#!/usr/bin/env python
# -*- coding: utf-8 -*-
# This file is part of the
# NIRDust Project (https://github.com/Gaiana/nirdust)
# Copyright (c) 2020, 2021 Gaia Gaspar, Jose Alacoria
# License: MIT
# Full Text: https://github.com/Gaiana/nirdust/LICENSE
# ==============================================================================
# DOCS
# ==============================================================================
"""Blackbody/temperature utilities."""
# ==============================================================================
# IMPORTS
# ==============================================================================
import warnings
from astropy import units as u
from astropy.modeling.models import BlackBody
import attr
from attr import validators
import matplotlib.pyplot as plt
import numpy as np
from scipy.optimize import OptimizeResult, basinhopping
from .core import NirdustSpectrum
# ==============================================================================
# EXCEPTIONS
# ==============================================================================
[docs]
class ConvergenceWarning(RuntimeWarning):
"""Raised when the fitting procedure failed to converge."""
pass
# ==============================================================================
# TARGET SPECTRUM MODEL
# ==============================================================================
[docs]
def target_model(external_spectrum, T, alpha, beta, gamma):
"""Compute the expected spectrum given a blackbody prediction.
Parameters
----------
external_spectrum: NirdustSpectrum object
Instance of NirdustSpectrum containing the external spectrum.
T: float
BlackBody temperature in Kelvin.
alpha: float
Multiplicative coefficient for external_flux.
beta: float
Multiplicative coefficient for blackbody.
gamma: float
Additive coefficient.
Return
------
prediction: `~numpy.ndarray`
Expected flux given the input parameters.
"""
spectral_axis = external_spectrum.spectral_axis
external_flux = external_spectrum.flux.value
# calculate the model
blackbody = BlackBody(u.Quantity(T, u.K))
bb_flux = blackbody(spectral_axis).value
prediction = alpha * external_flux + 10**beta * bb_flux + 10**gamma
return prediction
# ==============================================================================
# LIKELIHOOD FUNCTIONS
# ==============================================================================
[docs]
def negative_gaussian_log_likelihood(
theta, target_spectrum, external_spectrum
):
"""Negative Gaussian logarithmic likelihood.
Compute the negative likelihood of the model represented by the parameter
theta given the data. The negative sign is added for minimization
purposes, i.e. finding the maximum likelihood parameters is the same
as minimizing the negative likelihood.
Parameters
----------
theta: array-like
Parameter vector: (temperature, alpha, beta, gamma).
target_spectrum: NirdustSpectrum object
Instance of NirdustSpectrum containing the total central spectrum.
external_spectrum: NirdustSpectrum object
Instance of NirdustSpectrum containing the external spectrum.
Return
------
loglike: scalar
Negative logarithmic likelihood for parameter theta.
"""
prediction = target_model(external_spectrum, *theta)
noise = target_spectrum.noise
diff = target_spectrum.flux.value - prediction
loglike = np.sum(
-0.5 * np.log(2.0 * np.pi) - np.log(noise) - diff**2 / (2.0 * noise**2)
)
return -loglike
# ==============================================================================
# PHYSICAL CONSTRAINTS FUNCTIONS
# ==============================================================================
[docs]
def alpha_vs_beta(theta, target_spectrum, external_spectrum):
"""Alpha term positivity relative to beta term.
Here we assume that: alpha * ExternalSpectrum > 10**beta * BlackBody
in mean values.
Parameters
----------
theta: array-like
Parameter vector: (temperature, alpha, beta, gamma).
target_spectrum: NirdustSpectrum object
Instance of NirdustSpectrum containing the total central spectrum.
external_spectrum: NirdustSpectrum object
Instance of NirdustSpectrum containing the external spectrum.
Return
------
alpha_positivity: scalar
The difference between alpha term and beta term mean values, given the
data; i.e.: mean(alpha * ExternalSpectrum) - (10**beta * BlackBody)
"""
# we assume that alpha*ExternalSpectrum > beta*BlackBody, in mean values
T, alpha, beta, gamma = theta
prediction = target_model(external_spectrum, T, alpha, beta, gamma)
alpha_term = np.mean(alpha * external_spectrum.flux.value)
beta_term = np.mean(prediction - alpha_term - 10**gamma)
alpha_positivity = alpha_term - beta_term
# Positive output is True
return alpha_positivity
[docs]
def make_gamma_vs_target_flux(gamma_fraction):
"""Encapsulate gamma_fraction for gamma constraint function.
Parameters
----------
gamma_fraction: scalar
Value between [0, 1] representing the maximum fraction of Target
flux allowed for the gamma term.
