IR-eRdZddlZddlmZmZmZmZmZddl m Z ddl m Z m Z gdZGdd e ZGd d e ZGd d e ZGdde ZGdde ZGdde ZdS)z Power law model variants. N) MagnitudeQuantity UnitsErrordimensionless_unscaledmag)Fittable1DModel)InputParameterError Parameter) PowerLaw1DBrokenPowerLaw1DSmoothlyBrokenPowerLaw1DExponentialCutoffPowerLaw1D LogParabola1D Schechter1DceZdZdZeddZeddZeddZedZ edZ e d Z d Z d S) r a One dimensional power law model. Parameters ---------- amplitude : float Model amplitude at the reference point x_0 : float Reference point alpha : float Power law index See Also -------- BrokenPowerLaw1D, ExponentialCutoffPowerLaw1D, LogParabola1D Notes ----- Model formula (with :math:`A` for ``amplitude`` and :math:`\alpha` for ``alpha``): .. math:: f(x) = A (x / x_0) ^ {-\alpha} rz!Peak value at the reference pointdefault descriptionReference pointPower law indexc||z }||| zzS)z)One dimensional power law model function.)x amplitudex_0alphaxxs :lib/python3.11/site-packages/astropy/modeling/powerlaws.pyevaluatezPowerLaw1D.evaluate4sW25&>))cn||z }|| z}||z|z|z }| |ztj|z}|||gS)z@One dimensional power law derivative with respect to parameters.nplog)rrrrr d_amplituded_x_0d_alphas r fit_derivzPowerLaw1D.fit_deriv:sRWeVn E!K/#5*{*RVBZZ7UG,,r!cP|jjdS|jd|jjiSNrr input_unitinputsselfs r input_unitszPowerLaw1D.input_unitsE( 8  &4 A 344r!cP||jd||jddSNr)rrr.outputsr0 inputs_unit outputs_units r_parameter_units_for_data_unitsz*PowerLaw1D._parameter_units_for_data_unitsK-t{1~.%dl1o6   r!N)__name__ __module__ __qualname____doc__r rrr staticmethodr r)propertyr1r:rr!rr r s0 !1TUUUI )A+< = = =C Ia-> ? ? ?E**\* --\-55X5      r!r ceZdZdZeddZeddZeddZeddZe dZ e d Z e d Z d Zd S) r aV One dimensional power law model with a break. Parameters ---------- amplitude : float Model amplitude at the break point. x_break : float Break point. alpha_1 : float Power law index for x < x_break. alpha_2 : float Power law index for x > x_break. See Also -------- PowerLaw1D, ExponentialCutoffPowerLaw1D, LogParabola1D Notes ----- Model formula (with :math:`A` for ``amplitude`` and :math:`\alpha_1` for ``alpha_1`` and :math:`\alpha_2` for ``alpha_2``): .. math:: f(x) = \left \{ \begin{array}{ll} A (x / x_{break}) ^ {-\alpha_1} & : x < x_{break} \\ A (x / x_{break}) ^ {-\alpha_2} & : x > x_{break} \\ \end{array} \right. rPeak value at break pointr Break point"Power law index before break point!Power law index after break pointcRtj||k||}||z }||| zzS)z0One dimensional broken power law model function.)r$where)rrx_breakalpha_1alpha_2rrs rr zBrokenPowerLaw1D.evaluateys5Wgw77 [25&>))r!c tj||k||}||z }|| z}||z|z|z }| |ztj|z} tj||k| d} tj||k| d} ||| | gS)zGOne dimensional broken power law derivative with respect to parameters.r)r$rHr%) rrrIrJrKrrr& d_x_breakr( d_alpha_1 d_alpha_2s rr)zBrokenPowerLaw1D.fit_derivsWgw77 [eVn % 3g= *{*RVBZZ7HQ['155 HQ'\7A66 Y 9==r!cP|jjdS|jd|jjiSr+rIr-r.r/s rr1zBrokenPowerLaw1D.input_units( < " *4 A 