'''
res_decay:
submodule for residual function and gradient for fitting time delay scan with the
convolution of sum of exponential decay and instrumental response function
:copyright: 2021-2022 by pistack (Junho Lee).
:license: LGPL3.
'''
from typing import Optional, Sequence, Tuple
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, make_A_matrix_cauchy, fact_anal_A
from ..mathfun.exp_conv_irf import deriv_exp_conv_gau, deriv_exp_conv_cauchy
from ..mathfun.exp_conv_irf import deriv_exp_sum_conv_gau, deriv_exp_sum_conv_cauchy
# residual and gradient function for exponential decay model
[docs]def residual_decay(x0: np.ndarray, base: bool, irf: str,
t: Optional[Sequence[np.ndarray]] = None,
intensity: Optional[Sequence[np.ndarray]] = None, eps: Optional[Sequence[np.ndarray]] = None) -> np.ndarray:
'''
residual_decay
scipy.optimize.least_squares compatible vector residual function for fitting multiple set of time delay scan with the
sum of convolution of exponential decay 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}` to :math:`2+N_{scan}+N_{\\tau}`: time constant of each decay component
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}` to :math:`3+N_{scan}+N_{\\tau}`: time constant of each decay component
num_comp: number of exponential decay component (except base)
base: whether or not include baseline (i.e. very long lifetime 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
count = 0
for d in intensity:
num_t0 = d.shape[1] + num_t0
count = count + d.size
chi = np.empty(count)
tau = x0[num_irf+num_t0:]
if not base:
k = 1/tau
else:
k = np.empty(tau.size+1)
k[:-1] = 1/tau
k[-1] = 0
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(ti-t0, fwhm, k)
elif irf == 'c':
A = make_A_matrix_cauchy(ti-t0, fwhm, k)
else:
A_gau = make_A_matrix_gau(ti-t0, fwhm, k)
A_cauchy = make_A_matrix_cauchy(ti-t0, fwhm, k)
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_decay(x0: np.ndarray, num_comp: int, base: bool, 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_decay
scipy.optimize.minimize compatible scalar residual and its gradient function for fitting multiple set of time delay scan with the
sum of convolution of exponential decay 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}` to :math:`2+N_{scan}+N_{\\tau}`: time constant of each decay component
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}` to :math:`3+N_{scan}+N_{\\tau}`: time constant of each decay component
num_comp: number of exponential decay component (except base)
base: whether or not include baseline (i.e. very long lifetime 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
fix_param_idx: index for fixed parameter (masked array for `x0`)
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_fwhm(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
count = 0
for d in intensity:
num_t0 = num_t0 + d.shape[1]
count = count + d.size
tau = x0[num_irf+num_t0:]
if not base:
k = 1/tau
else:
k = np.empty(tau.size+1)
k[:-1] = 1/tau
k[-1] = 0
num_param = num_irf+num_t0+num_comp
chi = np.empty(count)
df = np.empty((count, tau.size+num_irf))
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(ti-t0, fwhm, k)
elif irf == 'c':
A = make_A_matrix_cauchy(ti-t0, fwhm, k)
else:
A_gau = make_A_matrix_gau(ti-t0, fwhm, k)
A_cauchy = make_A_matrix_cauchy(ti-t0, fwhm, k)
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]
if irf == 'g':
grad_tmp = deriv_exp_sum_conv_gau(ti-t0, fwhm, 1/tau, c, base)
elif irf == 'c':
grad_tmp = deriv_exp_sum_conv_cauchy(
ti-t0, fwhm, 1/tau, c, base)
else:
grad_gau = deriv_exp_sum_conv_gau(ti-t0, fwhm, 1/tau, c, base)
grad_cauchy = deriv_exp_sum_conv_cauchy(
ti-t0, fwhm, 1/tau, c, base)
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:] = np.einsum('j,ij->ij', -1/tau**2, grad_tmp[:, 2:])
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:
grad[fix_param_idx] = 0
return np.sum(chi**2)/2, grad
[docs]def residual_decay_same_t0(x0: np.ndarray, base: bool, irf: str,
t: Optional[Sequence[np.ndarray]] = None,
intensity: Optional[Sequence[np.ndarray]] = None,
eps: Optional[Sequence[np.ndarray]] = None) -> np.ndarray:
'''
residual_decay_same_t0
scipy.optimize.least_squares compatible vector residual function
for fitting multiple set of time delay scan with the
sum of convolution of exponential decay and instrumental response function
Set Time Zero of every time dset in same dataset same
Args:
x0: initial parameter,
if irf == 'g','c':
* 1st: fwhm_(G/L)
* 2nd to :math:`2+N_{dset}`: time zero of each data set
* :math:`2+N_{dset}` to :math:`2+N_{dset}+N_{\\tau}`: time constant of each decay component
if irf == 'pv':
* 1st and 2nd: fwhm_G, fwhm_L
* 3rd to :math:`3+N_{dset}`: time zero of each data set