Return
------
gamma_vs_target_flux: function
Function that computes the gamma term positivity relative to the
total target flux. Call signature:
gamma_vs_target_flux(theta, target_spectrum, external_spectrum)
"""
if not (0 <= gamma_fraction <= 1):
raise ValueError("Gamma fraction must be in the (0, 1) range.")
def gamma_vs_target_flux(theta, target_spectrum, external_spectrum):
# we assume that gamma can account for 5 percent or less of target flux
T, alpha, beta, gamma = theta
min_flux = target_spectrum.flux.value.min()
gamma_positivity = gamma_fraction * min_flux - 10**gamma
# Positive output is True
return gamma_positivity
return gamma_vs_target_flux
# ==============================================================================
# RESULT CLASSES
# ==============================================================================
[docs]
@attr.s(frozen=True)
class NirdustParameter:
"""Parameter representation.
Attributes
----------
name: str
Parameter name.
value: scalar, `~astropy.units.Quantity`
Expected value for parameter after fitting procedure.
uncertainty: scalar, `~astropy.units.Quantity`
Uncertainties associated to the fitted value.
"""
name = attr.ib(validator=validators.instance_of(str))
value = attr.ib()
uncertainty = attr.ib(default=None)
[docs]
@attr.s(frozen=True, repr=False)
class NirdustResults:
"""Create the class NirdustResults.
Storages the results obtained with BasinhoppingFitter plus the dust
spectrum. The method plot() can be called to plot the spectrum and
the blackbody model obtained in the fitting.
Attributes
----------
temperature: NirdustParameter
Parameter object with the expected blackbody temperature and
its uncertainty.
alpha: NirdustParameter
Parameter object with the expected alpha value and
its uncertainty. Note: No unit is provided as the intensity is in
arbitrary units.
beta: NirdustParameter
Parameter object with the expected beta value and
its uncertainty. Note: No unit is provided as the intensity is in
arbitrary units.
gamma: NirdustParameter
Parameter object with the expected gamma value and
its uncertainty. Note: No unit is provided as the intensity is in
arbitrary units.
fitted_blackbody: `~astropy.modeling.models.BlackBody`
BlackBody instance with the best fit value of temperature.
target_spectrum: NirdustSpectrum object
Instance of NirdustSpectrum containing the central spectrum.
external_spectrum: NirdustSpectrum object
Instance of NirdustSpectrum containing the external spectrum.
minimizer_results: OptimizeResult object
Instance of OptimizeResult that generates in the fitting procedure.
"""
temperature = attr.ib(validator=validators.instance_of(NirdustParameter))
alpha = attr.ib(validator=validators.instance_of(NirdustParameter))
beta = attr.ib(validator=validators.instance_of(NirdustParameter))
gamma = attr.ib(validator=validators.instance_of(NirdustParameter))
fitted_blackbody = attr.ib(validator=validators.instance_of(BlackBody))
target_spectrum = attr.ib(
validator=validators.instance_of(NirdustSpectrum),
)
external_spectrum = attr.ib(
validator=validators.instance_of(NirdustSpectrum),
)
minimizer_results = attr.ib(
validator=validators.instance_of(OptimizeResult),
)
def __repr__(self):
"""repr(x) <=> x.__repr__()."""
return (
f"NirdustResults("
f"temperature={self.temperature.value:.0f}, "
f"alpha={self.alpha.value:.2f}, "
f"beta={self.beta.value:.2f}, "
f"gamma={self.gamma.value:.2f},\n"
f"minimizer_results={self.minimizer_results})"
)
[docs]
def plot(
self, axes=None, data_kws=None, model_kws=None, show_components=False
):
"""Build a plot of the fitted spectrum and the fitted model.
Parameters
----------
axes: tuple of ``matplotlib.pyplot.Axis`` objects
Tuple with two objects of type Axes containing complete
information of the properties to generate the image, by default
it is None.
data_kws: ``dict``
Dictionaries of keyword arguments. Passed to the data plotting
function.
model_kws: ``dict``
Dictionaries of keyword arguments. Passed to the model plotting
function.
show_components: ``bool``
Flag to indicate if the three components of the model should be
plotted.
Return
------
out: ``matplotlib.pyplot.Axis`` :
The axis where the method draws.
