# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""
Power law model variants
"""
# pylint: disable=invalid-name
import numpy as np
from astropy.units import Magnitude, Quantity, UnitsError, dimensionless_unscaled, mag
from .core import Fittable1DModel
from .parameters import InputParameterError, Parameter
__all__ = [
"PowerLaw1D",
"BrokenPowerLaw1D",
"SmoothlyBrokenPowerLaw1D",
"ExponentialCutoffPowerLaw1D",
"LogParabola1D",
"Schechter1D",
]
[docs]class PowerLaw1D(Fittable1DModel):
"""
One dimensional power law model.
Parameters
----------
amplitude : float
Model amplitude at the reference point
x_0 : float
Reference point
alpha : float
Power law index
See Also
--------
BrokenPowerLaw1D, ExponentialCutoffPowerLaw1D, LogParabola1D
Notes
-----
Model formula (with :math:`A` for ``amplitude`` and :math:`\\alpha` for ``alpha``):
.. math:: f(x) = A (x / x_0) ^ {-\\alpha}
"""
amplitude = Parameter(default=1, description="Peak value at the reference point")
x_0 = Parameter(default=1, description="Reference point")
alpha = Parameter(default=1, description="Power law index")
[docs] @staticmethod
def evaluate(x, amplitude, x_0, alpha):
"""One dimensional power law model function"""
xx = x / x_0
return amplitude * xx ** (-alpha)
[docs] @staticmethod
def fit_deriv(x, amplitude, x_0, alpha):
"""One dimensional power law derivative with respect to parameters"""
xx = x / x_0
d_amplitude = xx ** (-alpha)
d_x_0 = amplitude * alpha * d_amplitude / x_0
d_alpha = -amplitude * d_amplitude * np.log(xx)
return [d_amplitude, d_x_0, d_alpha]
@property
def input_units(self):
if self.x_0.unit is None:
return None
return {self.inputs[0]: self.x_0.unit}
def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
return {
"x_0": inputs_unit[self.inputs[0]],
"amplitude": outputs_unit[self.outputs[0]],
}
[docs]class BrokenPowerLaw1D(Fittable1DModel):
"""
One dimensional power law model with a break.
Parameters
----------
amplitude : float
Model amplitude at the break point.
x_break : float
Break point.
alpha_1 : float
Power law index for x < x_break.
alpha_2 : float
Power law index for x > x_break.
See Also
--------
PowerLaw1D, ExponentialCutoffPowerLaw1D, LogParabola1D
Notes
-----
Model formula (with :math:`A` for ``amplitude`` and :math:`\\alpha_1`
for ``alpha_1`` and :math:`\\alpha_2` for ``alpha_2``):
.. math::
f(x) = \\left \\{
\\begin{array}{ll}
A (x / x_{break}) ^ {-\\alpha_1} & : x < x_{break} \\\\
A (x / x_{break}) ^ {-\\alpha_2} & : x > x_{break} \\\\
\\end{array}
\\right.
"""
amplitude = Parameter(default=1, description="Peak value at break point")
x_break = Parameter(default=1, description="Break point")
alpha_1 = Parameter(default=1, description="Power law index before break point")
alpha_2 = Parameter(default=1, description="Power law index after break point")
[docs] @staticmethod
def evaluate(x, amplitude, x_break, alpha_1, alpha_2):
"""One dimensional broken power law model function"""
alpha = np.where(x < x_break, alpha_1, alpha_2)
xx = x / x_break
return amplitude * xx ** (-alpha)
[docs] @staticmethod
def fit_deriv(x, amplitude, x_break, alpha_1, alpha_2):
"""One dimensional broken power law derivative with respect to parameters"""
alpha = np.where(x < x_break, alpha_1, alpha_2)
xx = x / x_break
d_amplitude = xx ** (-alpha)
d_x_break = amplitude * alpha * d_amplitude / x_break
d_alpha = -amplitude * d_amplitude * np.log(xx)
d_alpha_1 = np.where(x < x_break, d_alpha, 0)
d_alpha_2 = np.where(x >= x_break, d_alpha, 0)
return [d_amplitude, d_x_break, d_alpha_1, d_alpha_2]
@property
def input_units(self):
if self.x_break.unit is None:
return None
return {self.inputs[0]: self.x_break.unit}
def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
return {
"x_break": inputs_unit[self.inputs[0]],
"amplitude": outputs_unit[self.outputs[0]],
}
[docs]class SmoothlyBrokenPowerLaw1D(Fittable1DModel):
"""One dimensional smoothly broken power law model.
