a ߙfb@sdZddlZddlmZddlmZmZddl m Z m Z ddl m Z mZddlmZgd Zd ejZeeejjZd ed ed ZGd d d e ZGddde ZGddde ZGddde ZGddde Z Gddde Z!Gddde Z"Gddde Z#Gddde#Z$Gddde#Z%Gd d!d!e#Z&Gd"d#d#e#Z'Gd$d%d%e'Z(Gd&d'd'e'Z)Gd(d)d)e'Z*Gd*d+d+e Z+Gd,d-d-e Z,Gd.d/d/e Z-Gd0d1d1e Z.Gd2d3d3e Z/Gd4d5d5e Z0Gd6d7d7e Z1Gd8d9d9e Z2Gd:d;d;e Z3Gdd?d?e Z5Gd@dAdAe Z6GdBdCdCe Z7GdDdEdEe Z8GdFdGdGe Z9GdHdIdIe Z:GdJdKdKe Z;GdLdMdMe ZGdRdSdSe Z?GdTdUdUe Z@dS)VzMathematical models.N)units)Quantity UnitsError)Fittable1DModelFittable2DModel) ParameterInputParameterError)ellipse_extent)# AiryDisk2DMoffat1DMoffat2DBox1DBox2DConst1DConst2D Ellipse2DDisk2D Gaussian1D Gaussian2DLinear1D Lorentz1DRickerWavelet1DRickerWavelet2DRedshiftScaleFactorMultiplyPlanar2DScaleSersic1DSersic2DShiftSine1DCosine1D Tangent1D ArcSine1D ArcCosine1D ArcTangent1D Trapezoid1DTrapezoidDisk2DRing2DVoigt1DKingProjectedAnalytic1D Exponential1D Logarithmic1D@c@s|eZdZdZedddZedddZededfdd Zdd d Z e d dZ e ddZ e ddZe ddZddZdS)ra 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 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 rz&Amplitude (peak value) of the Gaussiandefault descriptionrzPosition of peak (Gaussian)Nz"Standard deviation of the Gaussianr1boundsr2@cCs |j}||j}||||fS)aU 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 `) 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' ) )meanstddevselffactorZx0dxr<Alib/python3.9/site-packages/astropy/modeling/functional_models.py bounding_box~s' zGaussian1D.bounding_boxcCs |jtS)z$Gaussian full width at half maximum.)r7GAUSSIAN_SIGMA_TO_FWHMr9r<r<r=fwhmszGaussian1D.fwhmcCs"|td||d|dS)z, Gaussian1D model function. r.npexp)x amplituder6r7r<r<r=evaluateszGaussian1D.evaluatecCs\td|d||d}|||||d}||||d|d}|||gS)z8 Gaussian1D model function derivatives. rBr.rC)rFrGr6r7 d_amplitudeZd_meanZd_stddevr<r<r= fit_derivszGaussian1D.fit_derivcCs"|jjdurdS|jd|jjiSNr)r6unitinputsr@r<r<r= input_unitss zGaussian1D.input_unitscCs*||jd||jd||jddS)Nr)r6r7rGrNoutputsr9Z inputs_unitZ outputs_unitr<r<r=_parameter_units_for_data_unitss   z*Gaussian1D._parameter_units_for_data_units)r5)__name__ __module__ __qualname____doc__rrGr6 FLOAT_EPSILONr7r>propertyrA staticmethodrHrKrOrSr<r<r<r=r"sT   ,    rcseZdZdZedddZedddZedddZedddZedd dZ ed d dZ ej ej ej d d d d ffd d Z e ddZe ddZdddZeddZeddZe ddZddZZS)raJ 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 rzAmplitude of the Gaussianr0rz(Peak position (along x axis) of Gaussianz(Peak position (along y axis) of Gaussianz1Standard deviation of the Gaussian (along x axis)z1Standard deviation of the Gaussian (along y axis)zNRotation angle either as a float (in radians) or a |Quantity| angle (optional)Nc s|dur@|dur|jjj}|dur,|jjj}|dur|jjj}n~|dusX|dusX|dur`tdt|}|jdkr|t dtj |\} } t | \}}| dddf} t | d| d}|di|ddtdf|ddtdftjf||||||d |dS) Nz3Cannot specify both cov_matrix and x/y_stddev/theta)r.r.zCovariance matrix must be 2x2rrr4x_stddevy_stddev)rGx_meany_meanr\r]theta) __class__r\r1r]r`r rDarrayshape ValueErrorZlinalgZeigsqrtZarctan2 setdefaultrXsuper__init__) r9rGr^r_r\r]r`Z cov_matrixkwargsZeig_valsZeig_vecsZy_vecrar<r=rh*s2      zGaussian2D.__init__cCs |jtS)z)Gaussian full width at half maximum in X.)r\r?r@r<r<r=x_fwhmOszGaussian2D.x_fwhmcCs |jtS)z)Gaussian full width at half maximum in Y.)r]r?r@r<r<r=y_fwhmTszGaussian2D.y_fwhmr5cCsj||j}||j}|jjdur*|jj}n|jj}t|||\}}|j||j|f|j||j|ffS)a 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 `) 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' ) N)r\r]r`Zquantityvaluer r_r^)r9r:abr`r;dyr<r<r=r>Ys,    zGaussian2D.bounding_boxcCst|d}t|d} td|} |d} |d} ||} ||}d|| | | }d| | | | }d| | || }|t|| d|| |||d S)z!Two dimensional Gaussian functionr.r/?rDcossinrE)rFyrGr^r_r\r]r`cost2sint2sin2txstd2ystd2xdiffydiffrnrocr<r<r=rHs zGaussian2D.evaluatec)Cst|}t|} t|d} t|d} td|} td|} |d}|d}|d}|d}||}||}|d}|d}d| || |}d| || |}d| || |}|t||||||| }| |d|d|}| |}| |}| || |}| |}| |}| } | |}!| |}"||}#|d||||}$|||d||}%|||||||!