def test_saveload_modelresult_roundtrip(method):
"""Test for modelresult.loads()/dumps() and repeating that"""
def mfunc(x, a, b):
return a * (x - b)
model = Model(mfunc)
params = model.make_params(a=0.1, b=3.0)
params['a'].set(min=.01, max=1, brute_step=0.01)
params['b'].set(min=.01, max=3.1, brute_step=0.01)
np.random.seed(2020)
xx = np.linspace(-5, 5, 201)
yy = 0.5 * (xx - 0.22) + np.random.normal(scale=0.01, size=len(xx))
result1 = model.fit(yy, params=params, x=xx, method=method)
result2 = ModelResult(model, Parameters())
result2.loads(result1.dumps(), funcdefs={'mfunc': mfunc})
result3 = ModelResult(model, Parameters())
result3.loads(result2.dumps(), funcdefs={'mfunc': mfunc})
assert result3 is not None
assert_allclose(result2.params['a'], 0.5, rtol=1.0e-2)
assert_allclose(result2.params['b'], 0.22, rtol=1.0e-2)
assert_allclose(result3.params['a'], 0.50, rtol=1.0e-2)
assert_allclose(result3.params['b'], 0.22, rtol=1.0e-2)
def fit_sine(x_data: np.ndarray,
y_data: np.ndarray,
initial_parameters=None,
positive_amplitude=True) -> Tuple[Dict[str, Any], Dict[str, Any]]:
""" Fit a sine wave for the inputted data; see sine function in functions.py for model
Args:
x_data: x data points
y_data: data to be fitted
initial_parameters: list of 4 floats with initial guesses for: amplitude, frequency, phase and offset
positive_amplitude: If True, then enforce the amplitude to be positive
Returns:
result_dict
"""
if initial_parameters is None:
initial_parameters = _estimate_initial_parameters_sine(x_data, y_data)
lmfit_model = Model(sine)
if positive_amplitude:
lmfit_model.set_param_hint('amplitude', min=0)
lmfit_result = lmfit_model.fit(y_data,
x=x_data,
**dict(
zip(lmfit_model.param_names,
initial_parameters)),
method='least_squares')
result_dict = extract_lmfit_parameters(lmfit_model, lmfit_result)
return result_dict['fitted_parameters'], result_dict
def test_lrtest(self):
rng = np.random.RandomState(RANDOM_SEED)
a, b = 1, 1
a_init, b_init = 2, 1
alfa = 0.05
noise = 0.03
t = np.linspace(0, 12)
def f(t, a, b):
return b + np.exp(-a * t)
y = f(t, a, b) + rng.normal(0, noise, len(t))
model = Model(f)
params = model.make_params(a=a_init, b=b_init)
two_var_fit = model.fit(y, t=t, params=params)
params['a'].set(vary=False)
params['b'].set(vary=True)
one_var_fit = model.fit(y, t=t, params=params)
prefer_m1, pval, D, ddf = curveball.models.lrtest(
one_var_fit, two_var_fit, alfa)
self.assertTrue(prefer_m1)
def _init_model(self):
#@todo make a separate class that subclasses Model.
# potentially allow users to change it.
self._model = Model(self._modelfunc)
for p, q in self._params.items():
self._model.set_param_hint(p,
value=q.value,
min=q.min,
max=q.max,
vary=q.vary)
self._model.make_params()
def test_saveload_modelresult_roundtrip():
"""Test for modelresult.loads()/dumps() and repeating that"""
def mfunc(x, a, b):
return a * (x - b)
model = Model(mfunc)
params = model.make_params(a=0.0, b=3.0)
xx = np.linspace(-5, 5, 201)
yy = 0.5 * (xx - 0.22) + np.random.normal(scale=0.01, size=len(xx))
result1 = model.fit(yy, params, x=xx)
result2 = ModelResult(model, Parameters())
result2.loads(result1.dumps(), funcdefs={'mfunc': mfunc})
result3 = ModelResult(model, Parameters())
result3.loads(result2.dumps(), funcdefs={'mfunc': mfunc})
assert result3 is not None
assert_param_between(result2.params['a'], 0.48, 0.52)
assert_param_between(result2.params['b'], 0.20, 0.25)
assert_param_between(result3.params['a'], 0.48, 0.52)
assert_param_between(result3.params['b'], 0.20, 0.25)
def test_saveload_modelresult_roundtrip():
"""Test for modelresult.loads()/dumps() and repeating that"""
def mfunc(x, a, b):
return a * (x-b)
model = Model(mfunc)
params = model.make_params(a=0.0, b=3.0)
xx = np.linspace(-5, 5, 201)
yy = 0.5 * (xx - 0.22) + np.random.normal(scale=0.01, size=len(xx))
result1 = model.fit(yy, params, x=xx)
result2 = ModelResult(model, Parameters())
result2.loads(result1.dumps(), funcdefs={'mfunc': mfunc})
result3 = ModelResult(model, Parameters())
result3.loads(result2.dumps(), funcdefs={'mfunc': mfunc})
assert result3 is not None
assert_param_between(result2.params['a'], 0.48, 0.52)
assert_param_between(result2.params['b'], 0.20, 0.25)
assert_param_between(result3.params['a'], 0.48, 0.52)
assert_param_between(result3.params['b'], 0.20, 0.25)
def fit_semiclassical(field,
conductance_data,
phase_relaxation_length_guess=100,
cubic_dresselhaus_guess=0.0,
linear_dresselhaus_guess=0.0,
rashba_guess=np.pi / 4):
print(r"Calculating best fit at zero in-plane field...")
start = time.time()
model = Model(magnetoconductivity_semiclassical)
# set parameter hints
model.set_param_hint('l_phi',
min=10,
max=1000,
value=phase_relaxation_length_guess,
vary=True)
model.set_param_hint('theta_alpha',
min=-2 * np.pi,
max=2 * np.pi,
value=rashba_guess,
vary=True)
model.set_param_hint('theta_beta1',
min=0,
max=2 * np.pi,
value=linear_dresselhaus_guess,
vary=False)
model.set_param_hint('theta_beta3',
value=cubic_dresselhaus_guess,
vary=False)
model.set_param_hint('B_magnitude', value=0.0, vary=False)
model.set_param_hint('B_angle', value=0.0, vary=False)
params = model.make_params()
# perform the fit
res = model.fit(conductance_data, params, field=field, method='nelder')
print(res.fit_report())
res = model.fit(conductance_data, res.params, field=field)
print(res.fit_report())
phi = res.best_values['l_phi']
phi_std = deepcopy(res.params['l_phi'].stderr)
alpha = res.best_values['theta_alpha']
alpha_std = deepcopy(res.params['theta_alpha'].stderr)
beta = res.best_values['theta_beta1']
beta_std = res.params['theta_beta1'].stderr
gamma = res.best_values['theta_beta3']
gamma_std = deepcopy(res.params['theta_beta3'].stderr)
print(r"done in {0:.1f} seconds.".format(time.time() - start))
return phi, phi_std, alpha, alpha_std, beta, beta_std, gamma, gamma_std, res.best_fit
def fit_semiclassical_inplane(field,
conductance_data,
phase_relaxation_length_fixed=680,
cubic_dresselhaus_fixed=0.0,
linear_dresselhaus_fixed=0.03,
rashba_fixed=1.17,
B_magnitude=0.0,
B_angle=0.0):
print(r"Calculating best fit in the presence of in-plane field...")
start = time.time()
model = Model(magnetoconductivity_semiclassical)
# set parameter hints
model.set_param_hint('l_phi',
value=phase_relaxation_length_fixed,
vary=False)
model.set_param_hint('theta_alpha', value=rashba_fixed, vary=False)
model.set_param_hint('theta_beta1',
value=linear_dresselhaus_fixed,
vary=False)
model.set_param_hint('theta_beta3',
value=cubic_dresselhaus_fixed,
vary=False)
model.set_param_hint('B_magnitude',
min=0.0,
max=0.2,
value=B_magnitude,
vary=True)
model.set_param_hint('B_angle', value=B_angle, vary=False)
params = model.make_params()
# perform the fit
res = model.fit(conductance_data, params, field=field, method='nelder')
print(res.fit_report())
res = model.fit(conductance_data, res.params, field=field)
print(res.fit_report())
r = res.best_values['B_magnitude']
theta = res.best_values['B_angle']
r_stderr = res.params['B_magnitude'].stderr
theta_stderr = res.params['B_angle'].stderr
print(r"done in {0:.1f} seconds.".format(time.time() - start))
return r, theta, r_stderr, theta_stderr, res.best_fit
# <examples/doc_model_savemodel.py>
import numpy as np
from lmfit.model import Model, save_model
def mysine(x, amp, freq, shift):
return amp * np.sin(x * freq + shift)
sinemodel = Model(mysine)
pars = sinemodel.make_params(amp=1, freq=0.25, shift=0)
save_model(sinemodel, 'sinemodel.sav')
# <end examples/doc_model_savemodel.py>
class H2ExcitationFit(ExcitationFit):
r"""Tool for fitting temperatures, column densities, and ortho-to-para ratio(`OPR`) from an :math:`H_2` excitation diagram. It takes as input a set of :math:`H_2` rovibrational line observations with errors represented as :class:`~pdrtpy.measurement.Measurement`.