788r!cP||jd||jddSNr)rIrr5r7s rr:z0BrokenPowerLaw1D._parameter_units_for_data_units-"4;q>2%dl1o6   r!N)r<r=r>r?r rrIrJrKr@r r)rAr1r:rr!rr r RsB !1LMMMIi}===Gi/STTTGi/RSSSG**\*  > >\ >99X9      r!r c eZdZdZeddddZeddZed d Zed d Zedd dZ dZ e e_ dZ e e _ e dZe dZedZdZdS)ra+ One dimensional smoothly broken power law model. Parameters ---------- amplitude : float Model amplitude at the break point. x_break : float Break point. alpha_1 : float Power law index for ``x << x_break``. alpha_2 : float Power law index for ``x >> x_break``. delta : float Smoothness parameter. See Also -------- BrokenPowerLaw1D Notes ----- Model formula (with :math:`A` for ``amplitude``, :math:`x_b` for ``x_break``, :math:`\alpha_1` for ``alpha_1``, :math:`\alpha_2` for ``alpha_2`` and :math:`\Delta` for ``delta``): .. math:: f(x) = A \left( \frac{x}{x_b} \right) ^ {-\alpha_1} \left\{ \frac{1}{2} \left[ 1 + \left( \frac{x}{x_b}\right)^{1 / \Delta} \right] \right\}^{(\alpha_1 - \alpha_2) \Delta} The change of slope occurs between the values :math:`x_1` and :math:`x_2` such that: .. math:: \log_{10} \frac{x_2}{x_b} = \log_{10} \frac{x_b}{x_1} \sim \Delta At values :math:`x \lesssim x_1` and :math:`x \gtrsim x_2` the model is approximately a simple power law with index :math:`\alpha_1` and :math:`\alpha_2` respectively. The two power laws are smoothly joined at values :math:`x_1 < x < x_2`, hence the :math:`\Delta` parameter sets the "smoothness" of the slope change. The ``delta`` parameter is bounded to values greater than 1e-3 (corresponding to :math:`x_2 / x_1 \gtrsim 1.002`) to avoid overflow errors. The ``amplitude`` parameter is bounded to positive values since this model is typically used to represent positive quantities. Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling import models x = np.logspace(0.7, 2.3, 500) f = models.SmoothlyBrokenPowerLaw1D(amplitude=1, x_break=20, alpha_1=-2, alpha_2=2) plt.figure() plt.title("amplitude=1, x_break=20, alpha_1=-2, alpha_2=2") f.delta = 0.5 plt.loglog(x, f(x), '--', label='delta=0.5') f.delta = 0.3 plt.loglog(x, f(x), '-.', label='delta=0.3') f.delta = 0.1 plt.loglog(x, f(x), label='delta=0.1') plt.axis([x.min(), x.max(), 0.1, 1.1]) plt.legend(loc='lower center') plt.grid(True) plt.show() rrrCT)rminrrrDrrErFMbP?zSmoothness Parameter)rrWrcTtj|dkrtddS)Nrzamplitude parameter must be > 0r$anyr r0values r_amplitude_validatorz-SmoothlyBrokenPowerLaw1D._amplitude_validators4 6%1*   I%&GHH H I Ir!cTtj|dkrtddS)NrZz delta parameter must be >= 0.001r\r^s r_delta_validatorz)SmoothlyBrokenPowerLaw1D._delta_validators4 6%%-  J%&HII I J Jr!c||z }tj|d}t|tr|j}|j}nd}tj||z } d} | | k} | r||| | zzd||z |zzz || <| | k} | r||| | zzd||z |zzz || <tj| | k} | r@tj | | } d| zdz } ||| | zz| ||z |zzz|| <|rt||ddS|S) z9One dimensional smoothly broken power law model function.F)subokN@?T)unitcopyrd) r$ zeros_like isinstancerrhr_r%maxabsexp)rrrIrJrKdeltarf return_unitlogt thresholditrs rr z!SmoothlyBrokenPowerLaw1D.evaluate s[ M"E * * * i * * #.K!IIK vbzzE! 