| }&|||||||"| }'||||||| | }(|#|$|%|&|'|(gS)zGTwo dimensional Gaussian function derivative with respect to parametersr.r/rIrq?rr))rFrurGr^r_r\r]r`costsintrvrwZcos2trxryrzZxstd3Zystd3r{r|Zxdiff2Zydiff2rnror}gZ da_dthetaZ da_dx_stddevZ da_dy_stddevZ db_dthetaZ db_dx_stddevZ db_dy_stddevZ dc_dthetaZ dc_dx_stddevZ dc_dy_stddevZdg_dAZ dg_dx_meanZ dg_dy_meanZ dg_dx_stddevZ dg_dy_stddevZ dg_dthetar<r<r=rKsd            zGaussian2D.fit_derivcCs<|jjdur|jjdurdS|jd|jj|jd|jjiSNrr)r^rMr_rNr@r<r<r=rOs zGaussian2D.input_unitscCsj||jd||jdkr$td||jd||jd||jd||jdtj||jddS)Nrr(Units of 'x' and 'y' inputs should match)r^r_r\r]r`rGrNruZradrQrRr<r<r=rSs     z*Gaussian2D._parameter_units_for_data_units)r5)rTrUrVrWrrGr^r_r\r]r`r1rhrYrkrlr>rZrHrKrOrS __classcell__r<r<rjr=rs.S      %   7  . rc@sheZdZdZedddZdZdZeddZ edd Z e d d Z e d d Z e ddZddZdS)r zv Shift a coordinate. Parameters ---------- offset : float Offset to add to a coordinate. rzOffset to add to a modelr0TcCs"|jjdurdS|jd|jjiSrL)offsetrMrNr@r<r<r=rOs zShift.input_unitscsR}|jd9_z jWnty2Yn0tfddjD|_|S)z,One dimensional inverse Shift model functionc3s|]}|jVqdSN)rHr.0rFr@r<r= z Shift.inverse..)copyrr>NotImplementedErrortupler9invr<r@r=inverses  z Shift.inversecCs||S)z$One dimensional Shift model functionr<)rFrr<r<r=rH szShift.evaluatecCs|S)z=Evaluate the implicit term (x) of one dimensional Shift modelr<)rFr<r<r=sum_of_implicit_termsszShift.sum_of_implicit_termscGst|}|gS)z@One dimensional Shift model derivative with respect to parameterrD ones_like)rFparamsZd_offsetr<r<r=rKs zShift.fit_derivcCsd||jdiS)NrrrQrRr<r<r=rSsz%Shift._parameter_units_for_data_unitsN)rTrUrVrWrrlinear_has_inverse_bounding_boxrYrOrrZrHrrKrSr<r<r<r=r s      r c@sheZdZdZedddZdZdZdZdZ dZ e ddZ e dd Z ed d Zed d ZddZdS)ra  Multiply a model by a dimensionless factor. Parameters ---------- factor : float Factor by which to scale a coordinate. Notes ----- If ``factor`` is a `~astropy.units.Quantity` then the units will be stripped before the scaling operation. rz Factor by which to scale a modelr0TcCs"|jjdurdS|jd|jjiSrL)r:rMrNr@r<r<r=rO9s zScale.input_unitscsT}dj|_z jWnty0Yn 0tfddjD|_|S)z,One dimensional inverse Scale model functionrc3s|]}|jVqdSrrHr:rr@r<r=rJrz Scale.inverse..rr:r>rrrr<r@r=r?s   z Scale.inversecCst|tjr|j}||S)z$One dimensional Scale model function) isinstancerrrmrFr:r<r<r=rHNs zScale.evaluatecGs |}|gS)z@One dimensional Scale model derivative with respect to parameterr<rFrZd_factorr<r<r=rKVszScale.fit_derivcCsd||jdiSNr:rrrRr<r<r=rS]sz%Scale._parameter_units_for_data_unitsN)rTrUrVrWrr:rfittableZ_input_units_strictZ _input_units_allow_dimensionlessrrYrOrrZrHrKrSr<r<r<r=rs      rc@sTeZdZdZedddZdZdZdZe ddZ e dd Z e d d Z d d ZdS)rz Multiply a model by a quantity or number. Parameters ---------- factor : float Factor by which to multiply a coordinate. rz#Factor by which to multiply a modelr0TcsT}dj|_z jWnty0Yn 0tfddjD|_|S)z/One dimensional inverse multiply model functionrc3s|]}|jVqdSrrrr@r<r=r|rz#Multiply.inverse..rrr<r@r=rqs   zMultiply.inversecCs||S)z'One dimensional multiply model functionr<rr<r<r=rHszMultiply.evaluatecGs |}|gS)zCOne dimensional multiply model derivative with respect to parameterr<rr<r<r=rKszMultiply.fit_derivcCsd||jdiSrrrRr<r<r=rSsz(Multiply._parameter_units_for_data_unitsN)rTrUrVrWrr:rrrrYrrZrHrKrSr<r<r<r=ras    rc@sDeZdZdZedddZdZeddZedd Z e d d Z d S) rz One dimensional redshift scale factor model. Parameters ---------- z : float Redshift value. Notes ----- Model formula: .. math:: f(x) = x (1 + z) ZRedshiftr)r2r1TcCs d||S)z2One dimensional RedshiftScaleFactor model functionrr<)rFzr<r<r=rHszRedshiftScaleFactor.evaluatecCs |}|gS)z4One dimensional RedshiftScaleFactor model derivativer<)rFrZd_zr<r<r=rKszRedshiftScaleFactor.fit_derivcs\}ddjd|_z jWnty8Yn 0tfddjD|_|S)z!Inverse RedshiftScaleFactor modelr~c3s|]}|jVqdSr)rHrrr@r<r=rrz.RedshiftScaleFactor.inverse..)rrr>rrrr<r@r=rs  zRedshiftScaleFactor.inverseN) rTrUrVrWrrrrZrHrKrYrr<r<r<r=rs   rc@sXeZdZdZedddZedddZedddZdZe d d Z e d d Z d dZ dS)ra 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 rSurface brightness at r_effr0Effective (half-light) radius