Often, excitation diagrams show evidence of both "hot" and "cold" gas components, where the cold gas dominates the intensity in the low `J` transitions and the hot gas dominates in the high `J` transitions. Given data over several transitions, one can fit for :math:`T_{cold}, T_{hot}, N_{total} = N_{cold}+ N_{hot}`, and optionally `OPR`. One needs at least 5 points to fit the temperatures and column densities (slope and intercept :math:`\times 2`), though one could compute (not fit) them with only 4 points. To additionally fit `OPR`, one should have 6 points (5 degrees of freedom).
Once the fit is done, :class:`~pdrtpy.plot.ExcitationPlot` can be used to view the results.
:param measurements: Input :math:`H_2` measurements to be fit.
:type measurements: list of :class:`~pdrtpy.measurement.Measurement`.
"""
def __init__(self,
measurements=None,
constantsfile="atomic_constants.tab"):
super().__init__(measurements, constantsfile)
self._canonical_opr = 3.0
self._opr = Measurement(data=[self._canonical_opr], uncertainty=None)
self._init_params()
self._init_model()
self._fitresult = None
self._temperature = None
self._total_colden = None
# position and size that was used for averaging/fit
self._position = None
self._size = None
def _init_params(self):
#fit input parameters
self._params = Parameters()
# we have to have opr max be greater than 3 so that fitting will work.
# the fit algorithm does not like when the initial value is pinned at one
# of the limits
self._params.add('opr', value=3.0, min=1.0, max=3.5, vary=False)
self._params.add('m1', value=0, min=-1, max=0)
self._params.add('n1', value=15, min=10, max=30)
self._params.add('m2', value=0, min=-1, max=0)
self._params.add('n2', value=15, min=10, max=30)
def _residual(self, params, x, data, error, idx):
# We assume that the column densities passed in have been normalized
# using the canonical OPR=3. Therefore what we are actually fitting is
# the ratio of the actual OPR to the canonical OPR.
# For odd J, input x = Nu/(3*(2J+1) where 3=canonical OPR.
#
# We want the model-data residual to be small, but if the opr
# is different from the canonical value of 3, then data[idx] will
# be low by a factor of 3/opr.
# So we must LOWER model[idx] artificially by dividing it by
# 3/opr, i.e. multiplying by opr/3. This is equivalent to addition in log-space.
p = params.valuesdict()
y1 = 10**(x * p['m1'] + p['n1'])
y2 = 10**(x * p['m2'] + p['n2'])
model = np.log10(y1 + y2)
if params['opr'].vary:
model += np.log10(p['opr'] / self._canonical_opr)
return (model - data) / error
def _modelfunc(self, x, m1, n1, m2, n2, opr, idx=[], fit_opr=False):
'''Function for fitting the excitation curve as sum of two linear functions
and allowing ortho-to-para ratio to vary. Para is even J, ortho is odd J.
:param x: independent axis array
:param m1: slope of first line
:type m1: float
:param n1: intercept of first line
:type n1: float
:param m2: slope of second line
:type m2: float
:param n2: intercept of second line
:type n2: float
:param opr: ortho-to-para ratio
:type opr: float
:type idx: np.ndarray
:param idx: list of indices that may have variable opr (odd J transitions)
:param fit_opr: indicate whether opr will be fit, default False (opr fixed)
:type fit_opr: False
:return: Sum of lines in log space:log10(10**(x*m1+n1) + 10**(x*m2+n2)) + log10(opr/3.0)
:rtype: :class:`numpy.ndarray`
'''
y1 = 10**(x * m1 + n1)
y2 = 10**(x * m2 + n2)
model = np.log10(y1 + y2)
# We assume that the column densities passed in have been normalized
# using the canonical OPR=3. Therefore what we are actually fitting is
# the ratio of the actual OPR to the canonical OPR.
# For odd J, input x = Nu/(3*(2J+1) where 3=canonical OPR.
#
# We want the model-data residual to be small, but if the opr
# is different from the canonical value of 3, then data[idx] will
# be low by a factor of 3/opr.
# So we must LOWER model[idx] artificially by dividing it by
# 3/opr, i.e. multiplying by opr/3. This is equivalent to addition in log-space.
if fit_opr:
model[idx] += np.log10(opr / self._canonical_opr)
return model
def _init_model(self):
#@todo make a separate class that subclasses Model.
# potentially allow users to change it.
self._model = Model(self._modelfunc)
for p, q in self._params.items():
self._model.set_param_hint(p,
value=q.value,
min=q.min,
max=q.max,
vary=q.vary)
self._model.make_params()
def _compute_quantities(self, fitmap):
"""Compute the temperatures and column densities for the hot and cold gas components. This method will set class variables `_temperature` and `_colden`.
:param params: The fit parameters returned from fit_excitation.
:type params: :class:`lmfit.Parameters`
"""
self._temperature = dict()
# N(J=0) column density = intercept on y axis
self._j0_colden = dict()
# total column density = N(J=0)*Z(T) where Z(T) is partition function
self._total_colden = dict()
size = fitmap.data.size
# create default arrays in which calculated values will be stored.
# Use nan as fill value because there may be nans in fitmapdata, in which
# case nothing need be done to arrays.
# tc, th = cold and hot temperatures
# utc, utc = uncertainties in cold and hot temperatures
# nc, nh = cold and hot column densities
# unc, unh = uncertainties in cold and hot temperatures
# opr = ortho to para ratio
# uopr = uncertainty in OPR
tc = np.full(shape=size, fill_value=np.nan, dtype=float)
th = np.full(shape=size, fill_value=np.nan, dtype=float)
utc = np.full(shape=size, fill_value=np.nan, dtype=float)
uth = np.full(shape=size, fill_value=np.nan, dtype=float)
nc = np.full(shape=size, fill_value=np.nan, dtype=float)
nh = np.full(shape=size, fill_value=np.nan, dtype=float)
unh = np.full(shape=size, fill_value=np.nan, dtype=float)
unc = np.full(shape=size, fill_value=np.nan, dtype=float)
opr = np.full(shape=size, fill_value=np.nan, dtype=float)
uopr = np.full(shape=size, fill_value=np.nan, dtype=float)
ff = fitmap.data.flatten()
ffmask = fitmap.mask.flatten()
for i in range(size):
if ffmask[i]:
continue
params = ff[i].params
for p in params:
if params[p].stderr is None:
print("AT pixel i [mask]", i, ffmask[i])
params.pretty_print()
raise Exception(
"Something went wrong with the fit and it was unable to calculate errors on the fitted parameters. It's likely that a two-temperature model is not appropriate for your data. Check the fit_result report and plot."
