'''
_transient_dmp_osc:
submodule 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 Optional, Union, Sequence, Tuple
import numpy as np
from ..mathfun.irf import calc_eta, calc_fwhm
from .transient_result import TransientResult
from ._ampgo import ampgo
from scipy.optimize import basinhopping
from scipy.optimize import least_squares
from ..mathfun.A_matrix import make_A_matrix_dmp_osc, fact_anal_A
from ..res.parm_bound import set_bound_t0, set_bound_tau
from ..res.res_osc import residual_dmp_osc, res_grad_dmp_osc
from ..res.res_osc import residual_dmp_osc_same_t0, res_grad_dmp_osc_same_t0
GLBSOLVER = {'basinhopping': basinhopping, 'ampgo': ampgo}
[docs]def fit_transient_dmp_osc(irf: str, fwhm_init: Union[float, np.ndarray],
t0_init: np.ndarray, tau_init: np.ndarray, period_init: np.ndarray,
method_glb: Optional[str] = None,
method_lsq: Optional[str] = 'trf',
kwargs_glb: Optional[dict] = None,
kwargs_lsq: Optional[dict] = None,
bound_fwhm: Optional[Sequence[Tuple[float, float]]] = None,
bound_t0: Optional[Sequence[Tuple[float, float]]] = None,
bound_tau: Optional[Sequence[Tuple[float, float]]] = None,
bound_period: Optional[Sequence[Tuple[float, float]]] = None,
name_of_dset: Optional[Sequence[str]] = None,
same_t0: Optional[bool] = False,
t: Optional[Sequence[np.ndarray]] = None,
intensity: Optional[Sequence[np.ndarray]] = None,
eps: Optional[Sequence[np.ndarray]] = None) -> TransientResult:
'''
driver routine for fitting multiple data set of time delay scan data with
sum of the convolution of dmped oscillation and instrumental response function.
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. (basinhopping)
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 obtain.
Thus, in this stage, it uses numerical jacobian.
Args:
irf ({'g', 'c', 'pv'}): shape of instrumental response function
'g': gaussian shape
'c': cauchy shape
'pv': pseudo voigt shape (kind: 2)
fwhm_init (float or np.ndarray): initial full width at half maximum for instrumental response function
* if irf in ['g', 'c'] then fwhm_init is float
* if irf == 'pv' then fwhm_init is the `numpy.ndarray` with [fwhm_G, fwhm_L]
t0_init (np.ndarray): time zeros for each scan
tau_init (np.ndarray): lifetime constant for each damped oscillation component
period_init (np.ndarray): period of each oscillation component
method_glb ({None, 'basinhopping', 'ampgo'}): Method for global optimization
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_fwhm (sequence of tuple): boundary for irf parameter. If upper and lower bound are same,
driver assumes that the parameter is fixed during optimization. If `bound_fwhm` is `None`,
the upper and lower bound are given as `(fwhm_init/2, 2*fwhm_init)`.
bound_t0 (sequence of tuple): boundary for time zero parameter.
If `bound_t0` is `None`, the upper and lower bound are given by `set_bound_t0`.
bound_tau (sequence of tuple): boundary for lifetime constant for damped oscillation component,
if `bound_tau` is `None`, the upper and lower bound are given by `set_bound_tau`.
bound_period (sequence of tuple): boundary for period of damped oscillation component,
if `bound_period` is `None`, the upper and lower bound are given by `set_bound_tau`.
name_of_dset (sequence of str): name of each dataset
same_t0 (bool): Whether or not time zero of every time delay scan in the same dataset should be same
t (sequence of np.narray): time scan range for each datasets
intensity (sequence of np.ndarray): sequence of intensity pf datasets for time delay scan (it should not contain time scan range)
eps (sequence of np.ndarray): sequence of estimated errors of each dataset
Returns:
TransientResult class object
'''
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 irf is not None and irf not in ['g', 'c', 'pv']:
raise Exception('Unsupported shape of instrumental response function Edge.')
