IR-e@0dZddlZddlZddlmZddlmZm Z ddl m Z ddl m Z ddlmZmZdd lmZmZdd lmZgd Zd ejzZeejejjZd ejd ejd zzZ GddeZ!GddeZ"GddeZ#GddeZ$GddeZ%GddeZ&GddeZ'GddeZ(Gdde(Z)Gd d!e(Z*Gd"d#e(Z+Gd$d%e(Z,Gd&d'e,Z-Gd(d)e,Z.Gd*d+e,Z/Gd,d-eZ0Gd.d/eZ1Gd0d1eZ2Gd2d3eZ3Gd4d5eZ4Gd6d7eZ5Gd8d9eZ6Gd:d;eZ7Gd<d=eZ8Gd>d?eZ9Gd@dAeZ:GdBdCeZ;GdDdEeZ<GdFdGeZ=GdHdIeZ>GdJdKeZ?GdLdMeZ@GdNdOeZAGdPdQeZBGdRdSeZCGdTdUeZDGdVdWeZEdS)XzMathematical models.N)units)Quantity UnitsError) HAS_SCIPY)AstropyDeprecationWarning)Fittable1DModelFittable2DModel)InputParameterError Parameter)ellipse_extent)# AiryDisk2DMoffat1DMoffat2DBox1DBox2DConst1DConst2D Ellipse2DDisk2D Gaussian1D Gaussian2DLinear1D Lorentz1DRickerWavelet1DRickerWavelet2DRedshiftScaleFactorMultiplyPlanar2DScaleSersic1DSersic2DShiftSine1DCosine1D Tangent1D ArcSine1D ArcCosine1D ArcTangent1D Trapezoid1DTrapezoidDisk2DRing2DVoigt1DKingProjectedAnalytic1D Exponential1D Logarithmic1D@ceZdZdZeddZeddZededfd Zdd Z e d Z e d Z e dZe dZdZdS)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 Gaussianr5boundsr6@c8|j}||jz}||z ||zfS)aV 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' ) )meanstddevselffactorx0dxs Blib/python3.11/site-packages/astropy/modeling/functional_models.py bounding_boxzGaussian1D.bounding_boxs+LY dk !Rb!!c |jtzS)z$Gaussian full width at half maximum.)r<GAUSSIAN_SIGMA_TO_FWHMr>s rBfwhmzGaussian1D.fwhms{333rDcN|tjd||z dzz|dzz zS)z, Gaussian1D model function. r1npexp)x amplituder;r<s rBevaluatezGaussian1D.evaluates/ 26$!d(q"8619"DEEEErDctjd|dzz ||z dzz}||z||z z|dzz }||z||z dzz|dzz }|||gS)z8 Gaussian1D model function derivatives. rJr1rK)rNrOr;r< d_amplituded_meand_stddevs rB fit_derivzGaussian1D.fit_derivsr fTFAI-Ta?@@ [(AH5 A{*a$h1_ inputs_unit outputs_units rB_parameter_units_for_data_unitsz*Gaussian1D._parameter_units_for_data_unitss; A/!$+a.1%dl1o6   rDr9)__name__ __module__ __qualname____doc__r rOr; FLOAT_EPSILONr<rCpropertyrH staticmethodrPrVr[rbrDrBrrAsRRh GI 9Q,I J J JDYt$8F )")")")"V44X4FF\F //\/66X6      rDrcheZdZdZeddZeddZeddZeddZedd Z ed d Z ej ej ej d d d d ffd Z e dZe dZddZedZedZe dZdZxZS)rap 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 Gaussianr4rz(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 |:||jjj}||jjj}||jjj}n|||t dt j|}|jdkrtdt j |\} } t j | \}}| dddf} t j | d| d}|di|ddtdf|ddtdft!jd ||||||d |dS) Nz3Cannot specify both cov_matrix and x/y_stddev/theta)r1r1zCovariance matrix must be 2x2rrr8x_stddevy_stddev)rOx_meany_meanrorpthetark) __class__ror5rprsr rLarrayshape ValueErrorlinalgeigsqrtarctan2 setdefaultrhsuper__init__) r>rOrqrrrorprs cov_matrixkwargseig_valseig_vecsy_vecrts rBr~zGaussian2D.__init__^s  >2:>2:},4#x';u?P)I*--J6)) !@AAA!#z!:!: Hh!#!2!2 HhQQQTNEJuQxq22E (B'''x##J0EFFFx##J0EFFF         rDc |jtzS)z)Gaussian full width at half maximum in X.)rorFrGs rBx_fwhmzGaussian2D.x_fwhm}555rDc |jtzS)z)Gaussian full width at half maximum in Y.)rprFrGs rBy_fwhmzGaussian2D.y_fwhmrrDr9c||jz}||jz}t|||j\}}|j|z |j|zf|j|z |j|zffS)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' ) )rorpr rsrrrq)r>r?abrAdys rBrCzGaussian2D.bounding_boxslV T] " T] "1dj11B[2 t{R/ 0 [2 t{R/ 0  rDchtj|dz}tj|dz} tjd|z} |dz} |dz} ||z } ||z }d|| z | | z zz}d| | z | | z z z}d| | z || z zz}|tj|| dzz|| z|zz||dzzz zS)z"Two dimensional Gaussian function.r1r2?rLcossinrM)rNyrOrqrrrorprscost2sint2sin2txstd2ystd2xdiffydiffrrcs rBrPzGaussian2D.evaluatesu "u "sU{##! ! F F  EEMeem4 5 EEMeem4 5 EEMeem4 5265!8|E E 12a%(lC D    rDcttj|}tj|} tj|dz} tj|dz} tjd|z} tjd|z} |dz}|dz}|dz}|dz}||z }||z }|dz}|dz}d| |z | |z zz}d| |z | |z z z}d| |z | |z zz}|tj||z||z|zz||zz z}| |zd|z d|z z z}| |z }| |z }| |z | |z z }| |z }| |z }| } | |z }!