Sersic IndexNcCsL|jdurddlm}||_|t|d|d ||d|dS)z(One dimensional Sersic profile function.Nr gammaincinvr.rqr) _gammaincinv scipy.specialrrDrE)clsrrGr_effnrr<r<r=rHs   $zSersic1D.evaluatecCs"|jjdurdS|jd|jjiSrL)rrMrNr@r<r<r=rOs zSersic1D.input_unitscCs||jd||jddS)Nr)rrGrPrRr<r<r=rSs  z(Sersic1D._parameter_units_for_data_units)rTrUrVrWrrGrrr classmethodrHrYrOrSr<r<r<r=rs?     rc@sHeZdZdZedddZedddZedddZedd Z d d Z d S) _Trigonometric1Da  Base class for one dimensional trigonometric and inverse trigonometric models Parameters ---------- amplitude : float Oscillation amplitude frequency : float Oscillation frequency phase : float Oscillation phase rzOscillation amplituder0zOscillation frequencyrzOscillation phasecCs&|jjdurdS|jdd|jjiS)Nrr~) frequencyrMrNr@r<r<r=rO/s z_Trigonometric1D.input_unitscCs"||jdd||jddSNrr)rrGrPrRr<r<r=rS5s z0_Trigonometric1D._parameter_units_for_data_unitsN) rTrUrVrWrrGrphaserYrOrSr<r<r<r=rs    rc@s4eZdZdZeddZeddZeddZdS) r!al One dimensional Sine model. Parameters ---------- amplitude : float Oscillation amplitude frequency : float Oscillation frequency phase : float Oscillation phase See Also -------- ArcSine1D, Cosine1D, Tangent1D, Const1D, Linear1D Notes ----- Model formula: .. math:: f(x) = A \sin(2 \pi f x + 2 \pi p) Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Sine1D plt.figure() s1 = Sine1D(amplitude=1, frequency=.25) r=np.arange(0, 10, .01) for amplitude in range(1,4): s1.amplitude = amplitude plt.plot(r, s1(r), color=str(0.25 * amplitude), lw=2) plt.axis([0, 10, -5, 5]) plt.show() cCs.t|||}t|tr |j}|t|S)z#One dimensional Sine model function)TWOPIrrrmrDrtrFrGrrargumentr<r<r=rHhs zSine1D.evaluatecCsltt||t|}t||tt||t|}t|tt||t|}|||gS)z%One dimensional Sine model derivative)rDrtrrsrFrGrrrJ d_frequencyd_phaser<r<r=rKus zSine1D.fit_derivcCst|j|j|jdS)zOne dimensional inverse of SinerGrr)r$rGrrr@r<r<r=rszSine1D.inverseN rTrUrVrWrZrHrKrYrr<r<r<r=r!:s-  r!c@s4eZdZdZeddZeddZeddZdS) r"ar 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() cCs.t|||}t|tr |j}|t|S)z%One dimensional Cosine model function)rrrrmrDrsrr<r<r=rHs zCosine1D.evaluatecCsptt||t|}t||tt||t| }t|tt||t| }|||gS)z'One dimensional Cosine model derivative)rDrsrrtrr<r<r=rKs zCosine1D.fit_derivcCst|j|j|jdS)z!One dimensional inverse of Cosiner)r%rGrrr@r<r<r=rszCosine1D.inverseNrr<r<r<r=r"s-  r"c@s<eZdZdZeddZeddZeddZdd Z d S) r#a 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() cCs.t|||}t|tr |j}|t|S)z&One dimensional Tangent model function)rrrrmrDtanrr<r<r=rH s zTangent1D.evaluatecCsbdtt||t|d}tt||t|}t|||}t||}|||gS)z(One dimensional Tangent model derivativerr.)rDrsrr)rFrGrrZsecrJrrr<r<r=rKs " zTangent1D.fit_derivcCst|j|j|jdS)z"One dimensional inverse of Tangentr)r&rGrrr@r<r<r=r#szTangent1D.inversecCs<d|j|jd|j|jg}|jjdur8||jj}|S)a Tuple defining the default ``bounding_box`` limits, ``(x_low, x_high)`` gпg?N)rrrM)r9Zbboxr<r<r=r>)s   zTangent1D.bounding_boxN) rTrUrVrWrZrHrKrYrr>r<r<r<r=r#s6   r#c@s$eZdZdZeddZddZdS)_InverseTrigonometric1DzE Base class for one dimensional inverse trigonometric models cCs"|jjdurdS|jd|jjiSrL)rGrMrNr@r<r<r=rO<s z#_InverseTrigonometric1D.input_unitscCs"||jdd||jddSrrQrNrRr<r<r=rSBs z7_InverseTrigonometric1D._parameter_units_for_data_unitsN)rTrUrVrWrYrOrSr<r<r<r=r7s rc@s<eZdZdZeddZeddZddZedd Z d S) r$aR 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() cCs2||}t|tr|j}t|t}|||S)z&One dimensional ArcSine model function)rrrmrDarcsinr)rFrGrrrZarc_siner<r<r=rH|s  zArcSine1D.evaluatecCsh| t||dtd||d}|t||t|d}d|t|j}|||gS)z(One dimensional ArcSine model derivativer.rr)rrDreronesrcrr<r<r=rKs,zArcSine1D.fit_derivcCsd|jd|jfSrrrrGr@r<r<r=r>szArcSine1D.bounding_boxcCst|j|j|jdS)z"One dimensional inverse of ArcSiner)r!rGrrr@r<r<r=rszArcSine1D.inverseN rTrUrVrWrZrHrKr>rYrr<r<r<r=r$Gs4  r$c@s<eZdZdZeddZeddZddZedd Z d S) r%aF 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() cCs2||}t|tr|j}t|t}|||S)z(One dimensional ArcCosine model function)rrrmrDarccosrrFrGrrrZarc_cosr<r<r=rHs  zArcCosine1D.evaluatecCsf|t||dtd||d}|t||t|d}d|t|j}|||gS)z*One