Parameters
----------
amplitude : float
Model amplitude at the break point.
x_break : float
Break point.
alpha_1 : float
Power law index for ``x << x_break``.
alpha_2 : float
Power law index for ``x >> x_break``.
delta : float
Smoothness parameter.
See Also
--------
BrokenPowerLaw1D
Notes
-----
Model formula (with :math:`A` for ``amplitude``, :math:`x_b` for
``x_break``, :math:`\\alpha_1` for ``alpha_1``,
:math:`\\alpha_2` for ``alpha_2`` and :math:`\\Delta` for
``delta``):
.. math::
f(x) = A \\left( \\frac{x}{x_b} \\right) ^ {-\\alpha_1}
\\left\\{
\\frac{1}{2}
\\left[
1 + \\left( \\frac{x}{x_b}\\right)^{1 / \\Delta}
\\right]
\\right\\}^{(\\alpha_1 - \\alpha_2) \\Delta}
The change of slope occurs between the values :math:`x_1`
and :math:`x_2` such that:
.. math::
\\log_{10} \\frac{x_2}{x_b} = \\log_{10} \\frac{x_b}{x_1}
\\sim \\Delta
At values :math:`x \\lesssim x_1` and :math:`x \\gtrsim x_2` the
model is approximately a simple power law with index
:math:`\\alpha_1` and :math:`\\alpha_2` respectively. The two
power laws are smoothly joined at values :math:`x_1 < x < x_2`,
hence the :math:`\\Delta` parameter sets the "smoothness" of the
slope change.
The ``delta`` parameter is bounded to values greater than 1e-3
(corresponding to :math:`x_2 / x_1 \\gtrsim 1.002`) to avoid
overflow errors.
The ``amplitude`` parameter is bounded to positive values since
this model is typically used to represent positive quantities.
Examples
--------
.. plot::
:include-source:
import numpy as np
import matplotlib.pyplot as plt
from astropy.modeling import models
x = np.logspace(0.7, 2.3, 500)
f = models.SmoothlyBrokenPowerLaw1D(amplitude=1, x_break=20,
alpha_1=-2, alpha_2=2)
plt.figure()
plt.title("amplitude=1, x_break=20, alpha_1=-2, alpha_2=2")
f.delta = 0.5
plt.loglog(x, f(x), '--', label='delta=0.5')
f.delta = 0.3
plt.loglog(x, f(x), '-.', label='delta=0.3')
f.delta = 0.1
plt.loglog(x, f(x), label='delta=0.1')
plt.axis([x.min(), x.max(), 0.1, 1.1])
plt.legend(loc='lower center')
plt.grid(True)
plt.show()
"""
amplitude = Parameter(
default=1, min=0, description="Peak value at break point", mag=True
)
x_break = Parameter(default=1, description="Break point")
alpha_1 = Parameter(default=-2, description="Power law index before break point")
alpha_2 = Parameter(default=2, description="Power law index after break point")
delta = Parameter(default=1, min=1.0e-3, description="Smoothness Parameter")
@amplitude.validator
def amplitude(self, value):
if np.any(value <= 0):
raise InputParameterError("amplitude parameter must be > 0")
@delta.validator
def delta(self, value):
if np.any(value < 0.001):
raise InputParameterError("delta parameter must be >= 0.001")
[docs] @staticmethod
def evaluate(x, amplitude, x_break, alpha_1, alpha_2, delta):
"""One dimensional smoothly broken power law model function"""
# Pre-calculate `x/x_b`
xx = x / x_break
# Initialize the return value
f = np.zeros_like(xx, subok=False)
if isinstance(amplitude, Quantity):
return_unit = amplitude.unit
amplitude = amplitude.value
else:
