Source code for astropy.modeling.functional_models

# Licensed under a 3-clause BSD style license - see LICENSE.rst

"""Mathematical models."""
# pylint: disable=line-too-long, too-many-lines, too-many-arguments, invalid-name
import numpy as np

from astropy import units as u
from astropy.units import Quantity, UnitsError
from astropy.utils.decorators import deprecated
from .core import (Fittable1DModel, Fittable2DModel)

from .parameters import Parameter, InputParameterError
from .utils import ellipse_extent


__all__ = ['AiryDisk2D', 'Moffat1D', 'Moffat2D', 'Box1D', 'Box2D', 'Const1D',
           'Const2D', 'Ellipse2D', 'Disk2D', 'Gaussian1D', 'Gaussian2D',
           'Linear1D', 'Lorentz1D', 'RickerWavelet1D', 'RickerWavelet2D',
           'RedshiftScaleFactor', 'Multiply', 'Planar2D', 'Scale',
           'Sersic1D', 'Sersic2D', 'Shift', 'Sine1D', 'Trapezoid1D',
           'TrapezoidDisk2D', 'Ring2D', 'Voigt1D', 'KingProjectedAnalytic1D',
           'Exponential1D', 'Logarithmic1D']

TWOPI = 2 * np.pi
FLOAT_EPSILON = float(np.finfo(np.float32).tiny)

# Note that we define this here rather than using the value defined in
# astropy.stats to avoid importing astropy.stats every time astropy.modeling
# is loaded.
GAUSSIAN_SIGMA_TO_FWHM = 2.0 * np.sqrt(2.0 * np.log(2.0))


[docs]class Gaussian1D(Fittable1DModel): """ One dimensional Gaussian model. Parameters ---------- amplitude : float or `~astropy.units.Quantity`. Amplitude (peak value) of the Gaussian - for a normalized profile (integrating to 1), set amplitude = 1 / (stddev * np.sqrt(2 * np.pi)) mean : float or `~astropy.units.Quantity`. Mean of the Gaussian. stddev : float or `~astropy.units.Quantity`. Standard deviation of the Gaussian with FWHM = 2 * stddev * np.sqrt(2 * np.log(2)). Notes ----- Either all or none of input ``x``, ``mean`` and ``stddev`` must be provided consistently with compatible units or as unitless numbers. Model formula: .. math:: f(x) = A e^{- \\frac{\\left(x - x_{0}\\right)^{2}}{2 \\sigma^{2}}} Examples -------- >>> from astropy.modeling import models >>> def tie_center(model): ... mean = 50 * model.stddev ... return mean >>> tied_parameters = {'mean': tie_center} Specify that 'mean' is a tied parameter in one of two ways: >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3, ... tied=tied_parameters) or >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3) >>> g1.mean.tied False >>> g1.mean.tied = tie_center >>> g1.mean.tied <function tie_center at 0x...> Fixed parameters: >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3, ... fixed={'stddev': True}) >>> g1.stddev.fixed True or >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3) >>> g1.stddev.fixed False >>> g1.stddev.fixed = True >>> g1.stddev.fixed True .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Gaussian1D plt.figure() s1 = Gaussian1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -1, 4]) plt.show() See Also -------- Gaussian2D, Box1D, Moffat1D, Lorentz1D """ amplitude = Parameter(default=1, description="Amplitude (peak value) of the Gaussian") mean = Parameter(default=0, description="Position of peak (Gaussian)") # Ensure stddev makes sense if its bounds are not explicitly set. # stddev must be non-zero and positive. stddev = Parameter(default=1, bounds=(FLOAT_EPSILON, None), description="Standard deviation of the Gaussian") def bounding_box(self, factor=5.5): """ Tuple defining the default ``bounding_box`` limits, ``(x_low, x_high)`` Parameters ---------- factor : float The multiple of `stddev` used to define the limits. The default is 5.5, corresponding to a relative error < 1e-7. Examples -------- >>> from astropy.modeling.models import Gaussian1D >>> model = Gaussian1D(mean=0, stddev=2) >>> model.bounding_box (-11.0, 11.0) This range can be set directly (see: `Model.bounding_box <astropy.modeling.Model.bounding_box>`) or by using a different factor, like: >>> model.bounding_box = model.bounding_box(factor=2) >>> model.bounding_box (-4.0, 4.0) """ x0 = self.mean dx = factor * self.stddev return (x0 - dx, x0 + dx) @property def fwhm(self): """Gaussian full width at half maximum.""" return self.stddev * GAUSSIAN_SIGMA_TO_FWHM
[docs] @staticmethod def evaluate(x, amplitude, mean, stddev): """ Gaussian1D model function. """ return amplitude * np.exp(- 0.5 * (x - mean) ** 2 / stddev ** 2)
[docs] @staticmethod def fit_deriv(x, amplitude, mean, stddev): """ Gaussian1D model function derivatives. """ d_amplitude = np.exp(-0.5 / stddev ** 2 * (x - mean) ** 2) d_mean = amplitude * d_amplitude * (x - mean) / stddev ** 2 d_stddev = amplitude * d_amplitude * (x - mean) ** 2 / stddev ** 3 return [d_amplitude, d_mean, d_stddev]
@property def input_units(self): if self.mean.unit is None: return None return {self.inputs[0]: self.mean.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'mean': inputs_unit[self.inputs[0]], 'stddev': inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Gaussian2D(Fittable2DModel): r""" Two dimensional Gaussian model. Parameters ---------- amplitude : float or `~astropy.units.Quantity`. Amplitude (peak value) of the Gaussian. x_mean : float or `~astropy.units.Quantity`. Mean of the Gaussian in x. y_mean : float or `~astropy.units.Quantity`. Mean of the Gaussian in y. x_stddev : float or `~astropy.units.Quantity` or None. Standard deviation of the Gaussian in x before rotating by theta. Must be None if a covariance matrix (``cov_matrix``) is provided. If no ``cov_matrix`` is given, ``None`` means the default value (1). y_stddev : float or `~astropy.units.Quantity` or None. Standard deviation of the Gaussian in y before rotating by theta. Must be None if a covariance matrix (``cov_matrix``) is provided. If no ``cov_matrix`` is given, ``None`` means the default value (1). theta : float or `~astropy.units.Quantity`, optional. Rotation angle (value in radians). The rotation angle increases counterclockwise. Must be None if a covariance matrix (``cov_matrix``) is provided. If no ``cov_matrix`` is given, ``None`` means the default value (0). cov_matrix : ndarray, optional A 2x2 covariance matrix. If specified, overrides the ``x_stddev``, ``y_stddev``, and ``theta`` defaults. Notes ----- Either all or none of input ``x, y``, ``[x,y]_mean`` and ``[x,y]_stddev`` must be provided consistently with compatible units or as unitless numbers. Model formula: .. math:: f(x, y) = A e^{-a\left(x - x_{0}\right)^{2} -b\left(x - x_{0}\right) \left(y - y_{0}\right) -c\left(y - y_{0}\right)^{2}} Using the following definitions: .. math:: a = \left(\frac{\cos^{2}{\left (\theta \right )}}{2 \sigma_{x}^{2}} + \frac{\sin^{2}{\left (\theta \right )}}{2 \sigma_{y}^{2}}\right) b = \left(\frac{\sin{\left (2 \theta \right )}}{2 \sigma_{x}^{2}} - \frac{\sin{\left (2 \theta \right )}}{2 \sigma_{y}^{2}}\right) c = \left(\frac{\sin^{2}{\left (\theta \right )}}{2 \sigma_{x}^{2}} + \frac{\cos^{2}{\left (\theta \right )}}{2 \sigma_{y}^{2}}\right) If using a ``cov_matrix``, the model is of the form: .. math:: f(x, y) = A e^{-0.5 \left(\vec{x} - \vec{x}_{0}\right)^{T} \Sigma^{-1} \left(\vec{x} - \vec{x}_{0}\right)} where :math:`\vec{x} = [x, y]`, :math:`\vec{x}_{0} = [x_{0}, y_{0}]`, and :math:`\Sigma` is the covariance matrix: .. math:: \Sigma = \left(\begin{array}{ccc} \sigma_x^2 & \rho \sigma_x \sigma_y \\ \rho \sigma_x \sigma_y & \sigma_y^2 \end{array}\right) :math:`\rho` is the correlation between ``x`` and ``y``, which should be between -1 and +1. Positive correlation corresponds to a ``theta`` in the range 0 to 90 degrees. Negative correlation corresponds to a ``theta`` in the range of 0 to -90 degrees. See [1]_ for more details about the 2D Gaussian function. See Also -------- Gaussian1D, Box2D, Moffat2D References ---------- .. [1] https://en.wikipedia.org/wiki/Gaussian_function """ amplitude = Parameter(default=1, description="Amplitude of the Gaussian") x_mean = Parameter(default=0, description="Peak position (along x axis) of Gaussian") y_mean = Parameter(default=0, description="Peak position (along y axis) of Gaussian") x_stddev = Parameter(default=1, description="Standard deviation of the Gaussian (along x axis)") y_stddev = Parameter(default=1, description="Standard deviation of the Gaussian (along y axis)") theta = Parameter(default=0.0, description="Rotation angle [in radians] (Optional parameter)") def __init__(self, amplitude=amplitude.default, x_mean=x_mean.default, y_mean=y_mean.default, x_stddev=None, y_stddev=None, theta=None, cov_matrix=None, **kwargs): if cov_matrix is None: if x_stddev is None: x_stddev = self.__class__.x_stddev.default if y_stddev is None: y_stddev = self.__class__.y_stddev.default if theta is None: theta = self.