"""
prediction = target_model(
self.external_spectrum,
self.temperature.value,
self.alpha.value,
self.beta.value,
self.gamma.value,
)
wave_axis = self.target_spectrum.spectral_axis.value
if axes is None:
gkw = {"height_ratios": [4, 1], "hspace": 0}
fig = plt.figure(figsize=(8, 6), tight_layout=True)
axes = fig.subplots(2, 1, sharex=True, gridspec_kw=gkw)
ax, axr = axes
# Target
data_kws = {} if data_kws is None else data_kws
data_kws.setdefault("color", "firebrick")
ax.plot(
wave_axis,
self.target_spectrum.flux.value,
label="target",
**data_kws,
)
# Prediction
model_kws = {} if model_kws is None else model_kws
model_kws.setdefault("color", "Navy")
ax.plot(
wave_axis,
prediction,
label="prediction",
**model_kws,
)
if show_components:
alpha_term = self.alpha.value * self.external_spectrum.flux.value
beta_term = (10**self.beta.value) * self.fitted_blackbody(
self.target_spectrum.spectral_axis
).value
gamma_term = (10**self.gamma.value) * np.ones_like(wave_axis)
ax.plot(
wave_axis,
alpha_term,
label=r"$\alpha$-term",
linestyle="--",
color="sandybrown",
)
ax.plot(
wave_axis,
beta_term,
label=r"$\beta$-term",
linestyle="--",
color="darkorchid",
)
ax.plot(
wave_axis,
gamma_term,
label=r"$\gamma$-term",
linestyle="--",
color="darkgreen",
)
residuals = (
self.target_spectrum.flux.value - prediction
) / self.target_spectrum.noise
axr.plot(
wave_axis,
residuals,
linestyle="solid",
color="gray",
)
# Ticks and Labels
ax.tick_params(axis="y", labelsize=12)
axr.tick_params(axis="both", labelsize=12)
axr.set_xlabel(r"Wavelength [$\AA$]", fontsize=12)
axr.set_ylabel("Residual", fontsize=12)
ax.set_ylabel("Intensity [arbitrary units]", fontsize=12)
ax.legend(fontsize=12)
return ax, axr
# ==============================================================================
# FITTER CLASSES
# ==============================================================================
[docs]
@attr.s
class BasinhoppingFitter:
"""Basinhopping fitter class.
Fit a BlackBody model to the data using scipy modeling methods.
Attributes
----------
target_spectrum: NirdustSpectrum object
Instance of NirdustSpectrum containing the central spectrum.
external_spectrum: NirdustSpectrum object
Instance of NirdustSpectrum containing the external spectrum.
basinhopping_kwargs: dict
Dictionary of keyword arguments to be passed to the scipy basinhopping
routine. Read the documentation for a detailed description:
docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.basinhopping.html
"""
target_spectrum = attr.ib(
validator=validators.instance_of(NirdustSpectrum)
)
external_spectrum = attr.ib(
validator=validators.instance_of(NirdustSpectrum)
)
basinhopping_kwargs = attr.ib(validator=validators.instance_of(dict))
total_noise_ = attr.ib(init=False)
# DEFINED
@total_noise_.default
def _total_noise__default(self):
"""Propagated noise."""
return np.sqrt(
self.target_spectrum.noise**2 + self.external_spectrum.noise**2
)
@property
def ndim_(self):
"""Return number of fittable parameters."""
return 4
[docs]
def fit(self, x0, minimizer_kwargs):
"""Start fitting computation.
Parameters
----------
x0: tuple
Vector indicating the initial guess values in order, i.e:
(T, alpha, beta, gamma). Default: (1000.0, 8.0, 9.0, -5.0)
minimizer_kwargs: dict
Extra keyword arguments to be passed to the local minimizer.
Return
------
results: NirdustResult object
Results of the fitting procedure.
"""
if x0 is None:
x0 = (1000.0, 8.0, 9.0, -5.0)
# x0 = make_initial_guess()
elif len(x0) != self.ndim_:
raise ValueError("Invalid initial parameters.")
res = self.run_model(x0, minimizer_kwargs)
# Now estimate parameters uncertainties
# res_with_errors = self.estimate_uncertainties(res)
# return result
temp, alpha, beta, gamma = res.x
temp = NirdustParameter("Temperature", temp * u.K, 0)
alpha = NirdustParameter("Alpha", alpha, 0)
beta = NirdustParameter("Beta", beta, 0)
gamma = NirdustParameter("Gamma", gamma, 0)
return NirdustResults(
temperature=temp,
alpha=alpha,
beta=beta,
gamma=gamma,
fitted_blackbody=BlackBody(temp.value),
target_spectrum=self.target_spectrum,
external_spectrum=self.external_spectrum,
minimizer_results=res,
)
[docs]
def run_model(self, x0, minimizer_kwargs):
"""Run fitter given an initial guess.
