# 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 .core import Fittable1DModel, Fittable2DModel
from .parameters import InputParameterError, Parameter
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",
"Cosine1D",
"Tangent1D",
"ArcSine1D",
"ArcCosine1D",
"ArcTangent1D",
"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
ModelBoundingBox(
intervals={
x: Interval(lower=-11.0, upper=11.0)
}
model=Gaussian1D(inputs=('x',))
order='C'
)
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
ModelBoundingBox(
intervals={
x: Interval(lower=-4.0, upper=4.0)
}
model=Gaussian1D(inputs=('x',))
order='C'
)
"""
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.
The rotation angle as an angular quantity
(`~astropy.units.Quantity` or `~astropy.coordinates.Angle`)
or a value in radians (as a float). 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 either as a "
"float (in radians) or a "
"|Quantity| angle (optional)"
),
)
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
ModelBoundingBox(
intervals={
x: Interval(lower=-5.5, upper=5.5)
y: Interval(lower=-11.0, upper=11.0)
}
model=Gaussian2D(inputs=('x', 'y'))
order='C'
)
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
ModelBoundingBox(
intervals={
x: Interval(lower=-2.0, upper=2.0)
y: Interval(lower=-4.0, upper=4.0)
}
model=Gaussian2D(inputs=('x', 'y'))
order='C'
)
"""
a = factor * self.x_stddev
b = factor * self.y_stddev
dx, dy = ellipse_extent(a, b, self.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(2.0 * 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(2.0 * theta)
sin2t = np.sin(2.0 * 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 * ((1.0 / ystd2) - (1.0 / 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 * ((2.0 * a * xdiff) + (b * ydiff))
dg_dy_mean = g * ((b * xdiff) + (2.0 * 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.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.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.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:
from scipy.special import gammaincinv
cls._gammaincinv = gammaincinv
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]],
}
class _Trigonometric1D(Fittable1DModel):
"""
Base class for one dimensional trigonometric and inverse trigonometric models
Parameters
----------
amplitude : float
Oscillation amplitude
frequency : float
Oscillation frequency
phase : float
Oscillation phase
"""
amplitude = Parameter(default=1, description="Oscillation amplitude")
frequency = Parameter(default=1, description="Oscillation frequency")
phase = Parameter(default=0, description="Oscillation phase")
@property
def input_units(self):
if self.frequency.unit is None:
return None
return {self.inputs[0]: 1.0 / 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 Sine1D(_Trigonometric1D):
"""
One dimensional Sine model.
Parameters
----------
amplitude : float
Oscillation amplitude
frequency : float
Oscillation frequency
phase : float
Oscillation phase
See Also
--------
ArcSine1D, Cosine1D, Tangent1D, 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()
"""
[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 inverse(self):
"""One dimensional inverse of Sine"""
return ArcSine1D(
amplitude=self.amplitude, frequency=self.frequency, phase=self.phase
)
[docs]class Cosine1D(_Trigonometric1D):
"""
One dimensional Cosine model.
Parameters
----------
amplitude : float
Oscillation amplitude
frequency : float
Oscillation frequency
phase : float
Oscillation phase
See Also
--------
ArcCosine1D, Sine1D, Tangent1D, Const1D, Linear1D
Notes
-----
Model formula:
.. math:: f(x) = A \\cos(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 Cosine1D
plt.figure()
s1 = Cosine1D(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()
"""
[docs] @staticmethod
def evaluate(x, amplitude, frequency, phase):
"""One dimensional Cosine 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.cos(argument)
[docs] @staticmethod
def fit_deriv(x, amplitude, frequency, phase):
"""One dimensional Cosine model derivative"""
d_amplitude = np.cos(TWOPI * frequency * x + TWOPI * phase)
d_frequency = -(
TWOPI * x * amplitude * np.sin(TWOPI * frequency * x + TWOPI * phase)
)
d_phase = -(TWOPI * amplitude * np.sin(TWOPI * frequency * x + TWOPI * phase))
return [d_amplitude, d_frequency, d_phase]
@property
def inverse(self):
"""One dimensional inverse of Cosine"""
return ArcCosine1D(
amplitude=self.amplitude, frequency=self.frequency, phase=self.phase
)
[docs]class Tangent1D(_Trigonometric1D):
"""
One dimensional Tangent model.
Parameters
----------
amplitude : float
Oscillation amplitude
frequency : float
Oscillation frequency
phase : float
Oscillation phase
See Also
--------
Sine1D, Cosine1D, Const1D, Linear1D
Notes
-----
Model formula:
.. math:: f(x) = A \\tan(2 \\pi f x + 2 \\pi p)
Note that the tangent function is undefined for inputs of the form
pi/2 + n*pi for all integers n. Thus thus the default bounding box
has been restricted to:
.. math:: [(-1/4 - p)/f, (1/4 - p)/f]
which is the smallest interval for the tangent function to be continuous
on.