return_unit = None
# The quantity `t = (x / x_b)^(1 / delta)` can become quite
# large. To avoid overflow errors we will start by calculating
# its natural logarithm:
logt = np.log(xx) / delta
# When `t >> 1` or `t << 1` we don't actually need to compute
# the `t` value since the main formula (see docstring) can be
# significantly simplified by neglecting `1` or `t`
# respectively. In the following we will check whether `t` is
# much greater, much smaller, or comparable to 1 by comparing
# the `logt` value with an appropriate threshold.
threshold = 30 # corresponding to exp(30) ~ 1e13
i = logt > threshold
if i.max():
# In this case the main formula reduces to a simple power
# law with index `alpha_2`.
f[i] = (
amplitude * xx[i] ** (-alpha_2) / (2.0 ** ((alpha_1 - alpha_2) * delta))
)
i = logt < -threshold
if i.max():
# In this case the main formula reduces to a simple power
# law with index `alpha_1`.
f[i] = (
amplitude * xx[i] ** (-alpha_1) / (2.0 ** ((alpha_1 - alpha_2) * delta))
)
i = np.abs(logt) <= threshold
if i.max():
# In this case the `t` value is "comparable" to 1, hence we
# we will evaluate the whole formula.
t = np.exp(logt[i])
r = (1.0 + t) / 2.0
f[i] = amplitude * xx[i] ** (-alpha_1) * r ** ((alpha_1 - alpha_2) * delta)
if return_unit:
return Quantity(f, unit=return_unit, copy=False, subok=True)
return f
[docs] @staticmethod
def fit_deriv(x, amplitude, x_break, alpha_1, alpha_2, delta):
"""One dimensional smoothly broken power law derivative with respect
to parameters"""
# Pre-calculate `x_b` and `x/x_b` and `logt` (see comments in
# SmoothlyBrokenPowerLaw1D.evaluate)
xx = x / x_break
logt = np.log(xx) / delta
# Initialize the return values
f = np.zeros_like(xx)
d_amplitude = np.zeros_like(xx)
d_x_break = np.zeros_like(xx)
d_alpha_1 = np.zeros_like(xx)
d_alpha_2 = np.zeros_like(xx)
d_delta = np.zeros_like(xx)
threshold = 30 # (see comments in SmoothlyBrokenPowerLaw1D.evaluate)
i = logt > threshold
if i.max():
f[i] = (
amplitude * xx[i] ** (-alpha_2) / (2.0 ** ((alpha_1 - alpha_2) * delta))
)
d_amplitude[i] = f[i] / amplitude
d_x_break[i] = f[i] * alpha_2 / x_break
d_alpha_1[i] = f[i] * (-delta * np.log(2))
d_alpha_2[i] = f[i] * (-np.log(xx[i]) + delta * np.log(2))
d_delta[i] = f[i] * (-(alpha_1 - alpha_2) * np.log(2))
i = logt < -threshold
if i.max():
f[i] = (
amplitude * xx[i] ** (-alpha_1) / (2.0 ** ((alpha_1 - alpha_2) * delta))
)
d_amplitude[i] = f[i] / amplitude
d_x_break[i] = f[i] * alpha_1 / x_break
d_alpha_1[i] = f[i] * (-np.log(xx[i]) - delta * np.log(2))
d_alpha_2[i] = f[i] * delta * np.log(2)
d_delta[i] = f[i] * (-(alpha_1 - alpha_2) * np.log(2))
i = np.abs(logt) <= threshold
if i.max():
t = np.exp(logt[i])
r = (1.0 + t) / 2.0
f[i] = amplitude * xx[i] ** (-alpha_1) * r ** ((alpha_1 - alpha_2) * delta)
d_amplitude[i] = f[i] / amplitude
d_x_break[i] = (
f[i] * (alpha_1 - (alpha_1 - alpha_2) * t / 2.0 / r) / x_break
)
d_alpha_1[i] = f[i] * (-np.log(xx[i]) + delta * np.log(r))
d_alpha_2[i] = f[i] * (-delta * np.log(r))
d_delta[i] = (
f[i]
* (alpha_1 - alpha_2)
* (np.log(r) - t / (1.0 + t) / delta * np.log(xx[i]))
)
return [d_amplitude, d_x_break, d_alpha_1, d_alpha_2, d_delta]
@property
def input_units(self):
if self.x_break.unit is None:
return None
return {self.inputs[0]: self.x_break.unit}
def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
return {
"x_break": inputs_unit[self.inputs[0]],
"amplitude": outputs_unit[self.outputs[0]],
}
[docs]class ExponentialCutoffPowerLaw1D(Fittable1DModel):
"""
One dimensional power law model with an exponential cutoff.
