Source code for gammapy.spectrum.models

# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""Spectral models for Gammapy."""
import operator
import numpy as np
from scipy.optimize import brentq
import astropy.units as u
from astropy.table import Table
from ..utils.energy import energy_logspace
from ..utils.scripts import make_path
from ..utils.fitting import Parameter, Parameters, Model
from ..utils.interpolation import ScaledRegularGridInterpolator
from .utils import integrate_spectrum

__all__ = [
    "SpectralModel",
    "ConstantModel",
    "CompoundSpectralModel",
    "PowerLaw",
    "PowerLaw2",
    "ExponentialCutoffPowerLaw",
    "ExponentialCutoffPowerLaw3FGL",
    "PLSuperExpCutoff3FGL",
    "PLSuperExpCutoff4FGL",
    "LogParabola",
    "TableModel",
    "AbsorbedSpectralModel",
    "Absorption",
    "NaimaModel",
    "SpectralGaussian",
    "SpectralLogGaussian",
    "ScaleModel",
]


[docs]class SpectralModel(Model): """Spectral model base class. Derived classes should store their parameters as `~gammapy.utils.modeling.Parameters` See for example return pardict of `~gammapy.spectrum.models.PowerLaw`. """
[docs] def __call__(self, energy): kwargs = {} for par in self.parameters.parameters: quantity = par.quantity if quantity.unit.physical_type == "energy": quantity = quantity.to(energy.unit) kwargs[par.name] = quantity return self.evaluate(energy, **kwargs)
def __mul__(self, model): if not isinstance(model, SpectralModel): model = ConstantModel(const=model) return CompoundSpectralModel(self, model, operator.mul) def __rmul__(self, model): # This is needed to support e.g. 5 * model return self.__mul__(model) def __add__(self, model): if not isinstance(model, SpectralModel): model = ConstantModel(const=model) return CompoundSpectralModel(self, model, operator.add) def __radd__(self, model): return self.__add__(model) def __sub__(self, model): if not isinstance(model, SpectralModel): model = ConstantModel(const=model) return CompoundSpectralModel(self, model, operator.sub) def __rsub__(self, model): return self.__sub__(model) def __truediv__(self, model): if not isinstance(model, SpectralModel): model = ConstantModel(const=model) return CompoundSpectralModel(self, model, operator.truediv) def __rtruediv__(self, model): return self.__div__(model) def _parse_uarray(self, uarray): from uncertainties import unumpy values = unumpy.nominal_values(uarray) errors = unumpy.std_devs(uarray) return values, errors def _convert_energy(self, energy): if "reference" in self.parameters.names: return energy.to(self.parameters["reference"].unit) elif "emin" in self.parameters.names: return energy.to(self.parameters["emin"].unit) else: return energy
[docs] def evaluate_error(self, energy): """Evaluate spectral model with error propagation. Parameters ---------- energy : `~astropy.units.Quantity` Energy at which to evaluate Returns ------- flux, flux_error : tuple of `~astropy.units.Quantity` Tuple of flux and flux error. """ energy = self._convert_energy(energy) unit = self(energy).unit upars = self.parameters._ufloats uarray = self.evaluate(energy.value, **upars) return self._parse_uarray(uarray) * unit
[docs] def integral(self, emin, emax, **kwargs): r"""Integrate spectral model numerically. .. math:: F(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}} \phi(E) dE If array input for ``emin`` and ``emax`` is given you have to set ``intervals=True`` if you want the integral in each energy bin. Parameters ---------- emin, emax : `~astropy.units.Quantity` Lower and upper bound of integration range. **kwargs : dict Keyword arguments passed to :func:`~gammapy.spectrum.integrate_spectrum` """ return integrate_spectrum(self, emin, emax, **kwargs)
[docs] def integral_error(self, emin, emax, **kwargs): """Integrate spectral model numerically with error propagation. Parameters ---------- emin, emax : `~astropy.units.Quantity` Lower adn upper bound of integration range. **kwargs : dict Keyword arguments passed to func:`~gammapy.spectrum.integrate_spectrum` Returns ------- integral, integral_error : tuple of `~astropy.units.Quantity` Tuple of integral flux and integral flux error. """ emin = self._convert_energy(emin) emax = self._convert_energy(emax) unit = self.integral(emin, emax, **kwargs).unit upars = self.parameters._ufloats def f(x): return self.evaluate(x, **upars) uarray = integrate_spectrum(f, emin.value, emax.value, **kwargs) return self._parse_uarray(uarray) * unit
[docs] def energy_flux(self, emin, emax, **kwargs): r"""Compute energy flux in given energy range. .. math:: G(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}} E \phi(E) dE Parameters ---------- emin, emax : `~astropy.units.Quantity` Lower and upper bound of integration range. **kwargs : dict Keyword arguments passed to func:`~gammapy.spectrum.integrate_spectrum` """ def f(x): return x * self(x) return integrate_spectrum(f, emin, emax, **kwargs)
[docs] def energy_flux_error(self, emin, emax, **kwargs): r"""Compute energy flux in given energy range with error propagation. .. math:: G(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}} E \phi(E) dE Parameters ---------- emin, emax : `~astropy.units.Quantity` Lower bound of integration range. **kwargs : dict Keyword arguments passed to :func:`~gammapy.spectrum.integrate_spectrum` Returns ------- energy_flux, energy_flux_error : tuple of `~astropy.units.Quantity` Tuple of energy flux and energy flux error. """ emin = self._convert_energy(emin) emax = self._convert_energy(emax) unit = self.energy_flux(emin, emax, **kwargs).unit upars = self.parameters._ufloats def f(x): return x * self.evaluate(x, **upars) uarray = integrate_spectrum(f, emin.value, emax.value, **kwargs) return self._parse_uarray(uarray) * unit
[docs] @classmethod def from_dict(cls, data): """Create from dict.""" data = data.copy() classname = data.pop("type") parameters = Parameters.from_dict(data) model = globals()[classname]() model.parameters = parameters model.parameters.covariance = parameters.covariance return model