9  5577 BqEwh//3GgG3IaLQ4E6BF1II#56IaLQ4BF2a5MM>EBF1II4E#EFIaL17W#4!5q !ABGAJ I:  5577 CBqEwh//3GgG3IaLQ4BF2a5MM>EBF1II4E#EFIaLQ4%<"&))3IaL17W#4!5q !ABGAJ F4LLI % 5577 tAwAqCAr!u'22QGgEBF1II4E#EFIaLQ4E6BF1II#56IaL!W$&6!99qC!G}u4rvbe}}DDF AJ Y 9gFFr!cP|jjdS|jd|jjiSr+rQr/s rr1z$SmoothlyBrokenPowerLaw1D.input_unitsrRr!cP||jd||jddSrTr5r7s rr:z8SmoothlyBrokenPowerLaw1D._parameter_units_for_data_unitsrUr!N)r<r=r>r?r rrIrJrKror` _validatorrbr@r r)rAr1r:rr!rrrs5ZZx q&AtIi}===Gi0TUUUGi/RSSSG IaV9O P P PEIII0IJJJ(E44\4l<G<G\<G|99X9      r!rceZdZdZeddZeddZeddZeddZe dZ e d Z e d Z d Zd S) ra One dimensional power law model with an exponential cutoff. Parameters ---------- amplitude : float Model amplitude x_0 : float Reference point alpha : float Power law index x_cutoff : float Cutoff point See Also -------- PowerLaw1D, BrokenPowerLaw1D, LogParabola1D Notes ----- Model formula (with :math:`A` for ``amplitude`` and :math:`\alpha` for ``alpha``): .. math:: f(x) = A (x / x_0) ^ {-\alpha} \exp (-x / x_{cutoff}) rPeak value of modelrrrz Cutoff pointcP||z }||| zztj| |z zS)z)BFA2=,A,AAAr!c||z }||z }|| ztj| z}||z|z|z }| |ztj|z} ||z|z|dzz } ||| | gS)ze One dimensional exponential cutoff power law derivative with respect to parameters. rY)r$rnr%) rrrrrrxcr&r'r( d_x_cutoffs rr)z%ExponentialCutoffPowerLaw1D.fit_derivs W \eVnrvrc{{2  !K/#5*{*RVBZZ7][08Q;> UGZ88r!cP|jjdS|jd|jjiSr+r,r/s rr1z'ExponentialCutoffPowerLaw1D.input_unitsr2r!ct||jd||jd||jddS)Nr)rrrr5r7s rr:z;ExponentialCutoffPowerLaw1D._parameter_units_for_data_unitss;t{1~.#DKN3%dl1o6   r!N)r<r=r>r?r rrrrr@r r)rAr1r:rr!rrrs4 !1FGGGI )A+< = = =C Ia-> ? ? ?Ey???HBB\B  9 9\ 955X5      r!rceZdZdZeddZeddZeddZeddZe d Z e d Z e d Z d Zd S)ra One dimensional log parabola model (sometimes called curved power law). Parameters ---------- amplitude : float Model amplitude x_0 : float Reference point alpha : float Power law index beta : float Power law curvature See Also -------- PowerLaw1D, BrokenPowerLaw1D, ExponentialCutoffPowerLaw1D Notes ----- Model formula (with :math:`A` for ``amplitude`` and :math:`\alpha` for ``alpha`` and :math:`\beta` for ``beta``): .. math:: f(x) = A \left( \frac{x}{x_{0}}\right)^{- \alpha - \beta \log{\left (\frac{x}{x_{0}} \right )}} rr}rrrrzPower law curvaturecR||z }| |tj|zz }|||zzS)z,One dimensional log parabola model function.r#)rrrrbetarexponents rr zLogParabola1D.evaluates5W6D26"::--2x<''r!c||z }tj|}| ||zz }||z}| |z|dzz} ||z||z|z ||z z z} | |z|z} || | | gS)zCOne dimensional log parabola derivative with respect to parameters.rYr#) rrrrrrlog_xxrr&d_betar'r(s rr)zLogParabola1D.fit_derivsW6D6M)(l k)FAI5K'4&=3+>C+OP*{*V3UGV44r!cP|jjdS|jd|jjiSr+r,r/s rr1zLogParabola1D.input_units r2r!cP||jd||jddSr4r5r7s rr:z-LogParabola1D._parameter_units_for_data_unitsr;r!N)r<r=r>r?r rrrrr@r r)rAr1r:rr!rrrs: !1FGGGI )A+< = = =C Ia-> ? ? ?E 9Q,A B