PowerLawSpectralModel¶

class gammapy.modeling.models.PowerLawSpectralModel(**kwargs)[source]¶

Bases: gammapy.modeling.models.SpectralModel

Spectral power-law model.

For more information see Power Law Spectral Model.

Parameters

indexQuantity: \(\Gamma\)
amplitudeQuantity: \(\phi_0\)
referenceQuantity: \(E_0\)

Attributes Summary

`amplitude`	A model parameter.
`covariance`
`default_parameters`
`index`	A model parameter.
`parameters`	Parameters (`Parameters`)
`pivot_energy`	The decorrelation energy is defined as:
`reference`	A model parameter.
`tag`

Methods Summary

`__call__`(self, energy)	Call self as a function.
`copy`(self)	A deep copy.
`create`(tag, \args, \\*kwargs)	Create a model instance.
`energy_flux`(self, emin, emax)	Compute energy flux in given energy range analytically.
`evaluate`(energy, index, amplitude, reference)	Evaluate the model (static function).
`evaluate_energy_flux`(emin, emax, index, …)	Evaluate the energy flux (static function)
`evaluate_error`(self, energy[, epsilon])	Evaluate spectral model with error propagation.
`evaluate_integral`(emin, emax, index, …)	Evaluate pwl integral (static function).
`from_dict`(data)
`from_parameters`(parameters, \\kwargs)	Create model from parameter list
`integral`(self, emin, emax, \\kwargs)	Integrate power law analytically.
`inverse`(self, value)	Return energy for a given function value of the spectral model.
`plot`(self, energy_range[, ax, energy_unit, …])	Plot spectral model curve.
`plot_error`(self, energy_range[, ax, …])	Plot spectral model error band.
`spectral_index`(self, energy[, epsilon])	Compute spectral index at given energy.
`to_dict`(self)	Create dict for YAML serialisation

Attributes Documentation

amplitude¶

A model parameter.

Note that the parameter value has been split into a factor and scale like this:

value = factor x scale

Users should interact with the value, quantity or min and max properties and consider the fact that there is a factor` and scale an implementation detail.

That was introduced for numerical stability in parameter and error estimation methods, only in the Gammapy optimiser interface do we interact with the factor, factor_min and factor_max properties, i.e. the optimiser “sees” the well-scaled problem.

Parameters

namestr: Name
valuefloat or Quantity: Value
scalefloat, optional: Scale (sometimes used in fitting)
unitUnit or str, optional: Unit
minfloat, optional: Minimum (sometimes used in fitting)
maxfloat, optional: Maximum (sometimes used in fitting)
frozenbool, optional: Frozen? (used in fitting)

covariance¶

default_parameters = <gammapy.modeling.parameter.Parameters object>¶

index¶

A model parameter.

Note that the parameter value has been split into a factor and scale like this:

value = factor x scale

Users should interact with the value, quantity or min and max properties and consider the fact that there is a factor` and scale an implementation detail.

That was introduced for numerical stability in parameter and error estimation methods, only in the Gammapy optimiser interface do we interact with the factor, factor_min and factor_max properties, i.e. the optimiser “sees” the well-scaled problem.

Parameters

namestr: Name
valuefloat or Quantity: Value
scalefloat, optional: Scale (sometimes used in fitting)
unitUnit or str, optional: Unit
minfloat, optional: Minimum (sometimes used in fitting)
maxfloat, optional: Maximum (sometimes used in fitting)
frozenbool, optional: Frozen? (used in fitting)

parameters¶: Parameters (Parameters)

pivot_energy¶

The decorrelation energy is defined as:

\[E_D = E_0 * \exp{cov(\phi_0, \Gamma) / (\phi_0 \Delta \Gamma^2)}\]

Formula (1) in https://arxiv.org/pdf/0910.4881.pdf

reference¶

A model parameter.

Note that the parameter value has been split into a factor and scale like this:

value = factor x scale

Users should interact with the value, quantity or min and max properties and consider the fact that there is a factor` and scale an implementation detail.

That was introduced for numerical stability in parameter and error estimation methods, only in the Gammapy optimiser interface do we interact with the factor, factor_min and factor_max properties, i.e. the optimiser “sees” the well-scaled problem.