[docs] def plot( self, energy_range, ax=None, energy_unit="TeV", flux_unit="cm-2 s-1 TeV-1", energy_power=0, n_points=100, **kwargs ): """Plot spectral model curve. kwargs are forwarded to `matplotlib.pyplot.plot` By default a log-log scaling of the axes is used, if you want to change the y axis scaling to linear you can use:: from gammapy.spectrum.models import ExponentialCutoffPowerLaw from astropy import units as u pwl = ExponentialCutoffPowerLaw() ax = pwl.plot(energy_range=(0.1, 100) * u.TeV) ax.set_yscale('linear') Parameters ---------- ax : `~matplotlib.axes.Axes`, optional Axis energy_range : `~astropy.units.Quantity` Plot range energy_unit : str, `~astropy.units.Unit`, optional Unit of the energy axis flux_unit : str, `~astropy.units.Unit`, optional Unit of the flux axis energy_power : int, optional Power of energy to multiply flux axis with n_points : int, optional Number of evaluation nodes Returns ------- ax : `~matplotlib.axes.Axes`, optional Axis """ import matplotlib.pyplot as plt ax = plt.gca() if ax is None else ax emin, emax = energy_range energy = energy_logspace(emin, emax, n_points, energy_unit) # evaluate model flux = self(energy).to(flux_unit) y = self._plot_scale_flux(energy, flux, energy_power) ax.plot(energy.value, y.value, **kwargs) self._plot_format_ax(ax, energy, y, energy_power) return ax
[docs] def plot_error( self, energy_range, ax=None, energy_unit="TeV", flux_unit="cm-2 s-1 TeV-1", energy_power=0, n_points=100, **kwargs ): """Plot spectral model error band. .. note:: This method calls ``ax.set_yscale("log", nonposy='clip')`` and ``ax.set_xscale("log", nonposx='clip')`` to create a log-log representation. The additional argument ``nonposx='clip'`` avoids artefacts in the plot, when the error band extends to negative values (see also https://github.com/matplotlib/matplotlib/issues/8623). When you call ``plt.loglog()`` or ``plt.semilogy()`` explicitely in your plotting code and the error band extends to negative values, it is not shown correctly. To circumvent this issue also use ``plt.loglog(nonposx='clip', nonposy='clip')`` or ``plt.semilogy(nonposy='clip')``. Parameters ---------- ax : `~matplotlib.axes.Axes`, optional Axis energy_range : `~astropy.units.Quantity` Plot range energy_unit : str, `~astropy.units.Unit`, optional Unit of the energy axis flux_unit : str, `~astropy.units.Unit`, optional Unit of the flux axis energy_power : int, optional Power of energy to multiply flux axis with n_points : int, optional Number of evaluation nodes **kwargs : dict Keyword arguments forwarded to `matplotlib.pyplot.fill_between` Returns ------- ax : `~matplotlib.axes.Axes`, optional Axis """ import matplotlib.pyplot as plt ax = plt.gca() if ax is None else ax kwargs.setdefault("facecolor", "black") kwargs.setdefault("alpha", 0.2) kwargs.setdefault("linewidth", 0) emin, emax = energy_range energy = energy_logspace(emin, emax, n_points, energy_unit) flux, flux_err = self.evaluate_error(energy).to(flux_unit) y_lo = self._plot_scale_flux(energy, flux - flux_err, energy_power) y_hi = self._plot_scale_flux(energy, flux + flux_err, energy_power) where = (energy >= energy_range[0]) & (energy <= energy_range[1]) ax.fill_between(energy.value, y_lo.value, y_hi.value, where=where, **kwargs) self._plot_format_ax(ax, energy, y_lo, energy_power) return ax
@staticmethod def _plot_format_ax(ax, energy, y, energy_power): ax.set_xlabel("Energy [{}]".format(energy.unit)) if energy_power > 0: ax.set_ylabel("E{} * Flux [{}]".format(energy_power, y.unit)) else: ax.set_ylabel("Flux [{}]".format(y.unit)) ax.set_xscale("log", nonposx="clip") ax.set_yscale("log", nonposy="clip") @staticmethod def _plot_scale_flux(energy, flux, energy_power): try: eunit = [_ for _ in flux.unit.bases if _.physical_type == "energy"][0] except IndexError: eunit = energy.unit y = flux * np.power(energy, energy_power) return y.to(flux.unit * eunit ** energy_power)
[docs] def spectral_index(self, energy, epsilon=1e-5): """Compute spectral index at given energy. Parameters ---------- energy : `~astropy.units.Quantity` Energy at which to estimate the index epsilon : float Fractional energy increment to use for determining the spectral index. Returns ------- index : float Estimated spectral index. """ f1 = self(energy) f2 = self(energy * (1 + epsilon)) return np.log(f1 / f2) / np.log(1 + epsilon)
[docs] def inverse(self, value, emin=0.1 * u.TeV, emax=100 * u.TeV): """Return energy for a given function value of the spectral model. Calls the `scipy.optimize.brentq` numerical root finding method. Parameters ---------- value : `~astropy.units.Quantity` Function value of the spectral model. emin : `~astropy.units.Quantity` Lower bracket value in case solution is not unique. emax : `~astropy.units.Quantity` Upper bracket value in case solution is not unique. Returns ------- energy : `~astropy.units.Quantity` Energies at which the model has the given ``value``. """ eunit = "TeV" energies = [] for val in np.atleast_1d(value): def f(x): # scale by 1e12 to achieve better precision energy = u.Quantity(x, eunit, copy=False) y = self(energy).to_value(value.unit) return 1e12 * (y - val.value) energy = brentq(f, emin.to_value(eunit), emax.to_value(eunit)) energies.append(energy) return u.Quantity(energies, eunit, copy=False)
[docs]class ConstantModel(SpectralModel): r"""Constant model. .. math:: \phi(E) = k Parameters ---------- const : `~astropy.units.Quantity` :math:`k` """ __slots__ = ["const"] def __init__(self, const): self.const = Parameter("const", const) super().__init__([self.const])
[docs] @staticmethod def evaluate(energy, const): """Evaluate the model (static function).""" return np.ones(np.atleast_1d(energy).shape) * const
[docs]class CompoundSpectralModel(SpectralModel): """Arithmetic combination of two spectral models. Itself again a spectral model. """ def __init__(self, model1, model2, operator): self.model1 = model1 self.model2 = model2 self.operator = operator parameters = ( self.model1.parameters.parameters + self.model2.parameters.parameters ) super().__init__(parameters) # TODO: Think about how to deal with covariance matrix def __str__(self): ss = self.__class__.__name__ ss += "\n Component 1 : {}".format(self.model1) ss += "\n Component 2 : {}".format(self.model2) ss += "\n Operator : {}".format(self.operator) return ss
[docs] def __call__(self, energy): val1 = self.model1(energy) val2 = self.model2(energy) return self.operator(val1, val2)
[docs] def to_dict(self): retval = dict() retval["model1"] = self.model1.to_dict() retval["model2"] = self.model2.to_dict() retval["operator"] = self.operator