dimensional ArcCosine model derivativer.rr)rrDrerrrcrr<r<r=rKs*zArcCosine1D.fit_derivcCsd|jd|jfSrrr@r<r<r=r>szArcCosine1D.bounding_boxcCst|j|j|jdS)z$One dimensional inverse of ArcCosiner)r"rGrrr@r<r<r=rszArcCosine1D.inverseNrr<r<r<r=r%s4  r%c@s4eZdZdZeddZeddZeddZdS) r&a 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() cCs2||}t|tr|j}t|t}|||S)z)One dimensional ArcTangent model function)rrrmrDarctanrrr<r<r=rH0s  zArcTangent1D.evaluatecCsb| t||dd||d}|t||t|d}d|t|j}|||gS)z+One dimensional ArcTangent model derivativer.rr)rrDrrrcrr<r<r=rK@s&zArcTangent1D.fit_derivcCst|j|j|jdS)z%One dimensional inverse of ArcTangentr)r#rGrrr@r<r<r=rIszArcTangent1D.inverseNrr<r<r<r=r&s.  r&c@sdeZdZdZedddZedddZdZedd Z ed d Z e d d Z e ddZ ddZdS)ra( 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 rzSlope of the straight liner0rzIntercept of the straight lineTcCs |||S)z#One dimensional Line model functionr<)rFslope interceptr<r<r=rHjszLinear1D.evaluatecGs|}t|}||gS)z@One dimensional Line model derivative with respect to parametersr)rFrZd_slope d_interceptr<r<r=rKps zLinear1D.fit_derivcCs&|jd}|j |j}|j||dS)Nr)rr)rrra)r9Z new_slopeZ new_interceptr<r<r=rxs zLinear1D.inversecCs6|jjdur|jjdurdS|jd|jj|jjiSrL)rrMrrNr@r<r<r=rO~szLinear1D.input_unitscCs,||jd||jd||jddS)Nr)rrrrRr<r<r=rSs z(Linear1D._parameter_units_for_data_unitsN)rTrUrVrWrrrrrZrHrKrYrrOrSr<r<r<r=rPs      rc@sXeZdZdZedddZedddZedddZdZe d d Z e d d Z d dZ dS)ra9 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 rzSlope of the plane in Xr0zSlope of the plane in YrzZ-intercept of the planeTcCs|||||S)z$Two dimensional Plane model functionr<)rFruslope_xslope_yrr<r<r=rHszPlanar2D.evaluatecGs|}|}t|}|||gS)zATwo dimensional Plane model derivative with respect to parametersr)rFrurZ d_slope_xZ d_slope_yrr<r<r=rKs zPlanar2D.fit_derivcCs(|d|d|d|d|ddS)NrrFru)rrrr<rRr<r<r=rSsz(Planar2D._parameter_units_for_data_unitsN) rTrUrVrWrrrrrrZrHrKrSr<r<r<r=rs     rc@sjeZdZdZedddZedddZedddZedd Z ed d Z dd dZ e ddZ ddZdS)ra 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() rz Peak valuer0rPosition of the peakzFull width at half maximumcCs(||dd||d|ddS)z)One dimensional Lorentzian model functionr/r.r<)rFrGx_0rAr<r<r=rHs zLorentz1D.evaluatecCsj|d|d||d}||d|d||d||d}d|||d|}|||gS)zFOne dimensional Lorentzian model derivative with respect to parametersr.rr<)rFrGrrArJd_x_0Zd_fwhmr<r<r=rKs zLorentz1D.fit_derivcCs |j}||j}||||fS)arTuple 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. )rrAr8r<r<r=r>s  zLorentz1D.bounding_boxcCs"|jjdurdS|jd|jjiSrLrrMrNr@r<r<r=rOs zLorentz1D.input_unitscCs*||jd||jd||jddS)Nr)rrArGrPrRr<r<r=rSs   z)Lorentz1D._parameter_units_for_data_unitsN)r)rTrUrVrWrrGrrArZrHrKr>rYrOrSr<r<r<r=rs4      rcseZdZdZedddZedddZedejddZ ee dd dZ e ejZ e e dZe e dejZejded Zejded Zd Zejeje je jd ffd d ZddZddZddZeddZddZedddZZ S)r*a 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() rrr0rzThe Lorentzian amplituder.z)The Lorentzian full width at half maximumz'The Gaussian full width at half maximum)ZdtypeN humlicek2c sxt|dvr$ddlm}||_n*t|dkr>|j|_ntd|d|jj|_t j f||||d|dS)N)wofzZscipyr)rrz2Not a valid method for Voigt1D Faddeeva function: .)r amplitude_Lfwhm_Lfwhm_G) strlowerrr _faddeeva _hum2zpf16crdrTmethodrgrh)r9rrrrrrirrjr<r=rhds   zVoigt1D.__init__cCs@|j|jjkr(tj||jdddr(|jS|||_||_|jS)zCall complex error (Faddeeva) function w(z) implemented by algorithm `method`; cache results for consecutive calls from `evaluate`, `fit_deriv`.g+=gV瞯<)ZrtolZatol)rc_last_zrDZallclose_last_wr)r9rr<r<r= _wrap_wofzrs zVoigt1D._wrap_wofzcCsBtd||d||j|}||j|j|||S)z@One dimensional Voigt function scaled to Lorentz peak amplitude.r.?)rD atleast_1dsqrt_ln2rreal sqrt_ln2pi)r9rFrrrrrr<r<r=rH~s$zVoigt1D.evaluatec Cs|j|}td||d||}||||||j}d||d|||} | j d||j||j|| j||j |d||| j|| j|gS)zLDerivative of the one dimensional Voigt function with respect to parameters.r.ry@)rrDrrsqrt_pirimag) r9rFrrrrsrwZdwdzr<r<r=rKs *zVoigt1D.fit_derivcCs"|jjdurdS|jd|jjiSrLrr@r<r<r=rOs zVoigt1D.input_unitscCs6||jd||jd||jd||jddS)Nr)rrrrrPrRr<r<r=rSs     