U e{j]R@sdZddlZddlmZmZmZmZmZddl m Z ddl m Z m Z ddd d d d gZGd dde ZGddde ZGdd d e ZGdd d e ZGdd d e ZGdd d e ZdS)z Power law model variants N) MagnitudeQuantity UnitsErrordimensionless_unscaledmag)Fittable1DModel)InputParameterError Parameter PowerLaw1DBrokenPowerLaw1DSmoothlyBrokenPowerLaw1DExponentialCutoffPowerLaw1D LogParabola1D Schechter1Dc@s`eZdZdZedddZedddZedddZeddZ ed d Z e d d Z d dZ dS)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 indexcCs||}||| S)z(One dimensional power law model function)x amplitudex_0alphaxxrrY/var/www/html/bwave/python/venv/lib/python3.8/site-packages/astropy/modeling/powerlaws.pyevaluate4szPowerLaw1D.evaluatecCs@||}|| }||||}| |t|}|||gS)z?One dimensional power law derivative with respect to parametersnplog)rrrrr d_amplituded_x_0d_alpharrr fit_deriv:s  zPowerLaw1D.fit_derivcCs"|jjdkrdS|jd|jjiSNrrunitinputsselfrrr input_unitsFs zPowerLaw1D.input_unitscCs||jd||jddSNr)rrr(outputsr*Z inputs_unitZ outputs_unitrrr_parameter_units_for_data_unitsLs  z*PowerLaw1D._parameter_units_for_data_unitsN)__name__ __module__ __qualname____doc__r rrr staticmethodrr$propertyr+r0rrrrr s      c@sleZdZdZedddZedddZedddZedddZe dd Z e d d Z e d d Z ddZdS)r aV One dimensional power law model with a break. Parameters ---------- amplitude : float Model amplitude at the break point. x_break : float Break point. alpha_1 : float Power law index for x < x_break. alpha_2 : float Power law index for x > x_break. See Also -------- PowerLaw1D, ExponentialCutoffPowerLaw1D, LogParabola1D Notes ----- Model formula (with :math:`A` for ``amplitude`` and :math:`\alpha_1` for ``alpha_1`` and :math:`\alpha_2` for ``alpha_2``): .. math:: f(x) = \left \{ \begin{array}{ll} A (x / x_{break}) ^ {-\alpha_1} & : x < x_{break} \\ A (x / x_{break}) ^ {-\alpha_2} & : x > x_{break} \\ \end{array} \right. rPeak value at break pointr Break point"Power law index before break point!Power law index after break pointcCs(t||k||}||}||| S)z/One dimensional broken power law model function)rwhere)rrx_breakalpha_1alpha_2rrrrrrzszBrokenPowerLaw1D.evaluatec Csxt||k||}||}|| }||||}| |t|} t||k| d} t||k| d} ||| | gS)zFOne dimensional broken power law derivative with respect to parametersr)rr;r ) rrr<r=r>rrr! d_x_breakr# d_alpha_1 d_alpha_2rrrr$s zBrokenPowerLaw1D.fit_derivcCs"|jjdkrdS|jd|jjiSr%r<r'r(r)rrrr+s zBrokenPowerLaw1D.input_unitscCs||jd||jddSNr)r<rr-r/rrrr0s  z0BrokenPowerLaw1D._parameter_units_for_data_unitsN)r1r2r3r4r rr<r=r>r5rr$r6r+r0rrrrr Ss!       c@seZdZdZedddddZedddZed d dZed d dZedd ddZ ej ddZe j ddZ e ddZ e ddZ eddZddZdS)r a+ 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() rrr7T)rminrrr8rr9r:MbP?zSmoothness Parameter)rrDrcCst|dkrtddS)Nrzamplitude parameter must be > 0ranyr r*valuerrrrsz"SmoothlyBrokenPowerLaw1D.amplitudecCst|dkrtddS)NrGz delta parameter must be >= 0.001rHrJrrrdeltaszSmoothlyBrokenPowerLaw1D.deltacCs$||}tj|dd}t|tr.