)
if params['m2'] < params['m1']:
cold = '2'
hot = '1'
else:
cold = '1'
hot = '2'
mcold = 'm' + cold
mhot = 'm' + hot
ncold = 'n' + cold
nhot = 'n' + hot
# cold and hot temperatures
utc[i] = params[mcold].stderr / params[mcold]
tc[i] = -utils.LOGE / params[mcold]
uth[i] = params[mhot].stderr / params[mhot]
th[i] = -utils.LOGE / params[mhot]
nc[i] = 10**params[ncold]
unc[i] = utils.LN10 * params[ncold].stderr * nc[i]
nh[i] = 10**params[nhot]
unh[i] = utils.LN10 * params[nhot].stderr * nh[i]
opr[i] = params['opr'].value
uopr[i] = params['opr'].stderr
# now reshape them all back to map shape
tc = tc.reshape(fitmap.data.shape)
th = th.reshape(fitmap.data.shape)
utc = utc.reshape(fitmap.data.shape)
uth = uth.reshape(fitmap.data.shape)
nc = nc.reshape(fitmap.data.shape)
nh = nh.reshape(fitmap.data.shape)
unh = unh.reshape(fitmap.data.shape)
unc = unc.reshape(fitmap.data.shape)
opr = opr.reshape(fitmap.data.shape)
uopr = uopr.reshape(fitmap.data.shape)
mask = fitmap.mask | np.logical_not(np.isfinite(tc))
ucc = StdDevUncertainty(np.abs(tc * utc))
self._temperature["cold"] = Measurement(data=tc,
unit=self._t_units,
uncertainty=ucc,
wcs=fitmap.wcs,
mask=mask)
mask = fitmap.mask | np.logical_not(np.isfinite(th))
uch = StdDevUncertainty(np.abs(th * uth))
self._temperature["hot"] = Measurement(data=th,
unit=self._t_units,
uncertainty=uch,
wcs=fitmap.wcs,
mask=mask)
# cold and hot total column density
ucn = StdDevUncertainty(np.abs(unc))
mask = fitmap.mask | np.logical_not(np.isfinite(nc))
self._j0_colden["cold"] = Measurement(nc,
unit=self._cd_units,
uncertainty=ucn,
wcs=fitmap.wcs,
mask=mask)
mask = fitmap.mask | np.logical_not(np.isfinite(nh))
uhn = StdDevUncertainty(np.abs(unh))
self._j0_colden["hot"] = Measurement(nh,
unit=self._cd_units,
uncertainty=uhn,
wcs=fitmap.wcs,
mask=mask)
#
self._total_colden["cold"] = self._j0_colden[
"cold"] * self._partition_function(self.tcold)
self._total_colden["hot"] = self._j0_colden[
"hot"] * self._partition_function(self.thot)
mask = fitmap.mask | np.logical_not(np.isfinite(opr))
self._opr = Measurement(opr,
unit=u.dimensionless_unscaled,
uncertainty=StdDevUncertainty(uopr),
wcs=fitmap.wcs,
mask=mask)
@property
def fit_result(self):
'''The result of the fitting procedure which includes fit statistics, variable values and uncertainties, and correlations between variables.
:rtype: :class:`lmfit.model.ModelResult`
'''
return self._fitresult
@property
def opr_fitted(self):
'''Was the ortho-to-para ratio fitted?
:returns: True if OPR was fitted, False if canonical LTE value was used
:rtype: bool
'''
if self._fitresult is None:
return False
return self._params['opr'].vary
@property
def opr(self):
'''The ortho-to-para ratio (OPR)
:returns: The fitted OPR is it was determined in the fit, otherwise the canonical LTE OPR
:rtype: :class:`~pdrtpy.measurement.Measurement`
'''
return self._opr
@property
def intensities(self):
'''The stored intensities. See :meth:`add_measurement`
:rtype: list of :class:`~pdrtpy.measurement.Measurement`
'''
return self._measurements
def colden(self, component): #,log=False):
'''The column density of hot or cold gas component, or total column density.
:param component: 'hot', 'cold', or 'total
:type component: str
:rtype: :class:`~pdrtpy.measurement.Measurement`
'''
#:param log: take the log10 of the column density
cl = component.lower()
if cl not in self._valid_components:
raise KeyError(
f"{cl} not a valid component. Must be one of {self._valid_components}"
)
#print(f'returning {cl}')
if cl == 'total':
return self.total_colden
else:
return self._total_colden[cl]
@property
def total_colden(self):
'''The fitted total column density
:rtype: :class:`~pdrtpy.measurement.Measurement`
'''
return self._total_colden["hot"] + self._total_colden["cold"]
@property
def hot_colden(self):
'''The fitted hot gas total column density
:rtype: :class:`~pdrtpy.measurement.Measurement`
'''
return self._total_colden["hot"]
@property
def cold_colden(self):
'''The fitted cold gas total column density
:rtype: :class:`~pdrtpy.measurement.Measurement`
'''
return self._total_colden["cold"]
@property
def tcold(self):
'''The fitted cold gas excitation temperature
:rtype: :class:`~pdrtpy.measurement.Measurement`
'''
return self._temperature['cold'] #self._fitparams.tcold
@property
def thot(self):
'''The fitted hot gas excitation temperature
:rtype: :class:`~pdrtpy.measurement.Measurement`
'''
return self._temperature['hot'] #self._fitparams.thot
@property
def temperature(self):
'''The fitted gas temperatures, returned in a dictionary with keys 'hot' and 'cold'.
:rtype: dict
'''
return self._temperature
def column_densities(self, norm=False, unit=utils._CM2, line=False):
r'''The computed upper state column densities of stored intensities
:param norm: if True, normalize the column densities by the
statistical weight of the upper state, :math:`g_u`.
Default: False
:type norm: bool
:param unit: The units in which to return the column density. Default: :math:`{\\rm }cm^{-2}`
:type unit: str or :class:`astropy.units.Unit`
:param line: if True, the dictionary index is the Line name,
otherwise it is the upper state :math:`J` number. Default: False
:type line: bool
:returns: dictionary of column densities indexed by upper state :math:`J` number or Line name. Default: False means return indexed by :math:`J`.
:rtype: dict
'''
# Compute column densities if needed.
# Note: this has a gotcha - if user changes an existing intensity
# Measurement in place, rather than replaceMeasurement(), the colden
# won't get recomputed. But we warned them!
#if not self._column_density or (len(self._column_density) != len(self._measurements)):
# screw it. just always compute them. Note to self: change this if it becomes computationally intensive
self._compute_column_densities(unit=unit, line=line)
if norm:
cdnorm = dict()
for cd in self._column_density:
if line:
denom = self._ac.loc[cd]["g_u"]
else:
denom = self._ac.loc['J_u', cd]["g_u"]
# This fails with complaints about units:
#self._column_density[cd] /= self._ac.loc[cd]["g_u"]
#gu = Measurement(self._ac.loc[cd]["g_u"],unit=u.dimensionless_unscaled)
cdnorm[cd] = self._column_density[cd] / denom
#return #self._column_density
return cdnorm
else:
return self._column_density
def energies(self, line=False):
'''Upper state energies of stored intensities, in K.
:param line: if True, the dictionary index is the Line name,
otherwise it is the upper state :math:`J` number. Default: False
:type line: bool
:returns: dictionary indexed by upper state :math:`J` number or Line name. Default: False means return indexed by :math:`J`.
:rtype: dict
'''
t = dict()
if line:
for m in self._measurements:
t[m] = self._ac.loc[m]["E_upper/k"]
else:
for m in self._measurements:
t[self._ac.loc[m]["J_u"]] = self._ac.loc[m]["E_upper/k"]
return t
def run(self, position=None, size=None, fit_opr=False, **kwargs):
r'''Fit the :math:`log N_u-E` diagram with two excitation temperatures,
a ``hot`` :math:`T_{ex}` and a ``cold`` :math:`T_{ex}`.
If ``position`` and ``size`` are given, the data will be averaged over a spatial box before fitting. The box is created using :class:`astropy.nddata.utils.Cutout2D`. If position or size is None, the data are averaged over all pixels. If the Measurements are single values, these arguments are ignored.
:param position: The position of the cutout array's center with respect to the data array. The position can be specified either as a `(x, y)` tuple of pixel coordinates.
:type position: tuple
:param size: The size of the cutout array along each axis in pixels. If size is a scalar number or a scalar :class:`~astropy.units.Quantity`, then a square cutout of size will be created. If `size` has two elements, they should be in `(nx, ny)` order [*this is the opposite of Cutout2D signature*]. Scalar numbers in size are assumed to be in units of pixels. Default value of None means use all pixels (position is ignored)
:type size: int, array_like`
:param fit_opr: Whether to fit the ortho-to-para ratio or not. If True, the OPR will be varied to determine the best value. If False, the OPR is fixed at the canonical LTE value of 3.
:type fit_opr: bool
'''
kwargs_opts = {
'mask': None,
'method': 'leastsq',
'nan_policy': 'raise',
'test': False,
'profile': False,
}
kwargs_opts.update(kwargs)
return self._fit_excitation(position, size, fit_opr, **kwargs_opts)
def intensity(self, colden):
'''Given an upper state column density :math:`N_u`, compute the intensity :math:`I`.