__class__.theta.default else: if x_stddev is not None or y_stddev is not None or theta is not None: raise InputParameterError("Cannot specify both cov_matrix and " "x/y_stddev/theta") # Compute principle coordinate system transformation cov_matrix = np.array(cov_matrix) if cov_matrix.shape != (2, 2): raise ValueError("Covariance matrix must be 2x2") eig_vals, eig_vecs = np.linalg.eig(cov_matrix) x_stddev, y_stddev = np.sqrt(eig_vals) y_vec = eig_vecs[:, 0] theta = np.arctan2(y_vec[1], y_vec[0]) # Ensure stddev makes sense if its bounds are not explicitly set. # stddev must be non-zero and positive. # TODO: Investigate why setting this in Parameter above causes # convolution tests to hang. kwargs.setdefault('bounds', {}) kwargs['bounds'].setdefault('x_stddev', (FLOAT_EPSILON, None)) kwargs['bounds'].setdefault('y_stddev', (FLOAT_EPSILON, None)) super().__init__( amplitude=amplitude, x_mean=x_mean, y_mean=y_mean, x_stddev=x_stddev, y_stddev=y_stddev, theta=theta, **kwargs) @property def x_fwhm(self): """Gaussian full width at half maximum in X.""" return self.x_stddev * GAUSSIAN_SIGMA_TO_FWHM @property def y_fwhm(self): """Gaussian full width at half maximum in Y.""" return self.y_stddev * GAUSSIAN_SIGMA_TO_FWHM def bounding_box(self, factor=5.5): """ Tuple defining the default ``bounding_box`` limits in each dimension, ``((y_low, y_high), (x_low, x_high))`` The default offset from the mean is 5.5-sigma, corresponding to a relative error < 1e-7. The limits are adjusted for rotation. Parameters ---------- factor : float, optional The multiple of `x_stddev` and `y_stddev` used to define the limits. The default is 5.5. Examples -------- >>> from astropy.modeling.models import Gaussian2D >>> model = Gaussian2D(x_mean=0, y_mean=0, x_stddev=1, y_stddev=2) >>> model.bounding_box ((-11.0, 11.0), (-5.5, 5.5)) This range can be set directly (see: `Model.bounding_box <astropy.modeling.Model.bounding_box>`) or by using a different factor like: >>> model.bounding_box = model.bounding_box(factor=2) >>> model.bounding_box ((-4.0, 4.0), (-2.0, 2.0)) """ a = factor * self.x_stddev b = factor * self.y_stddev theta = self.theta.value dx, dy = ellipse_extent(a, b, theta) return ((self.y_mean - dy, self.y_mean + dy), (self.x_mean - dx, self.x_mean + dx))
[docs] @staticmethod def evaluate(x, y, amplitude, x_mean, y_mean, x_stddev, y_stddev, theta): """Two dimensional Gaussian function""" cost2 = np.cos(theta) ** 2 sint2 = np.sin(theta) ** 2 sin2t = np.sin(2. * theta) xstd2 = x_stddev ** 2 ystd2 = y_stddev ** 2 xdiff = x - x_mean ydiff = y - y_mean a = 0.5 * ((cost2 / xstd2) + (sint2 / ystd2)) b = 0.5 * ((sin2t / xstd2) - (sin2t / ystd2)) c = 0.5 * ((sint2 / xstd2) + (cost2 / ystd2)) return amplitude * np.exp(-((a * xdiff ** 2) + (b * xdiff * ydiff) + (c * ydiff ** 2)))
[docs] @staticmethod def fit_deriv(x, y, amplitude, x_mean, y_mean, x_stddev, y_stddev, theta): """Two dimensional Gaussian function derivative with respect to parameters""" cost = np.cos(theta) sint = np.sin(theta) cost2 = np.cos(theta) ** 2 sint2 = np.sin(theta) ** 2 cos2t = np.cos(2. * theta) sin2t = np.sin(2. * theta) xstd2 = x_stddev ** 2 ystd2 = y_stddev ** 2 xstd3 = x_stddev ** 3 ystd3 = y_stddev ** 3 xdiff = x - x_mean ydiff = y - y_mean xdiff2 = xdiff ** 2 ydiff2 = ydiff ** 2 a = 0.5 * ((cost2 / xstd2) + (sint2 / ystd2)) b = 0.5 * ((sin2t / xstd2) - (sin2t / ystd2)) c = 0.5 * ((sint2 / xstd2) + (cost2 / ystd2)) g = amplitude * np.exp(-((a * xdiff2) + (b * xdiff * ydiff) + (c * ydiff2))) da_dtheta = (sint * cost * ((1. / ystd2) - (1. / xstd2))) da_dx_stddev = -cost2 / xstd3 da_dy_stddev = -sint2 / ystd3 db_dtheta = (cos2t / xstd2) - (cos2t / ystd2) db_dx_stddev = -sin2t / xstd3 db_dy_stddev = sin2t / ystd3 dc_dtheta = -da_dtheta dc_dx_stddev = -sint2 / xstd3 dc_dy_stddev = -cost2 / ystd3 dg_dA = g / amplitude dg_dx_mean = g * ((2. * a * xdiff) + (b * ydiff)) dg_dy_mean = g * ((b * xdiff) + (2. * c * ydiff)) dg_dx_stddev = g * (-(da_dx_stddev * xdiff2 + db_dx_stddev * xdiff * ydiff + dc_dx_stddev * ydiff2)) dg_dy_stddev = g * (-(da_dy_stddev * xdiff2 + db_dy_stddev * xdiff * ydiff + dc_dy_stddev * ydiff2)) dg_dtheta = g * (-(da_dtheta * xdiff2 + db_dtheta * xdiff * ydiff + dc_dtheta * ydiff2)) return [dg_dA, dg_dx_mean, dg_dy_mean, dg_dx_stddev, dg_dy_stddev, dg_dtheta]
@property def input_units(self): if self.x_mean.unit is None and self.y_mean.unit is None: return None return {self.inputs[0]: self.x_mean.unit, self.inputs[1]: self.y_mean.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): # Note that here we need to make sure that x and y are in the same # units otherwise this can lead to issues since rotation is not well # defined. if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]: raise UnitsError("Units of 'x' and 'y' inputs should match") return {'x_mean': inputs_unit[self.inputs[0]], 'y_mean': inputs_unit[self.inputs[0]], 'x_stddev': inputs_unit[self.inputs[0]], 'y_stddev': inputs_unit[self.inputs[0]], 'theta': u.rad, 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Shift(Fittable1DModel): """ Shift a coordinate. Parameters ---------- offset : float Offset to add to a coordinate. """ offset = Parameter(default=0, description="Offset to add to a model") linear = True _has_inverse_bounding_box = True @property def input_units(self): if self.offset.unit is None: return None return {self.inputs[0]: self.offset.unit} @property def inverse(self): """One dimensional inverse Shift model function""" inv = self.copy() inv.offset *= -1 try: self.bounding_box except NotImplementedError: pass else: inv.bounding_box = tuple(self.evaluate(x, self.offset) for x in self.bounding_box) return inv
[docs] @staticmethod def evaluate(x, offset): """One dimensional Shift model function""" return x + offset
[docs] @staticmethod def sum_of_implicit_terms(x): """Evaluate the implicit term (x) of one dimensional Shift model""" return x
[docs] @staticmethod def fit_deriv(x, *params): """One dimensional Shift model derivative with respect to parameter""" d_offset = np.ones_like(x) return [d_offset]
def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'offset': outputs_unit[self.outputs[0]]}
[docs]class Scale(Fittable1DModel): """ Multiply a model by a dimensionless factor. Parameters ---------- factor : float Factor by which to scale a coordinate. Notes ----- If ``factor`` is a `~astropy.units.Quantity` then the units will be stripped before the scaling operation. """ factor = Parameter(default=1, description="Factor by which to scale a model") linear = True fittable = True _input_units_strict = True _input_units_allow_dimensionless = True _has_inverse_bounding_box = True @property def input_units(self): if self.factor.unit is None: return None return {self.inputs[0]: self.factor.unit} @property def inverse(self): """One dimensional inverse Scale model function""" inv = self.copy() inv.factor = 1 / self.factor try: self.bounding_box except NotImplementedError: pass else: inv.bounding_box = tuple(self.evaluate(x, self.factor) for x in self.bounding_box) return inv
[docs] @staticmethod def evaluate(x, factor): """One dimensional Scale model function""" if isinstance(factor, u.Quantity): factor = factor.value return factor * x
[docs] @staticmethod def fit_deriv(x, *params): """One dimensional Scale model derivative with respect to parameter""" d_factor = x return [d_factor]
def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'factor': outputs_unit[self.outputs[0]]}
[docs]class Multiply(Fittable1DModel): """ Multiply a model by a quantity or number. Parameters ---------- factor : float Factor by which to multiply a coordinate. """ factor = Parameter(default=1, description="Factor by which to multiply a model") linear = True fittable = True _has_inverse_bounding_box = True @property def inverse(self): """One dimensional inverse multiply model function""" inv = self.copy() inv.factor = 1 / self.factor try: self.bounding_box except NotImplementedError: pass else: inv.bounding_box = tuple(self.evaluate(x, self.factor) for x in self.bounding_box) return inv
[docs] @staticmethod def evaluate(x, factor): """One dimensional multiply model function""" return factor * x
[docs] @staticmethod def fit_deriv(x, *params): """One dimensional multiply model derivative with respect to parameter""" d_factor = x return [d_factor]
def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'factor': outputs_unit[self.outputs[0]]}
[docs]class RedshiftScaleFactor(Fittable1DModel): """ One dimensional redshift scale factor model. Parameters ---------- z : float Redshift value. Notes ----- Model formula: .. math:: f(x) = x (1 + z) """ z = Parameter(description='Redshift', default=0) _has_inverse_bounding_box = True
[docs] @staticmethod def evaluate(x, z): """One dimensional RedshiftScaleFactor model function""" return (1 + z) * x
[docs] @staticmethod def fit_deriv(x, z): """One dimensional RedshiftScaleFactor model derivative""" d_z = x return [d_z]
