# PowerLawSpectralModel#

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

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 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 (Parameters) pivot_energy The decorrelation energy is defined as: reference A model parameter. tag type

Methods Summary

 __call__(energy) Call self as a function. copy(**kwargs) 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 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 all parameters from_dict(data) 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. to_dict([full_output]) Create dict for YAML serialisation 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 of 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.

is_normbool

Whether the parameter represents the flux norm of the model.

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 of 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.

is_normbool

Whether the parameter represents the flux norm of the model.

is_norm_spectral_model#

Whether model is a norm spectral model

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)

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 of 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.

is_normbool

Whether the parameter represents the flux norm of the model.

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

Methods Documentation

__call__(energy)#

Call self as a function.

copy(**kwargs)#
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 func: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

Step size of the gradient evaluation. Given as a fraction of the parameter error.

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

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(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)#
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

Step size of the gradient evaluation. Given as a fraction of the parameter error.

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

Lower energy bound of the roots finding

energy_maxQuantity

Upper energy bound of the roots finding

Returns
energylist of Quantity

each element contain 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.

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

Parameters
axAxes, optional

Axis

energy_boundsQuantity

Plot energy bounds passed to MapAxis.from_energy_bounds

sed_type{“dnde”, “flux”, “eflux”, “e2dnde”}

Evaluation methods of the model

energy_powerint, optional

Power of energy to multiply flux axis with

n_pointsint, optional

Number of evaluation nodes

**kwargsdict

Keyword arguments forwarded to plot

Returns
axAxes, optional

Axis

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 https://github.com/matplotlib/matplotlib/issues/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
axAxes, optional

Axis

energy_boundsQuantity

Plot energy bounds passed to MapAxis.from_energy_bounds

sed_type{“dnde”, “flux”, “eflux”, “e2dnde”}

Evaluation methods of the model

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

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

Fractional energy increment to use for determining the spectral index.

Returns
indexfloat

Estimated spectral index.

to_dict(full_output=False)#

Create dict for YAML serialisation

unfreeze()#

Restore parameters frozen status to default