.. math::
I = {A \Delta E~N_u \over 4\pi}
where :math:`A` is the Einstein A coefficient and :math:`\Delta E` is the energy of the transition.
:param colden: upper state column density
:type colden: :class:`~pdrtpy.measurement.Measurement`
:returns: optically thin intensity
:rtype: :class:`~pdrtpy.measurement.Measurement`
'''
# colden is N_upper
dE = self._ac.loc[
colden.id]["dE/k"] * constants.k_B.cgs * self._ac["dE/k"].unit
A = self._ac.loc[colden.id]["A"] * self._ac["A"].unit
v = A * dE / (4.0 * math.pi * u.sr)
val = Measurement(data=v.value, unit=v.unit, identifier=colden.id)
intensity = val * colden # error will get propagated
i = intensity.convert_unit_to(self._intensity_units)
i._identifier = val.id
return i
def upper_colden(self, intensity, unit):
'''Compute the column density in upper state :math:`N_u`, given an
intensity :math:`I` and assuming optically thin emission.
Units of :math:`I` need to be equivalent to
:math:`{\\rm erg~cm^{-2}~s^{-1}~sr^{-1}}`.
.. math::
I &= {A \Delta E~N_u \over 4\pi}
N_u &= 4\pi {I\over A\Delta E}
where :math:`A` is the Einstein A coefficient and :math:`\Delta E` is the energy of the transition.
:param intensity: A :class:`~pdrtpy.measurement.Measurement` instance containing intensity in units equivalent to :math:`{\\rm erg~cm^{-2}~s^{-1}~sr^{-1}}`
:type intensity: :class:`~pdrtpy.measurement.Measurement`
:param unit: The units in which to return the column density. Default: :math:`{\\rm }cm^{-2}`
:type unit: str or :class:`astropy.units.Unit`
:returns: a :class:`~pdrtpy.measurement.Measurement` of the column density.
:rtype: :class:`~pdrtpy.measurement.Measurement`
'''
dE = self._ac.loc[
intensity.id]["dE/k"] * constants.k_B.cgs * self._ac["dE/k"].unit
A = self._ac.loc[intensity.id]["A"] * self._ac["A"].unit
v = 4.0 * math.pi * u.sr / (A * dE)
val = Measurement(data=v.value, unit=v.unit)
N_upper = intensity * val # error will get propagated
return N_upper.convert_unit_to(unit)
def _compute_column_densities(self, unit=utils._CM2, line=False):
r'''Compute all upper level column densities for stored intensity measurements and puts them in a dictionary
:param unit: The units in which to return the column density. Default: :math:`{\\rm }cm^{-2}`
:type unit: str or :class:`astropy.units.Unit`
:param line: if True, the dictionary index is the Line name,
otherwise it is the upper state :math:`J` number. Default: False
:type line: bool
# should we reutrn something here or just compute them and never store.
# I'm beginning to think there is no reason to store them.
#:returns: dictionary of column densities as:class:`~pdrtpy.measurement.Measurement indexed by upper state :math:`J` number or Line name. Default: False means return indexed by :math:`J`.
#:returns: a :class:`~pdrtpy.measurement.Measurement` of the column density.
'''
self._column_density = dict()
for m in self._measurements:
if line:
index = m
else:
index = self._ac.loc[m]["J_u"]
self._column_density[index] = self.upper_colden(
self._measurements[m], unit)
def gu(self, id, opr):
r'''Get the upper state statistical weight $g_u$ for the given transition identifer, and, if the transition is odd-$J$, scale the result by the given ortho-to-para ratio. If the transition is even-$J$, the LTE value is returned.
:param id: the measurement identifier
:type id: str
:param opr:
:type opr: float
:raises KeyError: if id not in existing Measurements
:rtype: float
'''
if utils.is_even(self._ac.loc[id]["J_u"]):
return self._ac.loc[id]["g_u"]
else:
#print("Ju=%d scaling by [%.2f/%.2f]=%.2f"%(self._ac.loc[id]["J_u"],opr,self._canonical_opr,opr/self._canonical_opr))
return self._ac.loc[id]["g_u"] * opr / self._canonical_opr
def average_column_density(self,
position=None,
size=None,
norm=True,
unit=utils._CM2,
line=False,
clip=-1E40 * u.Unit("cm-2")):
r'''Compute the average column density over a spatial box. The box is created using :class:`astropy.nddata.utils.Cutout2D`.
:param position: The position of the cutout array's center with respect to the data array. The position can be specified either as a `(x, y)` tuple of pixel coordinates.
:type position: tuple
:param size: The size of the cutout array along each axis. If size is a scalar number or a scalar :class:`~astropy.units.Quantity`, then a square cutout of size will be created. If `size` has two elements, they should be in `(nx,ny)` order [*this is the opposite of Cutout2D signature*]. Scalar numbers in size are assumed to be in units of pixels. Default value of None means use all pixels (position is ignored)
:type size: int, array_like`
:param norm: if True, normalize the column densities by the
statistical weight of the upper state, :math:`g_u`. For ortho-$H_2$ $g_u = OPR \times (2J+1)$, for para-$H_2$ $g_u=2J+1$. In LTE, $OPR = 3$.
:type norm: bool
:param unit: The units in which to return the column density. Default: :math:`{\rm cm}^{-2}`
:type unit: str or :class:`astropy.units.Unit`
:param line: if True, the returned dictionary index is the Line name, otherwise it is the upper state :math:`J` number.
:type line: bool
:returns: dictionary of column density Measurements, with keys as :math:`J` number or Line name
:rtype: dict
:param clip: Column density value at which to clip pixels. Pixels with column densities below this value will not be used in the average. Default: a large negative number, which translates to no clipping.
:type clip: :class:`astropy.units.Quantity`
'''
#@todo
# - should default clip = None?
# Set norm=False because we normalize below if necessary.
if position is not None and size is None:
print("WARNING: ignoring position keyword since no size given")
if position is None and size is not None:
raise Exception(
"You must supply a position in addition to size for cutout")
if size is not None:
if np.isscalar(size):
size = np.array([size, size])
else:
#Cutout2D wants (ny,nx)
size = np.array([size[1], size[0]])
clip = clip.to("cm-2")
cdnorm = self.column_densities(norm=norm, unit=unit, line=line)
cdmeas = dict()
for cd in cdnorm:
ca = cdnorm[cd]
if size is not None:
if len(size) != len(ca.shape):
raise Exception(
f"Size dimensions [{len(size)}] don't match measurements [{len(ca.shape)}]"
)
#if size[0] > ca.shape[0] or size[1] > ca.shape[1]:
# raise Exception(f"Requested cutout size {size} exceeds measurement size {ca.shape}")
cutout = Cutout2D(ca.data,
position,
size,
ca.wcs,
mode='trim',
fill_value=np.nan)
w = Cutout2D(ca.uncertainty.array,
position,
size,
ca.wcs,
mode='trim',
fill_value=np.nan)
cddata = np.ma.masked_array(cutout.data,
mask=np.ma.mask_or(
np.isnan(cutout.data),
cutout.data < clip.value))
weights = np.ma.masked_array(w.data, np.isnan(w.data))
if False:
# save cutout as a test that we have the x,y correct in size param
t = Measurement(cddata,
unit=ca.unit,
uncertainty=StdDevUncertainty(weights),
identifier=ca.id)
#t.write("cutout.fits",overwrite=True)
else:
cddata = ca.data
# handle corner case of measurment.data is shape = (1,)
# and StdDevUncertainty.array is shape = ().
# They both have only one value but StdDevUncertainty stores
# its data in a peculiar way.
# alternative: check that type(ca.uncertainty.array) == np.ndarray would also work.
if np.shape(ca.data) == (1, ) and np.shape(
ca.uncertainty.array) == ():
weights = np.array([ca.uncertainty.array])
else:
weights = ca.uncertainty.array
cdavg = np.average(cddata, weights=weights)
error = np.nanmean(ca.error) / np.sqrt(ca.error.size) #-1
cdmeas[cd] = Measurement(data=cdavg,
uncertainty=StdDevUncertainty(error),
unit=ca.unit,
identifier=cd)
return cdmeas
def _get_ortho_indices(self, ids):
"""Given a list of J values, return the indices of those that are ortho
transitions (odd J)
:param ids:
:type ids: list of str
:returns: The array indices of the odd J values.