[docs]class PowerLaw(SpectralModel): r"""Spectral power-law model. .. math:: \phi(E) = \phi_0 \cdot \left( \frac{E}{E_0} \right)^{-\Gamma} Parameters ---------- index : `~astropy.units.Quantity` :math:`\Gamma` amplitude : `~astropy.units.Quantity` :math:`\phi_0` reference : `~astropy.units.Quantity` :math:`E_0` Examples -------- This is how to plot the default `PowerLaw` model:: from astropy import units as u from gammapy.spectrum.models import PowerLaw pwl = PowerLaw() pwl.plot(energy_range=[0.1, 100] * u.TeV) plt.show() """ __slots__ = ["index", "amplitude", "reference"] def __init__(self, index=2.0, amplitude="1e-12 cm-2 s-1 TeV-1", reference="1 TeV"): self.index = Parameter("index", index) self.amplitude = Parameter("amplitude", amplitude) self.reference = Parameter("reference", reference, frozen=True) super().__init__([self.index, self.amplitude, self.reference])
[docs] @staticmethod def evaluate(energy, index, amplitude, reference): """Evaluate the model (static function).""" return amplitude * np.power((energy / reference), -index)
[docs] @staticmethod def evaluate_integral(emin, emax, index, amplitude, reference): """Evaluate the model integral (static function).""" val = -1 * index + 1 prefactor = amplitude * reference / val upper = np.power((emax / reference), val) lower = np.power((emin / reference), val) integral = prefactor * (upper - lower) mask = np.isclose(val, 0) if mask.any(): integral[mask] = (amplitude * reference * np.log(emax / emin))[mask] return integral
[docs] @staticmethod def evaluate_energy_flux(emin, emax, index, amplitude, reference): """Evaluate the energy flux (static function)""" val = -1 * index + 2 prefactor = amplitude * reference ** 2 / val upper = (emax / reference) ** val lower = (emin / reference) ** val energy_flux = prefactor * (upper - lower) mask = np.isclose(val, 0) if mask.any(): # see https://www.wolframalpha.com/input/?i=a+*+x+*+(x%2Fb)+%5E+(-2) # for reference energy_flux[mask] = amplitude * reference ** 2 * np.log(emax / emin)[mask] return energy_flux
[docs] def integral(self, emin, emax, **kwargs): r"""Integrate power law analytically. .. math:: F(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}}\phi(E)dE = \left. \phi_0 \frac{E_0}{-\Gamma + 1} \left( \frac{E}{E_0} \right)^{-\Gamma + 1} \right \vert _{E_{min}}^{E_{max}} Parameters ---------- emin, emax : `~astropy.units.Quantity` Lower and upper bound of integration range """ emin = self._convert_energy(emin) emax = self._convert_energy(emax) kwargs = {p.name: p.quantity for p in self.parameters.parameters} return self.evaluate_integral(emin=emin, emax=emax, **kwargs)
[docs] def integral_error(self, emin, emax, **kwargs): r"""Integrate power law analytically with error propagation. Parameters ---------- emin, emax : `~astropy.units.Quantity` Lower and upper bound of integration range. Returns ------- integral, integral_error : tuple of `~astropy.units.Quantity` Tuple of integral flux and integral flux error. """ # kwargs are passed to this function but not used # this is to get a consistent API with SpectralModel.integral() emin = self._convert_energy(emin) emax = self._convert_energy(emax) unit = self.integral(emin, emax, **kwargs).unit upars = self.parameters._ufloats if np.isclose(upars["index"].nominal_value, 1): prefactor = upars["amplitude"] * upars["reference"] upper = np.log(emax.value) lower = np.log(emin.value) else: val = -1 * upars["index"] + 1 prefactor = upars["amplitude"] * upars["reference"] / val upper = np.power((emax.value / upars["reference"]), val) lower = np.power((emin.value / upars["reference"]), val) uarray = prefactor * (upper - lower) return self._parse_uarray(uarray) * unit
[docs] def energy_flux(self, emin, emax): r"""Compute energy flux in given energy range analytically. .. math:: G(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}}E \phi(E)dE = \left. \phi_0 \frac{E_0^2}{-\Gamma + 2} \left( \frac{E}{E_0} \right)^{-\Gamma + 2} \right \vert _{E_{min}}^{E_{max}} Parameters ---------- emin, emax : `~astropy.units.Quantity` Lower and upper bound of integration range. """ emin = self._convert_energy(emin) emax = self._convert_energy(emax) kwargs = {p.name: p.quantity for p in self.parameters.parameters} return self.evaluate_energy_flux(emin=emin, emax=emax, **kwargs)
[docs] def energy_flux_error(self, emin, emax, **kwargs): r"""Compute energy flux in given energy range analytically with error propagation. Parameters ---------- emin, emax : `~astropy.units.Quantity` Lower and upper bound of integration range. Returns ------- energy_flux, energy_flux_error : tuple of `~astropy.units.Quantity` Tuple of energy flux and energy flux error. """ emin = self._convert_energy(emin) emax = self._convert_energy(emax) unit = self.energy_flux(emin, emax, **kwargs).unit upars = self.parameters._ufloats val = -1 * upars["index"] + 2 if np.isclose(val.nominal_value, 0): # see https://www.wolframalpha.com/input/?i=a+*+x+*+(x%2Fb)+%5E+(-2) # for reference temp = upars["amplitude"] * upars["reference"] ** 2 uarray = temp * np.log(emax.value / emin.value) else: prefactor = upars["amplitude"] * upars["reference"] ** 2 / val upper = (emax.value / upars["reference"]) ** val lower = (emin.value / upars["reference"]) ** val uarray = prefactor * (upper - lower) return self._parse_uarray(uarray) * unit
[docs] def inverse(self, value): """Return energy for a given function value of the spectral model. Parameters ---------- value : `~astropy.units.Quantity` Function value of the spectral model. """ p = self.parameters base = value / p["amplitude"].quantity return p["reference"].quantity * np.power(base, -1.0 / p["index"].value)
@property def pivot_energy(self): r"""The decorrelation energy is defined as: .. math:: E_D = E_0 * \exp{cov(\phi_0, \Gamma) / (\phi_0 \Delta \Gamma^2)} Formula (1) in https://arxiv.org/pdf/0910.4881.pdf """ index_err = self.parameters.error("index") reference = self.reference.quantity amplitude = self.amplitude.quantity cov_index_ampl = self.parameters.covariance[0, 1] * amplitude.unit return reference * np.exp(cov_index_ampl / (amplitude * index_err ** 2))
