'''
exp_conv_irf:
submodule for fitting data with sum of exponential decay or damped oscillation convolved with irf
:copyright: 2021-2022 by pistack (Junho Lee).
:license: LGPL3.
'''
from typing import Tuple, Optional
import numpy as np
import scipy.linalg as LA
from .rate_eq import compute_signal_irf
from .A_matrix import make_A_matrix_exp, make_A_matrix_dmp_osc
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def exp_conv(t: np.ndarray, fwhm: float, tau: np.ndarray,
c: np.ndarray, base: Optional[bool] = True,
irf: Optional[str] = 'g', eta: Optional[float] = None) -> np.ndarray:
'''
Constructs the model for the convolution of n exponential and
instrumental response function
Supported instrumental response function are
* g: gaussian distribution
* c: cauchy distribution
* pv: pseudo voigt profile
Args:
t: time
fwhm: full width at half maximum of instrumental response function
tau: life time for each component
c: coefficient for each component
base: whether or not include baseline [default: True]
irf: shape of instrumental
response function [default: g]
* 'g': normalized gaussian distribution,
* 'c': normalized cauchy distribution,
* 'pv': pseudo voigt profile :math:`(1-\\eta)g(t, {fwhm}) + \\eta c(t, {fwhm})`
eta: mixing parameter for pseudo voigt profile
(only needed for pseudo voigt profile)
Returns:
Convolution of the sum of n exponential decays and instrumental
response function.
Note:
Size of weight `c` is `num_comp+1` when base is set to true.
Otherwise, its size is `num_comp`.
'''
A = make_A_matrix_exp(t, fwhm, tau, base, irf, eta)
y = c@A
return y
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def fact_anal_exp_conv(t: np.ndarray,
fwhm: float,
tau: np.ndarray,
base: Optional[bool] = True,
irf: Optional[str] = 'g',
eta: Optional[float] = None,
intensity: Optional[np.ndarray] = None,
eps: Optional[np.ndarray] = None
) -> np.ndarray:
'''
Estimate the best coefficiets when full width at half maximum fwhm
and life constant tau are given
When you fits your model to tscan data, you need to have
good initial guess for not only life time of
each component but also coefficients. To help this it solves
linear least square problem to find best coefficients when fwhm and
tau are given.
Supported instrumental response functions are
1. 'g': gaussian distribution
2. 'c': cauchy distribution
3. 'pv': pseudo voigt profile
Args:
t: time
fwhm: full width at half maximum of instrumental response function
tau: life time for each component
base: whether or not include baseline [default: True]
irf: shape of instrumental
response function [default: g]
* 'g': normalized gaussian distribution,
* 'c': normalized cauchy distribution,
* 'pv': pseudo voigt profile :math:`(1-\\eta)g(t, {fwhm}) + \\eta c(t, {fwhm})`
eta: mixing parameter for pseudo voigt profile
(only needed for pseudo voigt profile)
intensity: intensity of time scan data to fit
eps: standard error of data
Returns:
Best coefficient for given fwhm and tau, if base is set to `True` then
size of coefficient is `num_comp + 1`, otherwise is `num_comp`.
Note:
the dimension of the intensity must be one.
'''
A = make_A_matrix_exp(t, fwhm, tau, base, irf, eta)
if eps is None:
eps = np.ones_like(intensity)
y = intensity/eps
A = np.einsum('j,ij->ij', 1/eps, A)
c, _, _, _ = LA.lstsq(A.T, y, cond=1e-2)
return c
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def rate_eq_conv(t: np.ndarray, fwhm: float,
diff_abs: np.ndarray, eigval: np.ndarray, V: np.ndarray, c: np.ndarray,
irf: Optional[str] = 'g', eta: Optional[float] = None) -> np.ndarray:
'''
Constructs signal model rate equation with
instrumental response function
Supported instrumental response function are
* g: gaussian distribution
* c: cauchy distribution
* pv: pseudo voigt profile
Args:
t: time
fwhm: full width at half maximum of instrumental response function
diff_abs: coefficient for each excited state
eigval: eigenvalue of rate equation matrix
V: eigenvector of rate equation matrix
c: coefficient to match initial condition of rate equation
irf: shape of instrumental
response function [default: g]
* 'g': normalized gaussian distribution,
* 'c': normalized cauchy distribution,
* 'pv': pseudo voigt profile :math:`(1-\\eta)g(t, {fwhm}) + \\eta c(t, {fwhm})`
eta: mixing parameter for pseudo voigt profile
(only needed for pseudo voigt profile)
Returns:
Convolution of the solution of the rate equation and instrumental
response function.