num_comp = tau_init.size
num_irf = 1*(irf in ['g', 'c'])+2*(irf == 'pv')
num_param = num_irf+t0_init.size+2*num_comp
param = np.empty(num_param, dtype=float)
fix_param_idx = np.empty(num_param, dtype=bool)
param[:num_irf] = fwhm_init
param[num_irf:num_irf+t0_init.size] = t0_init
param[num_irf+t0_init.size:num_irf+t0_init.size+num_comp] = tau_init
param[num_irf+t0_init.size+num_comp:num_irf +
t0_init.size+2*num_comp] = period_init
bound = num_param*[None]
if bound_fwhm is None:
for i in range(num_irf):
bound[i] = (param[i]/2, 2*param[i])
else:
bound[:num_irf] = bound_fwhm
if bound_t0 is None:
for i in range(t0_init.size):
bound[i+num_irf] = set_bound_t0(t0_init[i], fwhm_init)
else:
bound[num_irf:num_irf+t0_init.size] = bound_t0
if bound_tau is None:
for i in range(num_comp):
bound[i+num_irf +
t0_init.size] = set_bound_tau(tau_init[i], fwhm_init)
else:
bound[num_irf+t0_init.size:num_irf+t0_init.size+num_comp] = bound_tau
if bound_period is None:
for i in range(num_comp):
bound[i+num_irf+t0_init.size +
num_comp] = set_bound_tau(period_init[i], fwhm_init)
else:
bound[num_irf+t0_init.size+num_comp:num_irf +
t0_init.size+2*num_comp] = bound_period
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_comp, irf, fix_param_idx, t, 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'] = min_go_kwargs['args']
minimizer_kwargs['jac'] = min_go_kwargs['jac']
minimizer_kwargs['bounds'] = min_go_kwargs['bounds']
kwargs_glb['minimizer_kwargs'] = minimizer_kwargs
else:
kwargs_glb = {'minimizer_kwargs': min_go_kwargs}
if same_t0:
res_go = GLBSOLVER[method_glb](res_grad_dmp_osc_same_t0, param, **kwargs_glb)
else:
res_go = GLBSOLVER[method_glb](res_grad_dmp_osc, param, **kwargs_glb)
else:
res_go = {}
res_go['x'] = param
res_go['message'] = None
res_go['nfev'] = 0
param_gopt = res_go['x']
args_lsq = (num_comp, irf, t, intensity, eps)
if kwargs_lsq is not None:
_ = kwargs_lsq.pop('args', None)
_ = kwargs_lsq.pop('kwargs', None)
kwargs_lsq['args'] = args_lsq
else:
kwargs_lsq = {'args': args_lsq}
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
if same_t0:
res_lsq = least_squares(residual_dmp_osc_same_t0, param_gopt,
method=method_lsq, bounds=bound_tuple, **kwargs_lsq)
else:
res_lsq = least_squares(residual_dmp_osc, param_gopt,
method=method_lsq, bounds=bound_tuple, **kwargs_lsq)
param_opt = res_lsq['x']
fwhm_opt = param_opt[:num_irf]
tau_opt = param_opt[num_irf+t0_init.size:num_irf+t0_init.size+num_comp]
period_opt = param_opt[num_irf+t0_init.size +
num_comp:num_irf+t0_init.size+2*num_comp]
fit = np.empty(len(t), dtype=object)
res = np.empty(len(t), dtype=object)
num_tot_scan = 0
for i in range(len(t)):
num_tot_scan = num_tot_scan + intensity[i].shape[1]
fit[i] = np.empty(intensity[i].shape)
res[i] = np.empty(intensity[i].shape)
# Calc individual chi2
chi = res_lsq['fun']
num_param_tot = 2*num_tot_scan*num_comp+num_param-np.sum(fix_param_idx)