Parameters

namestr: Name
valuefloat or Quantity: Value
scalefloat, optional: Scale (sometimes used in fitting)
unitUnit or str, optional: Unit
minfloat, optional: Minimum (sometimes used in fitting)
maxfloat, optional: Maximum (sometimes used in fitting)
frozenbool, optional: Frozen? (used in fitting)

tag = 'PowerLawSpectralModel'¶

Methods Documentation

__call__(self, energy)¶: Call self as a function.

copy(self)¶: A deep copy.

static create(tag, *args, **kwargs)¶

Create a model instance.

Examples

>>> from gammapy.modeling.models import Model
>>> spectral_model = Model.create("PowerLaw2SpectralModel", amplitude="1e-10 cm-2 s-1", index=3)
>>> type(spectral_model)
gammapy.modeling.models.spectral.PowerLaw2SpectralModel

energy_flux(self, emin, emax)[source]¶

Compute energy flux in given energy range analytically.

\[G(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}}E \phi(E)dE = \left. \phi_0 \frac{E_0^2}{-\Gamma + 2} \left( \frac{E}{E_0} \right)^{-\Gamma + 2} \right \vert _{E_{min}}^{E_{max}}\]

Parameters

emin, emaxQuantity: Lower and upper bound of integration range.

static evaluate(energy, index, amplitude, reference)[source]¶: Evaluate the model (static function).

static evaluate_energy_flux(emin, emax, index, amplitude, reference)[source]¶: Evaluate the energy flux (static function)

evaluate_error(self, energy, epsilon=0.0001)¶

Evaluate spectral model with error propagation.

Parameters

energyQuantity: Energy at which to evaluate
epsilonfloat: Step size of the gradient evaluation. Given as a fraction of the parameter error.

Returns

dnde, dnde_errortuple of Quantity: Tuple of flux and flux error.

static evaluate_integral(emin, emax, index, amplitude, reference)¶: Evaluate pwl integral (static function).

classmethod from_dict(data)¶

classmethod from_parameters(parameters, **kwargs)¶

Create model from parameter list

Parameters

parametersParameters: Parameters for init

Returns

modelModel: Model instance

integral(self, emin, emax, **kwargs)[source]¶

Integrate power law analytically.

\[F(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}}\phi(E)dE = \left. \phi_0 \frac{E_0}{-\Gamma + 1} \left( \frac{E}{E_0} \right)^{-\Gamma + 1} \right \vert _{E_{min}}^{E_{max}}\]

Parameters

emin, emaxQuantity: Lower and upper bound of integration range

inverse(self, value)[source]¶

Return energy for a given function value of the spectral model.

Parameters

valueQuantity: Function value of the spectral model.

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.modeling.models import ExpCutoffPowerLawSpectralModel
from astropy import units as u

pwl = ExpCutoffPowerLawSpectralModel()
ax = pwl.plot(energy_range=(0.1, 100) * u.TeV)
ax.set_yscale('linear')

Parameters

axAxes, optional: Axis
energy_rangeQuantity: Plot range
energy_unitstr, Unit, optional: Unit of the energy axis
flux_unitstr, Unit, optional: Unit of the flux axis
energy_powerint, optional: Power of energy to multiply flux axis with
n_pointsint, optional: Number of evaluation nodes

Returns

axAxes, optional: Axis

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

axAxes, optional: Axis
energy_rangeQuantity: Plot range
energy_unitstr, Unit, optional: Unit of the energy axis
flux_unitstr, Unit, optional: Unit of the flux axis
energy_powerint, optional: Power of energy to multiply flux axis with
n_pointsint, optional: Number of evaluation nodes
**kwargsdict: Keyword arguments forwarded to matplotlib.pyplot.fill_between

Returns

axAxes, optional: Axis

spectral_index(self, energy, epsilon=1e-05)¶

Compute spectral index at given energy.

Parameters

energyQuantity: Energy at which to estimate the index
epsilonfloat: Fractional energy increment to use for determining the spectral index.

Returns

indexfloat: Estimated spectral index.

to_dict(self)¶: Create dict for YAML serialisation