:rtype: list of int
"""
return np.where(self._ac.loc[ids]["J_u"] % 2 != 0)[0]
def _get_para_indices(self, ids):
"""Given a list of J values, return the indices of those that are para
transitions (even J)
:param ids:
:type ids: list of str
:returns: The array indices of the even J values.
:rtype: list of int
"""
return np.where(self._ac.loc[ids]["J_u"] % 2 == 0)[0]
# currently unused. in future may allow users to give first guesses at the temperatures, though not clear these will be better than _firstguess(). plus this does nothing for the intercepts
def _slopesfromguess(self, guess):
"""given a guess of two temperatures, compute slopes from them"""
if guess[0] < guess[1]:
thot = guess[1]
tcold = guess[2]
else:
tcold = guess[2]
thot = guess[1]
slope = []
slope[0] = -utils.LOGE / tcold
slope[1] = -utils.LOGE / thot
return slope
def _first_guess(self, x, y):
r"""The first guess at the fit parameters is done by finding the line between the first two (lowest energy) points to determine $T_{cold}and between the last two (highest energy) points to determine $T_{hot}. The first guess is needed to ensure the final fit converges. The guess doesn't need to be perfect, just in the ballpark.
:param x: array of energies, $E/k$
:type x: numpy array
:param y: array of normalized column densities $N_u/g_u$
:type y: numpy array
"""
slopecold = (y[1] - y[0]) / (x[1] - x[0])
slopehot = (y[-1] - y[-2]) / (x[-1] - x[-2])
intcold = y[1] - slopecold * x[1]
inthot = y[-1] - slopehot * x[-1]
#print("FG ",type(slopecold),type(slopehot),type(intcold),type(inthot))
return np.array([slopecold, intcold, slopehot, inthot])
def _fit_excitation(self, position, size, fit_opr=False, **kwargs):
"""Fit the :math:`log N_u-E` diagram with two excitation temperatures,
a ``hot`` :math:`T_{ex}` and a ``cold`` :math:`T_{ex}`. A first
pass guess is initially made using data partitioning and two
linear fits.
If ``position`` and ``size`` are given, the data will be averaged over a spatial box before fitting. The box is created using :class:`astropy.nddata.utils.Cutout2D`. If position or size is None, the data are averaged over all pixels. If the Measurements are single values, these arguments are ignored.
:param position: The position of the cutout array's center with respect to the data array. The position can be specified either as a `(x, y)` tuple of pixel coordinates or a :class:`~astropy.coordinates.SkyCoord`, which will use the :class:`~astropy.wcs.WCS` of the ::class:`~pdrtpy.measurement.Measurement`s added to this tool. See :class:`~astropy.nddata.utils.Cutout2D`.
:type position: tuple or :class:`astropy.coordinates.SkyCoord`
:param size: The size of the cutout array along each axis. If size is a scalar number or a scalar :class:`~astropy.units.Quantity`, then a square cutout of size will be created. If `size` has two elements, they should be in `(ny, nx)` order. Scalar numbers in size are assumed to be in units of pixels. `size` can also be a :class:`~astropy.units.Quantity` object or contain :class:`~astropy.units.Quantity` objects. Such :class:`~astropy.units.Quantity` objects must be in pixel or angular units. For all cases, size will be converted to an integer number of pixels, rounding the the nearest integer. See the mode keyword for additional details on the final cutout size. Default value of None means use all pixels (position is ignored)
:type size: int, array_like, or :class:`astropy.units.Quantity`
:param fit_opr: Whether to fit the ortho-to-para ratio or not. If True, the OPR will be varied to determine the best value. If False, the OPR is fixed at the canonical LTE value of 3.
:type fit_opr: bool
,dtype=object :returns: The fit result which contains slopes, intercepts, the ortho to para ratio (OPR), and fit statistics
:rtype: :class:`lmfit.model.ModelResult`
"""
profile = kwargs.pop('profile')
self._stats = None
if profile:
pr = cProfile.Profile()
pr.enable()
if fit_opr:
min_points = 5
else:
min_points = 4
self._opr = Measurement(data=[self._canonical_opr],
uncertainty=None)
self._params['opr'].vary = fit_opr
energy = self.energies(line=True)
_ee = np.array([c for c in energy.values()])
#@ todo: allow fitting of one-temperature model
if len(_ee) < min_points:
raise Exception(
f"You need at least {min_points:d} data points to determine two-temperature model"
)
if len(_ee) == min_points:
warnings.warn(
f"Number of data points is equal to number of free parameters ({min_points:d}). Fit will be over-constrained"
)
_energy = Measurement(_ee, unit="K")
_ids = list(energy.keys())
idx = self._get_ortho_indices(_ids)
# Get Nu/gu. Canonical opr will be used.
if position is None or size is None:
colden = self.column_densities(norm=True, line=True)
else:
colden = self.average_column_density(norm=True,
position=position,
size=size,
line=True)
# Need to stuff the data into a single vector
_cd = np.squeeze(np.array([c.data for c in colden.values()]))
_er = np.squeeze(np.array([c.error for c in colden.values()]))
_colden = Measurement(_cd,
uncertainty=StdDevUncertainty(_er),
unit="cm-2")
fk = utils.firstkey(colden)
x = _energy.data
y = np.log10(_colden.data)
#print("SHAPE Y LEN(SHAPE(Y) ",y.shape,len(y.shape))
#kwargs_opts = {"guess": self._first_guess(x,y)}
#kwargs_opts.update(kwargs)
sigma = utils.LOGE * _colden.error / _colden.data
slopecold, intcold, slopehot, inthot = self._first_guess(x, y)
#print(slopecold,slopehot,intcold,inthot)
tcold = (-utils.LOGE / slopecold)
thot = (-utils.LOGE / slopehot)
if np.shape(tcold) == ():
tcold = np.array([tcold])
thot = np.array([thot])
saveshape = tcold.shape
#print("TYPE COLD SIT",type(slopecold),type(intcold),type(tcold))
#print("SHAPES: colden/sigma/slope/int/temp/cd: ",np.shape(_colden),np.shape(sigma),np.shape(slopecold),np.shape(intcold),np.shape(tcold),np.shape(_cd))
#print("First guess at excitation temperatures:\n T_cold = %.1f K\n T_hot = %.1f K"%(tcold,thot))
fmdata = np.empty(tcold.shape, dtype=object).flatten()
#fm = FitMap(data=fmdata,wcs=colden[fk].wcs,uncertainty=None,unit=None)
tcold = tcold.flatten()
thot = thot.flatten()
slopecold = slopecold.flatten()
slopehot = slopehot.flatten()
inthot = inthot.flatten()
intcold = intcold.flatten()
#sigma = sigma.flatten()
# flatten any dimensions past 0
shp = y.shape
#print("NS ",shp[0],shp[1:])
if len(shp) == 1:
#print("adding new axis")
y = y[:, np.newaxis]
shp = y.shape
yr = y.reshape((shp[0], np.prod(shp[1:])))
sig = sigma.reshape((shp[0], np.prod(shp[1:])))
#print("YR, SIG SHAPE",yr.shape,sig.shape)
count = 0
#print("LEN(TCOLD)",len(tcold))
total = len(tcold)
fm_mask = np.full(shape=tcold.shape, fill_value=False)
# Suppress the incorrect warning about model parameters
warnings.simplefilter('ignore', category=UserWarning)
excount = 0
badfit = 0
# update whether opr is allowed to vary or not.
self._model.set_param_hint('opr', vary=fit_opr)
# use progress bar if more than one pixel
if total > 1:
progress = kwargs.pop("progress", True)
else:
progress = False
with get_progress_bar(progress, total, leave=True, position=0) as pbar:
for i in range(total):
if np.isfinite(yr[:, i]).all() and np.isfinite(sig[:,
i]).all():
# update Parameter hints based on first guess.
self._model.set_param_hint('m1',
value=slopecold[i],
vary=True)
self._model.set_param_hint('n1',
value=intcold[i],
vary=True)
self._model.set_param_hint('m2',
value=slopehot[i],
vary=True)
self._model.set_param_hint('n2',
value=inthot[i],
vary=True)
p = self._model.make_params()
wts = 1.0 / (sig[:, i] * sig[:, i])
try:
fmdata[i] = self._model.fit(
data=yr[:, i],
weights=wts,
x=x,
idx=idx,
fit_opr=fit_opr,
method=kwargs['method'],
nan_policy=kwargs['nan_policy'])
if fmdata[i].success and fmdata[i].errorbars:
count = count + 1
else:
fmdata[i] = None
fm_mask[i] = True
badfit = badfit + 1
except ValueError:
fmdata[i] = None
fm_mask[i] = True
excount = excount + 1
else:
fmdata[i] = None
fm_mask[i] = True
pbar.update(1)
warnings.resetwarnings()
fmdata = fmdata.reshape(saveshape)
fm_mask = fm_mask.reshape(saveshape)
self._fitresult = FitMap(fmdata,
wcs=colden[fk].wcs,
mask=fm_mask,
name="result")
# this will raise an exception if the fit was bad (fit errors == None)
self._compute_quantities(self._fitresult)
print(f"fitted {count} of {slopecold.size} pixels")
print(f'got {excount} exceptions and {badfit} bad fits')
#if successful, set the used position and size
self._position = position
self._size = size
if profile:
pr.disable()
s = io.StringIO()
sortby = SortKey.CUMULATIVE
ps = pstats.Stats(pr, stream=s).sort_stats(sortby)
ps.print_stats()
self._stats = s
def _two_lines(self, x, m1, n1, m2, n2):
'''This function is used to partition a fit to data using two lines and
an inflection point. Second slope is steeper because slopes are
negative in excitation diagram.