chi2 = 2*res_lsq['cost']
red_chi2 = chi2/(chi.size-num_param_tot)
start = 0
end = 0
chi2_ind = np.empty(len(t), dtype=object)
red_chi2_ind = np.empty(len(t), dtype=object)
num_param_ind = 4*num_comp+2+1*(irf == 'pv')
for i in range(len(t)):
step = intensity[i].shape[0]
chi2_ind_aux = np.empty(intensity[i].shape[1], dtype=float)
for j in range(intensity[i].shape[1]):
end = start + step
chi2_ind_aux[j] = np.sum(chi[start:end]**2)
start = end
chi2_ind[i] = chi2_ind_aux
red_chi2_ind[i] = chi2_ind[i]/(intensity[i].shape[0]-num_param_ind)
param_name = np.empty(param_opt.size, dtype=object)
c = np.empty(len(t), dtype=object)
phase = np.empty(len(t), dtype=object)
t0_idx = num_irf
if irf == 'g':
fwhm_pv = fwhm_opt[0]
eta = 0
param_name[0] = 'fwhm_G'
elif irf == 'c':
fwhm_pv = fwhm_opt[0]
eta = 1
param_name[0] = 'fwhm_L'
else:
fwhm_pv = calc_fwhm(fwhm_opt[0], fwhm_opt[1])
eta = calc_eta(fwhm_opt[0], fwhm_opt[1])
param_name[0] = 'fwhm_G'
param_name[1] = 'fwhm_L'
for i in range(len(t)):
c[i] = np.empty((num_comp, intensity[i].shape[1]))
phase[i] = np.empty((num_comp, intensity[i].shape[1]))
if same_t0:
A = \
make_A_matrix_dmp_osc(t[i]-param_opt[t0_idx],
fwhm_pv, tau_opt, period_opt, irf, eta)
for j in range(intensity[i].shape[1]):
if not same_t0:
A = \
make_A_matrix_dmp_osc(t[i]-param_opt[t0_idx],
fwhm_pv, tau_opt, period_opt, irf, eta)
tmp = fact_anal_A(A, intensity[i][:, j], eps[i][:, j])
c[i][:, j] = np.sqrt(tmp[:num_comp]**2+tmp[num_comp:]**2)
phase[i][:, j] = -np.arctan2(tmp[num_comp:], tmp[:num_comp])
fit[i][:, j] = tmp @ A
if not same_t0:
param_name[t0_idx] = f't_0_{i+1}_{j+1}'
t0_idx = t0_idx + 1
if same_t0:
param_name[t0_idx] = f't_0_{i}'
t0_idx = t0_idx + 1
res[i] = intensity[i] - fit[i]
for i in range(num_comp):
param_name[num_irf+t0_init.size+i] = f'tau_{i+1}'
param_name[num_irf+t0_init.size+num_comp+i] = f'period_{i+1}'
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 = TransientResult()
result['same_t0'] = same_t0
# save experimental fitting data
if name_of_dset is None:
name_of_dset = np.empty(len(t), dtype=object)
for i in range(len(t)):
name_of_dset[i] = f'dataset_{i+1}'
result['name_of_dset'] = name_of_dset
result['t'] = t
result['intensity'] = intensity
result['eps'] = eps
result['model'] = 'dmp_osc'
result['fit'] = fit
result['res'] = res
result['irf'] = irf
result['fwhm'] = fwhm_pv
result['eta'] = eta
result['param_name'] = param_name
result['x'] = param_opt
result['bounds'] = bound
result['base'] = False
result['c'] = c
result['phase'] = phase
result['chi2'] = chi2
result['chi2_ind'] = chi2_ind
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['red_chi2_ind'] = red_chi2_ind
result['nfev'] = res_go['nfev'] + res_lsq['nfev']
result['n_param'] = num_param_tot
result['n_param_ind'] = num_param_ind
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
result['n_osc'] = tau_init.size
result['n_decay'] = 0
return result