# Licensed under a 3-clause BSD style license - see LICENSE.rst """ Power law model variants """ from __future__ import (absolute_import, unicode_literals, division, print_function) import numpy as np from .core import Fittable1DModel from .parameters import Parameter __all__ = ['PowerLaw1D', 'BrokenPowerLaw1D', 'ExponentialCutoffPowerLaw1D', 'LogParabola1D'] 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) x_0 = Parameter(default=1) alpha = Parameter(default=1) @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] 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) x_break = Parameter(default=1) alpha_1 = Parameter(default=1) alpha_2 = Parameter(default=1) @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] 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) x_0 = Parameter(default=1) alpha = Parameter(default=1) x_cutoff = Parameter(default=1) @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] 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) x_0 = Parameter(default=1) alpha = Parameter(default=1) beta = Parameter(default=0) @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]