@property def inverse(self): """Inverse RedshiftScaleFactor model""" inv = self.copy() inv.z = 1.0 / (1.0 + self.z) - 1.0 try: self.bounding_box except NotImplementedError: pass else: inv.bounding_box = tuple(self.evaluate(x, self.z) for x in self.bounding_box) return inv
[docs]class Sersic1D(Fittable1DModel): r""" One dimensional Sersic surface brightness profile. Parameters ---------- amplitude : float Surface brightness at r_eff. r_eff : float Effective (half-light) radius n : float Sersic Index. See Also -------- Gaussian1D, Moffat1D, Lorentz1D Notes ----- Model formula: .. math:: I(r)=I_e\exp\left\{-b_n\left[\left(\frac{r}{r_{e}}\right)^{(1/n)}-1\right]\right\} The constant :math:`b_n` is defined such that :math:`r_e` contains half the total luminosity, and can be solved for numerically. .. math:: \Gamma(2n) = 2\gamma (b_n,2n) Examples -------- .. plot:: :include-source: import numpy as np from astropy.modeling.models import Sersic1D import matplotlib.pyplot as plt plt.figure() plt.subplot(111, xscale='log', yscale='log') s1 = Sersic1D(amplitude=1, r_eff=5) r=np.arange(0, 100, .01) for n in range(1, 10): s1.n = n plt.plot(r, s1(r), color=str(float(n) / 15)) plt.axis([1e-1, 30, 1e-2, 1e3]) plt.xlabel('log Radius') plt.ylabel('log Surface Brightness') plt.text(.25, 1.5, 'n=1') plt.text(.25, 300, 'n=10') plt.xticks([]) plt.yticks([]) plt.show() References ---------- .. [1] http://ned.ipac.caltech.edu/level5/March05/Graham/Graham2.html """ amplitude = Parameter(default=1, description="Surface brightness at r_eff") r_eff = Parameter(default=1, description="Effective (half-light) radius") n = Parameter(default=4, description="Sersic Index") _gammaincinv = None
[docs] @classmethod def evaluate(cls, r, amplitude, r_eff, n): """One dimensional Sersic profile function.""" if cls._gammaincinv is None: try: from scipy.special import gammaincinv cls._gammaincinv = gammaincinv except ValueError: raise ImportError('Sersic1D model requires scipy.') return (amplitude * np.exp( -cls._gammaincinv(2 * n, 0.5) * ((r / r_eff) ** (1 / n) - 1)))
@property def input_units(self): if self.r_eff.unit is None: return None return {self.inputs[0]: self.r_eff.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'r_eff': inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Sine1D(Fittable1DModel): """ One dimensional Sine model. Parameters ---------- amplitude : float Oscillation amplitude frequency : float Oscillation frequency phase : float Oscillation phase See Also -------- Const1D, Linear1D Notes ----- Model formula: .. math:: f(x) = A \\sin(2 \\pi f x + 2 \\pi p) Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Sine1D plt.figure() s1 = Sine1D(amplitude=1, frequency=.25) r=np.arange(0, 10, .01) for amplitude in range(1,4): s1.amplitude = amplitude plt.plot(r, s1(r), color=str(0.25 * amplitude), lw=2) plt.axis([0, 10, -5, 5]) plt.show() """ amplitude = Parameter(default=1, description="Oscillation amplitude") frequency = Parameter(default=1, description="Oscillation frequency") phase = Parameter(default=0, description="Oscillation phase")
[docs] @staticmethod def evaluate(x, amplitude, frequency, phase): """One dimensional Sine model function""" # Note: If frequency and x are quantities, they should normally have # inverse units, so that argument ends up being dimensionless. However, # np.sin of a dimensionless quantity will crash, so we remove the # quantity-ness from argument in this case (another option would be to # multiply by * u.rad but this would be slower overall). argument = TWOPI * (frequency * x + phase) if isinstance(argument, Quantity): argument = argument.value return amplitude * np.sin(argument)
[docs] @staticmethod def fit_deriv(x, amplitude, frequency, phase): """One dimensional Sine model derivative""" d_amplitude = np.sin(TWOPI * frequency * x + TWOPI * phase) d_frequency = (TWOPI * x * amplitude * np.cos(TWOPI * frequency * x + TWOPI * phase)) d_phase = (TWOPI * amplitude * np.cos(TWOPI * frequency * x + TWOPI * phase)) return [d_amplitude, d_frequency, d_phase]
@property def input_units(self): if self.frequency.unit is None: return None return {self.inputs[0]: 1. / self.frequency.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'frequency': inputs_unit[self.inputs[0]] ** -1, 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Linear1D(Fittable1DModel): """ One dimensional Line model. Parameters ---------- slope : float Slope of the straight line intercept : float Intercept of the straight line See Also -------- Const1D Notes ----- Model formula: .. math:: f(x) = a x + b """ slope = Parameter(default=1, description="Slope of the straight line") intercept = Parameter(default=0, description="Intercept of the straight line") linear = True
[docs] @staticmethod def evaluate(x, slope, intercept): """One dimensional Line model function""" return slope * x + intercept
[docs] @staticmethod def fit_deriv(x, *params): """One dimensional Line model derivative with respect to parameters""" d_slope = x d_intercept = np.ones_like(x) return [d_slope, d_intercept]
@property def inverse(self): new_slope = self.slope ** -1 new_intercept = -self.intercept / self.slope return self.__class__(slope=new_slope, intercept=new_intercept) @property def input_units(self): if self.intercept.unit is None and self.slope.unit is None: return None return {self.inputs[0]: self.intercept.unit / self.slope.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'intercept': outputs_unit[self.outputs[0]], 'slope': outputs_unit[self.outputs[0]] / inputs_unit[self.inputs[0]]}
[docs]class Planar2D(Fittable2DModel): """ Two dimensional Plane model. Parameters ---------- slope_x : float Slope of the plane in X slope_y : float Slope of the plane in Y intercept : float Z-intercept of the plane Notes ----- Model formula: .. math:: f(x, y) = a x + b y + c """ slope_x = Parameter(default=1, description="Slope of the plane in X") slope_y = Parameter(default=1, description="Slope of the plane in Y") intercept = Parameter(default=0, description="Z-intercept of the plane") linear = True
[docs] @staticmethod def evaluate(x, y, slope_x, slope_y, intercept): """Two dimensional Plane model function""" return slope_x * x + slope_y * y + intercept
[docs] @staticmethod def fit_deriv(x, y, *params): """Two dimensional Plane model derivative with respect to parameters""" d_slope_x = x d_slope_y = y d_intercept = np.ones_like(x) return [d_slope_x, d_slope_y, d_intercept]
def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'intercept': outputs_unit['z'], 'slope_x': outputs_unit['z'] / inputs_unit['x'], 'slope_y': outputs_unit['z'] / inputs_unit['y']}
[docs]class Lorentz1D(Fittable1DModel): """ One dimensional Lorentzian model. Parameters ---------- amplitude : float or `~astropy.units.Quantity`. Peak value - for a normalized profile (integrating to 1), set amplitude = 2 / (np.pi * fwhm) x_0 : float or `~astropy.units.Quantity`. Position of the peak fwhm : float or `~astropy.units.Quantity`. Full width at half maximum (FWHM) See Also -------- Gaussian1D, Box1D, RickerWavelet1D Notes ----- Either all or none of input ``x``, position ``x_0`` and ``fwhm`` must be provided consistently with compatible units or as unitless numbers. Model formula: .. math:: f(x) = \\frac{A \\gamma^{2}}{\\gamma^{2} + \\left(x - x_{0}\\right)^{2}} where :math:`\\gamma` is half of given FWHM. Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Lorentz1D plt.figure() s1 = Lorentz1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -1, 4]) plt.show() """ amplitude = Parameter(default=1, description="Peak value") x_0 = Parameter(default=0, description="Position of the peak") fwhm = Parameter(default=1, description="Full width at half maximum")
[docs] @staticmethod def evaluate(x, amplitude, x_0, fwhm): """One dimensional Lorentzian model function""" return (amplitude * ((fwhm / 2.) ** 2) / ((x - x_0) ** 2 + (fwhm / 2.) ** 2))
[docs] @staticmethod def fit_deriv(x, amplitude, x_0, fwhm): """One dimensional Lorentzian model derivative with respect to parameters""" d_amplitude = fwhm ** 2 / (fwhm ** 2 + (x - x_0) ** 2) d_x_0 = (amplitude * d_amplitude * (2 * x - 2 * x_0) / (fwhm ** 2 + (x - x_0) ** 2)) d_fwhm = 2 * amplitude * d_amplitude / fwhm * (1 - d_amplitude) return [d_amplitude, d_x_0, d_fwhm]