Examples
--------
.. plot::
:include-source:
import numpy as np
import matplotlib.pyplot as plt
from astropy.modeling.models import Tangent1D
plt.figure()
s1 = Tangent1D(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()
"""
[docs] @staticmethod
def evaluate(x, amplitude, frequency, phase):
"""One dimensional Tangent 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.tan(argument)
[docs] @staticmethod
def fit_deriv(x, amplitude, frequency, phase):
"""One dimensional Tangent model derivative"""
sec = 1 / (np.cos(TWOPI * frequency * x + TWOPI * phase)) ** 2
d_amplitude = np.tan(TWOPI * frequency * x + TWOPI * phase)
d_frequency = TWOPI * x * amplitude * sec
d_phase = TWOPI * amplitude * sec
return [d_amplitude, d_frequency, d_phase]
@property
def inverse(self):
"""One dimensional inverse of Tangent"""
return ArcTangent1D(
amplitude=self.amplitude, frequency=self.frequency, phase=self.phase
)
def bounding_box(self):
"""
Tuple defining the default ``bounding_box`` limits,
``(x_low, x_high)``
"""
bbox = [
(-1 / 4 - self.phase) / self.frequency,
(1 / 4 - self.phase) / self.frequency,
]
if self.frequency.unit is not None:
bbox = bbox / self.frequency.unit
return bbox
class _InverseTrigonometric1D(_Trigonometric1D):
"""
Base class for one dimensional inverse trigonometric models
"""
@property
def input_units(self):
if self.amplitude.unit is None:
return None
return {self.inputs[0]: self.amplitude.unit}
def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
return {
"frequency": outputs_unit[self.outputs[0]] ** -1,
"amplitude": inputs_unit[self.inputs[0]],
}
[docs]class ArcSine1D(_InverseTrigonometric1D):
"""
One dimensional ArcSine model returning values between -pi/2 and pi/2
only.
Parameters
----------
amplitude : float
Oscillation amplitude for corresponding Sine
frequency : float
Oscillation frequency for corresponding Sine
phase : float
Oscillation phase for corresponding Sine
See Also
--------
Sine1D, ArcCosine1D, ArcTangent1D
Notes
-----
Model formula:
.. math:: f(x) = ((arcsin(x / A) / 2pi) - p) / f
The arcsin function being used for this model will only accept inputs
in [-A, A]; otherwise, a runtime warning will be thrown and the result
will be NaN. To avoid this, the bounding_box has been properly set to
accommodate this; therefore, it is recommended that this model always
be evaluated with the ``with_bounding_box=True`` option.
Examples
--------
.. plot::
:include-source:
import numpy as np
import matplotlib.pyplot as plt
from astropy.modeling.models import ArcSine1D
plt.figure()
s1 = ArcSine1D(amplitude=1, frequency=.25)
r=np.arange(-1, 1, .01)
for amplitude in range(1,4):
s1.amplitude = amplitude
plt.plot(r, s1(r), color=str(0.25 * amplitude), lw=2)
plt.axis([-1, 1, -np.pi/2, np.pi/2])
plt.show()
"""
[docs] @staticmethod
def evaluate(x, amplitude, frequency, phase):
"""One dimensional ArcSine 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 = x / amplitude
if isinstance(argument, Quantity):
argument = argument.value
arc_sine = np.arcsin(argument) / TWOPI
return (arc_sine - phase) / frequency
[docs] @staticmethod
def fit_deriv(x, amplitude, frequency, phase):
"""One dimensional ArcSine model derivative"""
d_amplitude = -x / (
TWOPI * frequency * amplitude**2 * np.sqrt(1 - (x / amplitude) ** 2)
)
d_frequency = (phase - (np.arcsin(x / amplitude) / TWOPI)) / frequency**2
d_phase = -1 / frequency * np.ones(x.shape)
return [d_amplitude, d_frequency, d_phase]
def bounding_box(self):
"""
Tuple defining the default ``bounding_box`` limits,
``(x_low, x_high)``
"""
return -1 * self.amplitude, 1 * self.amplitude
@property
def inverse(self):
"""One dimensional inverse of ArcSine"""
return Sine1D(
amplitude=self.amplitude, frequency=self.frequency, phase=self.phase
)
[docs]class ArcCosine1D(_InverseTrigonometric1D):
"""
One dimensional ArcCosine returning values between 0 and pi only.