z'Voigt1D._parameter_units_for_data_units$@c Cstgd}tgd}dttj}||}d|||dd||d}t|j|krt|j|j|k}|t|d}||} |d ||d ||d ||d ||d ||d||d||d||d||d||d||d||d||d||d||d} | |d | |d | |d| |d| |d| |d| |d| |d} t ||| | |S)a 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 )gO@ytgˤo7 ypd,Ag"YYAygi"U&ymA6@gGb@y{3qglL}`syp@g:@qF@yJJe{LAgE|_yBP ?)g@r[gr[g@ Ar[g2 r[g@r[gSr[gT@r[gNr[r~r~rg]/M?g?g@y? rrIr.rr) rDrbrepianyrabsrwhereZplace) rrZAAZbbZ sqrt_piinvZzzrmaskZZZZZnumerZdenomr<r<r=rs  $:>zVoigt1D._hum2zpf16c)r)!rTrUrVrWrrrrDrrlogrrerrrzeroscomplexrfloatrrr1rhrrHrKrYrOrSrZrrr<r<rjr=r*!s<2      r*c@sLeZdZdZedddZdZeddZedd Z e d d Z d d Z dS)ra One dimensional Constant model. Parameters ---------- amplitude : float Value of the constant function See Also -------- Const2D Notes ----- Model formula: .. math:: f(x) = A Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Const1D plt.figure() s1 = Const1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -1, 4]) plt.show() rValue of the constant functionr0TcCsX|jdkr(tj|dd}||n|tj|dd}t|trTt||jddS|S)z'One dimensional Constant model functionrFZsubokrMr sizerDZ empty_likeZfillitemrrrrM)rFrGr<r<r=rHs  zConst1D.evaluatecCst|}|gS)zDOne dimensional Constant model derivative with respect to parametersr)rFrGrJr<r<r=rKs zConst1D.fit_derivcCsdSrr<r@r<r<r=rOszConst1D.input_unitscCsd||jdiSNrGrrrRr<r<r=rS sz'Const1D._parameter_units_for_data_unitsN) rTrUrVrWrrGrrZrHrKrYrOrSr<r<r<r=rs(    rc@s@eZdZdZedddZdZeddZe dd Z d d Z d S) rz Two dimensional Constant model. Parameters ---------- amplitude : float Value of the constant function See Also -------- Const1D Notes ----- Model formula: .. math:: f(x, y) = A rrr0TcCsX|jdkr(tj|dd}||n|tj|dd}t|trTt||jddS|S)z'Two dimensional Constant model functionrFrrr)rFrurGr<r<r=rH;s  zConst2D.evaluatecCsdSrr<r@r<r<r=rOLszConst2D.input_unitscCsd||jdiSrrrRr<r<r=rSPsz'Const2D._parameter_units_for_data_unitsN) rTrUrVrWrrGrrZrHrYrOrSr<r<r<r=r$s   rc@seZdZdZedddZedddZedddZedddZedd dZ edd dZ e d d Z e d dZe ddZddZdS)raA 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() rzValue of the ellipser0rz%X position of the center of the disk.z%Y position of the center of the disk.z The length of the semimajor axisz The length of the semiminor axiszQThe rotation angle in radians of the semimajor axis (Positive - counterclockwise)cCs||}||} t|} t|} || | | } ||  | | } | |d| |ddk}t|g|g}t|trt||jddS|S)z'Two dimensional Ellipse model function.r.r~Fr)rDrsrtselectrrrM)rFrurGry_0rnror`ZxxZyyrrZ numerator1Z numerator2Z in_ellipseresultr<r<r=rHs   zEllipse2D.evaluatecCsL|j}|j}|jj}t|||\}}|j||j|f|j||j|ffSzu Tuple defining the default ``bounding_box`` limits. ``((y_low, y_high), (x_low, x_high))`` )rnror`rmr rr)r9rnror`r;rpr<r<r=r>szEllipse2D.bounding_boxcCs0|jjdurdS|jd|jj|jd|jjiSrrrMrNrr@r<r<r=rOs  zEllipse2D.input_unitscCsj||jd||jdkr$td||jd||jd||jd||jdtj||jddS)Nrrr)rrrnror`rGrrRr<r<r=rSs     z)Ellipse2D._parameter_units_for_data_unitsN)rTrUrVrWrrGrrrnror`rZrHrYr>rOrSr<r<r<r=rTsE         rc@sleZdZdZedddZedddZedddZedddZe d d Z e d d Z e d dZ ddZdS)raf Two dimensional radial symmetric Disk model. Parameters ---------- amplitude : float Value of the disk function x_0 : float x position center of the disk y_0 : float y position center of the disk R_0 : float Radius of the disk See Also -------- Box2D, TrapezoidDisk2D Notes ----- Model formula: .. math:: f(r) = \left \{ \begin{array}{ll} A & : r \leq R_0 \\ 0 & : r > R_0 \end{array} \right. rzValue of disk functionr0rz X position of center of the diskz Y position of center of the diskzRadius of the diskcCsN||d||d}t||dkg|g}t|trJt||jddS|S)z#Two dimensional Disk model functionr.Fr)rDrrrrM)rFrurGrrR_0rrr r<r<r=rHs  zDisk2D.evaluatecCs0|j|j|j|jf|j|j|j|jffSr )rr rr@r<r<r=r>szDisk2D.bounding_boxcCs<|jjdur|jjdurdS|jd|jj|jd|jjiSrrrMrrNr@r<r<r=rOs zDisk2D.input_unitscCsZ||jd||jdkr$td||jd||jd||jd||jddS)Nrrr)rrr rGrNrrQrRr<r<r=rSs    z&Disk2D._parameter_units_for_data_unitsN)rTrUrVrWrrGrrr rZrHrYr>rOrSr<r<r<r=rs       rcseZdZdZedddZedddZedddZedddZedd dZ ej ej ej d d d ffd d Z e d dZ eddZeddZddZZS)r)a7 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}`. rzValue of the disk functionr0rzX position of center of disczY position of center of disczInner radius of the ringzWidth of the ringNc sn|dur*|dur*|dur*|jj}|jj}n|durL|durL|durL|jj}n|durv|durv|durv|jj}||}n|dur|dur|dur|jj}n~|dur|dur|dur||}n\|dur|dur|dur||}n:|dur|dur|durt|||krtdt|dks6t|dkrLtd|d|dtjf|||||d|dS)NzWidth must be r_out - r_inrzr_in=z and width=z must both be >=0)rGrrr_inwidth)rr1rrDrr rgrh)r9rGrrrrZr_outrirjr<r=rhSs0        zRing2D.