PowerLawSpectralModel#

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

Bases: SpectralModel

Spectral power-law model.

For more information see Power law spectral model.

Parameters:

indexQuantity: \(\Gamma\). Default is 2.0.
amplitudeQuantity: \(\phi_0\). Default is 1e-12 cm-2 s-1 TeV-1.
referenceQuantity: \(E_0\). Default is 1 TeV.

See also

PowerLaw2SpectralModel, PowerLawNormSpectralModel

Attributes Summary

`amplitude`	A model parameter.
`covariance`
`default_parameters`
`frozen`	Frozen status of a model, True if all parameters are frozen.
`index`	A model parameter.
`is_norm_spectral_model`	Whether model is a norm spectral model.
`parameters`	Parameters as a `Parameters` object.
`parameters_unique_names`
`pivot_energy`	The pivot or decorrelation energy is defined as:
`reference`	A model parameter.
`tag`
`type`

Methods Summary

`__call__`(energy)	Call self as a function.
`copy`(**kwargs)	Deep copy.
`energy_flux`(energy_min, energy_max, **kwargs)	Compute energy flux in given energy range.
`energy_flux_error`(energy_min, energy_max[, ...])	Evaluate the error of the energy flux of a given spectrum in a given energy range.
`evaluate`(energy, index, amplitude, reference)	Evaluate the model (static function).
`evaluate_energy_flux`(energy_min, energy_max, ...)	Compute energy flux in given energy range analytically (static function).
`evaluate_error`(energy[, epsilon])	Evaluate spectral model with error propagation.
`evaluate_integral`(energy_min, energy_max, ...)	Integrate power law analytically (static function).
`freeze`()	Freeze all parameters.
`from_dict`(data, **kwargs)
`from_parameters`(parameters, **kwargs)	Create model from parameter list.
`integral`(energy_min, energy_max, **kwargs)	Integrate spectral model numerically if no analytical solution defined.
`integral_error`(energy_min, energy_max[, epsilon])	Evaluate the error of the integral flux of a given spectrum in a given energy range.
`inverse`(value, *args)	Return energy for a given function value of the spectral model.
`inverse_all`(values[, energy_min, energy_max])	Return energies for multiple function values of the spectral model.
`plot`(energy_bounds[, ax, sed_type, ...])	Plot spectral model curve.
`plot_error`(energy_bounds[, ax, sed_type, ...])	Plot spectral model error band.
`reassign`(datasets_names, new_datasets_names)	Reassign a model from one dataset to another.
`reference_fluxes`(energy_axis)	Get reference fluxes for a given energy axis.
`spectral_index`(energy[, epsilon])	Compute spectral index at given energy.
`spectral_index_error`(energy[, epsilon])	Evaluate the error on spectral index at the given energy.
`to_dict`([full_output])	Create dictionary for YAML serialisation.
`unfreeze`()	Restore parameters frozen status to default.

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).
errorfloat: Parameter error.
scan_minfloat: Minimum value for the parameter scan. Overwrites scan_n_sigma.
scan_maxfloat: Minimum value for the parameter scan. Overwrites scan_n_sigma.
scan_n_values: int: Number of values to be used for the parameter scan.
scan_n_sigmaint: Number of sigmas to scan.
scan_values: `numpy.array`: Scan values. Overwrites all the scan keywords before.
scale_method{‘scale10’, ‘factor1’, None}: Method used to set factor and scale.
interp{“lin”, “sqrt”, “log”}: Parameter scaling to use for the scan.
priorPrior: Prior set on the parameter.

covariance#

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

frozen#: Frozen status of a model, True if all parameters are frozen.