Parameters
----------
amplitude : float
Model amplitude
x_0 : float
Reference point
alpha : float
Power law index
x_cutoff : float
Cutoff point
See Also
--------
PowerLaw1D, BrokenPowerLaw1D, LogParabola1D
Notes
-----
Model formula (with :math:`A` for ``amplitude`` and :math:`\\alpha` for ``alpha``):
.. math:: f(x) = A (x / x_0) ^ {-\\alpha} \\exp (-x / x_{cutoff})
"""
amplitude = Parameter(default=1, description="Peak value of model")
x_0 = Parameter(default=1, description="Reference point")
alpha = Parameter(default=1, description="Power law index")
x_cutoff = Parameter(default=1, description="Cutoff point")
[docs] @staticmethod
def evaluate(x, amplitude, x_0, alpha, x_cutoff):
"""One dimensional exponential cutoff power law model function"""
xx = x / x_0
return amplitude * xx ** (-alpha) * np.exp(-x / x_cutoff)
[docs] @staticmethod
def fit_deriv(x, amplitude, x_0, alpha, x_cutoff):
"""
One dimensional exponential cutoff power law derivative with respect to parameters
"""
xx = x / x_0
xc = x / x_cutoff
d_amplitude = xx ** (-alpha) * np.exp(-xc)
d_x_0 = alpha * amplitude * d_amplitude / x_0
d_alpha = -amplitude * d_amplitude * np.log(xx)
d_x_cutoff = amplitude * x * d_amplitude / x_cutoff**2
return [d_amplitude, d_x_0, d_alpha, d_x_cutoff]
@property
def input_units(self):
if self.x_0.unit is None:
return None
return {self.inputs[0]: self.x_0.unit}
def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
return {
"x_0": inputs_unit[self.inputs[0]],
"x_cutoff": inputs_unit[self.inputs[0]],
"amplitude": outputs_unit[self.outputs[0]],
}
[docs]class LogParabola1D(Fittable1DModel):
"""
One dimensional log parabola model (sometimes called curved power law).
Parameters
----------
amplitude : float
Model amplitude
x_0 : float
Reference point
alpha : float
Power law index
beta : float
Power law curvature
See Also
--------
PowerLaw1D, BrokenPowerLaw1D, ExponentialCutoffPowerLaw1D
Notes
-----
Model formula (with :math:`A` for ``amplitude`` and
:math:`\\alpha` for ``alpha`` and :math:`\\beta` for ``beta``):
.. math:: f(x) = A \\left(
\\frac{x}{x_{0}}\\right)^{- \\alpha - \\beta \\log{\\left (\\frac{x}{x_{0}}
\\right )}}
"""
amplitude = Parameter(default=1, description="Peak value of model")
x_0 = Parameter(default=1, description="Reference point")
alpha = Parameter(default=1, description="Power law index")
beta = Parameter(default=0, description="Power law curvature")
[docs] @staticmethod
def evaluate(x, amplitude, x_0, alpha, beta):
"""One dimensional log parabola model function"""
xx = x / x_0
exponent = -alpha - beta * np.log(xx)
return amplitude * xx**exponent
[docs] @staticmethod
def fit_deriv(x, amplitude, x_0, alpha, beta):
"""One dimensional log parabola derivative with respect to parameters"""
xx = x / x_0
log_xx = np.log(xx)
exponent = -alpha - beta * log_xx
d_amplitude = xx**exponent
d_beta = -amplitude * d_amplitude * log_xx**2
d_x_0 = amplitude * d_amplitude * (beta * log_xx / x_0 - exponent / x_0)
d_alpha = -amplitude * d_amplitude * log_xx
return [d_amplitude, d_x_0, d_alpha, d_beta]
@property
def input_units(self):
if self.x_0.unit is None:
return None
return {self.inputs[0]: self.x_0.unit}
def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
return {
"x_0": inputs_unit[self.inputs[0]],
"amplitude": outputs_unit[self.outputs[0]],
}
[docs]class Schechter1D(Fittable1DModel):
r"""
Schechter luminosity function (`Schechter 1976
<https://ui.adsabs.harvard.edu/abs/1976ApJ...203..297S/abstract>`_),
parameterized in terms of magnitudes.