| |z }"||z }#|d|z|z||zzz}$|||zd|z|zzz}%|||z||z|zz|!|zz z}&|||z||z|zz|"|zz z}'|||z||z|zz| |zz z}(|#|$|%|&|'|(gS)zHTwo dimensional Gaussian function derivative with respect to parameters.r1r2rRr?r))rNrrOrqrrrorprscostsintrrcos2trrrxstd3ystd3rrxdiff2ydiff2rrrg da_dtheta da_dx_stddev da_dy_stddev db_dtheta db_dx_stddev db_dy_stddev dc_dtheta dc_dx_stddev dc_dy_stddevdg_dA dg_dx_mean dg_dy_mean dg_dx_stddev dg_dy_stddev dg_dthetas) rBrVzGaussian2D.fit_derivsve}}ve}}u "u "sU{##sU{##! ! ! ! F F  EEMeem4 5 EEMeem4 5 EEMeem4 5 !f*UU1B!Cq6z!RSTT T4KC%KC%K#@A v~ v~ U]uu}5 v~ u} J v~ v~ I 37U?q5y9: 1u9q59: v%&./'(  v%&./'(  & 9u#4u#<x_unity_units rBr[zGaussian2D.input_unitss?'' >fn4 A A??rDc@||jd||jdkrtd||jd||jd||jd||jdtj||jddS)Nrr(Units of 'x' and 'y' inputs should match)rqrrrorprsrOrZruradr^r_s rBrbz*Gaussian2D._parameter_units_for_data_unitss t{1~ &+dk!n*E E EGHH H!$+a.1!$+a.1#DKN3#DKN3U%dl1o6    rDrc)rdrerfrgr rOrqrrrorprsr5r~rirrrCrjrPrVr[rb __classcell__rts@rBrrsSSj !1LMMMI YIFYIFyRHyRH I *   E#~~2 2 2 2 2 2 h66X666X62 2 2 2 h  \ 1V1V\1Vf@@X@        rDrceZdZdZeddZdZdZedZ edZ e dZ e d Z e d Zd Zd S) r#zv Shift a coordinate. Parameters ---------- offset : float Offset to add to a coordinate. rzOffset to add to a modelr4TcP|jjdS|jd|jjiSrX)offsetrYrZrGs rBr[zShift.input_units>( ; ! )4 A 677rDc}|xjdzc_ jtfdjD|_n#t$rYnwxYw|S)z-One dimensional inverse Shift model function.c3NK|]}|jV dSN)rPr.0rNr>s rB z Shift.inverse..OD%%23 a--%%%%%%rD)copyrrCtupleNotImplementedErrorr>invs` rBinversez Shift.inverseDsiikk b      %%%%%7;7H%%%  C  #    D  sA A! A!c ||zS)z%One dimensional Shift model function.rk)rNrs rBrPzShift.evaluateUs6zrDc|S)z>Evaluate the implicit term (x) of one dimensional Shift model.rk)rNs rBsum_of_implicit_termszShift.sum_of_implicit_termsZs rDc0tj|}|gS)zAOne dimensional Shift model derivative with respect to parameter.rL ones_like)rNparamsd_offsets rBrVzShift.fit_deriv_s<??zrDc,d||jdiS)Nrrr^r_s rBrbz%Shift._parameter_units_for_data_unitse,t|A788rDN)rdrerfrgr rlinear_has_inverse_bounding_boxrir[rrjrPrrVrbrkrDrBr#r#/sYq.H I I IF F $ 88X8 X \\\ 99999rDr#ceZdZdZeddZdZdZdZdZ dZ e dZ e dZ edZed Zd Zd S) r a 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 modelr4TcP|jjdS|jd|jjiSrX)r?rYrZrGs rBr[zScale.input_unitsrrDc}djz |_ jtfdjD|_n#t$rYnwxYw|S)z-One dimensional inverse Scale model function.rc3NK|]}|jV dSrrPr?rs rBrz Scale.inverse..rrDrr?rCrrrs` rBrz Scale.inverseiikk_      %%%%%7;7H7U7U7W7W%%%  C  #    D  A%% A21A2cNt|tjr|j}||zS)z%One dimensional Scale model function.) isinstancerrvaluerNr?s rBrPzScale.evaluates) faj ) ) "\FzrDc |}|gS)zAOne dimensional Scale model derivative with respect to parameter.rkrNrd_factors rBrVzScale.fit_derivzrDc,d||jdiSNr?rrr_s rBrbz%Scale._parameter_units_for_data_unitsrrDN)rdrerfrgr r?rfittable_input_units_strict _input_units_allow_dimensionlessrrir[rrjrPrVrbrkrDrBr r is  Yq.P Q Q QF FH'+$ $ 88X8 X \\ 99999rDr ceZdZdZeddZdZdZdZe dZ e dZ e dZ d Zd S) 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 modelr4Tc}djz |_ jtfdjD|_n#t$rYnwxYw|S)z0One dimensional inverse multiply model function.rc3NK|]}|jV dSrrrs rBrz#Multiply.inverse..rrDrrs` rBrzMultiply.inverserrc ||zS)z(One dimensional multiply model function.rkrs rBrPzMultiply.evaluateszrDc |}|gS)zDOne dimensional multiply model derivative with respect to parameter.rkrs rBrVzMultiply.fit_derivrrDc,d||jdiSrrr_s rBrbz(Multiply._parameter_units_for_data_unitsrrDN)rdrerfrgr r?rrrrirrjrPrVrbrkrDrBrrsYq.S T T TF FH $ X \\ 