:param x: array of x values
:type x: :class:`numpy.ndarray`
:param m1: slope of first line
:type m1: float
:param n1: intercept of first line
:type n1: float
:param m2: slope of second line
:type m2: float
:param n2: intercept of second line
:type n2: float
See https://stackoverflow.com/questions/48674558/how-to-implement-automatic-model-determination-and-two-state-model-fitting-in-py
'''
return np.max([m1 * x + n1, m2 * x + n2], axis=0)
def _one_line(self, x, m1, n1):
'''Return a line.
:param x: array of x values
:type x: :class:`numpy.ndarray`
:param m1: slope of first line
:type m1: float
:param n1: intercept of first line
:type n1: float
'''
return m1 * x + n1
def _partition_function(self, tex):
'''Calculate the H2 partition function given an excitation temperature
:param tex: the excitation temperature
:type tex: :class:`~pdrtpy.measurement.Measurement` or :class:`astropy.units.quantity.Quantity`
:returns: the partition function value
:rtype: numpy.ndarray
'''
# See Herbst et al 1996
# http://articles.adsabs.harvard.edu/pdf/1996AJ....111.2403H
# Z(T) = = 0.0247T * [1—exp(—6000/T)]^-1
# This is just being defensive. I know the temperatures used internally are in K.
t = np.ma.masked_invalid((tex.value * u.Unit(tex.unit)).to(
"K", equivalencies=u.temperature()).value)
t.mask = np.logical_or(t.mask, np.logical_not(np.isfinite(t)))
z = 0.0247 * t / (1.0 - np.exp(-6000.0 / t))
return z
def test_water(io_fix):
# Load data
res = load_nexus(io_fix['irs_res_f'])
dat = load_nexus(io_fix['irs_red_f'])
q_vals = io_fix['q_values']
# Define the fitting range
e_min = -0.4
e_max = 0.4
# Find indexes of dat['x'] with values in (e_min, e_max)
mask = np.intersect1d(np.where(dat['x'] > e_min),
np.where(dat['x'] < e_max))
# Drop data outside the fitting range
fr = dict() # fitting range. Use in place of 'dat'
fr['x'] = dat['x'][mask]
fr['y'] = np.asarray([y[mask] for y in dat['y']])
fr['e'] = np.asarray([e[mask] for e in dat['e']])
# Create the model
def generate_model_and_params(spectrum_index=None):
r"""Produce an LMFIT model and related set of fitting parameters"""
sp = '' if spectrum_index is None else '{}_'.format(
spectrum_index) # prefix if spectrum_index passed
# Model components
intensity = ConstantModel(prefix='I_' + sp) # I_amplitude
elastic = DeltaDiracModel(prefix='e_' + sp) # e_amplitude, e_center
# l_amplitude, l_center, l_sigma (also l_fwhm, l_height)
inelastic = LorentzianModel(prefix='l_' + sp)
# r_amplitude, r_center (both fixed)
resolution = TabulatedResolutionModel(res['x'],
res['y'],
prefix='r_' + sp)
background = LinearModel(prefix='b_' + sp) # b_slope, b_intercept
# Putting it all together
model = intensity * Convolve(resolution,
elastic + inelastic) + background
parameters = model.make_params() # model params are a separate entity
# Ties and constraints
parameters['e_' + sp + 'amplitude'].set(min=0.0, max=1.0)
parameters['l_' + sp + 'center'].set(expr='e_' + sp +
'center') # centers tied
parameters['l_' + sp + 'amplitude'].set(expr='1 - e_' + sp +
'amplitude')
# Some initial sensible values
init_vals = {
'I_' + sp + 'c': 1.0,
'e_' + sp + 'amplitude': 0.5,
'l_' + sp + 'sigma': 0.01,
'b_' + sp + 'slope': 0,
'b_' + sp + 'intercept': 0
}
for p, v in init_vals.items():
parameters[p].set(value=v)
return model, parameters
# Call the function
model, params = generate_model_and_params()
# Initial guess for first spectrum. Only set free parameters
for name, value in dict(I_c=4.0,
e_center=0,
e_amplitude=0.1,
l_sigma=0.03,
b_slope=0,
b_intercept=0).items():
params[name].set(value=value)
# Carry out the fit
fit = model.fit(fr['y'][0],
x=fr['x'],
params=params,
weights=1.0 / fr['e'][0])
assert_almost_equal(fit.redchi, 1.72, decimal=2)
# Carry out sequential fit
n_spectra = len(fr['y'])
fits = [
None,
] * n_spectra # store fits for all the tried spectra
fits[0] = fit # store previous fit
for i in range(1, n_spectra):
y_exp = fr['y'][i]
e_exp = fr['e'][i]
fit = model.fit(y_exp, x=fr['x'], params=params, weights=1.0 / e_exp)
fits[i] = fit # store fit results
assert_almost_equal(
[f.redchi for f in fits],
[1.72, 1.15, 0.81, 0.73, 0.73, 0.75, 0.81, 0.86, 0.75, 0.91],
decimal=2)
# Fit HWHM(Q^2) with Teixeira model
hwhms = 0.5 * np.asarray([fit.params['l_fwhm'].value for fit in fits])
def teixeira(q2s, difcoef, tau):
dq2 = difcoef * q2s
return hbar * dq2 / (1 + dq2 * tau)
teixeira_model = Model(teixeira) # create LMFIT Model instance
teixeira_model.set_param_hint('difcoef', min=0)
teixeira_model.set_param_hint('tau', min=0)
# Carry out the fit from an initial guess
teixeira_params = teixeira_model.make_params(difcoef=1.0, tau=1.0)
teixeira_fit = teixeira_model.fit(hwhms,
q2s=np.square(q_vals),
params=teixeira_params)
assert_almost_equal(
[teixeira_fit.best_values['difcoef'], teixeira_fit.best_values['tau']],
[0.16, 1.11],
decimal=2)
# Model for Simultaneous Fit of All Spectra with Teixeira Water Model
#
# create one model for each spectrum, but collect all parameters under
# a single instance of the Parameters class.
l_model = list()
g_params = lmfit.Parameters()
for i in range(n_spectra):
# model and parameters for one of the spectra
m, ps = generate_model_and_params(spectrum_index=i)
l_model.append(m)
[g_params.add(p) for p in ps.values()]
# Initialize parameter set with optimized parameters from sequential fit
for i in range(n_spectra):
optimized_params = fits[i].params # these are I_c, e_amplitude,...
for name in optimized_params:
# for instance, 'e_amplitude' splitted into 'e', and 'amplitude'
prefix, base = name.split('_')
# i_name is 'e_3_amplitude' for i=3
i_name = prefix + '_{}_'.format(i) + base
g_params[i_name].set(value=optimized_params[name].value)
# Introduce global parameters difcoef and tau.
# Use previous optimized values as initial guess
o_p = teixeira_fit.params
g_params.add('difcoef', value=o_p['difcoef'].value, min=0)
g_params.add('tau', value=o_p['tau'].value, min=0)
# Tie each lorentzian l_i_sigma to the teixeira expression
for i in range(n_spectra):
q2 = q_vals[i] * q_vals[i]
fmt = '{hbar}*difcoef*{q2}/(1+difcoef*{q2}*tau)'
teixeira_expression = fmt.format(hbar=hbar, q2=q2)
g_params['l_{}_sigma'.format(i)].set(expr=teixeira_expression)
# Carry out the Simultaneous Fit
def residuals(params):
l_residuals = list()
for i in range(n_spectra):
x = fr['x'] # fitting range of energies
y = fr['y'][i] # associated experimental intensities
e = fr['e'][i] # associated experimental errors
model_evaluation = l_model[i].eval(x=x, params=params)
l_residuals.append((model_evaluation - y) / e)
return np.concatenate(l_residuals)
# Minimizer object using the parameter set for all models and the
# function to calculate all the residuals.
minimizer = lmfit.Minimizer(residuals, g_params)
g_fit = minimizer.minimize()
assert_almost_equal(g_fit.redchi, 0.93, decimal=2)
def extract_soi_from_wal(field, r, reference_field, max_field,
model='full', truncation=1000,
guess_from_previous=True, guesses=None,
plot_fit=False, method='least_squares',
weigth_method='gauss', weight_stiffness=1.0,
htr=None, cubic_soi=None, density=None,
plot_path=None):
"""Extract the SOI parameters from fitting the wal conductance.