* :math:`3+N_{dset}` to :math:`3+N_{dset}+N_{\\tau}`: time constant of each decay component
num_comp: number of exponential decay component (except base)
base: whether or not include baseline (i.e. very long lifetime 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_dataset = len(t)
count = 0
for i in range(num_dataset):
count = count + intensity[i].size
chi = np.empty(count)
tau = x0[num_irf+num_dataset:]
if not base:
k = 1/tau
else:
k = np.empty(tau.size+1)
k[:-1] = 1/tau
k[-1] = 0
end = 0
t0_idx = num_irf
for ti, d, e in zip(t, intensity, eps):
t0 = x0[t0_idx]
if irf == 'g':
A = make_A_matrix_gau(ti-t0, fwhm, k)
elif irf == 'c':
A = make_A_matrix_cauchy(ti-t0, fwhm, k)
else:
A_gau = make_A_matrix_gau(ti-t0, fwhm, k)
A_cauchy = make_A_matrix_cauchy(ti-t0, fwhm, k)
A = A_gau + eta*(A_cauchy-A_gau)
for j in range(d.shape[1]):
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_decay_same_t0(x0: np.ndarray, num_comp: int, base: bool, 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_decay_same_t0
scipy.optimize.minimize compatible scalar residual
and its gradient function for fitting multiple set of time delay scan with the
sum of convolution of exponential decay and instrumental response function
Args:
x0: initial parameter,
if irf == 'g','c':
* 1st: fwhm_(G/L)
* 2nd to :math:`2+N_{dset}`: time zero of each dataset
* :math:`2+N_{dset}` to :math:`2+N_{dset}+N_{\\tau}`: time constant of each decay component
if irf == 'pv':
* 1st and 2nd: fwhm_G, fwhm_L
* 3rd to :math:`3+N_{dset}`: time zero of each dataset
* :math:`3+N_{dset}` to :math:`3+N_{dset}+N_{\\tau}`: time constant of each decay component
num_comp: number of exponential decay component (except base)
base: whether or not include baseline (i.e. very long lifetime 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
fix_param_idx: index for fixed parameter (masked array for `x0`)
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]
eta = None
else:
num_irf = 2
eta = calc_eta(x0[0], x0[1])
fwhm = calc_fwhm(x0[0], x0[1])
deta_G, deta_L = deriv_eta(x0[0], x0[1])
dfwhm_G, dfwhm_L = deriv_fwhm(x0[0], x0[1])
num_dataset = len(t)
count = 0
for i in range(num_dataset):
count = count + intensity[i].size
tau = x0[num_irf+num_dataset:num_irf+num_dataset+num_comp]
if base:
k = np.empty(tau.size+1)
k[:-1] = 1/tau
k[-1] = 0
else:
k = 1/tau
num_param = num_irf+num_dataset+num_comp
chi = np.empty(count)
df = np.zeros((count, num_irf+num_comp))
grad = np.zeros(num_param)
end = 0
t0_idx = num_irf
for ti, d, e in zip(t, intensity, eps):
step = d.shape[0]
A = np.empty((num_comp+1*base, step))
A_grad_decay = np.empty((num_comp+1*base, step, 3))
grad_decay = np.empty((step, num_comp+2))
t0 = x0[t0_idx]
if irf == 'g':
A[:num_comp+1*base, :] = make_A_matrix_gau(ti-t0, fwhm, k)
for i in range(num_comp):
A_grad_decay[i, :, :] = deriv_exp_conv_gau(ti-t0, fwhm, k[i])
if base:
A_grad_decay[-1, :, :] = deriv_exp_conv_gau(ti-t0, fwhm, 0)
elif irf == 'c':
A[:num_comp+1*base, :] = make_A_matrix_cauchy(ti-t0, fwhm, k)
for i in range(num_comp):
A_grad_decay[i, :, :] = deriv_exp_conv_cauchy(ti-t0, fwhm, k[i])
if base:
A_grad_decay[-1, :, :] = deriv_exp_conv_cauchy(ti-t0, fwhm, 0)
else:
tmp_gau = make_A_matrix_gau(ti-t0, fwhm, k)
tmp_cauchy = make_A_matrix_cauchy(ti-t0, fwhm, k)
diff = tmp_cauchy-tmp_gau
A[:num_comp+1*base, :] = tmp_gau + eta*diff
for i in range(num_comp):
tmp_grad_gau = deriv_exp_conv_gau(ti-t0, fwhm, k[i])
tmp_grad_cauchy = deriv_exp_conv_cauchy(ti-t0, fwhm, k[i])
A_grad_decay[i, :, :] = tmp_grad_gau + eta*(tmp_grad_cauchy-tmp_grad_gau)
if base:
tmp_grad_gau = deriv_exp_conv_gau(ti-t0, fwhm, 0)
tmp_grad_cauchy = deriv_exp_conv_cauchy(ti-t0, fwhm, 0)
A_grad_decay[-1, :, :] = tmp_grad_gau + eta*(tmp_grad_cauchy-tmp_grad_gau)
for j in range(d.shape[1]):
c = fact_anal_A(A, d[:, j], e[:, j])
chi[end:end+step] = (c@A-d[:, j])/e[:, j]
grad_decay[:, :2] = \
np.tensordot(c[:num_comp+1*base], A_grad_decay[:, :, :2], axes=1)
for i in range(num_comp):
grad_decay[:, 2+i] = c[i]*A_grad_decay[i, :, 2]
grad_decay = np.einsum('i,ij->ij', 1/e[:, j], grad_decay)
if irf in ['g', 'c']:
df[end:end+step, 0] = grad_decay[:, 1]
else:
cdiff = (c[:num_comp+1*base]@diff)/e[:, j]
df[end:end+step, 0] = dfwhm_G*grad_decay[:, 1]+deta_G*cdiff
df[end:end+step, 1] = dfwhm_L*grad_decay[:, 1]+deta_L*cdiff
grad[t0_idx] = grad[t0_idx] -chi[end:end+step]@grad_decay[:, 0]
df[end:end+step, num_irf:num_irf+num_comp] = \
np.einsum('j,ij->ij', -1/tau**2, grad_decay[:, 2:])
end = end + step
t0_idx = t0_idx + 1
mask = np.ones(num_param, dtype=bool)
mask[num_irf:num_irf+num_dataset] = False
grad[mask] = chi@df
if fix_param_idx is not None:
grad[fix_param_idx] = 0
return np.sum(chi**2)/2, grad