99999rDrcreZdZdZeddZdZedZedZ e dZ d S) rz One dimensional redshift scale factor model. Parameters ---------- z : float Redshift value. Notes ----- Model formula: .. math:: f(x) = x (1 + z) Redshiftr)r6r5Tcd|z|zS)z3One dimensional RedshiftScaleFactor model function.rrk)rNzs rBrPzRedshiftScaleFactor.evaluatesA{rDc |}|gS)z5One dimensional RedshiftScaleFactor model derivative.rk)rNrd_zs rBrVzRedshiftScaleFactor.fit_derivsu rDc}ddjzz dz |_ jtfdjD|_n#t$rYnwxYw|S)z"Inverse RedshiftScaleFactor model.rc3NK|]}|jV dSr)rPrrs rBrz.RedshiftScaleFactor.inverse..sD%%-. a((%%%%%%rD)rrrCrrrs` rBrzRedshiftScaleFactor.inversesiikksTV|$s*     %%%%%262C2P2P2R2R%%%  C  #    D  sA++ A87A8N) rdrerfrgr rrrjrPrVrirrkrDrBrrs    j!444A $\\ XrDrceZdZdZeddZeddZeddZdZe d Z e d Z d Z dS) r!a 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_effr4Effective (half-light) radius Sersic IndexNc|j ddlm}||_|tj|d|zd ||z d|z zdz zzS)z(One dimensional Sersic profile function.Nr gammaincinvr1rr) _gammaincinv scipy.specialr rLrM)clsrrOr_effnr s rBrPzSersic1D.evaluateQss   # 1 1 1 1 1 1*C 26   a!eS ) ) )a%iQU-Ca-G H    rDcP|jjdS|jd|jjiSrX)rrYrZrGs rBr[zSersic1D.input_units]s( : (4 A 566rDcP||jd||jddS)Nr)rrOr]r_s rBrbz(Sersic1D._parameter_units_for_data_unitscs- Q0%dl1o6   rD)rdrerfrgr rOrrr  classmethodrPrir[rbrkrDrBr!r! s==~ !1NOOOI Ia-L M M ME !888AL   [  77X7      rDr!c|eZdZdZeddZeddZeddZedZ 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 amplituder4zOscillation frequencyrzOscillation phasecV|jjdS|jdd|jjz iS)Nrr) frequencyrYrZrGs rBr[z_Trigonometric1D.input_units|s- > $ ,4 Adn&? ?@@rDcV||jddz||jddSNrr)rrOr]r_s rBrbz0_Trigonometric1D._parameter_units_for_data_unitss2$T[^4:%dl1o6   rDN) rdrerfrgr rOrphaserir[rbrkrDrBrrjs   !1HIIII !1HIIII Ia-@ A A AE AAXA      rDrcTeZdZdZedZedZedZdS)r$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() ct||z|zz}t|tr|j}|t j|zS)z$One dimensional Sine model function.)TWOPIrrrrLrrNrOrrarguments rBrPzSine1D.evaluateEIME12 h ) ) &~H26(++++rDc:tjt|z|zt|zz}t|z|ztjt|z|zt|zzz}t|ztjt|z|zt|zzz}|||gS)z&One dimensional Sine model derivative.)rLrrrrNrOrrrS d_frequencyd_phases rBrVzSine1D.fit_derivsfUY.2UU]BCC AI !BF59+-BUU]-R&S&SS['22rDcDt|j|j|jS)z One dimensional inverse of Sine.rOrr)r'rOrrrGs rBrzSine1D.inverse(ndj    rDN rdrerfrgrjrPrVrirrkrDrBr$r$p++Z , ,\ ,33\3  X   rDr$cTeZdZdZedZedZedZdS)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() ct||z|zz}t|tr|j}|t j|zS)z&One dimensional Cosine model function.)rrrrrLrrs rBrPzCosine1D.evaluaterrDc>tjt|z|zt|zz}t|z|ztjt|z|zt|zzz }t|ztjt|z|zt|zzz }|||gS)z(One dimensional Cosine model derivative.)rLrrrr!s rBrVzCosine1D.fit_derivsfUY.2UU]BCC AI !BF59+bboxs rBrCzTangent1D.bounding_boxxsN dj DN 2 TZ 4> 1  >  *$.--D rDN) rdrerfrgrjrPrVrirrCrkrDrBr&r&#s44l , ,\ ,33\3  X     rDr&c.eZdZdZedZdZdS)_InverseTrigonometric1DzF Base class for one dimensional inverse trigonometric models. cP|jjdS|jd|jjiSrX)rOrYrZrGs rBr[z#_InverseTrigonometric1D.input_unitss( > $ ,4 A 9::rDcV||jddz||jddSrr^rZr_s rBrbz7_InverseTrigonometric1D._parameter_units_for_data_unitss2%dl1o6"<$T[^4   rDN)rdrerfrgrir[rbrkrDrBr8r8sH;;X;      rDr8cZeZdZdZedZedZdZedZ dS)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() c||z }t|tr|j}tj|t z }||z |z S)z'One dimensional ArcSine model function.)rrrrLarcsinr)rNrOrrrarc_sines rBrPzArcSine1D.evaluatesJy= h ) ) &~H9X&&.5 I--rDc| t|z|dzztjd||z dzz zz }|tj||z tz z |dzz }d|z tj|jz}|||gS)z)One dimensional ArcSine model derivative.r1rr)rrLrzr>onesrvr!s rBrVzArcSine1D.fit_derivsb I  1 ,rwqA Ma;O7O/P/P P  !i- 8 85 @AYPQ\Q y.2717#3#33['22rDc*d|jzd|jzfSr4rrrOrGs rBrCzArcSine1D.bounding_box DN"A$666rDcDt|j|j|jS)z#One dimensional inverse of ArcSine.r%)r$rOrrrGs rBrzArcSine1D.inverses(ndj    rDN rdrerfrgrjrPrVrCrirrkrDrBr'r's22h . .