# 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 Quantity from .core import Fittable1DModel from .parameters import InputParameterError, Parameter __all__ = ['PowerLaw1D', 'BrokenPowerLaw1D', 'SmoothlyBrokenPowerLaw1D', 'ExponentialCutoffPowerLaw1D', 'LogParabola1D', 'Schechter1D'] 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") @staticmethod def evaluate(x, amplitude, x_0, alpha): """One dimensional power law model function""" xx = x / x_0 return amplitude * xx ** (-alpha) @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]]} 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") @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) @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]]} 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") 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.e-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") @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. ** ((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. ** ((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. + t) / 2. f[i] = amplitude * xx[i] ** (-alpha_1) * r ** ((alpha_1 - alpha_2) * delta) if return_unit: return Quantity(f, unit=return_unit, copy=False) return f @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. ** ((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. ** ((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. + t) / 2. 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. / 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. + 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]]} 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") @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) @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]]} 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") @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 @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]]} class Schechter1D(Fittable1DModel): r""" Schechter luminosity function (`Schechter 1976 `_), 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. Must not have units. 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 `_ """ phi_star = Parameter(default=1., description=('Normalization factor ' 'in units of number density')) m_star = Parameter(default=-20., description='Characteristic magnitude') alpha = Parameter(default=-1., description='Faint-end slope') @staticmethod def evaluate(mag, phi_star, m_star, alpha): """Schechter luminosity function model function.""" if isinstance(mag, Quantity) or isinstance(m_star, Quantity): raise ValueError('mag and m_star must not have units') factor = 10 ** (0.4 * (m_star - mag)) return (0.4 * np.log(10) * phi_star * factor**(alpha + 1) * np.exp(-factor)) @staticmethod def fit_deriv(mag, phi_star, m_star, alpha): """ Schechter luminosity function derivative with respect to parameters. """ if isinstance(mag, Quantity) or isinstance(m_star, Quantity): raise ValueError('mag and m_star must not have units') factor = 10 ** (0.4 * (m_star - mag)) 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]]}