'''
_static_voigt:
submodule for static spectrum with the
sum of voigt function, edge function and baseline function
:copyright: 2021-2022 by pistack (Junho Lee).
:license: LGPL3.
'''
from typing import Optional, Sequence, Tuple
import numpy as np
from numpy.polynomial.legendre import legval
from .static_result import StaticResult
from ._ampgo import ampgo
from scipy.optimize import basinhopping
from scipy.optimize import least_squares
from ..mathfun.peak_shape import voigt, edge_gaussian, edge_lorenzian
from ..mathfun.A_matrix import fact_anal_A
from ..res.parm_bound import set_bound_e0, set_bound_t0
from ..res.res_voigt import residual_voigt, res_grad_voigt
GLBSOLVER = {'basinhopping': basinhopping, 'ampgo': ampgo}
[docs]def fit_static_voigt(e0_init: np.ndarray, fwhm_G_init: np.ndarray, fwhm_L_init: np.ndarray,
edge: Optional[str] = None,
edge_pos_init: Optional[np.ndarray] = None,
edge_fwhm_init: Optional[np.ndarray] = None,
base_order: Optional[int] = None,
method_glb: Optional[str] = None,
method_lsq: Optional[str] = 'trf',
kwargs_glb: Optional[dict] = None,
kwargs_lsq: Optional[dict] = None,
bound_e0: Optional[Sequence[Tuple[float, float]]] = None,
bound_fwhm_G: Optional[Sequence[Tuple[float, float]]] = None,
bound_fwhm_L: Optional[Sequence[Tuple[float, float]]] = None,
bound_edge_pos: Optional[Sequence[Tuple[float, float]]] = None,
bound_edge_fwhm: Optional[Sequence[Tuple[float, float]]] = None,
e: Optional[np.ndarray] = None,
intensity: Optional[np.ndarray] = None,
eps: Optional[np.ndarray] = None) -> StaticResult:
'''
driver routine for fitting static spectrum with sum of voigt component, edge and
polynomial baseline.
It separates linear and non-linear part of the optimization problem to solve non linear least sequare
optimization problem efficiently.
Moreover this driver uses two step method to search best parameter, its covariance and
estimated parameter error.
Step 1. (method_glb)
Use global optimization to find rough global minimum of our objective function.
In this stage, it use analytic gradient for scalar residual function.
Step 2. (method_lsq)
Use least squares optimization algorithm to refine global minimum of objective function and approximate covariance matrix.
Because of linear and non-linear seperation scheme, the analytic jacobian for vector residual function is hard to optain.
Thus, in this stage, it uses numerical jacobian.
Args:
e0_init (np.ndarray): initial peak position of each voigt component
fwhm_G_init (np.ndarray): initial gaussian part of fwhm parameter of each voigt component
fwhm_L_init (np.ndarray): initial lorenzian part of fwhm parameter of each voigt component
edge ({'g', 'l'}): type of edge function. If edge is not set, edge feature is not included.
edge_pos_init (np.ndarray): initial edge position
edge_fwhm_init (np.ndarray): initial fwhm parameter of edge
method_glb ({None, 'basinhopping', 'ampgo'}): Method for global optimization Algorithm.
method_lsq ({'trf', 'dogbox', 'lm'}): method of local optimization for least_squares
minimization (refinement of global optimization solution)
kwargs_glb: keyward arguments for global optimization solver
kwargs_lsq: keyward arguments for least square optimization solver
bound_e0 (sequence of tuple): boundary for each voigt componet. If upper and lower bound are same,
driver assumes that the parameter is fixed during optimization. If `bound_e0` is `None`,
the upper and lower bound are given by `set_bound_e0`.
bound_fwhm_G (sequence of tuple): boundary for fwhm_G parameter.
If `bound_fwhm_G` is `None`, the upper and lower bound are given as `(fwhm_G/2, 2*fwhm_G)`.
bound_fwhm_L (sequence of tuple): boundary for fwhm_L parameter.
If `bound_fwhm_L` is `None`, the upper and lower bound are given as `(fwhm_L/2, 2*fwhm_L)`.
bound_edge_pos (sequence of tuple): boundary for edge position,
if `bound_edge_pos` is `None` and `edge` is set, the upper and lower bound are given by `set_bound_t0`.
bound_edge_fwhm (sequence of tuple): boundary for fwhm parameter of edge feature.