\ .33\3777  X   rDr'cZeZdZdZedZedZdZedZ dS)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() c||z }t|tr|j}tj|t z }||z |z S)z)One dimensional ArcCosine model function.)rrrrLarccosrrNrOrrrarc_coss rBrPzArcCosine1D.evaluate-Iy= h ) ) &~H)H%%-%9,,rDc|t|z|dzztjd||z dzz zz }|tj||z tz z |dzz }d|z tj|jz}|||gS)z+One dimensional ArcCosine model derivative.r1rr)rrLrzrJrArvr!s rBrVzArcCosine1D.fit_deriv=s I  1 ,rwqA Ma;O7O/P/P P  !i- 8 85 @AYPQ\Q y.2717#3#33['22rDc*d|jzd|jzfSrCrDrGs rBrCzArcCosine1D.bounding_boxGrErDcDt|j|j|jS)z%One dimensional inverse of ArcCosine.r%)r%rOrrrGs rBrzArcCosine1D.inverseNs(ndj    rDNrGrkrDrBr(r(s22h - -\ -33\3777  X   rDr(cTeZdZdZedZedZedZdS)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() c||z }t|tr|j}tj|t z }||z |z S)z*One dimensional ArcTangent model function.)rrrrLarctanrrKs rBrPzArcTangent1D.evaluaterMrDc| t|z|dzzd||z dzzzz }|tj||z tz z |dzz }d|z tj|jz}|||gS)z,One dimensional ArcTangent model derivative.r1rr)rrLrSrArvr!s rBrVzArcTangent1D.fit_derivsb I  1 ,Q]q4H0H I  !i- 8 85 @AYPQ\Q y.2717#3#33['22rDcDt|j|j|jS)z&One dimensional inverse of ArcTangent.r%)r&rOrrrGs rBrzArcTangent1D.inverser&rDNr'rkrDrBr)r)Vsp,,\ - -\ -33\3  X   rDr)ceZdZdZeddZeddZdZedZ ed Z e d Z e d Z d Zd S)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 liner4rzIntercept of the straight lineTc||z|zS)z$One dimensional Line model function.rk)rNslope intercepts rBrPzLinear1D.evaluatesqy9$$rDc6|}tj|}||gS)zAOne dimensional Line model derivative with respect to parameters.r)rNrd_slope d_intercepts rBrVzLinear1D.fit_derivs!l1oo %%rDcd|jdz}|j |jz }|||S)Nr)rXrY)rXrYrt)r> new_slope new_intercepts rBrzLinear1D.inverses3JN $*4 ~~I~GGGrDc|jj|jjdS|jd|jj|jjz iSrX)rYrYrXrZrGs rBr[zLinear1D.input_unitss< > $ ,1F1N4 A 9DJ!9:  #q1uq3w 747a#gRS^;S T Y,t3q;GUF++rDc8|j}||jz}||z ||zfS)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. )rmrHr=s rBrCzLorentz1D.bounding_boxYs*X di Rb!!rDcP|jjdS|jd|jjiSrXrmrYrZrGs rBr[zLorentz1D.input_unitsk( 8  &4 A 344rDct||jd||jd||jddS)Nr)rmrHrOr]r_s rBrbz)Lorentz1D._parameter_units_for_data_unitsqs;t{1~. A/%dl1o6   rDN)rq)rdrerfrgr rOrmrHrjrPrVrCrir[rbrkrDrBrrs22h !>>>I )A+A B B BC 9Q,H I I IDVV\V,,\,""""$55X5      rDrc.eZdZdZeddZeddZedejz dZ eej dd Z ej ejZ ej ej dZej ej dejzZejde Zejde Zd Zejeje je jd ffd Zd ZdZdZedZdZeddZxZ 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() rrkr4rzThe Lorentzian amplituder1z)The Lorentzian full width at half maximumz'The Gaussian full width at half maximum)dtypeNc t|dkr$trtj|dt | trd}nd}t|dvrddlm}||_nEt|dkr |j |_ntd|d|jj |_ tjd ||||d |dS) N humlicek2z has been deprecated since Astropy 5.3 and will be removed in a future version. It is recommended to always use the `~scipy.special.wofz` implementation when `scipy` is installed.wofz)r{scipyr)r{z2Not a valid method for Voigt1D Faddeeva function: .)rm amplitude_Lfwhm_Lfwhm_Grk)strlowerrwarningswarnrr r{ _faddeeva _hum2zpf16crwrdmethodr}r~) r>rmr~rrrrr{rts rBr~zVoigt1D.__init__s= v;;    + - -) - M---*     > %$ v;;    "3 3 3 * * * * * *!DNN [[   K / /!-DNNNVNNN n-  VF  NT     rDc@|j|jjkr$tj||jddr|jSt |t jr|t j n||_| |j|_|jS)zCall complex error (Faddeeva) function w(z) implemented by algorithm `method`; cache results for consecutive calls from `evaluate`, `fit_deriv`. g+=gV瞯<)rtolatol) rv_last_zrLallclose_last_wrrrto_valuedimensionless_unscaledr)r>rs rB _wrap_wofzzVoigt1D._wrap_wofzs 7dl( ( (R[ t|'. . . (< 5?q!