'''
res_osc:
submodule for residual function and gradient for fitting time delay scan with the
convolution of sum of damped oscillation and instrumental response function
:copyright: 2021-2022 by pistack (Junho Lee).
:license: LGPL3.
'''
from typing import Tuple, Optional, Sequence
import numpy as np
from ..mathfun.irf import calc_eta, deriv_eta
from ..mathfun.irf import calc_fwhm, deriv_fwhm
from ..mathfun.A_matrix import make_A_matrix_gau_osc, make_A_matrix_cauchy_osc, fact_anal_A
from ..mathfun.exp_conv_irf import deriv_dmp_osc_sum_conv_gau, deriv_dmp_osc_sum_conv_cauchy
# residual and gradient function for damped oscillation model
[docs]def residual_dmp_osc(x0: np.ndarray, num_comp: int, irf: str,
t: Optional[Sequence[np.ndarray]] = None,
intensity: Optional[Sequence[np.ndarray]] = None, eps: Optional[Sequence[np.ndarray]] = None) -> np.ndarray:
'''
residual_dmp_osc
`scipy.optimize.least_squares` compatible gradient of vector residual function for fitting multiple set of time delay scan with the
sum of convolution of damped oscillation and instrumental response function
Args:
x0: initial parameter,
if irf == 'g','c':
* 1st: fwhm_(G/L)
* 2nd to :math:`2+N_{scan}`: time zero of each scan
* :math:`2+N_{scan}+i`: time constant of each damped oscillation
* :math:`2+N_{scan}+N_{osc}+i`: period of each damped oscillation
if irf == 'pv':
* 1st and 2nd: fwhm_G, fwhm_L
* 3rd to :math:`3+N_{scan}`: time zero of each scan
* :math:`3+N_{scan}+i`: time constant of each damped oscillation
* :math:`3+N_{scan}+N_{osc}+i`: period of each damped oscillation
num_comp: number of damped oscillation component
irf: shape of instrumental response function
* 'g': normalized gaussian distribution,
* 'c': normalized cauchy distribution,
* 'pv': pseudo voigt profile :math:`(1-\\eta)g(f) + \\eta c(f)`
For pseudo voigt profile, the mixing parameter :math:`\\eta(f_G, f_L)` and
uniform fwhm paramter :math:`f(f_G, f_L)` are calculated by `calc_eta` and `calc_fwhm` routine
t: time points for each data set
intensity: sequence of intensity of datasets
eps: sequence of estimated error of datasets
Returns:
Residual vector
'''
x0 = np.atleast_1d(x0)
if irf in ['g', 'c']:
num_irf = 1
fwhm = x0[0]
else:
num_irf = 2
fwhm = calc_fwhm(x0[0], x0[1])
eta = calc_eta(x0[0], x0[1])
num_t0 = 0; sum = 0
for d in intensity:
num_t0 = d.shape[1] + num_t0
sum = sum + d.size
chi = np.empty(sum)
tau = x0[num_irf+num_t0:num_irf+num_t0+num_comp]
period = x0[num_irf+num_t0+num_comp:num_irf+num_t0+2*num_comp]
end = 0; t0_idx = num_irf
for ti,d,e in zip(t,intensity,eps):
for j in range(d.shape[1]):
t0 = x0[t0_idx]
if irf == 'g':
A = make_A_matrix_gau_osc(ti-t0, fwhm, 1/tau, period)
elif irf == 'c':
A = make_A_matrix_cauchy_osc(ti-t0, fwhm, 1/tau, period)
else:
A_gau = make_A_matrix_gau_osc(ti-t0, fwhm, 1/tau, period)
A_cauchy = make_A_matrix_cauchy_osc(ti-t0, fwhm, 1/tau, period)
A = A_gau + eta*(A_cauchy-A_gau)
c = fact_anal_A(A, d[:,j], e[:,j])
chi[end:end+d.shape[0]] = ((c@A) - d[:, j])/e[:, j]
end = end + d.shape[0]
t0_idx = t0_idx + 1
return chi
[docs]def res_grad_dmp_osc(x0: np.ndarray, num_comp: int, irf: str,
fix_param_idx: Optional[np.ndarray] = None,
t: Optional[Sequence[np.ndarray]] = None,
intensity: Optional[Sequence[np.ndarray]] = None,
eps: Optional[Sequence[np.ndarray]] = None) -> Tuple[np.ndarray, np.ndarray]:
'''
res_grad_dmp_osc
scipy.optimize.minimize compatible pair of scalar residual function and its gradient for fitting multiple set of time delay scan with the