If `bound_edge_fwhm` is `None`, the upper and lower bound are given as `(edge_fwhm/2, 2*edge_fwhm)`.
e (np.narray): energy range for data
intensity (np.ndarray): intensity of static spectrum data
eps (np.ndarray): estimated errors of static spectrum data
Returns:
StaticResult class object
Note:
* if initial fwhm_G is zero then such voigt component is treated as lorenzian component
* if initial fwhm_L is zero then such voigt component is treated as gaussian component
'''
if method_glb is not None and method_glb not in ['basinhopping', 'ampgo']:
raise Exception('Unsupported global optimization Method, Supported global optimization Methods are ampgo and basinhopping')
if method_lsq not in ['trf', 'lm', 'dogbox']:
raise Exception('Invalid local least square minimizer solver. It should be one of [trf, lm, dogbox]')
if edge is not None and edge not in ['g', 'l']:
raise Exception('Invalid Edge type.')
if e0_init is None:
num_voigt = 0
num_param = 0
else:
num_voigt = e0_init.size
num_param = 3*num_voigt
num_comp = num_voigt
num_edge = 0
if edge is not None:
num_edge = edge_pos_init.size
num_comp = num_comp+num_edge
num_param = num_param+2*num_edge
if base_order is not None:
num_comp = num_comp + base_order + 1
param = np.empty(num_param, dtype=float)
fix_param_idx = np.empty(num_param, dtype=bool)
param[:num_voigt] = e0_init
param[num_voigt:2*num_voigt] = fwhm_G_init
param[2*num_voigt:3*num_voigt] = fwhm_L_init
if edge is not None:
param[3*num_voigt:3*num_voigt+num_edge] = edge_pos_init
param[3*num_voigt+num_edge:] = edge_fwhm_init
bound = num_param*[None]
if bound_e0 is None:
for i in range(num_voigt):
bound[i] = set_bound_e0(e0_init[i], fwhm_G_init[i], fwhm_L_init[i])
else:
bound[:num_voigt] = bound_e0
if bound_fwhm_G is None:
for i in range(num_voigt):
bound[i+num_voigt] = (fwhm_G_init[i]/2, 2*fwhm_G_init[i])
else:
bound[num_voigt: 2*num_voigt] = bound_fwhm_G
if bound_fwhm_L is None:
for i in range(num_voigt):
bound[i+2*num_voigt] = (fwhm_L_init[i]/2, 2*fwhm_L_init[i])
else:
bound[2*num_voigt:3*num_voigt] = bound_fwhm_L
if edge is not None:
if bound_edge_pos is None:
for i in range(num_edge):
bound[3*num_voigt+i] = \
set_bound_t0(edge_pos_init[i], edge_fwhm_init[i])
else:
bound[3*num_voigt:3*num_voigt+num_edge] = bound_edge_pos
if bound_edge_fwhm is None:
for i in range(num_edge):
bound[3*num_voigt+num_edge+i] = \
(edge_fwhm_init[i]/2, 2*edge_fwhm_init[i])
else:
bound[3*num_voigt+num_edge:] = bound_edge_fwhm
for i in range(num_param):
fix_param_idx[i] = (bound[i][0] == bound[i][1])
if method_glb is not None:
go_args = (num_voigt, edge, num_edge, base_order, fix_param_idx,
e, intensity, eps)
min_go_kwargs = {'args': go_args, 'jac': True, 'bounds': bound}
if kwargs_glb is not None:
minimizer_kwargs = kwargs_glb.pop('minimizer_kwargs', None)
if minimizer_kwargs is None:
kwargs_glb['minimizer_kwargs'] = min_go_kwargs
else:
minimizer_kwargs['args'] = go_args
minimizer_kwargs['jac'] = True
minimizer_kwargs['bouns'] = bound
kwargs_glb['minimizer_kwargs'] = minimizer_kwargs
else:
kwargs_glb = {'minimizer_kwargs': min_go_kwargs}
res_go = GLBSOLVER[method_glb](res_grad_voigt, param, **kwargs_glb)
else:
res_go = {}
res_go['x'] = param
res_go['message'] = None
res_go['nfev'] = 0
param_gopt = res_go['x']
lsq_args = (num_voigt, edge, num_edge, base_order, e, intensity, eps)
if kwargs_lsq is not None:
_ = kwargs_lsq.pop('args', None)
_ = kwargs_lsq.pop('kwargs', None)
kwargs_lsq['args'] = lsq_args
else:
kwargs_lsq = {'args': lsq_args}
bound_tuple = (num_param*[None], num_param*[None])
for i in range(num_param):
bound_tuple[0][i] = bound[i][0]
bound_tuple[1][i] = bound[i][1]
if bound[i][0] == bound[i][1]:
if bound[i][0] > 0:
bound_tuple[1][i] = bound[i][1]*(1+1e-8)+1e-16
else:
bound_tuple[1][i] = bound[i][1]*(1-1e-8)+1e-16
# jacobian of vector residual function is inaccurate
res_lsq = least_squares(residual_voigt, param_gopt,
method=method_lsq, bounds=bound_tuple, **kwargs_lsq)
param_opt = res_lsq['x']
e0_opt = param_opt[:num_voigt]
fwhm_G_opt = param_opt[num_voigt:2*num_voigt]
fwhm_L_opt = param_opt[2*num_voigt:3*num_voigt]
e0_edge_opt = param_opt[3*num_voigt:3*num_voigt+num_edge]
fwhm_edge_opt = param_opt[3*num_voigt+num_edge:]
# Calc individual chi2
chi = res_lsq['fun']
num_param_tot = num_comp+num_param-np.sum(fix_param_idx)
chi2 = 2*res_lsq['cost']
red_chi2 = chi2/(chi.size-num_param_tot)
param_name = np.empty(param_opt.size, dtype=object)
for i in range(num_voigt):
param_name[i] = f'e0_{i+1}'
param_name[num_voigt+i] = f'fwhm_(G, {i+1})'
param_name[2*num_voigt+i] = f'fwhm_(L, {i+1})'
if edge is not None:
for i in range(num_edge):
param_name[3*num_voigt+i] = f'E0_({edge}, {i+1})'
if edge == 'g':
for i in range(num_edge):
param_name[3*num_voigt+num_edge+i] = \
f'fwhm_(G, edge, {i+1})'
elif edge == 'l':
for i in range(num_edge):
param_name[3*num_voigt+num_edge+i] = \
f'fwhm_(L, edge, {i+1})'
A = np.empty((num_comp, e.size))
for i in range(num_voigt):
A[i, :] = voigt(e-e0_opt[i], fwhm_G_opt[i], fwhm_L_opt[i])
base_start = num_voigt
if edge is not None:
base_start = base_start+num_edge
if edge == 'g':
for i in range(num_edge):
A[num_voigt+i, :] = \
edge_gaussian(e-e0_edge_opt[i], fwhm_edge_opt[i])
elif edge == 'l':
for i in range(num_edge):
A[num_voigt+i, :] = \
edge_lorenzian(e-e0_edge_opt[i], fwhm_edge_opt[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)
fit = c@A
if edge is not None:
fit_comp = np.einsum('i,ij->ij', c[:num_voigt+num_edge],
A[:num_voigt+num_edge, :])
else:
fit_comp = np.einsum('i,ij->ij', c[:num_voigt],
A[:num_voigt, :])
base = None
if base_order is not None:
base = c[base_start:]@A[base_start:, :]
res = intensity - fit
jac = res_lsq['jac']
hes = jac.T @ jac
cov = np.zeros_like(hes)
n_free_param = np.sum(~fix_param_idx)
mask_2d = np.einsum('i,j->ij', ~fix_param_idx, ~fix_param_idx)
cov[mask_2d] = np.linalg.inv(hes[mask_2d].reshape(
(n_free_param, n_free_param))).flatten()
cov_scaled = red_chi2*cov
param_eps = np.sqrt(np.diag(cov_scaled))
corr = cov_scaled.copy()
weight = np.einsum('i,j->ij', param_eps, param_eps)
corr[mask_2d] = corr[mask_2d]/weight[mask_2d]
result = StaticResult()
result['model'] = 'voigt'
result['e'] = e
result['intensity'] = intensity
result['eps'] = eps
result['fit'] = fit
result['fit_comp'] = fit_comp
result['res'] = res
result['base_order'] = base_order
result['edge'] = edge
result['n_voigt'] = num_voigt
result['n_edge'] = num_edge
result['param_name'] = param_name
result['x'] = param_opt
result['bounds'] = bound
result['base'] = base
result['c'] = c
result['chi2'] = chi2
result['aic'] = chi.size*np.log(chi2/chi.size)+2*num_param_tot
result['bic'] = chi.size * \
np.log(chi2/chi.size)+num_param_tot*np.log(chi.size)
result['red_chi2'] = red_chi2
result['nfev'] = res_go['nfev'] + res_lsq['nfev']
result['n_param'] = num_param_tot
result['num_pts'] = chi.size
result['jac'] = jac
result['cov'] = cov
result['corr'] = corr
result['cov_scaled'] = cov_scaled
result['x_eps'] = param_eps
result['method_lsq'] = method_lsq
result['message_lsq'] = res_lsq['message']
result['success_lsq'] = res_lsq['success']
if result['success_lsq']:
result['status'] = 0
else:
result['status'] = -1
if method_glb is not None:
result['method_glb'] = method_glb
result['message_glb'] = res_go['message'][0]
else:
result['method_glb'] = None
result['message_glb'] = None
return result