[docs]class PowerLaw2(SpectralModel): r"""Spectral power-law model with integral as amplitude parameter. See also: https://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/source_models.html .. math:: \phi(E) = F_0 \cdot \frac{\Gamma + 1}{E_{0, max}^{-\Gamma + 1} - E_{0, min}^{-\Gamma + 1}} \cdot E^{-\Gamma} Parameters ---------- index : `~astropy.units.Quantity` Spectral index :math:`\Gamma` amplitude : `~astropy.units.Quantity` Integral flux :math:`F_0`. emin : `~astropy.units.Quantity` Lower energy limit :math:`E_{0, min}`. emax : `~astropy.units.Quantity` Upper energy limit :math:`E_{0, max}`. Examples -------- This is how to plot the default `PowerLaw2` model:: from astropy import units as u from gammapy.spectrum.models import PowerLaw2 pwl2 = PowerLaw2() pwl2.plot(energy_range=[0.1, 100] * u.TeV) plt.show() """ __slots__ = ["index", "amplitude", "emin", "emax"] def __init__( self, amplitude="1e-12 cm-2 s-1", index=2, emin="0.1 TeV", emax="100 TeV" ): self.amplitude = Parameter("amplitude", amplitude) self.index = Parameter("index", index) self.emin = Parameter("emin", emin, frozen=True) self.emax = Parameter("emax", emax, frozen=True) super().__init__([self.index, self.amplitude, self.emin, self.emax])
[docs] @staticmethod def evaluate(energy, amplitude, index, emin, emax): """Evaluate the model (static function).""" top = -index + 1 # to get the energies dimensionless we use a modified formula bottom = emax - emin * (emin / emax) ** (-index) return amplitude * (top / bottom) * np.power(energy / emax, -index)
[docs] def integral(self, emin, emax, **kwargs): r"""Integrate power law analytically. .. math:: F(E_{min}, E_{max}) = F_0 \cdot \frac{E_{max}^{\Gamma + 1} \ - E_{min}^{\Gamma + 1}}{E_{0, max}^{\Gamma + 1} \ - E_{0, min}^{\Gamma + 1}} Parameters ---------- emin, emax : `~astropy.units.Quantity` Lower and upper bound of integration range. """ pars = self.parameters temp1 = np.power(emax, -pars["index"].value + 1) temp2 = np.power(emin, -pars["index"].value + 1) top = temp1 - temp2 temp1 = np.power(pars["emax"].quantity, -pars["index"].value + 1) temp2 = np.power(pars["emin"].quantity, -pars["index"].value + 1) bottom = temp1 - temp2 return pars["amplitude"].quantity * top / bottom
[docs] def integral_error(self, emin, emax, **kwargs): r"""Integrate power law analytically with error propagation. Parameters ---------- emin, emax : `~astropy.units.Quantity` Lower and upper bound of integration range. Returns ------- integral, integral_error : tuple of `~astropy.units.Quantity` Tuple of integral flux and integral flux error. """ emin = self._convert_energy(emin) emax = self._convert_energy(emax) unit = self.integral(emin, emax, **kwargs).unit upars = self.parameters._ufloats temp1 = np.power(emax.value, -upars["index"] + 1) temp2 = np.power(emin.value, -upars["index"] + 1) top = temp1 - temp2 temp1 = np.power(upars["emax"], -upars["index"] + 1) temp2 = np.power(upars["emin"], -upars["index"] + 1) bottom = temp1 - temp2 uarray = upars["amplitude"] * top / bottom return self._parse_uarray(uarray) * unit
[docs] def inverse(self, value): """Return energy for a given function value of the spectral model. Parameters ---------- value : `~astropy.units.Quantity` Function value of the spectral model. """ p = self.parameters amplitude, index, emin, emax = ( p["amplitude"].quantity, p["index"].value, p["emin"].quantity, p["emax"].quantity, ) # to get the energies dimensionless we use a modified formula top = -index + 1 bottom = emax - emin * (emin / emax) ** (-index) term = (bottom / top) * (value / amplitude) return np.power(term.to_value(""), -1.0 / index) * emax
[docs]class ExponentialCutoffPowerLaw(SpectralModel): r"""Spectral exponential cutoff power-law model. .. math:: \phi(E) = \phi_0 \cdot \left(\frac{E}{E_0}\right)^{-\Gamma} \exp(-\lambda E) Parameters ---------- index : `~astropy.units.Quantity` :math:`\Gamma` amplitude : `~astropy.units.Quantity` :math:`\phi_0` reference : `~astropy.units.Quantity` :math:`E_0` lambda_ : `~astropy.units.Quantity` :math:`\lambda` Examples -------- This is how to plot the default `ExponentialCutoffPowerLaw` model:: from astropy import units as u from gammapy.spectrum.models import ExponentialCutoffPowerLaw ecpl = ExponentialCutoffPowerLaw() ecpl.plot(energy_range=[0.1, 100] * u.TeV) plt.show() """ __slots__ = ["index", "amplitude", "reference", "lambda_"] def __init__( self, index=1.5, amplitude="1e-12 cm-2 s-1 TeV-1", reference="1 TeV", lambda_="0.1 TeV-1", ): self.index = Parameter("index", index) self.amplitude = Parameter("amplitude", amplitude) self.reference = Parameter("reference", reference, frozen=True) self.lambda_ = Parameter("lambda_", lambda_) super().__init__([self.index, self.amplitude, self.reference, self.lambda_])
[docs] @staticmethod def evaluate(energy, index, amplitude, reference, lambda_): """Evaluate the model (static function).""" pwl = amplitude * (energy / reference) ** (-index) try: cutoff = np.exp(-energy * lambda_) except AttributeError: from uncertainties.unumpy import exp cutoff = exp(-energy * lambda_) return pwl * cutoff
@property def e_peak(self): r"""Spectral energy distribution peak energy (`~astropy.utils.Quantity`). This is the peak in E^2 x dN/dE and is given by: .. math:: E_{Peak} = (2 - \Gamma) / \lambda """ p = self.parameters reference = p["reference"].quantity index = p["index"].quantity lambda_ = p["lambda_"].quantity if index >= 2: return np.nan * reference.unit else: return (2 - index) / lambda_
[docs]class ExponentialCutoffPowerLaw3FGL(SpectralModel): r"""Spectral exponential cutoff power-law model used for 3FGL. Note that the parametrization is different from `ExponentialCutoffPowerLaw`: .. math:: \phi(E) = \phi_0 \cdot \left(\frac{E}{E_0}\right)^{-\Gamma} \exp \left( \frac{E_0 - E}{E_{C}} \right) Parameters ---------- index : `~astropy.units.Quantity` :math:`\Gamma` amplitude : `~astropy.units.Quantity` :math:`\phi_0` reference : `~astropy.units.Quantity` :math:`E_0` ecut : `~astropy.units.Quantity` :math:`E_{C}` Examples -------- This is how to plot the default `ExponentialCutoffPowerLaw3FGL` model:: from astropy import units as u from gammapy.spectrum.models import ExponentialCutoffPowerLaw3FGL ecpl_3fgl = ExponentialCutoffPowerLaw3FGL() ecpl_3fgl.plot(energy_range=[0.1, 100] * u.TeV) plt.show() """ __slots__ = ["index", "amplitude", "reference", "ecut"] def __init__( self, index=1.5, amplitude="1e-12 cm-2 s-1 TeV-1", reference="1 TeV", ecut="10 TeV", ): self.index = Parameter("index", index) self.amplitude = Parameter("amplitude", amplitude) self.reference = Parameter("reference", reference, frozen=True) self.ecut = Parameter("ecut", ecut) super().__init__([self.index, self.amplitude, self.reference, self.ecut])
[docs] @staticmethod def evaluate(energy, index, amplitude, reference, ecut): """Evaluate the model (static function).""" pwl = amplitude * (energy / reference) ** (-index) try: cutoff = np.exp((reference - energy) / ecut) except AttributeError: from uncertainties.unumpy import exp cutoff = exp((reference - energy) / ecut) return pwl * cutoff