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).
errorfloat: Parameter error.
scan_minfloat: Minimum value for the parameter scan. Overwrites scan_n_sigma.
scan_maxfloat: Minimum value for the parameter scan. Overwrites scan_n_sigma.
scan_n_values: int: Number of values to be used for the parameter scan.
scan_n_sigmaint: Number of sigmas to scan.
scan_values: `numpy.array`: Scan values. Overwrites all the scan keywords before.
scale_method{‘scale10’, ‘factor1’, None}: Method used to set factor and scale.
interp{“lin”, “sqrt”, “log”}: Parameter scaling to use for the scan.
priorPrior: Prior set on the parameter.

is_norm_spectral_model#: Whether model is a norm spectral model.

parameters#: Parameters as a Parameters object.

parameters_unique_names#

pivot_energy#

The pivot or 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

Returns:

pivot energyQuantity: If no minimum is found, NaN will be returned.

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).
errorfloat: Parameter error.
scan_minfloat: Minimum value for the parameter scan. Overwrites scan_n_sigma.
scan_maxfloat: Minimum value for the parameter scan. Overwrites scan_n_sigma.
scan_n_values: int: Number of values to be used for the parameter scan.
scan_n_sigmaint: Number of sigmas to scan.
scan_values: `numpy.array`: Scan values. Overwrites all the scan keywords before.
scale_method{‘scale10’, ‘factor1’, None}: Method used to set factor and scale.
interp{“lin”, “sqrt”, “log”}: Parameter scaling to use for the scan.
priorPrior: Prior set on the parameter.

tag = ['PowerLawSpectralModel', 'pl']#

type#

Methods Documentation

__call__(energy)#: Call self as a function.

copy(**kwargs)#: Deep copy.

energy_flux(energy_min, energy_max, **kwargs)#

Compute energy flux in given energy range.

\[G(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}} E \phi(E) dE\]

Parameters:

energy_min, energy_maxQuantity: Lower and upper bound of integration range.
**kwargsdict: Keyword arguments passed to integrate_spectrum().

energy_flux_error(energy_min, energy_max, epsilon=0.0001, **kwargs)#

Evaluate the error of the energy flux of a given spectrum in a given energy range.

Parameters:

energy_min, energy_maxQuantity: Lower and upper bound of integration range.
epsilonfloat, optional: Step size of the gradient evaluation. Given as a fraction of the parameter error. Default is 1e-4.

Returns:

energy_flux, energy_flux_errtuple of Quantity: Energy flux and energy flux error between energy_min and energy_max.

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

static evaluate_energy_flux(energy_min, energy_max, index, amplitude, reference)[source]#

Compute energy flux in given energy range analytically (static function).

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

Parameters:

energy_min, energy_maxQuantity: Lower and upper bound of integration range.

evaluate_error(energy, epsilon=0.0001)#

Evaluate spectral model with error propagation.

Parameters:

energyQuantity: Energy at which to evaluate.
epsilonfloat, optional: Step size of the gradient evaluation. Given as a fraction of the parameter error. Default is 1e-4.

Returns:

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

static evaluate_integral(energy_min, energy_max, index, amplitude, reference)[source]#

Integrate power law analytically (static function).

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

Parameters:

energy_min, energy_maxQuantity: Lower and upper bound of integration range.

freeze()#: Freeze all parameters.

classmethod from_dict(data, **kwargs)#

classmethod from_parameters(parameters, **kwargs)#

Create model from parameter list.

Parameters:

parametersParameters: Parameters for init.

Returns:

modelModel: Model instance.

integral(energy_min, energy_max, **kwargs)#

Integrate spectral model numerically if no analytical solution defined.

\[F(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}} \phi(E) dE\]

Parameters:

energy_min, energy_maxQuantity: Lower and upper bound of integration range.
**kwargsdict: Keyword arguments passed to integrate_spectrum().

integral_error(energy_min, energy_max, epsilon=0.0001, **kwargs)#

Evaluate the error of the integral flux of a given spectrum in a given energy range.

Parameters:

energy_min, energy_maxQuantity: Lower and upper bound of integration range.
epsilonfloat, optional: Step size of the gradient evaluation. Given as a fraction of the parameter error. Default is 1e-4.

Returns:

flux, flux_errtuple of Quantity: Integral flux and flux error between energy_min and energy_max.

inverse(value, *args)[source]#

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

Parameters:

valueQuantity: Function value of the spectral model.

inverse_all(values, energy_min=<Quantity 0.1 TeV>, energy_max=<Quantity 100. TeV>)#

Return energies for multiple function values of the spectral model.