def bounding_box(self, factor=25): """Tuple defining the default ``bounding_box`` limits, ``(x_low, x_high)``. Parameters ---------- factor : float The multiple of FWHM used to define the limits. Default is chosen to include most (99%) of the area under the curve, while still showing the central feature of interest. """ x0 = self.x_0 dx = factor * self.fwhm return (x0 - dx, x0 + dx) @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]], 'fwhm': inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Voigt1D(Fittable1DModel): """ One dimensional model for the Voigt profile. Parameters ---------- x_0 : float or `~astropy.units.Quantity` Position of the peak amplitude_L : float or `~astropy.units.Quantity`. The Lorentzian amplitude (peak of the associated Lorentz function) - for a normalized profile (integrating to 1), set amplitude_L = 2 / (np.pi * fwhm_L) fwhm_L : float or `~astropy.units.Quantity` The Lorentzian full width at half maximum fwhm_G : float or `~astropy.units.Quantity`. The Gaussian full width at half maximum method : str, optional Algorithm for computing the complex error function; one of 'Humlicek2' (default, fast and generally more accurate than ``rtol=3.e-5``) or 'Scipy', alternatively 'wofz' (requires ``scipy``, almost as fast and reference in accuracy). See Also -------- Gaussian1D, Lorentz1D Notes ----- Either all or none of input ``x``, position ``x_0`` and the ``fwhm_*`` must be provided consistently with compatible units or as unitless numbers. Voigt function is calculated as real part of the complex error function computed from either Humlicek's rational approximations (JQSRT 21:309, 1979; 27:437, 1982) following Schreier 2018 (MNRAS 479, 3068; and ``hum2zpf16m`` from his cpfX.py module); or `~scipy.special.wofz` (implementing 'Faddeeva.cc'). Examples -------- .. plot:: :include-source: import numpy as np from astropy.modeling.models import Voigt1D import matplotlib.pyplot as plt plt.figure() x = np.arange(0, 10, 0.01) v1 = Voigt1D(x_0=5, amplitude_L=10, fwhm_L=0.5, fwhm_G=0.9) plt.plot(x, v1(x)) plt.show() """ x_0 = Parameter(default=0, description="Position of the peak") amplitude_L = Parameter(default=1, # noqa: N815 description="The Lorentzian amplitude") fwhm_L = Parameter(default=2/np.pi, # noqa: N815 description="The Lorentzian full width at half maximum") fwhm_G = Parameter(default=np.log(2), # noqa: N815 description="The Gaussian full width at half maximum") sqrt_pi = np.sqrt(np.pi) sqrt_ln2 = np.sqrt(np.log(2)) sqrt_ln2pi = np.sqrt(np.log(2) * np.pi) _last_z = np.zeros(1, dtype=complex) _last_w = np.zeros(1, dtype=float) _faddeeva = None def __init__(self, x_0=x_0.default, amplitude_L=amplitude_L.default, # noqa: N803 fwhm_L=fwhm_L.default, fwhm_G=fwhm_G.default, method='humlicek2', # noqa: N803 **kwargs): if str(method).lower() in ('wofz', 'scipy'): try: from scipy.special import wofz except (ValueError, ImportError) as err: raise ImportError(f'Voigt1D method {method} requires scipy: {err}.') from err self._faddeeva = wofz elif str(method).lower() == 'humlicek2': self._faddeeva = self._hum2zpf16c else: raise ValueError(f'Not a valid method for Voigt1D Faddeeva function: {method}.') self.method = self._faddeeva.__name__ super().__init__(x_0=x_0, amplitude_L=amplitude_L, fwhm_L=fwhm_L, fwhm_G=fwhm_G, **kwargs) def _wrap_wofz(self, z): """Call complex error (Faddeeva) function w(z) implemented by algorithm `method`; cache results for consecutive calls from `evaluate`, `fit_deriv`.""" if (z.shape == self._last_z.shape and np.allclose(z, self._last_z, rtol=1.e-14, atol=1.e-15)): return self._last_w self._last_w = self._faddeeva(z) self._last_z = z return self._last_w
[docs] def evaluate(self, x, x_0, amplitude_L, fwhm_L, fwhm_G): # noqa: N803 """One dimensional Voigt function scaled to Lorentz peak amplitude.""" z = np.atleast_1d(2 * (x - x_0) + 1j * fwhm_L) * self.sqrt_ln2 / fwhm_G # The normalised Voigt profile is w.real * self.sqrt_ln2 / (self.sqrt_pi * fwhm_G) * 2 ; # for the legacy definition we multiply with np.pi * fwhm_L / 2 * amplitude_L return self._wrap_wofz(z).real * self.sqrt_ln2pi / fwhm_G * fwhm_L * amplitude_L
[docs] def fit_deriv(self, x, x_0, amplitude_L, fwhm_L, fwhm_G): # noqa: N803 """Derivative of the one dimensional Voigt function with respect to parameters.""" s = self.sqrt_ln2 / fwhm_G z = np.atleast_1d(2 * (x - x_0) + 1j * fwhm_L) * s # V * constant from McLean implementation (== their Voigt function) w = self._wrap_wofz(z) * s * fwhm_L * amplitude_L * self.sqrt_pi # Schreier (2018) Eq. 6 == (dvdx + 1j * dvdy) / (sqrt(pi) * fwhm_L * amplitude_L) dwdz = -2 * z * w + 2j * s * fwhm_L * amplitude_L return [-dwdz.real * 2 * s, w.real / amplitude_L, w.real / fwhm_L - dwdz.imag * s, (-w.real - s * (2 * (x - x_0) * dwdz.real - fwhm_L * dwdz.imag)) / fwhm_G]
@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]], 'fwhm_L': inputs_unit[self.inputs[0]], 'fwhm_G': inputs_unit[self.inputs[0]], 'amplitude_L': outputs_unit[self.outputs[0]]} @staticmethod def _hum2zpf16c(z, s=10.0): """Complex error function w(z) for z = x + iy combining Humlicek's rational approximations: |x| + y > 10: Humlicek (JQSRT, 1982) rational approximation for region II; else: Humlicek (JQSRT, 1979) rational approximation with n=16 and delta=y0=1.35 Version using a mask and np.place; single complex argument version of Franz Schreier's cpfX.hum2zpf16m. Originally licensed under a 3-clause BSD style license - see https://atmos.eoc.dlr.de/tools/lbl4IR/cpfX.py """ # Optimized (single fraction) Humlicek region I rational approximation for n=16, delta=1.35 AA = np.array([+46236.3358828121, -147726.58393079657j, # noqa: N806 -206562.80451354137, 281369.1590631087j, +183092.74968253175, -184787.96830696272j, -66155.39578477248, 57778.05827983565j, +11682.770904216826, -9442.402767960672j, -1052.8438624933142, 814.0996198624186j, +45.94499030751872, -34.59751573708725j, -0.7616559377907136, 0.5641895835476449j]) # 1j/sqrt(pi) to the 12. digit bb = np.array([+7918.06640624997, 0.0, -126689.0625, 0.0, +295607.8125, 0.0, -236486.25, 0.0, +84459.375, 0.0, -15015.0, 0.0, +1365.0, 0.0, -60.0, 0.0, +1.0]) sqrt_piinv = 1.0 / np.sqrt(np.pi) zz = z * z w = 1j * (z * (zz * sqrt_piinv - 1.410474)) / (0.75 + zz*(zz - 3.0)) if np.any(z.imag < s): mask = abs(z.real) + z.imag < s # returns true for interior points # returns small complex array covering only the interior region Z = z[np.where(mask)] + 1.35j ZZ = Z * Z numer = (((((((((((((((AA[15]*Z + AA[14])*Z + AA[13])*Z + AA[12])*Z + AA[11])*Z + AA[10])*Z + AA[9])*Z + AA[8])*Z + AA[7])*Z + AA[6])*Z + AA[5])*Z + AA[4])*Z+AA[3])*Z + AA[2])*Z + AA[1])*Z + AA[0]) denom = (((((((ZZ + bb[14])*ZZ + bb[12])*ZZ + bb[10])*ZZ+bb[8])*ZZ + bb[6])*ZZ + bb[4])*ZZ + bb[2])*ZZ + bb[0] np.place(w, mask, numer / denom) return w
[docs]class Const1D(Fittable1DModel): """ One dimensional Constant model. Parameters ---------- amplitude : float Value of the constant function See Also -------- Const2D Notes ----- Model formula: .. math:: f(x) = A Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Const1D plt.figure() s1 = Const1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -1, 4]) plt.show() """ amplitude = Parameter(default=1, description="Value of the constant function") linear = True
[docs] @staticmethod def evaluate(x, amplitude): """One dimensional Constant model function""" if amplitude.size == 1: # This is slightly faster than using ones_like and multiplying x = np.empty_like(x, subok=False) x.fill(amplitude.item()) else: # This case is less likely but could occur if the amplitude # parameter is given an array-like value x = amplitude * np.ones_like(x, subok=False) if isinstance(amplitude, Quantity): return Quantity(x, unit=amplitude.unit, copy=False) return x
[docs] @staticmethod def fit_deriv(x, amplitude): """One dimensional Constant model derivative with respect to parameters""" d_amplitude = np.ones_like(x) return [d_amplitude]
@property def input_units(self): return None def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Const2D(Fittable2DModel): """ Two dimensional Constant model. Parameters ---------- amplitude : float Value of the constant function See Also -------- Const1D Notes ----- Model formula: .. math:: f(x, y) = A """ amplitude = Parameter(default=1, description="Value of the constant function") linear = True
[docs] @staticmethod def evaluate(x, y, amplitude): """Two dimensional Constant model function""" if amplitude.size == 1: # This is slightly faster than using ones_like and multiplying x = np.empty_like(x, subok=False) x.fill(amplitude.item()) else: # This case is less likely but could occur if the amplitude # parameter is given an array-like value x = amplitude * np.ones_like(x, subok=False) if isinstance(amplitude, Quantity): return Quantity(x, unit=amplitude.unit, copy=False) return x