[docs]class PLSuperExpCutoff3FGL(SpectralModel): r"""Spectral super exponential cutoff power-law model used for 3FGL. .. math:: \phi(E) = \phi_0 \cdot \left(\frac{E}{E_0}\right)^{-\Gamma_1} \exp \left( \left(\frac{E_0}{E_{C}} \right)^{\Gamma_2} - \left(\frac{E}{E_{C}} \right)^{\Gamma_2} \right) Parameters ---------- index_1 : `~astropy.units.Quantity` :math:`\Gamma_1` index_2 : `~astropy.units.Quantity` :math:`\Gamma_2` amplitude : `~astropy.units.Quantity` :math:`\phi_0` reference : `~astropy.units.Quantity` :math:`E_0` ecut : `~astropy.units.Quantity` :math:`E_{C}` Examples -------- This is how to plot the default `PLSuperExpCutoff3FGL` model:: from astropy import units as u from gammapy.spectrum.models import PLSuperExpCutoff3FGL secpl_3fgl = PLSuperExpCutoff3FGL() secpl_3fgl.plot(energy_range=[0.1, 100] * u.TeV) plt.show() """ __slots__ = ["index_1", "index_2", "amplitude", "reference", "ecut"] def __init__( self, index_1=1.5, index_2=2, amplitude="1e-12 cm-2 s-1 TeV-1", reference="1 TeV", ecut="10 TeV", ): self.index_1 = Parameter("index_1", index_1) self.index_2 = Parameter("index_2", index_2) self.amplitude = Parameter("amplitude", amplitude) self.reference = Parameter("reference", reference, frozen=True) self.ecut = Parameter("ecut", ecut) super().__init__( [self.index_1, self.index_2, self.amplitude, self.reference, self.ecut] )
[docs] @staticmethod def evaluate(energy, amplitude, reference, ecut, index_1, index_2): """Evaluate the model (static function).""" pwl = amplitude * (energy / reference) ** (-index_1) try: cutoff = np.exp((reference / ecut) ** index_2 - (energy / ecut) ** index_2) except AttributeError: from uncertainties.unumpy import exp cutoff = exp((reference / ecut) ** index_2 - (energy / ecut) ** index_2) return pwl * cutoff
[docs]class PLSuperExpCutoff4FGL(SpectralModel): r"""Spectral super exponential cutoff power-law model used for 4FGL. This model parametrisation is very similar, but slightly different from `PLSuperExpCutoff3FGL` or `ExponentialCutoffPowerLaw3FGL`. See Equation (3) in https://arxiv.org/pdf/1902.10045.pdf .. math:: \phi(E) = \phi_0 \cdot \left(\frac{E}{E_0}\right)^{-\Gamma_1} \exp \left( a \left( E_0 ^{\Gamma_2} - E^{\Gamma_2} \right) \right) Parameters ---------- index_1 : `~astropy.units.Quantity` :math:`\Gamma_1` index_2 : `~astropy.units.Quantity` :math:`\Gamma_2` amplitude : `~astropy.units.Quantity` :math:`\phi_0` reference : `~astropy.units.Quantity` :math:`E_0` expfactor : `~astropy.units.Quantity` :math: `a` Examples -------- This is how to plot the default `PLSuperExpCutoff4FGL` model:: from astropy import units as u from gammapy.spectrum.models import PLSuperExpCutoff4FGL model = PLSuperExpCutoff4FGL() model.plot(energy_range=[0.1, 100] * u.TeV) plt.show() """ __slots__ = ["index_1", "index_2", "amplitude", "reference", "expfactor"] def __init__( self, index_1=1.5, index_2=2, amplitude="1e-12 cm-2 s-1 TeV-1", reference="1 TeV", expfactor="1e-2 TeV-2", ): self.index_1 = Parameter("index_1", index_1) self.index_2 = Parameter("index_2", index_2) self.amplitude = Parameter("amplitude", amplitude) self.reference = Parameter("reference", reference, frozen=True) self.expfactor = Parameter("expfactor", expfactor) super().__init__( [self.index_1, self.index_2, self.amplitude, self.reference, self.expfactor] )
[docs] @staticmethod def evaluate(energy, amplitude, reference, expfactor, index_1, index_2): """Evaluate the model (static function).""" pwl = amplitude * (energy / reference) ** (-index_1) try: cutoff = np.exp(expfactor * (reference ** index_2 - energy ** index_2)) except AttributeError: from uncertainties.unumpy import exp cutoff = exp(expfactor * (reference ** index_2 - energy ** index_2)) return pwl * cutoff
[docs]class LogParabola(SpectralModel): r"""Spectral log parabola model. .. math:: \phi(E) = \phi_0 \left( \frac{E}{E_0} \right) ^ { - \alpha - \beta \log{ \left( \frac{E}{E_0} \right) } } Note that :math:`log` refers to the natural logarithm. This is consistent with the `Fermi Science Tools <https://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/source_models.html>`_ and `ctools <http://cta.irap.omp.eu/ctools-devel/users/user_manual/getting_started/models.html#log-parabola>`_. The `Sherpa <http://cxc.harvard.edu/sherpa/ahelp/logparabola.html_ package>`_ package, however, uses :math:`log_{10}`. If you have parametrization based on :math:`log_{10}` you can use the :func:`~gammapy.spectrum.models.LogParabola.from_log10` method. Parameters ---------- amplitude : `~astropy.units.Quantity` :math:`\phi_0` reference : `~astropy.units.Quantity` :math:`E_0` alpha : `~astropy.units.Quantity` :math:`\alpha` beta : `~astropy.units.Quantity` :math:`\beta` Examples -------- This is how to plot the default `LogParabola` model:: from astropy import units as u from gammapy.spectrum.models import LogParabola log_parabola = LogParabola() log_parabola.plot(energy_range=[0.1, 100] * u.TeV) plt.show() """ __slots__ = ["amplitude", "reference", "alpha", "beta"] def __init__( self, amplitude="1e-12 cm-2 s-1 TeV-1", reference="10 TeV", alpha=2, beta=1 ): self.amplitude = Parameter("amplitude", amplitude) self.reference = Parameter("reference", reference, frozen=True) self.alpha = Parameter("alpha", alpha) self.beta = Parameter("beta", beta) super().__init__([self.amplitude, self.reference, self.alpha, self.beta])
[docs] @classmethod def from_log10(cls, amplitude, reference, alpha, beta): """Construct from :math:`log_{10}` parametrization.""" beta_ = beta / np.log(10) return cls(amplitude=amplitude, reference=reference, alpha=alpha, beta=beta_)
[docs] @staticmethod def evaluate(energy, amplitude, reference, alpha, beta): """Evaluate the model (static function).""" try: xx = (energy / reference).to("") exponent = -alpha - beta * np.log(xx) except AttributeError: from uncertainties.unumpy import log xx = energy / reference exponent = -alpha - beta * log(xx) return amplitude * np.power(xx, exponent)
@property def e_peak(self): r"""Spectral energy distribution peak energy (`~astropy.units.Quantity`). This is the peak in E^2 x dN/dE and is given by: .. math:: E_{Peak} = E_{0} \exp{ (2 - \alpha) / (2 * \beta)} """ p = self.parameters reference = p["reference"].quantity alpha = p["alpha"].quantity beta = p["beta"].quantity return reference * np.exp((2 - alpha) / (2 * beta))