Parameters
----------
phi_star : float
The normalization factor in units of number density.
m_star : float
The characteristic magnitude where the power-law form of the
function cuts off.
alpha : float
The power law index, also known as the faint-end slope. Must not
have units.
See Also
--------
PowerLaw1D, ExponentialCutoffPowerLaw1D, BrokenPowerLaw1D
Notes
-----
Model formula (with :math:`\phi^{*}` for ``phi_star``, :math:`M^{*}`
for ``m_star``, and :math:`\alpha` for ``alpha``):
.. math::
n(M) \ dM = (0.4 \ln 10) \ \phi^{*} \
[{10^{0.4 (M^{*} - M)}}]^{\alpha + 1} \
\exp{[-10^{0.4 (M^{*} - M)}]} \ dM
``phi_star`` is the normalization factor in units of number density.
``m_star`` is the characteristic magnitude where the power-law form
of the function cuts off into the exponential form. ``alpha`` is
the power-law index, defining the faint-end slope of the luminosity
function.
Examples
--------
.. plot::
:include-source:
from astropy.modeling.models import Schechter1D
import astropy.units as u
import matplotlib.pyplot as plt
import numpy as np
phi_star = 4.3e-4 * (u.Mpc ** -3)
m_star = -20.26
alpha = -1.98
model = Schechter1D(phi_star, m_star, alpha)
mag = np.linspace(-25, -17)
fig, ax = plt.subplots()
ax.plot(mag, model(mag))
ax.set_yscale('log')
ax.set_xlim(-22.6, -17)
ax.set_ylim(1.e-7, 1.e-2)
ax.set_xlabel('$M_{UV}$')
ax.set_ylabel('$\phi$ [mag$^{-1}$ Mpc$^{-3}]$')
References
----------
.. [1] Schechter 1976; ApJ 203, 297
(https://ui.adsabs.harvard.edu/abs/1976ApJ...203..297S/abstract)
.. [2] `Luminosity function <https://en.wikipedia.org/wiki/Luminosity_function_(astronomy)>`_
"""
phi_star = Parameter(
default=1.0, description="Normalization factor in units of number density"
)
m_star = Parameter(default=-20.0, description="Characteristic magnitude", mag=True)
alpha = Parameter(default=-1.0, description="Faint-end slope")
@staticmethod
def _factor(magnitude, m_star):
factor_exp = magnitude - m_star
if isinstance(factor_exp, Quantity):
if factor_exp.unit == mag:
factor_exp = Magnitude(factor_exp.value, unit=mag)
return factor_exp.to(dimensionless_unscaled)
else:
raise UnitsError(
"The units of magnitude and m_star must be a magnitude"
)
else:
return 10 ** (-0.4 * factor_exp)
[docs] def evaluate(self, mag, phi_star, m_star, alpha):
"""Schechter luminosity function model function."""
factor = self._factor(mag, m_star)
return 0.4 * np.log(10) * phi_star * factor ** (alpha + 1) * np.exp(-factor)
[docs] def fit_deriv(self, mag, phi_star, m_star, alpha):
"""
Schechter luminosity function derivative with respect to
parameters.
"""
factor = self._factor(mag, m_star)
d_phi_star = 0.4 * np.log(10) * factor ** (alpha + 1) * np.exp(-factor)
func = phi_star * d_phi_star
d_m_star = (alpha + 1) * 0.4 * np.log(10) * func - (
0.4 * np.log(10) * func * factor
)
d_alpha = func * np.log(factor)
return [d_phi_star, d_m_star, d_alpha]
@property
def input_units(self):
if self.m_star.unit is None:
return None
return {self.inputs[0]: self.m_star.unit}
def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
return {
"m_star": inputs_unit[self.inputs[0]],
"phi_star": outputs_unit[self.outputs[0]],
}