@property def input_units(self): return None def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Ellipse2D(Fittable2DModel): """ A 2D Ellipse model. Parameters ---------- amplitude : float Value of the ellipse. x_0 : float x position of the center of the disk. y_0 : float y position of the center of the disk. a : float The length of the semimajor axis. b : float The length of the semiminor axis. theta : float The rotation angle in radians of the semimajor axis. The rotation angle increases counterclockwise from the positive x axis. See Also -------- Disk2D, Box2D Notes ----- Model formula: .. math:: f(x, y) = \\left \\{ \\begin{array}{ll} \\mathrm{amplitude} & : \\left[\\frac{(x - x_0) \\cos \\theta + (y - y_0) \\sin \\theta}{a}\\right]^2 + \\left[\\frac{-(x - x_0) \\sin \\theta + (y - y_0) \\cos \\theta}{b}\\right]^2 \\leq 1 \\\\ 0 & : \\mathrm{otherwise} \\end{array} \\right. Examples -------- .. plot:: :include-source: import numpy as np from astropy.modeling.models import Ellipse2D from astropy.coordinates import Angle import matplotlib.pyplot as plt import matplotlib.patches as mpatches x0, y0 = 25, 25 a, b = 20, 10 theta = Angle(30, 'deg') e = Ellipse2D(amplitude=100., x_0=x0, y_0=y0, a=a, b=b, theta=theta.radian) y, x = np.mgrid[0:50, 0:50] fig, ax = plt.subplots(1, 1) ax.imshow(e(x, y), origin='lower', interpolation='none', cmap='Greys_r') e2 = mpatches.Ellipse((x0, y0), 2*a, 2*b, theta.degree, edgecolor='red', facecolor='none') ax.add_patch(e2) plt.show() """ amplitude = Parameter(default=1, description="Value of the ellipse") x_0 = Parameter(default=0, description="X position of the center of the disk.") y_0 = Parameter(default=0, description="Y position of the center of the disk.") a = Parameter(default=1, description="The length of the semimajor axis") b = Parameter(default=1, description="The length of the semiminor axis") theta = Parameter(default=0, description="The rotation angle in radians of the semimajor axis (Positive - counterclockwise)")
[docs] @staticmethod def evaluate(x, y, amplitude, x_0, y_0, a, b, theta): """Two dimensional Ellipse model function.""" xx = x - x_0 yy = y - y_0 cost = np.cos(theta) sint = np.sin(theta) numerator1 = (xx * cost) + (yy * sint) numerator2 = -(xx * sint) + (yy * cost) in_ellipse = (((numerator1 / a) ** 2 + (numerator2 / b) ** 2) <= 1.) result = np.select([in_ellipse], [amplitude]) if isinstance(amplitude, Quantity): return Quantity(result, unit=amplitude.unit, copy=False) return result
@property def bounding_box(self): """ Tuple defining the default ``bounding_box`` limits. ``((y_low, y_high), (x_low, x_high))`` """ a = self.a b = self.b theta = self.theta.value dx, dy = ellipse_extent(a, b, theta) return ((self.y_0 - dy, self.y_0 + dy), (self.x_0 - dx, self.x_0 + dx)) @property def input_units(self): if self.x_0.unit is None: return None return {self.inputs[0]: self.x_0.unit, self.inputs[1]: self.y_0.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): # Note that here we need to make sure that x and y are in the same # units otherwise this can lead to issues since rotation is not well # defined. if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]: raise UnitsError("Units of 'x' and 'y' inputs should match") return {'x_0': inputs_unit[self.inputs[0]], 'y_0': inputs_unit[self.inputs[0]], 'a': inputs_unit[self.inputs[0]], 'b': inputs_unit[self.inputs[0]], 'theta': u.rad, 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Disk2D(Fittable2DModel): """ Two dimensional radial symmetric Disk model. Parameters ---------- amplitude : float Value of the disk function x_0 : float x position center of the disk y_0 : float y position center of the disk R_0 : float Radius of the disk See Also -------- Box2D, TrapezoidDisk2D Notes ----- Model formula: .. math:: f(r) = \\left \\{ \\begin{array}{ll} A & : r \\leq R_0 \\\\ 0 & : r > R_0 \\end{array} \\right. """ amplitude = Parameter(default=1, description="Value of disk function") x_0 = Parameter(default=0, description="X position of center of the disk") y_0 = Parameter(default=0, description="Y position of center of the disk") R_0 = Parameter(default=1, description="Radius of the disk")
[docs] @staticmethod def evaluate(x, y, amplitude, x_0, y_0, R_0): """Two dimensional Disk model function""" rr = (x - x_0) ** 2 + (y - y_0) ** 2 result = np.select([rr <= R_0 ** 2], [amplitude]) if isinstance(amplitude, Quantity): return Quantity(result, unit=amplitude.unit, copy=False) return result
@property def bounding_box(self): """ Tuple defining the default ``bounding_box`` limits. ``((y_low, y_high), (x_low, x_high))`` """ return ((self.y_0 - self.R_0, self.y_0 + self.R_0), (self.x_0 - self.R_0, self.x_0 + self.R_0)) @property def input_units(self): if self.x_0.unit is None and self.y_0.unit is None: return None return {self.inputs[0]: self.x_0.unit, self.inputs[1]: self.y_0.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): # Note that here we need to make sure that x and y are in the same # units otherwise this can lead to issues since rotation is not well # defined. if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]: raise UnitsError("Units of 'x' and 'y' inputs should match") return {'x_0': inputs_unit[self.inputs[0]], 'y_0': inputs_unit[self.inputs[0]], 'R_0': inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Ring2D(Fittable2DModel): """ Two dimensional radial symmetric Ring model. Parameters ---------- amplitude : float Value of the disk function x_0 : float x position center of the disk y_0 : float y position center of the disk r_in : float Inner radius of the ring width : float Width of the ring. r_out : float Outer Radius of the ring. Can be specified instead of width. See Also -------- Disk2D, TrapezoidDisk2D Notes ----- Model formula: .. math:: f(r) = \\left \\{ \\begin{array}{ll} A & : r_{in} \\leq r \\leq r_{out} \\\\ 0 & : \\text{else} \\end{array} \\right. Where :math:`r_{out} = r_{in} + r_{width}`. """ amplitude = Parameter(default=1, description="Value of the disk function") x_0 = Parameter(default=0, description="X position of center of disc") y_0 = Parameter(default=0, description="Y position of center of disc") r_in = Parameter(default=1, description="Inner radius of the ring") width = Parameter(default=1, description="Width of the ring") def __init__(self, amplitude=amplitude.default, x_0=x_0.default, y_0=y_0.default, r_in=r_in.default, width=width.default, r_out=None, **kwargs): # If outer radius explicitly given, it overrides default width. if r_out is not None: if width != self.width.default: raise InputParameterError( "Cannot specify both width and outer radius separately.") width = r_out - r_in elif width is None: width = self.width.default super().__init__( amplitude=amplitude, x_0=x_0, y_0=y_0, r_in=r_in, width=width, **kwargs)
[docs] @staticmethod def evaluate(x, y, amplitude, x_0, y_0, r_in, width): """Two dimensional Ring model function.""" rr = (x - x_0) ** 2 + (y - y_0) ** 2 r_range = np.logical_and(rr >= r_in ** 2, rr <= (r_in + width) ** 2) result = np.select([r_range], [amplitude]) if isinstance(amplitude, Quantity): return Quantity(result, unit=amplitude.unit, copy=False) return result
@property def bounding_box(self): """ Tuple defining the default ``bounding_box``. ``((y_low, y_high), (x_low, x_high))`` """ dr = self.r_in + self.width return ((self.y_0 - dr, self.y_0 + dr), (self.x_0 - dr, self.x_0 + dr)) @property def input_units(self): if self.x_0.unit is None: return None return {self.inputs[0]: self.x_0.unit, self.inputs[1]: self.y_0.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): # Note that here we need to make sure that x and y are in the same # units otherwise this can lead to issues since rotation is not well # defined. if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]: raise UnitsError("Units of 'x' and 'y' inputs should match") return {'x_0': inputs_unit[self.inputs[0]], 'y_0': inputs_unit[self.inputs[0]], 'r_in': inputs_unit[self.inputs[0]], 'width': inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Box1D(Fittable1DModel): """ One dimensional Box model. Parameters ---------- amplitude : float Amplitude A x_0 : float Position of the center of the box function width : float Width of the box See Also -------- Box2D, TrapezoidDisk2D Notes ----- Model formula: .. math:: f(x) = \\left \\{ \\begin{array}{ll} A & : x_0 - w/2 \\leq x \\leq x_0 + w/2 \\\\ 0 & : \\text{else} \\end{array} \\right. Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Box1D plt.figure() s1 = Box1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor s1.width = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -1, 4]) plt.show() """ amplitude = Parameter(default=1, description="Amplitude A") x_0 = Parameter(default=0, description="Position of center of box function") width = Parameter(default=1, description="Width of the box")
[docs] @staticmethod def evaluate(x, amplitude, x_0, width): """One dimensional Box model function""" inside = np.logical_and(x >= x_0 - width / 2., x <= x_0 + width / 2.) return np.select([inside], [amplitude], 0)