[docs]class TableModel(SpectralModel): """A model generated from a table of energy and value arrays. the units returned will be the units of the values array provided at initialization. The model will return values interpolated in log-space, returning 0 for energies outside of the limits of the provided energy array. Class implementation follows closely what has been done in `naima.models.TableModel` Parameters ---------- energy : `~astropy.units.Quantity` array Array of energies at which the model values are given values : array Array with the values of the model at energies ``energy``. norm : float Model scale that is multiplied to the supplied arrays. Defaults to 1. values_scale : {'log', 'lin', 'sqrt'} Interpolation scaling applied to values. If the values vary over many magnitudes a 'log' scaling is recommended. interp_kwargs : dict Interpolation keyword arguments pass to `scipy.interpolate.interp1d`. By default all values outside the interpolation range are set to zero. If you want to apply linear extrapolation you can pass `interp_kwargs={'fill_value': 'extrapolate', 'kind': 'linear'}` meta : dict, optional Meta information, meta['filename'] will be used for serialization """ __slots__ = ["energy", "values", "norm", "meta", "_evaluate"] def __init__( self, energy, values, norm=1, values_scale="log", interp_kwargs=None, meta=None ): self.norm = Parameter("norm", norm, unit="") self.energy = energy self.values = values self.meta = dict() if meta is None else meta interp_kwargs = interp_kwargs or {} interp_kwargs.setdefault("values_scale", "log") interp_kwargs.setdefault("points_scale", ("log",)) self._evaluate = ScaledRegularGridInterpolator( points=(energy,), values=values, **interp_kwargs ) super().__init__([self.norm])
[docs] @classmethod def read_xspec_model(cls, filename, param, **kwargs): """Read XSPEC table model. The input is a table containing absorbed values from a XSPEC model as a function of energy. TODO: Format of the file should be described and discussed in https://gamma-astro-data-formats.readthedocs.io/en/latest/index.html Parameters ---------- filename : str File containing the XSPEC model param : float Model parameter value Examples -------- Fill table from an EBL model (Franceschini, 2008) >>> from gammapy.spectrum.models import TableModel >>> filename = '$GAMMAPY_DATA/ebl/ebl_franceschini.fits.gz' >>> table_model = TableModel.read_xspec_model(filename=filename, param=0.3) """ filename = str(make_path(filename)) # Check if parameter value is in range table_param = Table.read(filename, hdu="PARAMETERS") pmin = table_param["MINIMUM"] pmax = table_param["MAXIMUM"] if param < pmin or param > pmax: raise ValueError( "Out of range: param={}, min={}, max={}".format(param, pmin, pmax) ) # Get energy values table_energy = Table.read(filename, hdu="ENERGIES") energy_lo = table_energy["ENERG_LO"] energy_hi = table_energy["ENERG_HI"] # set energy to log-centers energy = np.sqrt(energy_lo * energy_hi) # Get spectrum values (no interpolation, take closest value for param) table_spectra = Table.read(filename, hdu="SPECTRA") idx = np.abs(table_spectra["PARAMVAL"] - param).argmin() values = u.Quantity(table_spectra[idx][1], "", copy=False) # no dimension kwargs.setdefault("values_scale", "lin") return cls(energy=energy, values=values, **kwargs)
[docs] @classmethod def read_fermi_isotropic_model(cls, filename, **kwargs): """Read Fermi isotropic diffuse model. See `LAT Background models <https://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html>`_ Parameters ---------- filename : str filename """ filename = str(make_path(filename)) vals = np.loadtxt(filename) energy = u.Quantity(vals[:, 0], "MeV", copy=False) values = u.Quantity(vals[:, 1], "MeV-1 s-1 cm-2 sr-1", copy=False) return cls(energy=energy, values=values, **kwargs)
[docs] def evaluate(self, energy, norm): """Evaluate the model (static function).""" values = self._evaluate((energy,), clip=True) return norm * values
[docs] def to_dict(self, selection="all"): return { "type": self.__class__.__name__, "parameters": self.parameters.to_dict(selection)["parameters"], "energy": { "data": self.energy.data.tolist(), "unit": str(self.energy.unit), }, "values": { "data": self.values.data.tolist(), "unit": str(self.values.unit), }, }
[docs]class ScaleModel(SpectralModel): """Wrapper to scale another spectral model by a norm factor. Parameters ---------- model : `SpectralModel` Spectral model to wrap. norm : float Multiplicative norm factor for the model value. """ __slots__ = ["norm", "model"] def __init__(self, model, norm=1): self.norm = Parameter("norm", norm, unit="") self.model = model super().__init__([self.norm])
[docs] def evaluate(self, energy, norm): return norm * self.model(energy)
[docs]class Absorption: r"""Gamma-ray absorption models. Parameters ---------- energy_lo, energy_hi : `~astropy.units.Quantity` Lower and upper bin edges of energy axis param_lo, param_hi : `~astropy.units.Quantity` Lower and upper bin edges of parameter axis data : `~astropy.units.Quantity` Model value Examples -------- Create and plot EBL absorption models for a redshift of 0.5: .. plot:: :include-source: import matplotlib.pyplot as plt import astropy.units as u from gammapy.spectrum.models import Absorption # Load tables for z=0.5 redshift = 0.5 dominguez = Absorption.read_builtin('dominguez').table_model(redshift) franceschini = Absorption.read_builtin('franceschini').table_model(redshift) finke = Absorption.read_builtin('finke').table_model(redshift) # start customised plot energy_range = [0.08, 3] * u.TeV ax = plt.gca() opts = dict(energy_range=energy_range, energy_unit='TeV', ax=ax, flux_unit='') franceschini.plot(label='Franceschini 2008', **opts) finke.plot(label='Finke 2010', **opts) dominguez.plot(label='Dominguez 2011', **opts) # tune plot ax.set_ylabel(r'Absorption coefficient [$\exp{(-\tau(E))}$]') ax.set_xlim(energy_range.value) # we get ride of units ax.set_ylim([1.e-4, 2.]) ax.set_yscale('log') ax.set_title('EBL models (z=' + str(redshift) + ')') plt.grid(which='both') plt.legend(loc='best') # legend # show plot plt.show() """ def __init__( self, energy_lo, energy_hi, param_lo, param_hi, data, interp_kwargs=None ): self.data = data # set values log centers self.energy = np.sqrt(energy_lo * energy_hi) self.param = (param_hi + param_lo) / 2 interp_kwargs = interp_kwargs or {} interp_kwargs.setdefault("points_scale", ("log", "lin")) self._evaluate = ScaledRegularGridInterpolator( points=(self.param, self.energy), values=data, **interp_kwargs )
