
101.54236607 - 8.62233908 <= t_0_1_4 <= 101.54236607 + 8.62233908
497.19877854 - 37.40493731 <= tau_1 <= 497.19877854 + 37.40493731
9901.14794048 - 557.94642507 <= tau_2 <= 9901.14794048 + 557.94642507
However, as you can see, in many case, ASE does not much different from more sophisticated f-test based error estimation.