@property def bounding_box(self): """ Tuple defining the default ``bounding_box`` limits. ``(x_low, x_high))`` """ dx = self.width / 2 return (self.x_0 - dx, self.x_0 + dx) @property def input_units(self): if self.x_0.unit is None: return None return {self.inputs[0]: self.x_0.unit} @property def return_units(self): if self.amplitude.unit is None: return None return {self.outputs[0]: self.amplitude.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'x_0': inputs_unit[self.inputs[0]], 'width': inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Box2D(Fittable2DModel): """ Two dimensional Box model. Parameters ---------- amplitude : float Amplitude x_0 : float x position of the center of the box function x_width : float Width in x direction of the box y_0 : float y position of the center of the box function y_width : float Width in y direction of the box See Also -------- Box1D, Gaussian2D, Moffat2D Notes ----- Model formula: .. math:: f(x, y) = \\left \\{ \\begin{array}{ll} A : & x_0 - w_x/2 \\leq x \\leq x_0 + w_x/2 \\text{ and} \\\\ & y_0 - w_y/2 \\leq y \\leq y_0 + w_y/2 \\\\ 0 : & \\text{else} \\end{array} \\right. """ amplitude = Parameter(default=1, description="Amplitude") x_0 = Parameter(default=0, description="X position of the center of the box function") y_0 = Parameter(default=0, description="Y position of the center of the box function") x_width = Parameter(default=1, description="Width in x direction of the box") y_width = Parameter(default=1, description="Width in y direction of the box")
[docs] @staticmethod def evaluate(x, y, amplitude, x_0, y_0, x_width, y_width): """Two dimensional Box model function""" x_range = np.logical_and(x >= x_0 - x_width / 2., x <= x_0 + x_width / 2.) y_range = np.logical_and(y >= y_0 - y_width / 2., y <= y_0 + y_width / 2.) result = np.select([np.logical_and(x_range, y_range)], [amplitude], 0) if isinstance(amplitude, Quantity): return Quantity(result, unit=amplitude.unit, copy=False) return result
@property def bounding_box(self): """ Tuple defining the default ``bounding_box``. ``((y_low, y_high), (x_low, x_high))`` """ dx = self.x_width / 2 dy = self.y_width / 2 return ((self.y_0 - dy, self.y_0 + dy), (self.x_0 - dx, self.x_0 + dx)) @property def input_units(self): if self.x_0.unit is None: return None return {self.inputs[0]: self.x_0.unit, self.inputs[1]: self.y_0.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'x_0': inputs_unit[self.inputs[0]], 'y_0': inputs_unit[self.inputs[1]], 'x_width': inputs_unit[self.inputs[0]], 'y_width': inputs_unit[self.inputs[1]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Trapezoid1D(Fittable1DModel): """ One dimensional Trapezoid model. Parameters ---------- amplitude : float Amplitude of the trapezoid x_0 : float Center position of the trapezoid width : float Width of the constant part of the trapezoid. slope : float Slope of the tails of the trapezoid See Also -------- Box1D, Gaussian1D, Moffat1D Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Trapezoid1D plt.figure() s1 = Trapezoid1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor s1.width = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -1, 4]) plt.show() """ amplitude = Parameter(default=1, description="Amplitude of the trapezoid") x_0 = Parameter(default=0, description="Center position of the trapezoid") width = Parameter(default=1, description="Width of constant part of the trapezoid") slope = Parameter(default=1, description="Slope of the tails of trapezoid")
[docs] @staticmethod def evaluate(x, amplitude, x_0, width, slope): """One dimensional Trapezoid model function""" # Compute the four points where the trapezoid changes slope # x1 <= x2 <= x3 <= x4 x2 = x_0 - width / 2. x3 = x_0 + width / 2. x1 = x2 - amplitude / slope x4 = x3 + amplitude / slope # Compute model values in pieces between the change points range_a = np.logical_and(x >= x1, x < x2) range_b = np.logical_and(x >= x2, x < x3) range_c = np.logical_and(x >= x3, x < x4) val_a = slope * (x - x1) val_b = amplitude val_c = slope * (x4 - x) result = np.select([range_a, range_b, range_c], [val_a, val_b, val_c]) if isinstance(amplitude, Quantity): return Quantity(result, unit=amplitude.unit, copy=False) return result
@property def bounding_box(self): """ Tuple defining the default ``bounding_box`` limits. ``(x_low, x_high))`` """ dx = self.width / 2 + self.amplitude / self.slope return (self.x_0 - dx, self.x_0 + dx) @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]], 'width': inputs_unit[self.inputs[0]], 'slope': outputs_unit[self.outputs[0]] / inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class TrapezoidDisk2D(Fittable2DModel): """ Two dimensional circular Trapezoid model. Parameters ---------- amplitude : float Amplitude of the trapezoid x_0 : float x position of the center of the trapezoid y_0 : float y position of the center of the trapezoid R_0 : float Radius of the constant part of the trapezoid. slope : float Slope of the tails of the trapezoid in x direction. See Also -------- Disk2D, Box2D """ amplitude = Parameter(default=1, description="Amplitude of the trapezoid") x_0 = Parameter(default=0, description="X position of the center of the trapezoid") y_0 = Parameter(default=0, description="Y position of the center of the trapezoid") R_0 = Parameter(default=1, description="Radius of constant part of trapezoid") slope = Parameter(default=1, description="Slope of tails of trapezoid in x direction")
[docs] @staticmethod def evaluate(x, y, amplitude, x_0, y_0, R_0, slope): """Two dimensional Trapezoid Disk model function""" r = np.sqrt((x - x_0) ** 2 + (y - y_0) ** 2) range_1 = r <= R_0 range_2 = np.logical_and(r > R_0, r <= R_0 + amplitude / slope) val_1 = amplitude val_2 = amplitude + slope * (R_0 - r) result = np.select([range_1, range_2], [val_1, val_2]) if isinstance(amplitude, Quantity): return Quantity(result, unit=amplitude.unit, copy=False) return result
@property def bounding_box(self): """ Tuple defining the default ``bounding_box``. ``((y_low, y_high), (x_low, x_high))`` """ dr = self.R_0 + self.amplitude / self.slope return ((self.y_0 - dr, self.y_0 + dr), (self.x_0 - dr, self.x_0 + dr)) @property def input_units(self): if self.x_0.unit is None and self.y_0.unit is None: return None return {self.inputs[0]: self.x_0.unit, self.inputs[1]: self.y_0.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): # Note that here we need to make sure that x and y are in the same # units otherwise this can lead to issues since rotation is not well # defined. if inputs_unit['x'] != inputs_unit['y']: raise UnitsError("Units of 'x' and 'y' inputs should match") return {'x_0': inputs_unit[self.inputs[0]], 'y_0': inputs_unit[self.inputs[0]], 'R_0': inputs_unit[self.inputs[0]], 'slope': outputs_unit[self.outputs[0]] / inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class RickerWavelet1D(Fittable1DModel): """ One dimensional Ricker Wavelet model (sometimes known as a "Mexican Hat" model). .. note:: See https://github.com/astropy/astropy/pull/9445 for discussions related to renaming of this model. Parameters ---------- amplitude : float Amplitude x_0 : float Position of the peak sigma : float Width of the Ricker wavelet See Also -------- RickerWavelet2D, Box1D, Gaussian1D, Trapezoid1D Notes ----- Model formula: .. math:: f(x) = {A \\left(1 - \\frac{\\left(x - x_{0}\\right)^{2}}{\\sigma^{2}}\\right) e^{- \\frac{\\left(x - x_{0}\\right)^{2}}{2 \\sigma^{2}}}} Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import RickerWavelet1D plt.figure() s1 = RickerWavelet1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor s1.width = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -2, 4]) plt.show() """ amplitude = Parameter(default=1, description="Amplitude (peak) value") x_0 = Parameter(default=0, description="Position of the peak") sigma = Parameter(default=1, description="Width of the Ricker wavelet")
[docs] @staticmethod def evaluate(x, amplitude, x_0, sigma): """One dimensional Ricker Wavelet model function""" xx_ww = (x - x_0) ** 2 / (2 * sigma ** 2) return amplitude * (1 - 2 * xx_ww) * np.exp(-xx_ww)
def bounding_box(self, factor=10.0): """Tuple defining the default ``bounding_box`` limits, ``(x_low, x_high)``. Parameters ---------- factor : float The multiple of sigma used to define the limits. """ x0 = self.x_0 dx = factor * self.sigma return (x0 - dx, x0 + dx) @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]], 'sigma': inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class RickerWavelet2D(Fittable2DModel): """ Two dimensional Ricker Wavelet model (sometimes known as a "Mexican Hat" model). .. note:: See https://github.com/astropy/astropy/pull/9445 for discussions related to renaming of this model. Parameters ---------- amplitude : float Amplitude x_0 : float x position of the peak y_0 : float y position of the peak sigma : float Width of the Ricker wavelet See Also -------- RickerWavelet1D, Gaussian2D Notes ----- Model formula: .. math:: f(x, y) = A \\left(1 - \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\sigma^{2}}\\right) e^{\\frac{- \\left(x - x_{0}\\right)^{2} - \\left(y - y_{0}\\right)^{2}}{2 \\sigma^{2}}} """ amplitude = Parameter(default=1, description="Amplitude (peak) value") x_0 = Parameter(default=0, description="X position of the peak") y_0 = Parameter(default=0, description="Y position of the peak") sigma = Parameter(default=1, description="Width of the Ricker wavelet")