'''
A = compute_signal_irf(t, eigval, V, c, fwhm, irf, eta)
y = diff_abs@A
return y
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def fact_anal_rate_eq_conv(t: np.ndarray, fwhm: float,
eigval: np.ndarray, V: np.ndarray, c: np.ndarray,
exclude: Optional[str] = None, irf: Optional[str] = 'g',
eta: Optional[float] = None, intensity: Optional[np.ndarray] = None,
eps: Optional[np.ndarray] = None) -> np.ndarray:
'''
Estimate the best coefficiets when full width at half maximum fwhm
and eigenvector and eigenvalue of rate equation matrix are given
Supported instrumental response functions are
1. 'g': gaussian distribution
2. 'c': cauchy distribution
3. 'pv': pseudo voigt profile
Args:
t: time
fwhm: full width at half maximum of instrumental response function
eigval: eigenvalue of rate equation matrix
V: eigenvector of rate equation matrix
c: coefficient to match initial condition of rate equation
exclude: exclude either 'first' or 'last' element or both 'first' and 'last' element.
* 'first' : exclude first element
* 'last' : exclude last element
* 'first_and_last' : exclude both first and last element
* None : Do not exclude any element [default]
irf: shape of instrumental
response function [default: g]
* 'g': normalized gaussian distribution,
* 'c': normalized cauchy distribution,
* 'pv': pseudo voigt profile :math:`(1-\\eta)g(t, {fwhm}) + \\eta c(t, {fwhm})`
eta: mixing parameter for pseudo voigt profile
(only needed for pseudo voigt profile)
intensity: intensity of time scan data to fit
eps: standard error of data
Returns:
Best coefficient for each component.
Note:
1. eigval, V, c should be obtained from solve_model
2. The dimension of the intensity should be one.
'''
A = compute_signal_irf(t, eigval, V, c, fwhm, irf, eta)
diff_abs = np.zeros(A.shape[0])
if eps is None:
eps = np.ones_like(intensity)
y = intensity/eps
if exclude == 'first_and_last':
B = np.einsum('j,ij->ij', 1/eps, A[1:-1, :])
elif exclude == 'first':
B = np.einsum('j,ij->ij', 1/eps, A[1:, :])
elif exclude == 'last':
B = np.einsum('j,ij->ij', 1/eps, A[:-1, :])
else:
B = np.einsum('j,ij->ij', 1/eps, A)
coeff, _, _, _ = LA.lstsq(B.T, y, cond=1e-2)
if exclude == 'first_and_last':
diff_abs[1:-1] = coeff
elif exclude == 'first':
diff_abs[1:] = coeff
elif exclude == 'last':
diff_abs[:-1] = coeff
else:
diff_abs = coeff
return diff_abs
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def dmp_osc_conv(t: np.ndarray, fwhm: float,
tau: np.ndarray,
T: np.ndarray,
phase: np.ndarray,
c: np.ndarray,
irf: Optional[str] = 'g',
eta: Optional[float] = None
) -> np.ndarray:
'''
Constructs convolution of sum of damped oscillation and
instrumental response function
Supported instrumental response function are
* g: gaussian distribution
* c: cauchy distribution
* pv: pseudo voigt profile
Args:
t: time
fwhm: full width at half maximum of instrumental response function
tau: lifetime of vibration
T: period of vibration
phase: phase factor
c: coefficient for each damping oscillation component
irf: shape of instrumental
response function [default: g]
* 'g': normalized gaussian distribution,
* 'c': normalized cauchy distribution,
* 'pv': pseudo voigt profile :math:`(1-\\eta)g(t, {fwhm}) + \\eta c(t, {fwhm})`
eta: mixing parameter for pseudo voigt profile
(only needed for pseudo voigt profile)
Returns:
Convolution of sum of damped oscillation and instrumental
response function.
'''
c_osc = np.zeros(2*c.size)
for i in range(c.size):
c_osc[i] = c[i]*np.cos(phase[i])
c_osc[i+c.size] = -c[i]*np.sin(phase[i])
A = make_A_matrix_dmp_osc(t, fwhm, tau, T, irf, eta)
y = c_osc@A
return y
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def fact_anal_dmp_osc_conv(t: np.ndarray,
fwhm: float,
tau: np.ndarray, T: np.ndarray,
irf: Optional[str] = 'g',
eta: Optional[float] = None,
intensity: Optional[np.ndarray] = None,
eps: Optional[np.ndarray] = None
) -> Tuple[np.ndarray, np.ndarray]:
'''
Estimate the best coefficiets and phase factor
when full width at half maximum fwhm
, life constant tau and period of vibration T are given
Supported instrumental response functions are
1. 'g': gaussian distribution
2. 'c': cauchy distribution
3. 'pv': pseudo voigt profile
Args:
t: time
fwhm: full width at half maximum of instrumental response function
tau: life time for each component
T: period of vibration of each component
phase: phase factor for each component
irf: shape of instrumental
response function [default: g]
* 'g': normalized gaussian distribution,
* 'c': normalized cauchy distribution,
* 'pv': pseudo voigt profile :math:`(1-\\eta)g(t, {fwhm}) + \\eta c(t, {fwhm})`
eta: mixing parameter for pseudo voigt profile
(only needed for pseudo voigt profile)
intensity: intensity of time scan data to fit
eps: standard error of data
Returns:
Pair of phase factor and Best coefficient for given damped oscillation component.
Note:
the dimension of the intensity should be one.
