'''
res_thy:
submodule for residual function and dfient for fitting static spectrum with the
sum of voigt broadened theoretical spectrum, edge function and base function
:copyright: 2021-2022 by pistack (Junho Lee).
:license: LGPL3.
'''
from typing import Optional, Sequence
import numpy as np
from numpy.polynomial.legendre import legval
from ..mathfun.A_matrix import fact_anal_A
from ..mathfun.peak_shape import voigt_thy, edge_gaussian, edge_lorenzian
from ..mathfun.peak_shape import deriv_voigt_thy, deriv_edge_gaussian, deriv_edge_lorenzian
[docs]def residual_thy(x0: np.ndarray, policy: str, thy_peak: Sequence[np.ndarray],
edge: Optional[str] = None, num_edge: Optional[int] = 0,
base_order: Optional[int] = None,
e: np.ndarray = None,
intensity: np.ndarray = None, eps: np.ndarray = None) -> np.ndarray:
'''
residaul_thy
`scipy.optimize.least_squares` compatible vector residual function for fitting static spectrum with the
sum of voigt broadend theoretical spectrum, edge function base function
Args:
x0: initial parameter
* 1st and 2nd: fwhm_G and fwhm_L
if policy == 'scale':
* 3rd to :math:`2+{num}_{thy}` : peak_scale
if policy == 'shift':
* 3rd to :math:`2+{num}_{thy}` : peak_shift
if policy == 'both':
* 3rd to :math:`2+{num}_{thy}` : peak_shift
* :math:`2+{num}_{thy}` to :math:`2+2 {num}_{thy}`: peak_scale
if edge is not None:
* :math:`2+{num}_{thy}+{num}_{edge}+i` or :math:`2+2{num_thy}+{num}_{edge}+i`: i th edge position
* :math:`2+{num}_{thy}+2{num}_{edge}+i` or :math:`2+2{num_thy}+2{num}_{edge}+i`: fwhm of i th edge
policy ({'shift', 'scale', 'both'}): Policy to match discrepency
between experimental data and theoretical spectrum.
* 'shift' : Default option, shift peak position by peak_factor
* 'scale' : scale peak position by peak_factor
* 'both' : both shift and scale peak postition, peak_factor should be a tuple of `shift_factor` and `scale_factor`.
thy_peak: theoretically calculated peak position and intensity
edge ({'g', 'l'}): type of edge shape function
if edge is not set, it does not include edge function.
num_edge: the number of edge feature
base_order (int): polynomial order of baseline function
if base_order is not set, it does not include baseline function.
fix_param_idx: idx for fixed parameter (masked array for `x0`)
e: 1d array of energy points of data (n,)
intensity: intensity of static data (n,)
eps: estimated error of data (n,)
Returns:
Tuple of scalar residual function :math:`(\\frac{1}{2}\\sum_i {res}^2_i)` and its gradient
Note:
* If fwhm_G of ith voigt component is zero then it is treated as lorenzian function with fwhm_L
* If fwhm_L of ith voigt component is zero then it is treated as gaussian function with fwhm_G
'''
x0 = np.atleast_1d(x0)
thy_comp = len(thy_peak)
tot_comp = thy_comp
if policy in ['scale', 'shift']:
peak_factor = x0[2:2+thy_comp]
elif policy == 'both':
peak_factor = np.empty(thy_comp, dtype=object)
for i in range(thy_comp):
peak_factor[i] = np.array([x0[2+i], x0[2+thy_comp+i]])
if edge is not None:
tot_comp = tot_comp+num_edge
if base_order is not None:
tot_comp = tot_comp+base_order+1
A = np.empty((tot_comp, e.size))
for i in range(thy_comp):
A[i, :] = \
voigt_thy(e, thy_peak[i], x0[0], x0[1],
peak_factor[i], policy)/np.sum(thy_peak[i][:, 1])
base_start = thy_comp
if edge is not None:
base_start = base_start+num_edge
if policy in ['shift', 'scale']:
param_edge_start = 2+thy_comp
else:
param_edge_start = 2+2*thy_comp
if edge == 'g':
for i in range(num_edge):
A[thy_comp+i, :] = edge_gaussian(e-x0[param_edge_start+i],
x0[param_edge_start+num_edge+i])
elif edge == 'l':
for i in range(num_edge):
A[thy_comp+i, :] = edge_lorenzian(e-x0[param_edge_start+i],
x0[param_edge_start+num_edge+i])
if base_order is not None:
e_max = np.max(e)
e_min = np.min(e)
e_norm = 2*(e-(e_max+e_min)/2)/(e_max-e_min)
tmp = np.eye(base_order+1)
A[base_start:, :] = legval(e_norm, tmp, tensor=True)
c = fact_anal_A(A, intensity, eps)
chi = (c@A - intensity)/eps
return chi
[docs]def res_grad_thy(x0: np.ndarray, policy: str, thy_peak: Sequence[np.ndarray],
edge: Optional[str] = None, num_edge: Optional[int] = 0,
base_order: Optional[int] = None,
fix_param_idx: Optional[np.ndarray] = None,
e: np.ndarray = None,
intensity: np.ndarray = None, eps: np.ndarray = None) -> np.ndarray:
'''
res_grad_thy
`scipy.optimize.minimize` compatible scalar residual function and its gradient for fitting static spectrum with the
sum of voigt broadend theoretical spectrum, edge function base function
Args:
x0: initial parameter
* 1st and 2nd: fwhm_G and fwhm_L
if policy == 'scale':
* 3rd to :math:`2+{num}_{thy}` : peak_scale
if policy == 'shift':
* 3rd to :math:`2+{num}_{thy}` : peak_shift
if policy == 'both':
* 3rd to :math:`2+{num}_{thy}` : peak_shift
* :math:`2+{num}_{thy}` to :math:`2+2 {num}_{thy}`: peak_scale
if edge is not None:
* :math:`2+{num}_{thy}+{num}_{edge}+i` or :math:`2+2{num_thy}+{num}_{edge}+i`: i th edge position
* :math:`2+{num}_{thy}+2{num}_{edge}+i` or :math:`2+2{num_thy}+2{num}_{edge}+i`: fwhm of i th edge
policy ({'shift', 'scale', 'both'}): Policy to match discrepency
between experimental data and theoretical spectrum.