__init__c Csf||d||d}t||dk|||dk}t|g|g} t|trbt| |jddS| S)z$Two dimensional Ring model function.r.FrrD logical_andrrrrM) rFrurGrrrrr Zr_ranger r<r<r=rHos   zRing2D.evaluatecCs4|j|j}|j||j|f|j||j|ffSn Tuple defining the default ``bounding_box``. ``((y_low, y_high), (x_low, x_high))`` )rrrrr9Zdrr<r<r=r>{s zRing2D.bounding_boxcCs0|jjdurdS|jd|jj|jd|jjiSrr r@r<r<r=rOs  zRing2D.input_unitscCsf||jd||jdkr$td||jd||jd||jd||jd||jddS)Nrrr)rrrrrGrrRr<r<r=rSs     z&Ring2D._parameter_units_for_data_units)rTrUrVrWrrGrrrrr1rhrZrHrYr>rOrSrr<r<rjr=r)&s"&        r)c@sleZdZdZedddZedddZedddZedd Z e d d Z e d d Z e ddZ ddZdS)ra 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() rz Amplitude Ar0rz"Position of center of box functionzWidth of the boxcCs6t|||dk|||dk}t|g|gdS)z"One dimensional Box model functionr/r)rDrr)rFrGrrZinsider<r<r=rHs$zBox1D.evaluatecCs|jd}|j||j|fSzc Tuple defining the default ``bounding_box`` limits. ``(x_low, x_high))`` r.)rrr9r;r<r<r=r>s zBox1D.bounding_boxcCs"|jjdurdS|jd|jjiSrLrr@r<r<r=rOs zBox1D.input_unitscCs"|jjdurdS|jd|jjiSrL)rGrMrQr@r<r<r= return_unitss zBox1D.return_unitscCs*||jd||jd||jddS)Nr)rrrGrPrRr<r<r=rSs   z%Box1D._parameter_units_for_data_unitsN)rTrUrVrWrrGrrrZrHrYr>rOrrSr<r<r<r=rs4       rc@sxeZdZdZedddZedddZedddZedddZedd dZ e d d Z e d d Z e ddZddZdS)ra Two dimensional Box model. Parameters ---------- amplitude : float Amplitude x_0 : float x position of the center of the box function x_width : float Width in x direction of the box y_0 : float y position of the center of the box function y_width : float Width in y direction of the box See Also -------- Box1D, Gaussian2D, Moffat2D Notes ----- Model formula: .. math:: f(x, y) = \left \{ \begin{array}{ll} A : & x_0 - w_x/2 \leq x \leq x_0 + w_x/2 \text{ and} \\ & y_0 - w_y/2 \leq y \leq y_0 + w_y/2 \\ 0 : & \text{else} \end{array} \right. rZ Amplituder0rz,X position of the center of the box functionz,Y position of the center of the box functionzWidth in x direction of the boxzWidth in y direction of the boxc Cst|||dk|||dk}t|||dk|||dk}tt||g|gd} t|tr|t| |jddS| S)z"Two dimensional Box model functionr/rFrr) rFrurGrrx_widthy_widthZx_rangeZy_ranger r<r<r=rH% s zBox2D.evaluatecCs<|jd}|jd}|j||j|f|j||j|ffS)rr.)rrrr)r9r;rpr<r<r=r>4 s   zBox2D.bounding_boxcCs0|jjdurdS|jd|jj|jd|jjiSrr r@r<r<r=rOB s  zBox2D.input_unitscCsB||jd||jd||jd||jd||jddS)Nrr)rrrrrGrPrRr<r<r=rSI s      z%Box2D._parameter_units_for_data_unitsN)rTrUrVrWrrGrrrrrZrHrYr>rOrSr<r<r<r=rs$        rc@sleZdZdZedddZedddZedddZedddZe d d Z e d d Z e d dZ ddZdS)r'ae 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() rAmplitude of the trapezoidr0rz Center position of the trapezoidz'Width of constant part of the trapezoidzSlope of the tails of trapezoidcCs||d}||d}|||}|||}t||k||k} t||k||k} t||k||k} |||} |} |||}t| | | g| | |g}t|trt||jddS|S)z(One dimensional Trapezoid model functionr/Frr)rFrGrrrZx2Zx3Zx1Zx4Zrange_aZrange_bZrange_cZval_aZval_bZval_cr r<r<r=rH s       zTrapezoid1D.evaluatecCs*|jd|j|j}|j||j|fSr)rrGrrrr<r<r=r> szTrapezoid1D.bounding_boxcCs"|jjdurdS|jd|jjiSrLrr@r<r<r=rO s zTrapezoid1D.input_unitscCsD||jd||jd||jd||jd||jddS)Nr)rrrrGrPrRr<r<r=rS s    z+Trapezoid1D._parameter_units_for_data_unitsN)rTrUrVrWrrGrrrrZrHrYr>rOrSr<r<r<r=r'Q s)       r'c@sxeZdZdZedddZedddZedddZedddZedd dZ e d d Z e d d Z e ddZddZdS)r(a 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 rrr0rz)X position of the center of the trapezoidz)Y position of the center of the trapezoidz$Radius of constant part of