*4M4M TAJJq/ 0 0 0ST ~~dl33 |rDctjd||z zd|zz|jz|z }||j|jz|z |z|zS)z@One dimensional Voigt function scaled to Lorentz peak amplitude.r1?)rL atleast_1dsqrt_ln2rreal sqrt_ln2pi)r>rNrmr~rrrs rBrPzVoigt1D.evaluates[ M!q3w-"v+5 6 6 F Oq!!&86AFJ[XXrDcz|j|z }tjd||z zd|zz|z}|||z|z|z|jz}d|z|zd|z|z|zz} | j dz|z|j|z |j|z | j|zz |j |d||z z| jz|| jzz zz |z gS)z^ Derivative of the one dimensional Voigt function with respect to parameters. r1ry@)rrLrrsqrt_pirimag) r>rNrmr~rrsrwdwdzs rBrVzVoigt1D.fit_derivs MF " M!q3w-"v+5 6 6 : OOA   "V +k 9DL HAvzBFVOk99YJNQ  F[ FVOdi!m +fWqASMDI58JJK Kv U   rDcP|jjdS|jd|jjiSrXrtrGs rBr[zVoigt1D.input_units rurDc||jd||jd||jd||jddS)Nr)rmrrr~r]r_s rBrbz'Voigt1D._parameter_units_for_data_unitssIt{1~.!$+a.1!$+a.1' Q8    rD$@ctjgd}tjgd}dtjtjz }||z}d|||zdz zzd||dz zzz }tj|j|krvt |j|jz|k}|tj|dz}||z} |d |z|d z|z|d z|z|d z|z|d z|z|dz|z|dz|z|dz|z|dz|z|dz|z|dz|z|dz|z|dz|z|dz|z|dz|z|dz} | |dz| z|dz| z|dz| z|dz| z|dz| z|dz| z|dz| z|dz} tj ||| | z |S)aComplex error function w(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@gg@ Ag2 g@gSgT@gNrrrg]/M?g?g@y? rrRr1rr) rLrurzpianyrabsrwhereplace) rrAAbb sqrt_piinvzzrmaskZZZnumerdenoms rBrzVoigt1D._hum2zpf16csX      X      2725>>) U !rJ12 3tbBHo7M N 6!&1*   -qv;;'!+D"(4..!E)AQB$&b6!8bf#4a"7"R&"@!!Cbf!La ORTUWRX XZ[[!"v &'()+-a5123468e<=>?ACAGHIJLNqERSTUQ% !"#%'U+,-..0e45679;A?@ABDFqEJKLMOQRSuUE1:r/BqE1251=rA"Q%GKbQReSUWWe "$Q%()+,.0e4E HQeem , , ,rDr)!rdrerfrgr rmr~rLrrlogrrzrrrzeroscomplexrfloatrrr5r~rrPrVrir[rbrjrrrs@rBr-r-ys00d )A+A B B BC)A3MNNNK YBE 'RFYq 'PFbgbennGrwvrvayy!!HRU*++Jbhq(((Gbhq&&&GI K'~~ % % % % % % N YYY   &55X5    JJJ\JJJJJrDr-czeZdZdZedddZdZedZedZ e dZ d Z d S) 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 functionTr5r6magc:|jdkrItj||j|j}||n|tj|dz}t|trt||j ddS|S)z(One dimensional Constant model function.rrvrxFsubokTr5rr sizerL empty_likervrxfillitemrrrr5)rNrOs rBrPzConst1D.evaluate >Q   iqwagFFFA FF9>>## $ $ $ $BL%8888A i * * LAINdKKK KrDc0tj|}|gS)zEOne dimensional Constant model derivative with respect to parameters.r)rNrOrSs rBrVzConst1D.fit_derivsl1oo }rDcdSrrkrGs rBr[zConst1D.input_unitstrDc,d||jdiSNrOrrr_s rBrbz'Const1D._parameter_units_for_data_units\$,q/:;;rDN) rdrerfrgr rOrrjrPrVrir[rbrkrDrBrris&&P ?TIF  \ \ X<<<<rrrsrArs rBrCzEllipse2D.bounding_boxIs[ F F 1e,,BB2 .B2 0NOOrDc~|jjdS|jd|jj|jd|jjiSrrmrYrZrrGs rBr[zEllipse2D.input_unitsW< 8  &4 KNDH/ KNDH/  rDc@||jd||jdkrtd||jd||jd||jd||jdtj||jddS)Nrrr)rmrrrrsrOrr_s rBrbz)Ellipse2D._parameter_units_for_data_units`s t{1~ &+dk!n*E E EGHH Ht{1~.t{1~.T[^,T[^,U%dl1o6    rDN)rdrerfrgr rOrmrrrrsrjrPrirCr[rbrkrDrBrrs DDL !1GTRRRI )A+R S S SC )A+R S S SC !)KLLLA !)KLLLA I Q   E  \  P PX P  X       rDrceZdZdZedddZeddZedd Zedd Ze d Z e d Z e d Z dZdS)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 functionTrrz X position of center of the diskr4z Y position of center of the diskzRadius of the diskc||z dz||z dzz}tj||dzkg|g}t|trt||jddS|S)z$Two dimensional Disk model function.r1FTr)rLrrrr5)rNrrOrmrR_0rrrs rBrPzDisk2D.evaluatesl#g!^q3w1n ,B#q&L>I;77 i * * QFe4PPP P rDcz|j|jz |j|jzf|j|jz |j|jzffSr)rrrmrGs rBrCzDisk2D.bounding_boxsBX $(TX"5 6 X $(TX"5 6  rDcv|jj}|jj}||dS|jd||jd|iSrrmrYrrZrs rBr[zDisk2D.input_unitss?$$ >fn4 A A??rDc||jd||jdkrtd||jd||jd||jd||jddS)Nrrr)rmrrrOrZrr^r_s rBrbz&Disk2D._parameter_units_for_data_unitss{ t{1~ &+dk!n*E E EGHH Ht{1~.t{1~.t{1~.