'''
A = make_A_matrix_dmp_osc(t, fwhm, tau, T, irf, eta)
if eps is None:
eps = np.ones_like(intensity)
y = intensity/eps
A = np.einsum('j,ij->ij', 1/eps, A)
c, _, _, _ = LA.lstsq(A.T, y, cond=1e-2)
c_cosine = c[:tau.size]
c_sine = c[tau.size:]
c_amp = np.sqrt(c_cosine**2+c_sine**2)
phase_factor = -np.arctan2(c_sine, c_cosine)
return phase_factor, c_amp
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def sum_exp_dmp_osc_conv(t: np.ndarray, fwhm: float,
tau: np.ndarray,
tau_osc: np.ndarray,
T: np.ndarray,
phase: np.ndarray,
c: np.ndarray,
c_osc: np.ndarray,
base: Optional[bool] = True,
irf: Optional[str] = 'g',
eta: Optional[float] = None
) -> np.ndarray:
'''
Constructs convolution of sum of exponential decay and damped oscillation and
instrumental response function
Supported instrumental response function are
* g: gaussian distribution
* c: cauchy distribution
* pv: pseudo voigt profile
Args:
t: time
fwhm: full width at half maximum of instrumental response function
tau: lifetime of decay
tau_osc: lifetime of vibration
T: period of vibration
phase: phase factor
c: coefficient for each decay component
c_osc: coefficient for each vibrational component
base: Whether or not use baseline feature for exponential decay component [default: True]
irf: shape of instrumental
response function [default: g]
* 'g': normalized gaussian distribution,
* 'c': normalized cauchy distribution,
* 'pv': pseudo voigt profile :math:`(1-\\eta)g(t, {fwhm}) + \\eta c(t, {fwhm})`
eta: mixing parameter for pseudo voigt profile
(only needed for pseudo voigt profile)
Returns:
Convolution of sum of exponential decay and damped oscillation and instrumental
response function.
'''
c_osc_2 = np.zeros(2*c_osc.size)
for i in range(c_osc.size):
c_osc_2[i] = c_osc[i]*np.cos(phase[i])
c_osc_2[i+c_osc.size] = -c_osc[i]*np.sin(phase[i])
A = make_A_matrix_exp(t, fwhm, tau, base, irf, eta)
A_osc = make_A_matrix_dmp_osc(t, fwhm, tau_osc, T, irf, eta)
y = c@A + c_osc_2@A_osc
return y
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def fact_anal_sum_exp_dmp_osc_conv(t: np.ndarray, fwhm: float,
tau: np.ndarray, tau_osc: np.ndarray,
T: np.ndarray,
base: Optional[bool] = True,
irf: Optional[str] = 'g',
eta: Optional[float] = None,
intensity: Optional[np.ndarray] = None,
eps: Optional[np.ndarray] = None) -> \
Tuple[np.ndarray, np.ndarray, np.ndarray]:
'''
Estimate the best coefficiets and phase factor of oscillation part
when full width at half maximum fwhm
, lifetime constant of decay component tau, lifetime constant of
oscillation component tau_osc and period of vibration T are given
Supported instrumental response functions are
1. 'g': gaussian distribution
2. 'c': cauchy distribution
3. 'pv': pseudo voigt profile
Args:
t: time
fwhm: full width at half maximum of instrumental response function
tau: life time for each decay component
tau_osc: life time for each vibration component
base: Whether or not use baseline feature for exponential decay component
[default: True]
T: period of vibration of each component
phase: phase factor for each component
irf: shape of instrumental
response function [default: g]
* 'g': normalized gaussian distribution,
* 'c': normalized cauchy distribution,
* 'pv': pseudo voigt profile :math:`(1-\\eta)g(t, {fwhm}) + \\eta c(t, {fwhm})`
eta: mixing parameter for pseudo voigt profile
(only needed for pseudo voigt profile)
intensity: intensity of time scan data to fit
eps: standard error of data
Returns:
Tuple of best coefficient and phase_factor for given decay and damped oscillation component.
(c_(decay), phase_(osc), c_(osc))
Note:
the dimension of the intensity should be one.
'''
A = np.empty((tau.size+1*base+2*tau_osc.size, t.size))
A[:tau.size+1*base, :] = make_A_matrix_exp(t, fwhm, tau, base, irf, eta)
A[tau.size+1*base:,
:] = make_A_matrix_dmp_osc(t, fwhm, tau_osc, T, irf, eta)
if eps is None:
eps = np.ones_like(intensity)
y = intensity/eps
A = np.einsum('j,ij->ij', 1/eps, A)
c, _, _, _ = LA.lstsq(A.T, y, cond=1e-2)
c_cosine = c[tau.size+1*base:tau.size+1*base+tau_osc.size]
c_sine = c[tau.size+1*base+tau_osc.size:]
c_osc = np.sqrt(c_cosine**2+c_sine**2)
phase_osc = -np.arctan2(c_sine, c_cosine)
return c[:tau.size+1*base], phase_osc, c_osc