Parameters
----------
x0: tuple
Vector indicating the initial guess values in order, i.e:
(T, alpha, beta, gamma). Default: (1000.0, 1.0, 1.0, 1.0)
minimizer_kwargs: dict
Extra keyword arguments to be passed to the local minimizer.
Return
------
results: OptimizeResult object
Results of the local minimizer.
"""
res = basinhopping(
negative_gaussian_log_likelihood,
x0,
minimizer_kwargs=minimizer_kwargs,
**self.basinhopping_kwargs,
)
if not res.success:
warnings.warn(
"Fitting procedure failed to converge.", ConvergenceWarning
)
return res
# ==============================================================================
# FITTER FUNCTION WRAPPER
# ==============================================================================
[docs]
def make_constraints(args, gamma_fraction):
"""Make scipy minimizer constraints.
Parameters
----------
args: tuple
Extra arguments to be passed to likelihood and model functions.
args = (target_spectrum, external_spectrum)
gamma_fraction: float
Maximum fraction allowed to constraint the gamma value in the fitting
procedure.
Return
------
contraints: tuple
Constraints as required by the scipy SLSQP minimizer.
"""
gamma_vs_target_flux = make_gamma_vs_target_flux(gamma_fraction)
constraints = (
{"type": "ineq", "fun": alpha_vs_beta, "args": args},
{"type": "ineq", "fun": gamma_vs_target_flux, "args": args},
)
return constraints
[docs]
def make_minimizer_kwargs(args, bounds, constraints, options=None):
"""Make scipy minimizer keyword arguments.
Parameters
----------
args: tuple
Extra arguments to be passed to likelihood and model functions.
args = (target_spectrum, external_spectrum)
bounds: tuple
Tuple of 4 pairs of values indicating the minimum and maximum allowed
values of the fitted parameters. The order is: T, alpha, beta, gamma.
Example: bounds = ((0, 2000), (0, 20), (6, 10), (-10, 0))
contraints: tuple
Constraints as required by the scipy SLSQP minimizer.
options: dict
Extra options to be passed to the local minimizer through the
`options` keyword. Default: {"maxiter": 1000}
"""
if options is None:
options = {"maxiter": 1000}
minimizer_kwargs = {
"method": "SLSQP",
"args": args,
"bounds": bounds,
"constraints": constraints,
"options": options,
"jac": "3-point",
}
return minimizer_kwargs
# bounds
BOUNDS = ((0.0, 2000.0), (0, 20), (6, 10), (-10, 0))
[docs]
def fit_blackbody(
target_spectrum,
external_spectrum,
x0=None,
bounds=None,
gamma_target_fraction=0.05,
seed=None,
niter=200,
stepsize=1,
):
"""Fitter function.
Fit a BlackBody model to the data using Markov Chain Monte Carlo (MCMC)
sampling of the parameter space using the emcee implementation.
This function serves as a wrapper around the NirdustFitter class.
Parameters
----------
target_spectrum: NirdustSpectrum object
Instance of NirdustSpectrum containing the nuclear spectrum.
external_spectrum: NirdustSpectrum object
Instance of NirdustSpectrum containing the external spectrum.
x0: tuple, optional
Vector indicating the initial guess values of temperature, alpha, beta
and gamma.
bounds: tuple
Tuple of 4 pairs of values indicating the minimum and maximum allowed
values of the fitted parameters. The order is: T, alpha, beta, gamma.
Example: bounds = ((0, 2000), (0, 20), (6, 10), (-10, 0))
gamma_target_fraction: float
Maximum fraction of gamma vs target flux allowed to constraint the
gamma value in the fitting procedure. Default: 0.05.
seed: int
Random number generation seed for the basinhopping algorithm.
niter: int
Number of basinhopping iterations. This numbers represents how many
times the local minimizer will be excecuted. Default: 200.
stepsize: float
Maximum step size for use in the random displacement of x0 for each
basinhopping iteration. Default: 1.0.
Return
------
result: NirdustResults object
Instance of NirdustResults after the fitting procedure.
"""
# Check defaults
if bounds is None:
bounds = BOUNDS
basinhopping_kwargs = {
"niter": niter,
"T": 100,
"stepsize": stepsize,
"seed": seed,
"niter_success": None,
"interval": 50,
}
fitter = BasinhoppingFitter(
target_spectrum=target_spectrum,
external_spectrum=external_spectrum,
basinhopping_kwargs=basinhopping_kwargs,
)
args = (target_spectrum, external_spectrum)
constraints = make_constraints(args, gamma_target_fraction)
minimizer_kwargs = make_minimizer_kwargs(args, bounds, constraints)
result = fitter.fit(x0, minimizer_kwargs=minimizer_kwargs)
return result