Parameters
----------
amplitude : float
Oscillation amplitude for corresponding Cosine
frequency : float
Oscillation frequency for corresponding Cosine
phase : float
Oscillation phase for corresponding Cosine
See Also
--------
Cosine1D, ArcSine1D, ArcTangent1D
Notes
-----
Model formula:
.. math:: f(x) = ((arccos(x / A) / 2pi) - p) / f
The arccos function being used for this model will only accept inputs
in [-A, A]; otherwise, a runtime warning will be thrown and the result
will be NaN. To avoid this, the bounding_box has been properly set to
accommodate this; therefore, it is recommended that this model always
be evaluated with the ``with_bounding_box=True`` option.
Examples
--------
.. plot::
:include-source:
import numpy as np
import matplotlib.pyplot as plt
from astropy.modeling.models import ArcCosine1D
plt.figure()
s1 = ArcCosine1D(amplitude=1, frequency=.25)
r=np.arange(-1, 1, .01)
for amplitude in range(1,4):
s1.amplitude = amplitude
plt.plot(r, s1(r), color=str(0.25 * amplitude), lw=2)
plt.axis([-1, 1, 0, np.pi])
plt.show()
"""
[docs] @staticmethod
def evaluate(x, amplitude, frequency, phase):
"""One dimensional ArcCosine 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 = x / amplitude
if isinstance(argument, Quantity):
argument = argument.value
arc_cos = np.arccos(argument) / TWOPI
return (arc_cos - phase) / frequency
[docs] @staticmethod
def fit_deriv(x, amplitude, frequency, phase):
"""One dimensional ArcCosine model derivative"""
d_amplitude = x / (
TWOPI * frequency * amplitude**2 * np.sqrt(1 - (x / amplitude) ** 2)
)
d_frequency = (phase - (np.arccos(x / amplitude) / TWOPI)) / frequency**2
d_phase = -1 / frequency * np.ones(x.shape)
return [d_amplitude, d_frequency, d_phase]
def bounding_box(self):
"""
Tuple defining the default ``bounding_box`` limits,
``(x_low, x_high)``
"""
return -1 * self.amplitude, 1 * self.amplitude
@property
def inverse(self):
"""One dimensional inverse of ArcCosine"""
return Cosine1D(
amplitude=self.amplitude, frequency=self.frequency, phase=self.phase
)
[docs]class ArcTangent1D(_InverseTrigonometric1D):
"""
One dimensional ArcTangent model returning values between -pi/2 and
pi/2 only.
Parameters
----------
amplitude : float
Oscillation amplitude for corresponding Tangent
frequency : float
Oscillation frequency for corresponding Tangent
phase : float
Oscillation phase for corresponding Tangent
See Also
--------
Tangent1D, ArcSine1D, ArcCosine1D
Notes
-----
Model formula:
.. math:: f(x) = ((arctan(x / A) / 2pi) - p) / f
Examples
--------
.. plot::
:include-source:
import numpy as np
import matplotlib.pyplot as plt
from astropy.modeling.models import ArcTangent1D
plt.figure()
s1 = ArcTangent1D(amplitude=1, frequency=.25)
r=np.arange(-10, 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([-10, 10, -np.pi/2, np.pi/2])
plt.show()
"""
[docs] @staticmethod
def evaluate(x, amplitude, frequency, phase):
"""One dimensional ArcTangent 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 = x / amplitude
if isinstance(argument, Quantity):
argument = argument.value
arc_cos = np.arctan(argument) / TWOPI
return (arc_cos - phase) / frequency
[docs] @staticmethod
def fit_deriv(x, amplitude, frequency, phase):
"""One dimensional ArcTangent model derivative"""
d_amplitude = -x / (
TWOPI * frequency * amplitude**2 * (1 + (x / amplitude) ** 2)
)
d_frequency = (phase - (np.arctan(x / amplitude) / TWOPI)) / frequency**2
d_phase = -1 / frequency * np.ones(x.shape)
return [d_amplitude, d_frequency, d_phase]
@property
def inverse(self):
"""One dimensional inverse of ArcTangent"""
return Tangent1D(
amplitude=self.amplitude, frequency=self.frequency, phase=self.phase
)
[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 / 2.0) ** 2) / ((x - x_0) ** 2 + (fwhm / 2.0) ** 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, description="The Lorentzian amplitude")
fwhm_L = Parameter(
default=2 / np.pi, description="The Lorentzian full width at half maximum"
)
fwhm_G = Parameter(
default=np.log(2), 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,
fwhm_L=fwhm_L.default,
fwhm_G=fwhm_G.default,
method="humlicek2",
**kwargs,
):
if str(method).lower() in ("wofz", "scipy"):
from scipy.special import wofz
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=1.0e-14, atol=1.0e-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):
"""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):
"""
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
# fmt: off
AA = np.array(
[
+46236.3358828121, -147726.58393079657j,
-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,
-126689.0625,
+295607.8125,
-236486.25,
+84459.375,
-15015.0,
+1365.0,
-60.0,
+1.0,
]
)
# fmt: on
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
# fmt: off
# Recursive algorithms for the polynomials in Z with coefficients AA, bb
# numer = 0.0
# for A in AA[::-1]:
# numer = numer * Z + A
# Explicitly unrolled above loop for speed
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 = 0.0