%dl1o6    rDN)rdrerfrgr rOrmrrrjrPrirCr[rbrkrDrBrrps@ !1ItTTTI )A+M N N NC )A+M N N NC )A+? @ @ @C\   X  @@X@      rDrceZdZdZedddZeddZedd Zedd Zedd Z ej ej ej d d d ffd Z e dZ edZedZdZxZS)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 functionTrrzX position of center of discr4zY position of center of disczInner radius of the ringzWidth of the ringNc ||||jj}|jj}n||| |jj}ns||||jj}||z }n[||| |jj}nH| ||||z }n<| ||||z }n0|.|,|*tj|||z krt dtj|dkstj|dkrt d|d|dt jd|||||d|dS)NzWidth must be r_out - r_inrzr_in=z and width=z must both be >=0)rOrmrr_inwidthrk)rr5rrLrr r}r~) r>rOrmrrrr_outrrts rBr~zRing2D.__init__s Lu}5=9$DJ&EEU]J&EEl!29$DDLEElU5F9$DDU%6U]DLEEl!29J5=DDU%6U=Nve -.. H)*FGGG 6$(   Prveai00 P%&N&N&NU&N&N&NOO O ScE  MS     rDc||z dz||z dzz}tj||dzk|||zdzk}tj|g|g} t|trt | |jddS| S)z$Two dimensional Ring model function.r1FTrrL logical_andrrrr5) rNrrOrmrrrrr_rangers rBrPzRing2D.evaluate s#g!^q3w1n ,.tQwte|6I0IJJG9yk22 i * * QFe4PPP P rDcp|j|jz}|j|z |j|zf|j|z |j|zffSn Tuple defining the default ``bounding_box``. ``((y_low, y_high), (x_low, x_high))`` )rrrrmr>drs rBrCzRing2D.bounding_box s?Y #B2 .B2 0NOOrDc~|jjdS|jd|jj|jd|jjiSrrrGs rBr[zRing2D.input_units) rrDc*||jd||jdkrtd||jd||jd||jd||jd||jddS)Nrrr)rmrrrrOrr_s rBrbz&Ring2D._parameter_units_for_data_units2 s t{1~ &+dk!n*E E EGHH Ht{1~.t{1~. A/ Q0%dl1o6    rD)rdrerfrgr rOrmrrrr5r~rjrPrirCr[rbrrs@rBr,r,s5$$L !1MSWXXXI )A+I J J JC )A+I J J JC 9Q,F G G GD Ia-@ A A AE# K K ! ! ! ! ! ! F\PPXP  X         rDr,ceZdZdZedddZeddZedd Zed Z e d Z e d Z e d Z dZdS)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 ATrrz"Position of center of box functionr4zWidth of the boxctj|||dz z k|||dz zk}tj|g|gdS)z#One dimensional Box model function.r2r)rLrr)rNrOrmrinsides rBrPzBox1D.evaluatez sIS53;%6 6S53;=N8NOOy&I;222rDc>|jdz }|j|z |j|zfSzc Tuple defining the default ``bounding_box`` limits. ``(x_low, x_high))`` r1)rrmr>rAs rBrCzBox1D.bounding_box s&Z!^2 tx"}--rDcP|jjdS|jd|jjiSrXrtrGs rBr[zBox1D.input_units rurDcP|jjdS|jd|jjiSrX)rOr5r^rGs rB return_unitszBox1D.return_units s( >  &4 Q!455rDct||jd||jd||jddS)Nr)rmrrOr]r_s rBrbz%Box1D._parameter_units_for_data_units ;t{1~. Q0%dl1o6   rDN)rdrerfrgr rOrmrrjrPrirCr[r rbrkrDrBrrA s22h !DIIII )A+O P P PC Ia-? @ @ @E33\3 ..X.55X5 66X6      rDrceZdZdZedddZeddZedd Zedd Zedd Z e d Z e d Z e dZdZdS)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. r AmplitudeTrrz,X position of the center of the box functionr4z,Y position of the center of the box functionzWidth in x direction of the boxzWidth in y direction of the boxcZtj|||dz z k|||dz zk}tj|||dz z k|||dz zk}tjtj||g|gd} t|trt | |jddS| S)z#Two dimensional Box model function.r2rFTrr) rNrrOrmrx_widthy_widthx_rangey_rangers rBrPzBox2D.evaluate s.cGcM&9!91gPSm@S;STT.cGcM&9!91gPSm@S;STTBN7G<<= {ANN i * * QFe4PPP P rDcz|jdz }|jdz }|j|z |j|zf|j|z |j|zffS)rr1)rrrrm)r>rArs rBrCzBox2D.bounding_box sI\A  \A B2 .B2 0NOOrDc~|jjdS|jd|jj|jd|jjiSrrrGs rBr[zBox2D.input_units rrDc||jd||jd||jd||jd||jddS)Nrr)rmrrrrOr]r_s rBrbz%Box2D._parameter_units_for_data_units sWt{1~.t{1~."4;q>2"4;q>2%dl1o6    rDN)rdrerfrgr rOrmrrrrjrPrirCr[rbrkrDrBrr s""H !$GGGI )M   C )M   Ci/PQQQGi/PQQQG  \  P PX P  X      rDrceZdZdZeddZeddZeddZeddZe d Z e d Z e d Z d Zd S)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 trapezoidr4rz Center position of the trapezoidz'Width of constant part of the trapezoidzSlope of the tails of trapezoidc||dz z }||dz z}|||z z }|||z z}tj||k||k} tj||k||k} tj||k||k} |||z z} |} |||z z}tj| | | g| | |g}t|trt ||jddS|S)z)One dimensional Trapezoid model function.r2FTrr)rNrOrmrrXx2x3x1x4range_arange_brange_cval_aval_bval_crs rBrPzTrapezoid1D.evaluate( s 53;  53;  )e# # )e# #.b!b&11.b!b&11.b!b&11R a GWg6u8MNN i * * QFe4PPP P rDc^|jdz |j|jz z}|j|z |j|zfSr)rrOrXrmrs rBrCzTrapezoid1D.bounding_box? s4Z!