[docs] @classmethod def read(cls, filename): """Build object from an XSPEC model. Todo: Format of XSPEC binary files should be referenced at https://gamma-astro-data-formats.readthedocs.io/en/latest/ Parameters ---------- filename : str File containing the model. """ # Create EBL data array filename = str(make_path(filename)) table_param = Table.read(filename, hdu="PARAMETERS") par_min = table_param["MINIMUM"] par_max = table_param["MAXIMUM"] par_array = table_param[0]["VALUE"] par_delta = np.diff(par_array) * 0.5 param_lo, param_hi = par_array, par_array # initialisation param_lo[0] = par_min - par_delta[0] param_lo[1:] -= par_delta param_hi[:-1] += par_delta param_hi[-1] = par_max # Get energy values table_energy = Table.read(filename, hdu="ENERGIES") energy_lo = u.Quantity( table_energy["ENERG_LO"], "keV", copy=False ) # unit not stored in file energy_hi = u.Quantity( table_energy["ENERG_HI"], "keV", copy=False ) # unit not stored in file # Get spectrum values table_spectra = Table.read(filename, hdu="SPECTRA") data = table_spectra["INTPSPEC"].data return cls( energy_lo=energy_lo, energy_hi=energy_hi, param_lo=param_lo, param_hi=param_hi, data=data, )
[docs] @classmethod def read_builtin(cls, name): """Read one of the built-in absorption models. Parameters ---------- name : {'franceschini', 'dominguez', 'finke'} name of one of the available model in gammapy-data References ---------- .. [1] Franceschini et al., "Extragalactic optical-infrared background radiation, its time evolution and the cosmic photon-photon opacity", `Link <https://ui.adsabs.harvard.edu/abs/2008A%26A...487..837F>`__ .. [2] Dominguez et al., " Extragalactic background light inferred from AEGIS galaxy-SED-type fractions" `Link <https://ui.adsabs.harvard.edu/abs/2011MNRAS.410.2556D>`__ .. [3] Finke et al., "Modeling the Extragalactic Background Light from Stars and Dust" `Link <https://ui.adsabs.harvard.edu/abs/2010ApJ...712..238F>`__ """ models = dict() models["franceschini"] = "$GAMMAPY_DATA/ebl/ebl_franceschini.fits.gz" models["dominguez"] = "$GAMMAPY_DATA/ebl/ebl_dominguez11.fits.gz" models["finke"] = "$GAMMAPY_DATA/ebl/frd_abs.fits.gz" return cls.read(models[name])
[docs] def table_model(self, parameter, unit="TeV"): """Table model for a given parameter (`~gammapy.spectrum.models.TableModel`). Parameters ---------- parameter : float Parameter value. unit : str, (optional) desired value for energy axis """ energy = self.energy.to(unit) values = self.evaluate(energy=energy, parameter=parameter) return TableModel(energy=energy, values=values, values_scale="lin")
[docs] def evaluate(self, energy, parameter): """Evaluate model for energy and parameter value.""" return self._evaluate((parameter, energy))
[docs]class AbsorbedSpectralModel(SpectralModel): """Spectral model with EBL absorption. Parameters ---------- spectral_model : `~gammapy.spectrum.models.SpectralModel` Spectral model. absorption : `~gammapy.spectrum.models.Absorption` Absorption model. parameter : float parameter value for absorption model parameter_name : str, optional parameter name """ __slots__ = ["spectral_model", "absorption", "parameter", "parameter_name"] def __init__( self, spectral_model, absorption, parameter, parameter_name="redshift" ): self.spectral_model = spectral_model self.absorption = absorption self.parameter = parameter self.parameter_name = parameter_name min_ = self.absorption.param.min() max_ = self.absorption.param.max() par = Parameter(parameter_name, parameter, min=min_, max=max_, frozen=True) parameters = spectral_model.parameters.parameters.copy() parameters.append(par) super().__init__(parameters)
[docs] def evaluate(self, energy, **kwargs): """Evaluate the model at a given energy.""" # assign redshift value and remove it from dictionnary # since it does not belong to the spectral model parameter = kwargs[self.parameter_name] del kwargs[self.parameter_name] flux = self.spectral_model.evaluate(energy=energy, **kwargs) absorption = self.absorption.evaluate(energy=energy, parameter=parameter) return flux * absorption
[docs]class NaimaModel(SpectralModel): r"""A wrapper for Naima models. This class provides an interface with the models defined in the `~naima.models` module. The model accepts as a positional argument a `Naima <https://naima.readthedocs.io/en/latest/>`_ radiative model instance, used to compute the non-thermal emission from populations of relativistic electrons or protons due to interactions with the ISM or with radiation and magnetic fields. One of the advantages provided by this class consists in the possibility of performing a maximum likelihood spectral fit of the model's parameters directly on observations, as opposed to the MCMC `fit to flux points <https://naima.readthedocs.io/en/latest/mcmc.html>`_ featured in Naima. All the parameters defining the parent population of charged particles are stored as `~gammapy.utils.modeling.Parameter` and left free by default. In case that the radiative model is ` ~naima.radiative.Synchrotron`, the magnetic field strength may also be fitted. Parameters can be freezed/unfreezed before the fit, and maximum/minimum values can be set to limit the parameters space to the physically interesting region. Parameters ---------- radiative_model : `~naima.models.BaseRadiative` An instance of a radiative model defined in `~naima.models` distance : `~astropy.units.Quantity`, optional Distance to the source. If set to 0, the intrinsic differential luminosity will be returned. Default is 1 kpc seed : str or list of str, optional Seed photon field(s) to be considered for the `radiative_model` flux computation, in case of a `~naima.models.InverseCompton` model. It can be a subset of the `seed_photon_fields` list defining the `radiative_model`. Default is the whole list of photon fields Examples -------- Create and plot a spectral model that convolves an `ExponentialCutoffPowerLaw` electron distribution with an `InverseCompton` radiative model, in the presence of multiple seed photon fields. .. plot:: :include-source: import naima from gammapy.spectrum.models import NaimaModel import astropy.units as u import matplotlib.pyplot as plt particle_distribution = naima.models.ExponentialCutoffPowerLaw(1e30 / u.eV, 10 * u.TeV, 3.0, 30 * u.TeV) radiative_model = naima.radiative.InverseCompton( particle_distribution, seed_photon_fields=[ "CMB", ["FIR", 26.5 * u.K, 0.415 * u.eV / u.cm ** 3], ], Eemin=100 * u.GeV, ) model = NaimaModel(radiative_model, distance=1.5 * u.kpc) opts = { "energy_range" : [10 * u.GeV, 80 * u.TeV], "energy_power" : 2, "flux_unit" : "erg-1 cm-2 s-1", } # Plot the total inverse Compton emission model.plot(label='IC (total)', **opts) # Plot the separate contributions from each seed photon field for seed, ls in zip(['CMB','FIR'], ['-','--']): model = NaimaModel(radiative_model, seed=seed, distance=1.5 * u.kpc) model.plot(label="IC ({})".format(seed), ls=ls, color="gray", **opts) plt.legend(loc='best') plt.show() """ # TODO: prevent users from setting new attributes after init def __init__(self, radiative_model, distance=1.0 * u.kpc, seed=None): import naima self.radiative_model = radiative_model self._particle_distribution = self.radiative_model.particle_distribution self.distance = Parameter("distance", distance, frozen=True) self.seed = seed # This ensures the support of naima.models.TableModel if isinstance(self._particle_distribution, naima.models.TableModel): param_names = ["amplitude"] else: param_names = self._particle_distribution.param_names parameters = [] for name in param_names: value = getattr(self._particle_distribution, name) setattr(self, name, Parameter(name, value)) parameters.append(getattr(self, name)) # In case of a synchrotron radiative model, append B to the fittable parameters if "B" in self.radiative_model.param_names: B = getattr(self.radiative_model, "B") setattr(self, "B", Parameter("B", B)) parameters.append(getattr(self, "B")) super().__init__(parameters)
[docs] def evaluate_error(self, energy): # This method will need to be overridden here, since the radiative models in naima don't # support the evaluation on energy values that is performed in the base class method raise NotImplementedError( "Error evaluation for naima models currently not supported." )
[docs] def evaluate(self, energy, **kwargs): """Evaluate the model.""" for name, value in kwargs.items(): setattr(self._particle_distribution, name, value) distance = self.distance.quantity # Flattening the input energy list and later reshaping the flux list # prevents some radiative models from displaying broadcasting problems. if self.seed is None: dnde = self.radiative_model.flux(energy.flatten(), distance=distance) else: dnde = self.radiative_model.flux( energy.flatten(), seed=self.seed, distance=distance ) dnde = dnde.reshape(energy.shape) unit = 1 / (energy.unit * u.cm ** 2 * u.s) return dnde.to(unit)
[docs]class SpectralGaussian(SpectralModel): r"""Gaussian spectral model. .. math:: \phi(E) = \frac{N_0}{\sigma \sqrt{2\pi}} \exp{ \frac{\left( E-\bar{E} \right)^2 }{2 \sigma^2} } Parameters ---------- norm : `~astropy.units.Quantity` :math:`N_0` mean : `~astropy.units.Quantity` :math:`\bar{E}` sigma : `~astropy.units.Quantity` :math:`\sigma` Examples -------- This is how to plot the default `Gaussian` spectral model: .. code:: python from astropy import units as u from gammapy.spectrum.models import SpectralGaussian gaussian = SpectralGaussian() gaussian.plot(energy_range=[0.1, 100] * u.TeV) plt.show() """ def __init__( self, norm=1e-12 * u.Unit("cm-2 s-1"), mean=1 * u.TeV, sigma=2 * u.TeV ): self.norm = Parameter("norm", norm) self.mean = Parameter("mean", mean) self.sigma = Parameter("sigma", sigma) super().__init__([self.norm, self.mean, self.sigma])
[docs] @staticmethod def evaluate(energy, norm, mean, sigma): return ( norm / (sigma * np.sqrt(2 * np.pi)) * np.exp((energy - mean) ** 2 / (2 * sigma ** 2)) )
[docs] def integral(self, emin, emax, **kwargs): r"""Integrate Gaussian analytically. .. math:: F(E_{min}, E_{max}) = \frac{N_0}{2} \left[ erf(\frac{E - \bar{E}}{\sqrt{2} \sigma})\right]_{E_{min}}^{E_{max}} Parameters ---------- emin, emax : `~astropy.units.Quantity` Lower and upper bound of integration range """ from scipy.special import erf # kwargs are passed to this function but not used # this is to get a consistent API with SpectralModel.integral() pars = self.parameters u_min = ( (emin - pars["mean"].quantity) / (np.sqrt(2) * pars["sigma"].quantity) ).to_value("") u_max = ( (emax - pars["mean"].quantity) / (np.sqrt(2) * pars["sigma"].quantity) ).to_value("") return pars["norm"].quantity / (2) * (erf(u_max) - erf(u_min))
[docs] def energy_flux(self, emin, emax): r"""Compute energy flux in given energy range analytically. .. math:: G(E_{min}, E_{max}) = \frac{N_0 \sigma}{\sqrt{2*\pi}}* \left[ - \exp(\frac{E_{min}-\bar{E}}{\sqrt{2} \sigma}) \right]_{E_{min}}^{E_{max}} + \frac{N_0 * \bar{E}}{2} \left[ erf(\frac{E - \bar{E}}{\sqrt{2} \sigma}) \right]_{E_{min}}^{E_{max}} Parameters ---------- emin, emax : `~astropy.units.Quantity` Lower and upper bound of integration range. """ from scipy.special import erf pars = self.parameters u_min = ( (emin - pars["mean"].quantity) / (np.sqrt(2) * pars["sigma"].quantity) ).to_value("") u_max = ( (emax - pars["mean"].quantity) / (np.sqrt(2) * pars["sigma"].quantity) ).to_value("") a = pars["norm"].quantity * pars["sigma"].quantity / np.sqrt(2 * np.pi) b = pars["norm"].quantity * pars["mean"].quantity / 2 return a * (np.exp(-u_min ** 2) - np.exp(-u_max ** 2)) + b * ( erf(u_max) - erf(u_min) )
[docs]class SpectralLogGaussian(SpectralModel): r"""Gaussian Log spectral model. .. math:: \phi(E) = \frac{N_0}{E \, \sigma \sqrt{2\pi}} \exp{ \frac{\left( \ln(\frac{E}{\bar{E}}) \right)^2 }{2 \sigma^2} } This model was used in this CTA study for the electron spectrum: Table 3 in https://ui.adsabs.harvard.edu/abs/2013APh....43..171B Parameters ---------- norm : `~astropy.units.Quantity` :math:`N_0` mean : `~astropy.units.Quantity` :math:`\bar{E}` sigma : `float` :math:`\sigma` Examples -------- This is how to plot a Gaussian Log spectral model. Very similar from the `SpectralGaussian` model but the Gaussian is based on the logarithm of the energy .. code:: python from astropy import units as u from gammapy.spectrum.models import SpectralLogGaussian gaussian = SpectralLogGaussian() gaussian.plot(energy_range=[0.1, 100] * u.TeV) plt.show() """ def __init__(self, norm=1e-12 * u.Unit("cm-2 s-1"), mean=1 * u.TeV, sigma=2): self.norm = Parameter("norm", norm) self.mean = Parameter("mean", mean) self.sigma = Parameter("sigma", sigma) super().__init__([self.norm, self.mean, self.sigma])
[docs] @staticmethod def evaluate(energy, norm, mean, sigma): return ( norm / (energy * sigma * np.sqrt(2 * np.pi)) * np.exp(-(np.log(energy / mean)) ** 2 / (2 * sigma ** 2)) )