* 'shift' : Default option, shift peak position by peak_factor
* 'scale' : scale peak position by peak_factor
* 'both' : both shift and scale peak postition, peak_factor should be a tuple of `shift_factor` and `scale_factor`.
thy_peak: theoretically calculated peak position and intensity
edge ({'g', 'l'}): type of edge shape function
if edge is not set, it does not include edge function.
num_edge: the number of edge feature
base_order (int): polynomial order of baseline function
if base_order is not set, it does not include baseline function.
fix_param_idx: idx for fixed parameter (masked array for `x0`)
e: 1d array of energy points of data (n,)
intensity: intensity of static data (n,)
eps: estimated error of data (n,)
Returns:
Tuple of scalar residual function :math:`(\\frac{1}{2}\\sum_i {res}^2_i)` and its gradient
Note:
* If fwhm_G of ith voigt component is zero then it is treated as lorenzian function with fwhm_L
* If fwhm_L of ith voigt component is zero then it is treated as gaussian function with fwhm_G
'''
x0 = np.atleast_1d(x0)
thy_comp = len(thy_peak)
tot_comp = thy_comp
area = np.empty(thy_comp)
for i in range(thy_comp):
area[i] = np.sum(thy_peak[i][:, 1])
if policy in ['scale', 'shift']:
peak_factor = x0[2:2+thy_comp]
elif policy == 'both':
peak_factor = np.empty(thy_comp, dtype=object)
for i in range(thy_comp):
peak_factor[i] = np.array([x0[2+i], x0[2+thy_comp+i]])
if edge is not None:
tot_comp = tot_comp+num_edge
if base_order is not None:
tot_comp = tot_comp+base_order+1
A = np.empty((tot_comp, e.size))
for i in range(thy_comp):
A[i, :] = \
voigt_thy(e, thy_peak[i], x0[0], x0[1],
peak_factor[i], policy)/area[i]
base_start = thy_comp
if edge is not None:
base_start = base_start+num_edge
if policy in ['shift', 'scale']:
param_edge_start = 2+thy_comp
else:
param_edge_start = 2+2*thy_comp
if edge == 'g':
for i in range(num_edge):
A[thy_comp+i, :] = edge_gaussian(e-x0[param_edge_start+i],
x0[param_edge_start+num_edge+i])
elif edge == 'l':
for i in range(num_edge):
A[thy_comp+i, :] = edge_lorenzian(e-x0[param_edge_start+i],
x0[param_edge_start+num_edge+i])
if base_order is not None:
e_max = np.max(e)
e_min = np.min(e)
e_norm = 2*(e-(e_max+e_min)/2)/(e_max-e_min)
tmp = np.eye(base_order+1)
A[base_start:, :] = legval(e_norm, tmp, tensor=True)
c = fact_anal_A(A, intensity, eps)
chi = (c@A - intensity)/eps
df = np.zeros((intensity.size, x0.size))
for i in range(thy_comp):
deriv_thy = (c[i]/area[i]) *\
deriv_voigt_thy(e, thy_peak[i], x0[0],
x0[1], peak_factor[i], policy)
df[:, :2] = df[:, :2] + deriv_thy[:, :2]
if policy in ['scale', 'shift']:
df[:, 2+i] = deriv_thy[:, 2]
else:
df[:, 2+i] = deriv_thy[:, 2]
df[:, 2+thy_comp+i] = deriv_thy[:, 3]
if edge is not None:
if edge == 'g':
for i in range(num_edge):
df_edge = c[thy_comp+i]*deriv_edge_gaussian(e-x0[param_edge_start+i],
x0[param_edge_start+num_edge+i])
df[:, param_edge_start+i] = -df_edge[:, 0]
df[:, param_edge_start+num_edge+i] = df_edge[:, 1]
elif edge == 'l':
for i in range(num_edge):
df_edge = c[thy_comp+i]*deriv_edge_lorenzian(e-x0[param_edge_start+i],
x0[param_edge_start+num_edge+i])
df[:, param_edge_start+i] = -df_edge[:, 0]
df[:, param_edge_start+num_edge+i] = df_edge[:, 1]
df = np.einsum('i,ij->ij', 1/eps, df)
df[:, fix_param_idx] = 0
return np.sum(chi**2)/2, chi@df