^dntz9 92 tx"}--rDcP|jjdS|jd|jjiSrXrtrGs rBr[zTrapezoid1D.input_unitsJ rurDc||jd||jd||jd||jdz ||jddS)Nr)rmrrXrOr]r_s rBrbz+Trapezoid1D._parameter_units_for_data_unitsP sYt{1~. Q0!$,q/2[Q5PP%dl1o6    rDN)rdrerfrgr rOrmrrXrjrPrirCr[rbrkrDrBr*r* s''R !1MNNNI )A+M N N NC Ia-V W W WE Ia-N O O OE\,..X.55X5      rDr*ceZdZdZeddZeddZeddZeddZedd Z e d Z e d Z e d Zd ZdS)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 rrr4rz)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 directioncDtj||z dz||z dzz}||k}tj||k||||z zk} |} ||||z zz} tj|| g| | g} t |t rt | |jddS| S)z.Two dimensional Trapezoid Disk model function.r1FTr)rLrzrrrrr5) rNrrOrmrrrXr range_1range_2val_1val_2rs rBrPzTrapezoidDisk2D.evaluatew s GQWNa#g!^3 4 4s(.S!sY5F/F*FGGES1W--GW-u~>> i * * QFe4PPP P rDc|j|j|jz z}|j|z |j|zf|j|z |j|zffSr)rrOrXrrmrs rBrCzTrapezoidDisk2D.bounding_box sFX3 3B2 .B2 0NOOrDcv|jj}|jj}||dS|jd||jd|iSrrrs rBr[zTrapezoidDisk2D.input_units s?$$ >fn4 A A??rDc&|d|dkrtd||jd||jd||jd||jd||jdz ||jddS)NrNrrr)rmrrrXrO)rrZr^r_s rBrbz/TrapezoidDisk2D._parameter_units_for_data_units s s {3/ / /GHH Ht{1~.t{1~.t{1~.!$,q/2[Q5PP%dl1o6    rDN)rdrerfrgr rOrmrrrXrjrPrirCr[rbrkrDrBr+r+Y s* !1MNNNI )A+V W W WC )A+V W W WC )A+Q R R RC IK   E  \ PPXP@@X@      rDr+ceZdZdZeddZeddZeddZedZ dd Z e d Z d Z d S)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) valuer4rrkWidth of the Ricker waveletcf||z dzd|dzzz }|dd|zz ztj| zS)z.One dimensional Ricker Wavelet model function.r1rrK)rNrOrmsigmaxx_wws rBrPzRickerWavelet1D.evaluate s@SQ!eQh,/AE M*RVUF^^;;rDrc8|j}||jz}||z ||zfS)zTuple defining the default ``bounding_box`` limits, ``(x_low, x_high)``. Parameters ---------- factor : float The multiple of sigma used to define the limits. )rmr5r=s rBrCzRickerWavelet1D.bounding_box s*X dj Rb!!rDcP|jjdS|jd|jjiSrXrtrGs rBr[zRickerWavelet1D.input_units rurDct||jd||jd||jddS)Nr)rmr5rOr]r_s rBrbz/RickerWavelet1D._parameter_units_for_data_units r rDNr)rdrerfrgr rOrmr5rjrPrCrir[rbrkrDrBrr s44l !1IJJJI )A+A B B BC Ia-J K K KE<<\< " " " "55X5      rDrceZdZdZeddZeddZeddZeddZe d Z e d Z d Z d S) 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}}} rr2r4rX position of the peakY position of the peakr3cr||z dz||z dzzd|dzzz }|d|z ztj| zS)z.Two dimensional Ricker Wavelet model function.r1rrK)rNrrOrmrr5rr_wws rBrPzRickerWavelet2D.evaluate1 sIc'a1s7q.0Q\BAI&77rDc~|jjdS|jd|jj|jd|jjiSrrrGs rBr[zRickerWavelet2D.input_units7 rrDc||jd||jdkrtd||jd||jd||jd||jddS)Nrrr)rmrr5rOrr_s rBrbz/RickerWavelet2D._parameter_units_for_data_units@ { t{1~ &+dk!n*E E EGHH Ht{1~.t{1~. Q0%dl1o6    rDN)rdrerfrgr rOrmrr5rjrPrir[rbrkrDrBrr s""H !1IJJJI )A+C D D DC )A+C D D DC Ia-J K K KE88\8   X       rDrceZdZdZeddZeddZeddZeddZd Z d Z e d Z e d Zd Zd S) ra 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 functionr4rr;r<z;The radius of the Airy disk (radius of first zero crossing)NcZ|j3ddlm}m}|dddtjz |_||_t j||z dz||z dzz||jz z } t| tr| tj } t j | j} tj| | dkz} d|| z| z dz| | dk<t|trt| tj dd } | |z} | S) z$Two dimensional Airy model function.Nr)j1jn_zerosrr1r2FT)rr)_rzr rDrErLr_j1rzrrrrrrArv) r rNrrOrmrradiusrDrEr rrts rBrPzAiryDisk2D.evaluate s$ 7? 2 2 2 2 2 2 2 2hq!nnQ'"%/CGCG GQWNa#g!^3 4 48H I a " " 5 1344A GAG   UQq1uX #''"++%*q0!a% i * * NA45MMMA YrDc~|jjdS|jd|jj|jd|jjiSrrrGs rBr[zAiryDisk2D.input_units rrDc||jd||jdkrtd||jd||jd||jd||jddS)Nrrr)rmrrHrOrr_s rBrbz*AiryDisk2D._parameter_units_for_data_units s{ t{1~ &+dk!n*E E EGHH Ht{1~.t{1~.!