trapezoidz*Slope of tails of trapezoid in x directionc Cst||d||d}||k}t||k||||k} |} ||||} t|| g| | g} t|trt| |jddS| S)z-Two dimensional Trapezoid Disk model functionr.Fr)rDrerrrrrM) rFrurGrrr rrZrange_1Zrange_2Zval_1Zval_2r r<r<r=rH s zTrapezoidDisk2D.evaluatecCs:|j|j|j}|j||j|f|j||j|ffSr)r rGrrrrr<r<r=r> szTrapezoidDisk2D.bounding_boxcCs<|jjdur|jjdurdS|jd|jj|jd|jjiSrrr@r<r<r=rO s zTrapezoidDisk2D.input_unitscCsh|d|dkrtd||jd||jd||jd||jd||jd||jddS)NrFrurr)rrr rrG)rrNrQrRr<r<r=rS s    z/TrapezoidDisk2D._parameter_units_for_data_unitsN)rTrUrVrWrrGrrr rrZrHrYr>rOrSr<r<r<r=r( s        r(c@s^eZdZdZedddZedddZedddZedd Z dd d Z e d dZ ddZ dS)ra 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() rAmplitude (peak) valuer0rrWidth of the Ricker waveletcCs4||dd|d}|dd|t| S)z-One dimensional Ricker Wavelet model functionr.rrC)rFrGrsigmaZxx_wwr<r<r=rH8 szRickerWavelet1D.evaluatercCs |j}||j}||||fS)zTuple defining the default ``bounding_box`` limits, ``(x_low, x_high)``. Parameters ---------- factor : float The multiple of sigma used to define the limits. )rrr8r<r<r=r>? s  zRickerWavelet1D.bounding_boxcCs"|jjdurdS|jd|jjiSrLrr@r<r<r=rON s zRickerWavelet1D.input_unitscCs*||jd||jd||jddS)Nr)rrrGrPrRr<r<r=rST s   z/RickerWavelet1D._parameter_units_for_data_unitsN)r)rTrUrVrWrrGrrrZrHr>rYrOrSr<r<r<r=r s6      rc@s`eZdZdZedddZedddZedddZedddZe d d Z e d d Z d dZ dS)ra  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}}} rrr0rX position of the peakY position of the peakrcCs<||d||dd|d}|d|t| S)z-Two dimensional Ricker Wavelet model functionr.rrC)rFrurGrrrZrr_wwr<r<r=rH s$zRickerWavelet2D.evaluatecCs0|jjdurdS|jd|jj|jd|jjiSrr r@r<r<r=rO s  zRickerWavelet2D.input_unitscCsZ||jd||jdkr$td||jd||jd||jd||jddS)Nrrr)rrrrGrrRr<r<r=rS s    z/RickerWavelet2D._parameter_units_for_data_unitsN)rTrUrVrWrrGrrrrZrHrYrOrSr<r<r<r=rZ s$      rc@sheZdZdZedddZedddZedddZedddZd Z d Z e d d Z e d d ZddZd S)r a 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 rz+Amplitude (peak value) of the Airy functionr0rr r!z;The radius of the Airy disk (radius of first zero crossing)Nc Cs|jdur6ddlm}m}|dddtj|_||_t||d||d||j} t| t rt| t j } t | j} tj| | dk} d|| | d| | dk<t|t rt | t j dd} | |9} | S) z#Two dimensional Airy model functionNr)j1jn_zerosrr.r/F)r)_rzrr"r#rDr_j1rerrZto_valuerZdimensionless_unscaledrrc) rrFrurGrrradiusr"r#rrZrtr<r<r=rH s (    zAiryDisk2D.evaluatecCs0|jjdurdS|jd|jj|jd|jjiSrr r@r<r<r=rO s  zAiryDisk2D.input_unitscCsZ||jd||jdkr$td||jd||jd||jd||jddS)Nrrr)rrr&rGrrRr<r<r=rS s    z*AiryDisk2D._parameter_units_for_data_units)rTrUrVrWrrGrrr&r$r%rrHrYrOrSr<r<r<r=r s*     r c@sxeZdZdZedddZedddZedddZedddZe d d Z e d d Z e d dZ e ddZddZdS)r a 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() rzAmplitude of the modelr0rz%X position of maximum of Moffat modelzCore width of Moffat modelPower index of the Moffat modelcCs(dt|jtdd|jdSz Moffat full width at half maximum. Derivation of the formula is available in `this notebook by Yoonsoo Bach `_. r/r~rDrgammarealphar@r<r<r=rA7 sz Moffat1D.fwhmcCs|d|||d| S)z%One dimensional Moffat model functionrr.r<)rFrGrr*r+r<r<r=rH@ szMoffat1D.evaluatec Csd||d|d}|| }d|||||||d}d||||d|||d}| |t|} |||| gS)zBOne dimensional Moffat model derivative with respect to parametersrr.rIrDr) rFrGrr*r+Zfacd_Ard_gammad_alphar<r<r=rKF s $ zMoffat1D.fit_derivcCs"|jjdurdS|jd|jjiSrLrr@r<r<r=rOR s zMoffat1D.input_unitscCs*||jd||jd||jddS)Nr)rr*rGrPrRr<r<r=rSX s   z(Moffat1D._parameter_units_for_data_unitsN)rTrUrVrWrrGrr*r+rYrArZrHrKrOrSr<r<r<r=r s1        r c@seZdZdZedddZedddZedddZedddZedd dZ e d d Z e d d Z e ddZe ddZddZdS)r ak 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} rz#Amplitude (peak value) of the modelr0rz-X position of the maximum of the Moffat modelz-Y position of the maximum of the Moffat modelzCore width of the Moffat modelr'cCs(dt|jtdd|jdSr(r)r@r<r<r=rA sz Moffat2D.fwhmcCs2||d||d|d}|d|| S)z%Two dimensional Moffat model functionr.rr<)rFrurGrrr*r+rr_ggr<r<r=rH s zMoffat2D.evaluatec Cs||d||d|d}d|| }d||||||dd|} d||||||dd|} | |td|} d|||||d|} || | | | gS)zBTwo