Calls the scipy.optimize.brentq numerical root finding method.

Parameters:

valuesQuantity: Function values of the spectral model.
energy_minQuantity, optional: Lower energy bound of the roots finding. Default is 0.1 TeV.
energy_maxQuantity, optional: Upper energy bound of the roots finding. Default is 100 TeV.

Returns:

energylist of Quantity: Each element contains the energies at which the model has corresponding value of values.

plot(energy_bounds, ax=None, sed_type='dnde', energy_power=0, n_points=100, **kwargs)#

Plot spectral model curve.

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_bounds=(0.1, 100) * u.TeV)
>>> ax.set_yscale('linear')

Parameters:

energy_boundsQuantity, list of Quantity or MapAxis: Energy bounds between which the model is to be plotted. Or an axis defining the energy bounds between which the model is to be plotted.
axAxes, optional: Matplotlib axes. Default is None.
sed_type{“dnde”, “flux”, “eflux”, “e2dnde”}: Evaluation methods of the model. Default is “dnde”.
energy_powerint, optional: Power of energy to multiply flux axis with. Default is 0.
n_pointsint, optional: Number of evaluation nodes. Default is 100.
**kwargsdict: Keyword arguments forwarded to plot.

Returns:

axAxes, optional: Matplotlib axes.

Notes

If energy_bounds is supplied as a list, tuple, or Quantity, an energy_axis is created internally with n_points bins between the given bounds.

plot_error(energy_bounds, ax=None, sed_type='dnde', energy_power=0, n_points=100, **kwargs)#

Plot spectral model error band.

Note

This method calls ax.set_yscale("log", nonpositive='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 matplotlib/matplotlib#8623).

When you call plt.loglog() or plt.semilogy() explicitly 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', nonpositive='clip') or plt.semilogy(nonpositive='clip').

Parameters:

energy_boundsQuantity, list of Quantity or MapAxis: Energy bounds between which the model is to be plotted. Or an axis defining the energy bounds between which the model is to be plotted.
axAxes, optional: Matplotlib axes. Default is None.
sed_type{“dnde”, “flux”, “eflux”, “e2dnde”}: Evaluation methods of the model. Default is “dnde”.
energy_powerint, optional: Power of energy to multiply flux axis with. Default is 0.
n_pointsint, optional: Number of evaluation nodes. Default is 100.
**kwargsdict: Keyword arguments forwarded to matplotlib.pyplot.fill_between.

Returns:

axAxes, optional: Matplotlib axes.

Notes

If energy_bounds is supplied as a list, tuple, or Quantity, an energy_axis is created internally with n_points bins between the given bounds.

reassign(datasets_names, new_datasets_names)#

Reassign a model from one dataset to another.

Parameters:

datasets_namesstr or list: Name of the datasets where the model is currently defined.
new_datasets_namesstr or list: Name of the datasets where the model should be defined instead. If multiple names are given the two list must have the save length, as the reassignment is element-wise.

Returns:

modelModel: Reassigned model.

reference_fluxes(energy_axis)#

Get reference fluxes for a given energy axis.

Parameters:

energy_axisMapAxis: Energy axis.

Returns:

fluxesdict of Quantity: Reference fluxes.

spectral_index(energy, epsilon=1e-05)#

Compute spectral index at given energy.

Parameters:

energyQuantity: Energy at which to estimate the index.
epsilonfloat, optional: Fractional energy increment to use for determining the spectral index. Default is 1e-5.

Returns:

indexfloat: Estimated spectral index.

spectral_index_error(energy, epsilon=1e-05)#

Evaluate the error on spectral index at the given energy.

Parameters:

energyQuantity: Energy at which to estimate the index.
epsilonfloat, optional: Fractional energy increment to use for determining the spectral index. Default is 1e-5.

Returns:

index, index_errortuple of float: Estimated spectral index and its error.

to_dict(full_output=False)#: Create dictionary for YAML serialisation.

unfreeze()#: Restore parameters frozen status to default.