# for b in bb[::-1]:
# denom = denom * ZZ + b
# Explicitly unrolled above loop for speed
denom = (((((((ZZ + bb[7])*ZZ + bb[6])*ZZ + bb[5])*ZZ+bb[4])*ZZ + bb[3])*ZZ +
bb[2])*ZZ + bb[1])*ZZ + bb[0]
# fmt: on
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", mag=True
)
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(amplitude, shape=x.shape, dtype=x.dtype)
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, subok=True)
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", mag=True
)
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(amplitude, shape=x.shape, dtype=x.dtype)
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, subok=True)
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 or `~astropy.units.Quantity`, optional
The rotation angle as an angular quantity
(`~astropy.units.Quantity` or `~astropy.coordinates.Angle`)
or a value in radians (as a float). 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", mag=True)
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=0.0,
description=(
"Rotation angle either as a float (in radians) or a |Quantity| angle"
),
)
[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) <= 1.0
result = np.select([in_ellipse], [amplitude])
if isinstance(amplitude, Quantity):
return Quantity(result, unit=amplitude.unit, copy=False, subok=True)
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
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", mag=True)
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, subok=True)
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", mag=True)
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=None,
width=None,
r_out=None,
**kwargs,
):
if (r_in is None) and (r_out is None) and (width is None):
r_in = self.r_in.default
width = self.width.default
elif (r_in is not None) and (r_out is None) and (width is None):
width = self.width.default
elif (r_in is None) and (r_out is not None) and (width is None):
r_in = self.r_in.default
width = r_out - r_in
elif (r_in is None) and (r_out is None) and (width is not None):
r_in = self.r_in.default
elif (r_in is not None) and (r_out is not None) and (width is None):
width = r_out - r_in
elif (r_in is None) and (r_out is not None) and (width is not None):
r_in = r_out - width
elif (r_in is not None) and (r_out is not None) and (width is not None):
if np.any(width != (r_out - r_in)):
raise InputParameterError("Width must be r_out - r_in")
if np.any(r_in < 0) or np.any(width < 0):
raise InputParameterError(f"{r_in=} and {width=} must both be >=0")
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, subok=True)
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", mag=True)
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 / 2.0, x <= x_0 + width / 2.0)
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", mag=True)
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 / 2.0, x <= x_0 + x_width / 2.0)
y_range = np.logical_and(y >= y_0 - y_width / 2.0, y <= y_0 + y_width / 2.0)
result = np.select([np.logical_and(x_range, y_range)], [amplitude], 0)
if isinstance(amplitude, Quantity):
return Quantity(result, unit=amplitude.unit, copy=False, subok=True)
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 / 2.0
x3 = x_0 + width / 2.0
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, subok=True)
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, subok=True)
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:
from scipy.special import j1, jn_zeros
cls._rz = jn_zeros(1, 1)[0] / np.pi
cls._j1 = j1
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, subok=True)
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 or `~astropy.units.Quantity`, optional
The rotation angle as an angular quantity
(`~astropy.units.Quantity` or `~astropy.coordinates.Angle`)
or a value in radians (as a float). The rotation angle
increases 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=0.0,
description=(
"Rotation angle either as a float (in radians) or a |Quantity| angle"
),
)
_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:
from scipy.special import gammaincinv
cls._gammaincinv = gammaincinv
bn = cls._gammaincinv(2.0 * 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) parameter < 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=f"c ~ {mod.concentration:0.2f}")
plt.xlabel("r")
plt.ylabel(r"$\\sigma/\\sigma_0$")
plt.legend()
plt.show()
References
----------
.. [1] https://ui.adsabs.harvard.edu/abs/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] * 0.0
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)
)
* (
1.0 / np.sqrt(r_core**2 + x**2)
- 1.0 / np.sqrt(r_core**2 + r_tide**2)
)
+ 2
* amplitude
* r_core
* (
1.0 / np.sqrt(r_core**2 + x**2)
- 1.0 / np.sqrt(r_core**2 + r_tide**2)
)
** 2
)
d_r_tide = (
2
* amplitude
* r_core**2
* r_tide
* (
1.0 / np.sqrt(r_core**2 + x**2)
- 1.0 / 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 np.all(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 np.all(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]],
}