dimensional Moffat model derivative with respect to parametersr.rr,) rFrurGrrr*r+r0r-rZd_y_0r/r.r<r<r=rK s  zMoffat2D.fit_derivcCs4|jjdurdS|jd|jj|jd|jjiSdSrr r@r<r<r=rO s  zMoffat2D.input_unitscCsZ||jd||jdkr$td||jd||jd||jd||jddS)Nrrr)rrr*rGrrRr<r<r=rS s    z(Moffat2D._parameter_units_for_data_unitsN)rTrUrVrWrrGrrr*r+rYrArZrHrKrOrSr<r<r<r=r ^ s         r c@seZdZdZedddZedddZedddZedd dZedd dZ edd dZ edd dZ d Z e ddZeddZddZd S)ra 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 rrr0rrrrzX position of the centerzY position of the centerZ Ellipticityz5Rotation angle in radians (counterclockwise-positive)Nc Cs|jdurddlm} | |_|d|d} |d||} } t| t| }}||||||}|| ||||}t|| d|| d}|t| |d|dS)z(Two dimensional Sersic profile function.Nrrr/rqrr.)rrrrDrsrtrerE)rrFrurGrrrrellipr`rZbnrnroZ cos_thetaZ sin_thetaZx_majZx_minrr<r<r=rH s  zSersic2D.evaluatecCs0|jjdurdS|jd|jj|jd|jjiSrr r@r<r<r=rO s  zSersic2D.input_unitscCs^||jd||jdkr$td||jd||jd||jdtj||jddS)Nrrr)rrrr`rGrrRr<r<r=rS s    z(Sersic2D._parameter_units_for_data_units)rTrUrVrWrrGrrrrr1r`rrrHrYrOrSr<r<r<r=r sH         rc@seZdZdZededfddZededfddZededfddZe d d Z e d d Z e d dZ e ddZe ddZddZdS)r+a 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='c ~ {:0.2f}'.format(mod.concentration)) plt.xlabel("r") plt.ylabel(r"$\sigma/\sigma_0$") plt.legend() plt.show() References ---------- .. [1] https://ui.adsabs.harvard.edu/abs/1962AJ.....67..471K rNzAmplitude or scaling factorr3z Core Radiusr.z Tidal RadiuscCstt|j|jS)z)Concentration parameter of the king model)rDZlog10rr_tider_corer@r<r<r= concentrationm sz%KingProjectedAnalytic1D.concentrationcCsh||ddt|d|ddt|d|dd}||k|dkB}||d||<|S)z/ Analytic King model function. r.rrr[rDre)rFrGr3r2r r4r<r<r=rHr s"z KingProjectedAnalytic1D.evaluatec Cs|ddt|d|ddt|d|dd}d||d||d|dd||d|dddt|d|ddt|d|dd||dt|d|ddt|d|dd}d||d|dt|d|ddt|d|d|d|dd}||k|dkB}||d||<||d||<||d||<|||gS)z; Analytic King model function derivatives. r.rg?r~rr5)rFrGr3r2rJZd_r_coreZd_r_tider4r<r<r=rK s8$2"z!KingProjectedAnalytic1D.fit_derivcCsd|jd|jfS)z Tuple defining the default ``bounding_box`` limits. The model is not defined for r > r_tide. ``(r_low, r_high)`` rr)r2r@r<r<r=r> s z$KingProjectedAnalytic1D.bounding_boxcCs"|jjdurdS|jd|jjiSrL)r3rMrNr@r<r<r=rO s z#KingProjectedAnalytic1D.input_unitscCs*||jd||jd||jddS)Nr)r3r2rGrPrRr<r<r=rS s   z7KingProjectedAnalytic1D._parameter_units_for_data_units)rTrUrVrWrrXrGr3r2rYr4rZrHrKr>rOrSr<r<r<r=r+- s;     r+c@sjeZdZdZeddZeddZeddZeddZ e dd Z ej d d Ze d d Z ddZdS)r-z One dimensional logarithmic model. Parameters ---------- amplitude : float, optional tau : float, optional See Also -------- Exponential1D, Gaussian1D rr1cCs|t||Srr,rFrGtaur<r<r=rH szLogarithmic1D.evaluatecCs*t||}t|j||}||gSr)rDrrrcrFrGr8rJZd_taur<r<r=rK szLogarithmic1D.fit_derivcCs|j}|j}t||dSN)rGr8)r8rGr,r9Z new_amplitudeZnew_taur<r<r=r szLogarithmic1D.inversecCst|dkrtddS)Nr!0 is not an allowed value for taurDallrdr9valr<r<r=r8 szLogarithmic1D.taucCs"|jjdurdS|jd|jjiSrLr8rMrNr@r<r<r=rO s zLogarithmic1D.input_unitscCs||jd||jddSNr)r8rGrPrRr<r<r=rS s  z-Logarithmic1D._parameter_units_for_data_unitsNrTrUrVrWrrGr8rZrHrKrYrZ validatorrOrSr<r<r<r=r- s       r-c@sjeZdZdZeddZeddZeddZeddZ e dd Z ej d d Ze d d Z ddZdS)r,z One dimensional exponential model. Parameters ---------- amplitude : float, optional tau : float, optional See Also -------- Logarithmic1D, Gaussian1D rr6cCs|t||SrrCr7r<r<r=rH szExponential1D.evaluatecCs6t||}| ||dt||}||gS)z& Derivative with respect to parametersr.rCr9r<r<r=rK s zExponential1D.fit_derivcCs|j}|j}t||dSr:)r8rGr-r;r<r<r=r szExponential1D.inversecCst|dkrtddS)z tau cannot be 0rr<Nr=r?r<r<r=r8 szExponential1D.taucCs"|jjdurdS|jd|jjiSrLrAr@r<r<r=rO s zExponential1D.input_unitscCs||jd||jddSrBrPrRr<r<r=rS s  z-Exponential1D._parameter_units_for_data_unitsNrCr<r<r<r=r, s       r,)ArWZnumpyrDZastropyrrZ astropy.unitsrrcorerr parametersrr Zutilsr __all__rrrZfinfoZfloat32ZtinyrXrerr?rrr rrrrrr!r"r#rr$r%r&rrrr*rrrrr)rrr'r(rrr r r rr+r-r,r<r<r<r=sj   ,:B/2[MMc]]O90h8L0Ov^W`L]Db^Xw1