$+a.1%dl1o6    rD)rdrerfrgr rOrmrrHrFrGrrPrir[rbrkrDrBrrN s**X LI )A+C D D DC )A+C D D DC YQF C C[6  X       rDrceZdZdZeddZeddZeddZeddZe d Z e d Z e d Z e d Zd ZdS)ra 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 modelr4rz%X position of maximum of Moffat modelzCore width of Moffat modelPower index of the Moffat modelcdtj|jztjdd|jz zdz zSa Moffat full width at half maximum. Derivation of the formula is available in `this notebook by Yoonsoo Bach `_. r2rrLrgammarzalpharGs rBrHz Moffat1D.fwhm ;RVDJ'''"'## :J2Kc2Q*R*RRRrDc,|d||z |z dzz| zzS)z&One dimensional Moffat model function.rr1rk)rNrOrmrQrRs rBrPzMoffat1D.evaluate s)A!c'U!2q 88ufEEErDcd||z dz|dzz z}|| z}d|z|z||z z|z||dzzz }d|z|z||z dzz|z||dzzz }| |ztj|z} |||| gS)zCOne dimensional Moffat model derivative with respect to parameters.rr1rRrLr) rNrOrmrQrRfacd_Arod_gammad_alphas rBrVzMoffat1D.fit_deriv s1s7q.5!8++ufoI %S1C73>Ji-%'1s7q.83>#q.Q*s"RVC[[0UGW--rDcP|jjdS|jd|jjiSrXrtrGs rBr[zMoffat1D.input_units rurDct||jd||jd||jddS)Nr)rmrQrOr]r_s rBrbz(Moffat1D._parameter_units_for_data_units r rDN)rdrerfrgr rOrmrQrRrirHrjrPrVr[rbrkrDrBrr s//b !1IJJJI )A+R S S SC Ia-I J J JE Ia-N O O OE SSXSFF\F..\.55X5      rDrceZdZdZeddZeddZeddZeddZedd Z e d Z e d Z e d Ze d ZdZdS)rak 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 modelr4rz-X position of the maximum of the Moffat modelz-Y position of the maximum of the Moffat modelzCore width of the Moffat modelrMcdtj|jztjdd|jz zdz zSrOrPrGs rBrHz Moffat2D.fwhmA rSrDcH||z dz||z dzz|dzz }|d|z| zzS)z&Two dimensional Moffat model function.r1rrk)rNrrOrmrrQrRrr_ggs rBrPzMoffat2D.evaluateK s=c'a1s7q.0E1H<AIE6222rDc.||z dz||z dzz|dzz }d|z| z}d|z|z|z||z z|dzd|zzz } d|z|z|z||z z|dzd|zzz } | |ztjd|zz} d|z|z|z|z|d|zzz } || | | | gS)zCTwo dimensional Moffat model derivative with respect to parameters.r1rrV) rNrrOrmrrQrRr`rXrod_y_0rZrYs rBrVzMoffat2D.fit_derivQ sc'a1s7q.0E1H<5yuf%I %+q3w75!8q5y;QRI %+q3w75!8q5y;QR*s"RVAI%6%66i-%'#-5!e)9LMUE7G44rDc~|jjdS|jd|jj|jd|jjiSrrrGs rBr[zMoffat2D.input_units\ s> 8  &4 A 3 A 3 rDc||jd||jdkrtd||jd||jd||jd||jddS)Nrrr)rmrrQrOrr_s rBrbz(Moffat2D._parameter_units_for_data_unitsf rArDN)rdrerfrgr rOrmrrQrRrirHrjrPrVr[rbrkrDrBrr s< !1VWWWI )N   C )N   C Ia-M N N NE Ia-N O O OE SSXS33\3 55\5X      rDrceZdZdZeddZeddZeddZedd Zedd Z edd Z ed d Z dZ e dZedZdZdS)r"a 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]) plt.show() References ---------- .. [1] http://ned.ipac.caltech.edu/level5/March05/Graham/Graham2.html rrr4rrrrzX position of the centerzY position of the center EllipticityrmrNc |j ddlm} | |_|d|zd} |d|z |z} } tj| tj| }}||z |z||z |zz}||z |z||z |zz}tj|| z dz|| z dzz}|tj| |d|z zdz zzS)z(Two dimensional Sersic profile function.Nrrr2rrr1)r r r rLrrrzrM)r rNrrOrrrmrelliprsr bnrr cos_theta sin_thetax_majx_minrs rBrPzSersic2D.evaluate s   # 1 1 1 1 1 1*C    cAgs + +q5yE)1!ve}}bfUmm9 SI%SI(==c' Y&!c'Y)>> GUQY1$ a'77 8 8262#q1u)9":;;;;rDc~|jjdS|jd|jj|jd|jjiSrrrGs rBr[zSersic2D.input_units rrDc||jd||jdkrtd||jd||jd||jdtj||jddS)Nrrr)rmrrrsrOrr_s rBrbz(Sersic2D._parameter_units_for_data_units s t{1~ &+dk!n*E E EGHH Ht{1~.t{1~. Q0U%dl1o6    rD)rdrerfrgr rOrrrmrrhrsr rrPrir[rbrkrDrBr"r"t sJJX !1NOOOI Ia-L M M ME !888A )A+E F F FC )A+E F F FC Ia] ; ; ;E I Q   E L<<[<   X       rDr"ceZdZdZededfdZededfdZededfdZe d Z e dd Z e d Z d Zd Ze dZe dZdZdS)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=f"c ~ {mod.concentration:0.2f}") plt.xlabel("r") plt.ylabel(r"$\sigma/\sigma_0$") plt.legend() plt.show() References ---------- .. 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