This algorithm assumes that the data are properly centered.
The fitted values are expressed in the unit of the input field (the guesses
should use the same convention).
Parameters
----------
field : np.ndarray
Magnetic field values for which the the resistance was measured.
This can be a multidimensional array in which case the last
dimension will be considered as the swept dimension.
r : np.ndarray
Resistance values in Ω which were measured.
This can be a multidimensional array in which case the last dimension
will be considered as the swept dimension.
reference_field : float
Field used a reference to eliminate possible experimental offsets.
max_field : float
Maximum field to consider when fitting the data since we know that the
theory breaks down at high field.
model : {'full', 'simplified'}, optional
Model used to describe the WAL. 'simplified' corresponds to the
situation in which either the Rashba term or the linear Dresselhaus
term can be neglected. 'full' corresponds to a more complete model,
however each evaluation of the fitting function requires to find the
eigenvalues of a large and as a consequence may be slow.
truncation : int, optional
Both models imply a truncation of the number of Landau levels
considered: in the 'simplified' case this enters the evaluation of a
series, in the 'full' case the size of the matrix whose eigenvalues
need to be computed.
guess_from_previous : bool, optionnal
When we perform a fit for multiple sweeps should the fitted values of
the previous sweep be used as guess for the next sweep.
guesses :
Guessed fit parameters to use. Those should include the dephasing
field, both linear fields (for simplified the second one is ignored),
the cubic term field.
plot_fit : bool, optional
Should each fit be plotted to allow manual verification.
method : str, optional
Algorithm to use to perform the fit. See lmfit.minimize documentation
for acceptable values.
# XXX
Returns
-------
dephasing_field : np.ndarray
Dephasing field and standard deviation.
linear_soi : np.ndarray
Fields and standard deviations corresponding to the linear SOI terms.
This is returned as a ...2x2 array in which the first line is the
Rashba term and the second the Dresselhaus.
cubic_soi_field : np.ndarray
Field and standard deviation corresponding to the cubic Dresselhaus
term.
"""
# Identify the shape of the data and make them suitable for the following
# treatment.
if len(field.shape) >= 2:
original_shape = field.shape[:-1]
trace_number = np.prod(original_shape)
field = field.reshape((trace_number, -1))
r = r.reshape((trace_number, -1))
if guesses is not None and len(guesses) == 4:
g = np.empty(original_shape + (4,))
for i in range(4):
g[..., i] = guesses[i]
guesses = g
if guesses is not None:
guesses = guesses.reshape((trace_number, -1))
else:
guesses = np.array([None]*trace_number)
else:
trace_number = 1
field = np.array((field,))
r = np.array((r,))
guesses = np.array((guesses,))
results = np.zeros((4, 2, trace_number))
# Express the conductance in usual WAL normalization. (e^2/(2πh))
# W. Knap et al.,
# Weak Antilocalization and Spin Precession in Quantum Wells.
# Physical Review B. 53, 3912–3924 (1996).
sigma = (1/r) / (cs.e**2/(2*np.pi*cs.Planck))
# Create the fitting model
if model == 'full':
raise ValueError('Unsupported model')
model_obj = Model(full_wal)
else:
model_obj = Model(simple_wal)
model_obj.set_param_hint('series_truncation',
value=truncation,
vary=False)
model_obj.set_param_hint('low_field_reference',
value=reference_field,
vary=False)
names = (('dephasing_field', 'linear_soi_rashba', 'linear_soi_dressel',
'cubic_soi') if model == 'full' else
('dephasing_field', 'linear_soi', '', 'cubic_soi'))
for name in [n for n in names if n]:
if name == 'cubic_soi' and cubic_soi is not None:
model_obj.set_param_hint(name, value=cubic_soi,
vary=cubic_soi is None)
elif name == 'dephasing_field':
model_obj.set_param_hint(name, min=0, value=0.0003)
else:
model_obj.set_param_hint(name, min=0, value=0.01)
if cubic_soi is None:
model_obj.set_param_hint('soi', min=0, value=0.01)
model_obj.set_param_hint('rashba_fraction', min=0, max=1, value=1)
model_obj.set_param_hint('linear_soi', expr='soi*rashba_fraction')
model_obj.set_param_hint('cubic_soi', expr='soi*(1-rashba_fraction)')
params = model_obj.make_params()
# Perform a fit for each magnetic field sweep
for i in range(trace_number):
print(f'Treating WAL trace {i+1}/{trace_number}')
# Conserve only the data for positive field since we symmetrized the
# data
mask = np.where(np.logical_and(np.greater(field[i], 0.0002),
np.less(field[i], max_field)))
f, s = field[i][mask], sigma[i][mask]
# Find the conductance at the reference field and compute Δσ
ref_ind = np.argmin(np.abs(f - reference_field))
reference_field = f[ref_ind]
dsigma = s - s[ref_ind]
# Build the weights
weights = weight_wal_data(f, dsigma, mask=weigth_method,
stiffness=weight_stiffness)
# Set the initial values for the parameters
if i != 0 and guess_from_previous:
params = res.params
else:
if guesses[i] is not None:
for n, v in zip(names, guesses[i]):
if n and (n != 'cubic_soi' or cubic_soi is None):
params[n].value = v
params['low_field_reference'].value = reference_field
# Perform the fit
res = model_obj.fit(dsigma, params, field=f, method='nelder',
weights=weights)
res = model_obj.fit(dsigma, res.params, field=f, method=method,
weights=weights)
for j, n in enumerate(names):
if not n:
continue
results[j, 0, i] = res.best_values[n]
results[j, 1, i] = res.params[n].stderr
# If requested plot the result.
if plot_fit:
fig, ax = plt.subplots(constrained_layout=True)
if density is not None:
fig.suptitle(f'Density {density[i]/1e4:.1e} (cm$^2$)')
ax.plot(field[i]*1e3, sigma[i] - s[ref_ind], '+')
ax.plot(np.concatenate((-f[::-1], f))*1e3,
np.concatenate((res.best_fit[::-1], res.best_fit)))
ax.set_xlabel('Magnetic field B (mT)')
ax.set_ylabel(r'Δσ(B) - Δσ(0) ($\frac{e^2}{2\,π\,\hbar})$')
amp = abs(np.max(s - s[ref_ind]) - np.min(s - s[ref_ind]))
ax.set_ylim((None, np.max(s - s[ref_ind]) + 0.1*amp))
if htr is None:
ax.set_xlim((-max_field*1e3, max_field*1e3))
else:
# ax.set_xlim((-5*htr[i]*1e3, 5*htr[i]*1e3))
ax.set_xlim((-50, 50))
if htr is not None:
ax.axvline(htr[i]*1e3, color='k', label='H$_{tr}$')
ax.legend()
# if plot_path:
# path = os.path.join(plot_path,
# f'fit_{i}_n_{density[i]}.pickle')
# with open(path, 'wb') as fig_pickle:
# pickle.dump(fig, fig_pickle)
# ax2 = ax.twinx()
# ax2.plot(f*1e3, weights, color='C2')
# plt.show()
if results.shape[0] == 1:
return results[0]
else:
results = results.reshape((4, 2) + original_shape)
return results[0], results[1:3], results[3]
def fit_double_gaussian(x_data,
y_data,
maxiter=None,
maxfun=None,
verbose=1,
initial_params=None):
""" Fitting of double gaussian
Fitting the Gaussians and finding the split between the up and the down state,
separation between the max of the two gaussians measured in the sum of the std.