B BD((\(  5 5\ 555X5      r!rceZdZdZeddZedddZed d Zed Z d Z d Z e dZ dZdS)ra@ Schechter luminosity function (`Schechter 1976 `_), parameterized in terms of magnitudes. Parameters ---------- phi_star : float The normalization factor in units of number density. m_star : float The characteristic magnitude where the power-law form of the function cuts off. alpha : float The power law index, also known as the faint-end slope. Must not have units. See Also -------- PowerLaw1D, ExponentialCutoffPowerLaw1D, BrokenPowerLaw1D Notes ----- Model formula (with :math:`\phi^{*}` for ``phi_star``, :math:`M^{*}` for ``m_star``, and :math:`\alpha` for ``alpha``): .. math:: n(M) \ dM = (0.4 \ln 10) \ \phi^{*} \ [{10^{0.4 (M^{*} - M)}}]^{\alpha + 1} \ \exp{[-10^{0.4 (M^{*} - M)}]} \ dM ``phi_star`` is the normalization factor in units of number density. ``m_star`` is the characteristic magnitude where the power-law form of the function cuts off into the exponential form. ``alpha`` is the power-law index, defining the faint-end slope of the luminosity function. Examples -------- .. plot:: :include-source: from astropy.modeling.models import Schechter1D import astropy.units as u import matplotlib.pyplot as plt import numpy as np phi_star = 4.3e-4 * (u.Mpc ** -3) m_star = -20.26 alpha = -1.98 model = Schechter1D(phi_star, m_star, alpha) mag = np.linspace(-25, -17) fig, ax = plt.subplots() ax.plot(mag, model(mag)) ax.set_yscale('log') ax.set_xlim(-22.6, -17) ax.set_ylim(1.e-7, 1.e-2) ax.set_xlabel('$M_{UV}$') ax.set_ylabel('$\phi$ [mag$^{-1}$ Mpc$^{-3}]$') References ---------- .. [1] Schechter 1976; ApJ 203, 297 (https://ui.adsabs.harvard.edu/abs/1976ApJ...203..297S/abstract) .. [2] `Luminosity function `_ rgz/Normalization factor in units of number densityrg4zCharacteristic magnitudeT)rrrgzFaint-end slopec||z }t|trT|jtkr5t |jt}|tStddd|zzS)N)rhz5The units of magnitude and m_star must be a magnitude gٿ) rkrrhrrr_torr) magnitudem_star factor_exps r_factorzSchechter1D._factordsz' j( + + -#%%&z'7cBBB !}}%;<<< K$+, ,r!c|||}dtjdz|z||dzzztj| zS)z-Schechter luminosity function model function.皙?rrrr$r%rn)r0rphi_starrrfactors rr zSchechter1D.evaluatetsJc6**RVBZZ(*V -BBRVVG__TTr!cX|||}dtjdz||dzzztj| z}||z}|dzdztjdz|zdtjdz|z|zz }|tj|z} ||| gS)z^ Schechter luminosity function derivative with respect to parameters. rrrr) r0rrrrr d_phi_starfuncd_m_starr(s rr)zSchechter1D.fit_derivzs c6**26"::%519(==wO *$AI$rvbzz1D8 "&** t #f , 'Hg..r!cP|jjdS|jd|jjiSr+)rr-r.r/s rr1zSchechter1D.input_unitss( ; ! )4 A 677r!cP||jd||jddS)Nr)rrr5r7s rr:z+Schechter1D._parameter_units_for_data_unitss-!$+a.1$T\!_5   r!N)r<r=r>r?r rrrr@rr r)rAr1r:rr!rrrsEENy!RHYu2LRV W W WF Id0A B B BE - -\ -UUU /// 88X8      r!r)r?numpyr$ astropy.unitsrrrrrcorer parametersr r __all__r r rrrrrr!rrsVVVVVVVVVVVVVV!!!!!!66666666   8 8 8 8 8 8 8 8 vF F F F F F F F Rq q q q q q q q h@ @ @ @ @ /@ @ @ FA A A A A OA A A H~ ~ ~ ~ ~ /~ ~ ~ ~ ~ r!