[docs] @staticmethod def evaluate(x, y, amplitude, x_0, y_0, sigma): """Two dimensional Ricker Wavelet model function""" rr_ww = ((x - x_0) ** 2 + (y - y_0) ** 2) / (2 * sigma ** 2) return amplitude * (1 - rr_ww) * np.exp(- rr_ww)
@property def input_units(self): if self.x_0.unit is None: return None return {self.inputs[0]: self.x_0.unit, self.inputs[1]: self.y_0.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): # Note that here we need to make sure that x and y are in the same # units otherwise this can lead to issues since rotation is not well # defined. if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]: raise UnitsError("Units of 'x' and 'y' inputs should match") return {'x_0': inputs_unit[self.inputs[0]], 'y_0': inputs_unit[self.inputs[0]], 'sigma': inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class AiryDisk2D(Fittable2DModel): """ Two dimensional Airy disk model. Parameters ---------- amplitude : float Amplitude of the Airy function. x_0 : float x position of the maximum of the Airy function. y_0 : float y position of the maximum of the Airy function. radius : float The radius of the Airy disk (radius of the first zero). See Also -------- Box2D, TrapezoidDisk2D, Gaussian2D Notes ----- Model formula: .. math:: f(r) = A \\left[\\frac{2 J_1(\\frac{\\pi r}{R/R_z})}{\\frac{\\pi r}{R/R_z}}\\right]^2 Where :math:`J_1` is the first order Bessel function of the first kind, :math:`r` is radial distance from the maximum of the Airy function (:math:`r = \\sqrt{(x - x_0)^2 + (y - y_0)^2}`), :math:`R` is the input ``radius`` parameter, and :math:`R_z = 1.2196698912665045`). For an optical system, the radius of the first zero represents the limiting angular resolution and is approximately 1.22 * lambda / D, where lambda is the wavelength of the light and D is the diameter of the aperture. See [1]_ for more details about the Airy disk. References ---------- .. [1] https://en.wikipedia.org/wiki/Airy_disk """ amplitude = Parameter(default=1, description="Amplitude (peak value) of the Airy function") x_0 = Parameter(default=0, description="X position of the peak") y_0 = Parameter(default=0, description="Y position of the peak") radius = Parameter(default=1, description="The radius of the Airy disk (radius of first zero crossing)") _rz = None _j1 = None
[docs] @classmethod def evaluate(cls, x, y, amplitude, x_0, y_0, radius): """Two dimensional Airy model function""" if cls._rz is None: try: from scipy.special import j1, jn_zeros cls._rz = jn_zeros(1, 1)[0] / np.pi cls._j1 = j1 except ValueError: raise ImportError('AiryDisk2D model requires scipy.') r = np.sqrt((x - x_0) ** 2 + (y - y_0) ** 2) / (radius / cls._rz) if isinstance(r, Quantity): # scipy function cannot handle Quantity, so turn into array. r = r.to_value(u.dimensionless_unscaled) # Since r can be zero, we have to take care to treat that case # separately so as not to raise a numpy warning z = np.ones(r.shape) rt = np.pi * r[r > 0] z[r > 0] = (2.0 * cls._j1(rt) / rt) ** 2 if isinstance(amplitude, Quantity): # make z quantity too, otherwise in-place multiplication fails. z = Quantity(z, u.dimensionless_unscaled, copy=False) z *= amplitude return z
@property def input_units(self): if self.x_0.unit is None: return None return {self.inputs[0]: self.x_0.unit, self.inputs[1]: self.y_0.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): # Note that here we need to make sure that x and y are in the same # units otherwise this can lead to issues since rotation is not well # defined. if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]: raise UnitsError("Units of 'x' and 'y' inputs should match") return {'x_0': inputs_unit[self.inputs[0]], 'y_0': inputs_unit[self.inputs[0]], 'radius': inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Moffat1D(Fittable1DModel): """ One dimensional Moffat model. Parameters ---------- amplitude : float Amplitude of the model. x_0 : float x position of the maximum of the Moffat model. gamma : float Core width of the Moffat model. alpha : float Power index of the Moffat model. See Also -------- Gaussian1D, Box1D Notes ----- Model formula: .. math:: f(x) = A \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha} Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Moffat1D plt.figure() s1 = Moffat1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor s1.width = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -1, 4]) plt.show() """ amplitude = Parameter(default=1, description="Amplitude of the model") x_0 = Parameter(default=0, description="X position of maximum of Moffat model") gamma = Parameter(default=1, description="Core width of Moffat model") alpha = Parameter(default=1, description="Power index of the Moffat model") @property def fwhm(self): """ Moffat full width at half maximum. Derivation of the formula is available in `this notebook by Yoonsoo Bach <https://nbviewer.jupyter.org/github/ysbach/AO_2017/blob/master/04_Ground_Based_Concept.ipynb#1.2.-Moffat>`_. """ return 2.0 * np.abs(self.gamma) * np.sqrt(2.0 ** (1.0 / self.alpha) - 1.0)
[docs] @staticmethod def evaluate(x, amplitude, x_0, gamma, alpha): """One dimensional Moffat model function""" return amplitude * (1 + ((x - x_0) / gamma) ** 2) ** (-alpha)
[docs] @staticmethod def fit_deriv(x, amplitude, x_0, gamma, alpha): """One dimensional Moffat model derivative with respect to parameters""" fac = (1 + (x - x_0) ** 2 / gamma ** 2) d_A = fac ** (-alpha) d_x_0 = (2 * amplitude * alpha * (x - x_0) * d_A / (fac * gamma ** 2)) d_gamma = (2 * amplitude * alpha * (x - x_0) ** 2 * d_A / (fac * gamma ** 3)) d_alpha = -amplitude * d_A * np.log(fac) return [d_A, d_x_0, d_gamma, 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]], 'gamma': inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Moffat2D(Fittable2DModel): """ Two dimensional Moffat model. Parameters ---------- amplitude : float Amplitude of the model. x_0 : float x position of the maximum of the Moffat model. y_0 : float y position of the maximum of the Moffat model. gamma : float Core width of the Moffat model. alpha : float Power index of the Moffat model. See Also -------- Gaussian2D, Box2D Notes ----- Model formula: .. math:: f(x, y) = A \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha} """ amplitude = Parameter(default=1, description="Amplitude (peak value) of the model") x_0 = Parameter(default=0, description="X position of the maximum of the Moffat model") y_0 = Parameter(default=0, description="Y position of the maximum of the Moffat model") gamma = Parameter(default=1, description="Core width of the Moffat model") alpha = Parameter(default=1, description="Power index of the Moffat model") @property def fwhm(self): """ Moffat full width at half maximum. Derivation of the formula is available in `this notebook by Yoonsoo Bach <https://nbviewer.jupyter.org/github/ysbach/AO_2017/blob/master/04_Ground_Based_Concept.ipynb#1.2.-Moffat>`_. """ return 2.0 * np.abs(self.gamma) * np.sqrt(2.0 ** (1.0 / self.alpha) - 1.0)
[docs] @staticmethod def evaluate(x, y, amplitude, x_0, y_0, gamma, alpha): """Two dimensional Moffat model function""" rr_gg = ((x - x_0) ** 2 + (y - y_0) ** 2) / gamma ** 2 return amplitude * (1 + rr_gg) ** (-alpha)
[docs] @staticmethod def fit_deriv(x, y, amplitude, x_0, y_0, gamma, alpha): """Two dimensional Moffat model derivative with respect to parameters""" rr_gg = ((x - x_0) ** 2 + (y - y_0) ** 2) / gamma ** 2 d_A = (1 + rr_gg) ** (-alpha) d_x_0 = (2 * amplitude * alpha * d_A * (x - x_0) / (gamma ** 2 * (1 + rr_gg))) d_y_0 = (2 * amplitude * alpha * d_A * (y - y_0) / (gamma ** 2 * (1 + rr_gg))) d_alpha = -amplitude * d_A * np.log(1 + rr_gg) d_gamma = (2 * amplitude * alpha * d_A * rr_gg / (gamma * (1 + rr_gg))) return [d_A, d_x_0, d_y_0, d_gamma, d_alpha]