Args:
x_data (array): x values of the data
y_data (array): y values of the data
maxiter (int): Legacy argument, not used any more
maxfun (int): Legacy argument, not used any more
verbose (int): set to >0 to print convergence messages
initial_params (None or array): optional, initial guess for the fit parameters:
[A_dn, A_up, sigma_dn, sigma_up, mean_dn, mean_up]
Returns:
par_fit (array): fit parameters of the double gaussian: [A_dn, A_up, sigma_dn, sigma_up, mean_dn, mean_up]
result_dict (dict): dictionary with results of the fit. Fields guaranteed in the dictionary:
parameters (array): Fitted parameters
parameters initial guess (array): initial guess for the fit parameters, either the ones give to the
function, or generated by the function: [A_dn, A_up, sigma_dn, sigma_up, mean_dn, mean_up]
reduced_chi_squared (float): Reduced chi squared value of the fit
separation (float): separation between the max of the two gaussians measured in the sum of the std
split (float): value that separates the up and the down level
left (array), right (array): Parameters of the left and right fitted Gaussian
"""
if maxiter is not None:
warnings.warn('argument maxiter is not used any more')
if maxfun is not None:
warnings.warn('argument maxfun is not used any more')
if initial_params is None:
initial_params = _estimate_double_gaussian_parameters(x_data, y_data)
def _double_gaussian(x, A_dn, A_up, sigma_dn, sigma_up, mean_dn, mean_up):
""" Double Gaussian helper function for lmfit """
gauss_dn = gaussian(x, mean_dn, sigma_dn, A_dn)
gauss_up = gaussian(x, mean_up, sigma_up, A_up)
double_gauss = gauss_dn + gauss_up
return double_gauss
lmfit_method = 'least_squares'
double_gaussian_model = Model(_double_gaussian)
delta_x = x_data.max() - x_data.min()
bounds = [x_data.min() - .1 * delta_x, x_data.max() + .1 * delta_x]
double_gaussian_model.set_param_hint('mean_up',
min=bounds[0],
max=bounds[1])
double_gaussian_model.set_param_hint('mean_dn',
min=bounds[0],
max=bounds[1])
double_gaussian_model.set_param_hint('A_up', min=0)
double_gaussian_model.set_param_hint('A_dn', min=0)
param_names = double_gaussian_model.param_names
result = double_gaussian_model.fit(y_data,
x=x_data,
**dict(zip(param_names,
initial_params)),
verbose=False,
method=lmfit_method)
par_fit = np.array([result.best_values[p] for p in param_names])
if par_fit[4] > par_fit[5]:
par_fit = np.take(par_fit, [1, 0, 3, 2, 5, 4])
# separation is the difference between the max of the gaussians divided by the sum of the std of both gaussians
separation = (par_fit[5] - par_fit[4]) / (abs(par_fit[2]) +
abs(par_fit[3]))
# split equal distant to both peaks measured in std from the peak
weigthed_distance_split = par_fit[4] + separation * abs(par_fit[2])
result_dict = {
'parameters': par_fit,
'parameters initial guess': initial_params,
'separation': separation,
'split': weigthed_distance_split,
'reduced_chi_squared': result.redchi,
'left': np.take(par_fit, [4, 2, 0]),
'right': np.take(par_fit, [5, 3, 1]),
'type': 'fitted double gaussian'
}
return par_fit, result_dict
def fit_gaussian(x_data,
y_data,
maxiter=None,
maxfun=None,
verbose=0,
initial_parameters=None,
initial_params=None,
estimate_offset=True):
""" Fitting of a gaussian, see function 'gaussian' for the model that is fitted
The final optimization of the fit is performed with `lmfit <https://lmfit.github.io/lmfit-py/>`
using the `least_squares` method.
Args:
x_data (array): x values of the data
y_data (array): y values of the data
verbose (int): set positive for verbose fit
initial_parameters (None or array): optional, initial guess for the
fit parameters: [mean, s, amplitude, offset]
estimate_offset (bool): If True then include offset in the Gaussian parameters
maxiter (int): Legacy argument, not used
maxfun (int): Legacy argument, not used
Returns:
par_fit (array): fit parameters of the gaussian: [mean, s, amplitude, offset]
result_dict (dict): result dictonary containging the fitparameters and the initial guess parameters
"""
if initial_params is not None:
warnings.warn('use initial_parameters instead of initial_params')
initial_parameters = initial_params
if maxiter is not None:
warnings.warn('argument maxiter is not used any more')
if maxfun is not None:
warnings.warn('argument maxfun is not used any more')
if initial_parameters is None:
initial_parameters = _estimate_initial_parameters_gaussian(
x_data, y_data, include_offset=estimate_offset)
if estimate_offset:
def gaussian_model(x, mean, sigma, amplitude, offset):
""" Gaussian helper function for lmfit """
y = gaussian(x, mean, sigma, amplitude, offset)
return y
else:
def gaussian_model(x, mean, sigma, amplitude): # type: ignore
""" Gaussian helper function for lmfit """
y = gaussian(x, mean, sigma, amplitude)
return y
lmfit_method = 'least_squares'
lmfit_model = Model(gaussian_model)
lmfit_model.set_param_hint('amplitude', min=0)
lmfit_result = lmfit_model.fit(y_data,
x=x_data,
**dict(
zip(lmfit_model.param_names,
initial_parameters)),
verbose=verbose,
method=lmfit_method)
result_dict = extract_lmfit_parameters(lmfit_model, lmfit_result)
result_dict['parameters fitted gaussian'] = result_dict[
'fitted_parameters']
result_dict['parameters initial guess'] = result_dict['initial_parameters']
return result_dict['fitted_parameters'], result_dict
def fit_basic(
x,
y,
dy=None,
model="line",
init={},
fix=None,
method="leastsq",
emcee=False,
plot_corner=False,
**kwargs,
):
if model in "linear":
func = linear
elif model in "power":
func = power
elif model in "quadratic":
func = quadratic
elif model in "exponential":
func = exponential
else:
raise ValueError("Model {} not defined.".format(model))
model = Model(func, nan_policy="omit")
pars = init_pars(model, init, x, y)
if fix is not None:
for vn in fix:
pars[vn].set(value=fix[vn], vary=0)
if dy is not None:
dy = np.abs(dy)
wgt = np.array(
[1.0 / dy[i] if dy[i] > 0 else 0 for i in range(len(y))])
is_weighted = True
else:
wgt = None
is_weighted = False
if emcee:
mi = lmfit.minimize(
residual,
pars,
args=(x, model),
kws={
"data": y,
"eps": wgt
},
method="nelder",
nan_policy="omit",
)
# mi.params.add('f', value=1, min=0.001, max=2)
mini = lmfit.Minimizer(residual,
mi.params,
fcn_args=(x, model),
fcn_kws={
"data": y,
"eps": wgt
})
out = mini.emcee(burn=300,
steps=1000,
thin=20,
params=mi.params,
is_weighted=is_weighted)
out, fit_report = get_ml_solution(out, fix)
print(list(out.params.valuesdict().values()))
if plot_corner:
corner.corner(
out.flatchain,
labels=out.var_names,
truths=list(out.params.valuesdict().values()),
)
else:
out = lmfit.minimize(
residual,
pars,
args=(x, model),
method=method,
kws={
"data": y,
"eps": wgt
},
nan_policy="omit",
**kwargs,
)
fit_report = lmfit.fit_report(out)
pars_arr = np.zeros((len(pars), 2))
for i, vn in enumerate(pars):
pars_arr[i, 0] = out.params[vn].value
pars_arr[i, 1] = out.params[vn].stderr if out.params[vn].stderr else 0
if not emcee:
gof = np.array([out.chisqr, out.redchi, out.bic, out.aic])
else:
gof = 0
return pars_arr, gof, out, fit_report, model
#!/usr/bin/env python
# <examples/doc_mode_savemodel.py>
import numpy as np
from lmfit.model import Model, save_model
def mysine(x, amp, freq, shift):
return amp * np.sin(x*freq + shift)
sinemodel = Model(mysine)
pars = sinemodel.make_params(amp=1, freq=0.25, shift=0)
save_model(sinemodel, 'sinemodel.sav')
# <end examples/doc_model_savemodel.py>