sum of convolution of damped oscillation and instrumental response function
Args:
x0: initial parameter,
if irf == 'g','c':
* 1st: fwhm_(G/L)
* 2nd to :math:`2+N_{scan}`: time zero of each scan
* :math:`2+N_{scan}+i`: time constant of each damped oscillation
* :math:`2+N_{scan}+N_{osc}+i`: period of each damped oscillation
if irf == 'pv':
* 1st and 2nd: fwhm_G, fwhm_L
* 3rd to :math:`3+N_{scan}`: time zero of each scan
* :math:`3+N_{scan}+i`: time constant of each damped oscillation
* :math:`3+N_{scan}+N_{osc}+i`: period of each damped oscillation
num_comp: number of damped oscillation component
irf: shape of instrumental response function
* 'g': normalized gaussian distribution,
* 'c': normalized cauchy distribution,
* 'pv': pseudo voigt profile :math:`(1-\\eta)g(f) + \\eta c(f)`
For pseudo voigt profile, the mixing parameter :math:`\\eta(f_G, f_L)` and
uniform fwhm paramter :math:`f(f_G, f_L)` are calculated by `calc_eta` and `calc_fwhm` routine
fix_param_idx: index for fixed parameter (masked array for `x0`)
t: time points for each data set
intensity: sequence of intensity of datasets
eps: sequence of estimated error of datasets
Returns:
Tuple of scalar residual function :math:`(\\frac{1}{2}\\sum_i {res}^2_i)` and its gradient
'''
x0 = np.atleast_1d(x0)
if irf in ['g', 'c']:
num_irf = 1
fwhm = x0[0]
else:
num_irf = 2
eta = calc_eta(x0[0], x0[1])
fwhm = calc_eta(x0[0], x0[1])
dfwhm_G, dfwhm_L = deriv_fwhm(x0[0], x0[1])
deta_G, deta_L = deriv_eta(x0[0], x0[1])
num_t0 = 0; sum = 0
for d in intensity:
num_t0 = num_t0 + d.shape[1]
sum = sum + d.size
tau = x0[num_irf+num_t0:num_irf+num_t0+num_comp]
period = x0[num_irf+num_t0+num_comp:num_irf+num_t0+2*num_comp]
num_param = num_irf+num_t0+2*num_comp
chi = np.empty(sum)
df = np.zeros((sum, num_irf+2*num_comp))
grad = np.empty(num_param)
end = 0; t0_idx = num_irf
for ti,d,e in zip(t, intensity, eps):
step = d.shape[0]
for j in range(d.shape[1]):
t0 = x0[t0_idx]
if irf == 'g':
A = make_A_matrix_gau_osc(ti-t0, fwhm, 1/tau, period)
elif irf == 'c':
A = make_A_matrix_cauchy_osc(ti-t0, fwhm, 1/tau, period)
else:
A_gau = make_A_matrix_gau_osc(ti-t0, fwhm, 1/tau, period)
A_cauchy = make_A_matrix_cauchy_osc(ti-t0, fwhm, 1/tau, period)
diff = A_cauchy-A_gau
A = A_gau + eta*diff
c = fact_anal_A(A, d[:,j], e[:,j])
chi[end:end+step] = (c@A-d[:,j])/e[:, j]
c_amp = np.sqrt(c[:num_comp]**2+c[num_comp:]**2)
phase = -np.arctan2(c[num_comp:], c[:num_comp])
if irf == 'g':
grad_tmp = deriv_dmp_osc_sum_conv_gau(ti-t0, fwhm, 1/tau, period, phase, c_amp)
elif irf == 'c':
grad_tmp = deriv_dmp_osc_sum_conv_cauchy(ti-t0, fwhm, 1/tau, period, phase, c_amp)
else:
grad_gau = deriv_dmp_osc_sum_conv_gau(ti-t0, fwhm[0], 1/tau, period, phase, c_amp)
grad_cauchy = deriv_dmp_osc_sum_conv_cauchy(ti-t0, fwhm[1], 1/tau, period, phase, c_amp)
grad_tmp = grad_gau + eta*(grad_cauchy-grad_gau)
grad_tmp = np.einsum('i,ij->ij', 1/e[:, j], grad_tmp)
if irf in ['g', 'c']:
df[end:end+step, 0] = grad_tmp[:, 1]
else:
cdiff = (c@diff)/e[:, j]
df[end:end+step, 0] = dfwhm_G*grad_tmp[:, 1]+deta_G*cdiff
df[end:end+step, 1] = dfwhm_L*grad_tmp[:, 1]+deta_L*cdiff
grad[t0_idx] = -chi[end:end+step]@grad_tmp[:, 0]
df[end:end+step, :num_irf+num_comp] = \
np.einsum('j,ij->ij', -1/tau**2, grad_tmp[:, 2:2+num_comp])
df[end:end+step, num_irf+num_comp:] = grad_tmp[:, 2+num_comp:2+2*num_comp]
end = end + step
t0_idx = t0_idx + 1
mask = np.ones(num_param, dtype=bool)
mask[num_irf:num_irf+num_t0] = False
grad[mask] = chi@df
if fix_param_idx is not None:
df[:, fix_param_idx] = 0
return np.sum(chi**2)/2, grad