@property def input_units(self): if self.x_0.unit is None: return None else: return {self.inputs[0]: self.x_0.unit, self.inputs[1]: self.y_0.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): # Note that here we need to make sure that x and y are in the same # units otherwise this can lead to issues since rotation is not well # defined. if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]: raise UnitsError("Units of 'x' and 'y' inputs should match") return {'x_0': inputs_unit[self.inputs[0]], 'y_0': inputs_unit[self.inputs[0]], 'gamma': inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Sersic2D(Fittable2DModel): r""" Two dimensional Sersic surface brightness profile. Parameters ---------- amplitude : float Surface brightness at r_eff. r_eff : float Effective (half-light) radius n : float Sersic Index. x_0 : float, optional x position of the center. y_0 : float, optional y position of the center. ellip : float, optional Ellipticity. theta : float, optional Rotation angle in radians, counterclockwise from the positive x-axis. See Also -------- Gaussian2D, Moffat2D Notes ----- Model formula: .. math:: I(x,y) = I(r) = I_e\exp\left\{-b_n\left[\left(\frac{r}{r_{e}}\right)^{(1/n)}-1\right]\right\} The constant :math:`b_n` is defined such that :math:`r_e` contains half the total luminosity, and can be solved for numerically. .. math:: \Gamma(2n) = 2\gamma (2n,b_n) Examples -------- .. plot:: :include-source: import numpy as np from astropy.modeling.models import Sersic2D import matplotlib.pyplot as plt x,y = np.meshgrid(np.arange(100), np.arange(100)) mod = Sersic2D(amplitude = 1, r_eff = 25, n=4, x_0=50, y_0=50, ellip=.5, theta=-1) img = mod(x, y) log_img = np.log10(img) plt.figure() plt.imshow(log_img, origin='lower', interpolation='nearest', vmin=-1, vmax=2) plt.xlabel('x') plt.ylabel('y') cbar = plt.colorbar() cbar.set_label('Log Brightness', rotation=270, labelpad=25) cbar.set_ticks([-1, 0, 1, 2], update_ticks=True) plt.show() References ---------- .. [1] http://ned.ipac.caltech.edu/level5/March05/Graham/Graham2.html """ amplitude = Parameter(default=1, description="Surface brightness at r_eff") r_eff = Parameter(default=1, description="Effective (half-light) radius") n = Parameter(default=4, description="Sersic Index") x_0 = Parameter(default=0, description="X position of the center") y_0 = Parameter(default=0, description="Y position of the center") ellip = Parameter(default=0, description="Ellipticity") theta = Parameter(default=0, description="Rotation angle in radians (counterclockwise-positive)") _gammaincinv = None
[docs] @classmethod def evaluate(cls, x, y, amplitude, r_eff, n, x_0, y_0, ellip, theta): """Two dimensional Sersic profile function.""" if cls._gammaincinv is None: try: from scipy.special import gammaincinv cls._gammaincinv = gammaincinv except ValueError: raise ImportError('Sersic2D model requires scipy.') bn = cls._gammaincinv(2. * n, 0.5) a, b = r_eff, (1 - ellip) * r_eff cos_theta, sin_theta = np.cos(theta), np.sin(theta) x_maj = (x - x_0) * cos_theta + (y - y_0) * sin_theta x_min = -(x - x_0) * sin_theta + (y - y_0) * cos_theta z = np.sqrt((x_maj / a) ** 2 + (x_min / b) ** 2) return amplitude * np.exp(-bn * (z ** (1 / n) - 1))
@property def input_units(self): if self.x_0.unit is None: return None return {self.inputs[0]: self.x_0.unit, self.inputs[1]: self.y_0.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): # Note that here we need to make sure that x and y are in the same # units otherwise this can lead to issues since rotation is not well # defined. if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]: raise UnitsError("Units of 'x' and 'y' inputs should match") return {'x_0': inputs_unit[self.inputs[0]], 'y_0': inputs_unit[self.inputs[0]], 'r_eff': inputs_unit[self.inputs[0]], 'theta': u.rad, 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class KingProjectedAnalytic1D(Fittable1DModel): """ Projected (surface density) analytic King Model. Parameters ---------- amplitude : float Amplitude or scaling factor. r_core : float Core radius (f(r_c) ~ 0.5 f_0) r_tide : float Tidal radius. Notes ----- This model approximates a King model with an analytic function. The derivation of this equation can be found in King '62 (equation 14). This is just an approximation of the full model and the parameters derived from this model should be taken with caution. It usually works for models with a concentration (c = log10(r_t/r_c) paramter < 2. Model formula: .. math:: f(x) = A r_c^2 \\left(\\frac{1}{\\sqrt{(x^2 + r_c^2)}} - \\frac{1}{\\sqrt{(r_t^2 + r_c^2)}}\\right)^2 Examples -------- .. plot:: :include-source: import numpy as np from astropy.modeling.models import KingProjectedAnalytic1D import matplotlib.pyplot as plt plt.figure() rt_list = [1, 2, 5, 10, 20] for rt in rt_list: r = np.linspace(0.1, rt, 100) mod = KingProjectedAnalytic1D(amplitude = 1, r_core = 1., r_tide = rt) sig = mod(r) plt.loglog(r, sig/sig[0], label='c ~ {:0.2f}'.format(mod.concentration)) plt.xlabel("r") plt.ylabel(r"$\\sigma/\\sigma_0$") plt.legend() plt.show() References ---------- .. [1] http://articles.adsabs.harvard.edu/pdf/1962AJ.....67..471K """ amplitude = Parameter(default=1, bounds=(FLOAT_EPSILON, None), description="Amplitude or scaling factor") r_core = Parameter(default=1, bounds=(FLOAT_EPSILON, None), description="Core Radius") r_tide = Parameter(default=2, bounds=(FLOAT_EPSILON, None), description="Tidal Radius") @property def concentration(self): """Concentration parameter of the king model""" return np.log10(np.abs(self.r_tide/self.r_core))
[docs] @staticmethod def evaluate(x, amplitude, r_core, r_tide): """ Analytic King model function. """ result = amplitude * r_core ** 2 * (1/np.sqrt(x ** 2 + r_core ** 2) - 1/np.sqrt(r_tide ** 2 + r_core ** 2)) ** 2 # Set invalid r values to 0 bounds = (x >= r_tide) | (x < 0) result[bounds] = result[bounds] * 0. return result
[docs] @staticmethod def fit_deriv(x, amplitude, r_core, r_tide): """ Analytic King model function derivatives. """ d_amplitude = r_core ** 2 * (1/np.sqrt(x ** 2 + r_core ** 2) - 1/np.sqrt(r_tide ** 2 + r_core ** 2)) ** 2 d_r_core = 2 * amplitude * r_core ** 2 * (r_core/(r_core ** 2 + r_tide ** 2) ** (3/2) - r_core/(r_core ** 2 + x ** 2) ** (3/2)) * \ (1./np.sqrt(r_core ** 2 + x ** 2) - 1./np.sqrt(r_core ** 2 + r_tide ** 2)) + \ 2 * amplitude * r_core * (1./np.sqrt(r_core ** 2 + x ** 2) - 1./np.sqrt(r_core ** 2 + r_tide ** 2)) ** 2 d_r_tide = (2 * amplitude * r_core ** 2 * r_tide * (1./np.sqrt(r_core ** 2 + x ** 2) - 1./np.sqrt(r_core ** 2 + r_tide ** 2)))/(r_core ** 2 + r_tide ** 2) ** (3/2) # Set invalid r values to 0 bounds = (x >= r_tide) | (x < 0) d_amplitude[bounds] = d_amplitude[bounds]*0 d_r_core[bounds] = d_r_core[bounds]*0 d_r_tide[bounds] = d_r_tide[bounds]*0 return [d_amplitude, d_r_core, d_r_tide]
@property def bounding_box(self): """ Tuple defining the default ``bounding_box`` limits. The model is not defined for r > r_tide. ``(r_low, r_high)`` """ return (0 * self.r_tide, 1 * self.r_tide) @property def input_units(self): if self.r_core.unit is None: return None return {self.inputs[0]: self.r_core.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'r_core': inputs_unit[self.inputs[0]], 'r_tide': inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Logarithmic1D(Fittable1DModel): """ One dimensional logarithmic model. Parameters ---------- amplitude : float, optional tau : float, optional See Also -------- Exponential1D, Gaussian1D """ amplitude = Parameter(default=1) tau = Parameter(default=1)
[docs] @staticmethod def evaluate(x, amplitude, tau): return amplitude * np.log(x / tau)
[docs] @staticmethod def fit_deriv(x, amplitude, tau): d_amplitude = np.log(x / tau) d_tau = np.zeros(x.shape) - (amplitude / tau) return [d_amplitude, d_tau]
@property def inverse(self): new_amplitude = self.tau new_tau = self.amplitude return Exponential1D(amplitude=new_amplitude, tau=new_tau) @tau.validator def tau(self, val): if val == 0: raise ValueError("0 is not an allowed value for tau") @property def input_units(self): if self.tau.unit is None: return None return {self.inputs[0]: self.tau.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'tau': inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
[docs]class Exponential1D(Fittable1DModel): """ One dimensional exponential model. Parameters ---------- amplitude : float, optional tau : float, optional See Also -------- Logarithmic1D, Gaussian1D """ amplitude = Parameter(default=1) tau = Parameter(default=1)
[docs] @staticmethod def evaluate(x, amplitude, tau): return amplitude * np.exp(x / tau)
[docs] @staticmethod def fit_deriv(x, amplitude, tau): ''' Derivative with respect to parameters''' d_amplitude = np.exp(x / tau) d_tau = -amplitude * (x / tau**2) * np.exp(x / tau) return [d_amplitude, d_tau]
@property def inverse(self): new_amplitude = self.tau new_tau = self.amplitude return Logarithmic1D(amplitude=new_amplitude, tau=new_tau) @tau.validator def tau(self, val): ''' tau cannot be 0''' if val == 0: raise ValueError("0 is not an allowed value for tau") @property def input_units(self): if self.tau.unit is None: return None return {self.inputs[0]: self.tau.unit} def _parameter_units_for_data_units(self, inputs_unit, outputs_unit): return {'tau': inputs_unit[self.inputs[0]], 'amplitude': outputs_unit[self.outputs[0]]}
@deprecated('4.0', alternative='RickerWavelet1D') class MexicanHat1D(RickerWavelet1D): """ Deprecated.""" @deprecated('4.0', alternative='RickerWavelet2D') class MexicanHat2D(RickerWavelet2D): """ Deprecated."""