|j}|j}nd}t||} d} | | k} | rz||| | d||||| <| | k} | r||| | d||||| <t| | k} | r t | | } d| d} ||| | | ||||| <|r t||dddS|S) z8One dimensional smoothly broken power law model functionF)subokN@?T)r'copyrM) r zeros_like isinstancerr'rKr maxabsexp)rrr<r=r>rLrfZ return_unitlogt thresholditrrrrr s0      &z!SmoothlyBrokenPowerLaw1D.evaluatecCs||}t||}t|}t|} t|} t|} t|} t|} d}||k}|r|||| d|||||<|||| |<||||| |<||| td| |<||t|| |td| |<|||| td| |<|| k}|r|||| d|||||<|||| |<||||| |<||t|| |td| |<|||td| |<|||| td| |<t||k}|rt||}d|d}|||| ||||||<|||| |<||||||d||| |<||t|| |t|| |<||| t|| |<||||t||d||t||| |<| | | | | gS)zWOne dimensional smoothly broken power law derivative with respect to parametersrNrOrFrP)rr rRrTrUrV)rrr<r=r>rLrrXrWr!r?r@rAZd_deltarYrZr[r\rrrr$EsX        *    *   &"*&z"SmoothlyBrokenPowerLaw1D.fit_derivcCs"|jjdkrdS|jd|jjiSr%rBr)rrrr+s z$SmoothlyBrokenPowerLaw1D.input_unitscCs||jd||jddSrCr-r/rrrr0s  z8SmoothlyBrokenPowerLaw1D._parameter_units_for_data_unitsN)r1r2r3r4r rr<r=r>rL validatorr5rr$r6r+r0rrrrr s,\      7 > c@sleZdZdZedddZedddZedddZedddZe dd Z e d d Z e d d Z ddZdS)ra One dimensional power law model with an exponential cutoff. Parameters ---------- amplitude : float Model amplitude x_0 : float Reference point alpha : float Power law index x_cutoff : float Cutoff point See Also -------- PowerLaw1D, BrokenPowerLaw1D, LogParabola1D Notes ----- Model formula (with :math:`A` for ``amplitude`` and :math:`\alpha` for ``alpha``): .. math:: f(x) = A (x / x_0) ^ {-\alpha} \exp (-x / x_{cutoff}) rPeak value of modelrrrz Cutoff pointcCs&||}||| t| |S)z;One dimensional exponential cutoff power law model function)rrV)rrrrx_cutoffrrrrrsz$ExponentialCutoffPowerLaw1D.evaluatec Csj||}||}|| t| }||||}| |t|} ||||d} ||| | gS)zd One dimensional exponential cutoff power law derivative with respect to parameters rF)rrVr ) rrrrr_rZxcr!r"r#Z d_x_cutoffrrrr$sz%ExponentialCutoffPowerLaw1D.fit_derivcCs"|jjdkrdS|jd|jjiSr%r&r)rrrr+s z'ExponentialCutoffPowerLaw1D.input_unitscCs*||jd||jd||jddS)Nr)rr_rr-r/rrrr0s   z;ExponentialCutoffPowerLaw1D._parameter_units_for_data_unitsN)r1r2r3r4r rrrr_r5rr$r6r+r0rrrrrs       c@sleZdZdZedddZedddZedddZedddZe d d Z e d d Z e d dZ ddZdS)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 curvaturecCs(||}| |t|}|||S)z+One dimensional log parabola model functionr)rrrrbetarexponentrrrrszLogParabola1D.evaluatec Csp||}t|}| ||}||}| ||d} |||||||} | ||} || | | gS)zBOne dimensional log parabola derivative with respect to parametersrFr) rrrrr`rZlog_xxrar!Zd_betar"r#rrrr$s zLogParabola1D.fit_derivcCs"|jjdkrdS|jd|jjiSr%r&r)rrrr+s zLogParabola1D.input_unitscCs||jd||jddSr,r-r/rrrr0s  z-LogParabola1D._parameter_units_for_data_unitsN)r1r2r3r4r rrrr`r5rr$r6r+r0rrrrrs       c@sfeZdZdZedddZeddddZed d dZed d Z d dZ ddZ e ddZ ddZdS)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 `_ rPz/Normalization factor in units of number densityrg4zCharacteristic magnitudeT)rrrgzFaint-end slopecCsN||}t|tr>|jtkr4t|jtd}|tStdn dd|SdS)N)r'z5The units of magnitude and m_star must be a magnitude gٿ) rSrr'rrrKtorr)Z magnitudem_starZ factor_exprrr_factorjs   zSchechter1D._factorcCs6|||}dtd|||dt| S)z-Schechter luminosity function model function.皙?rbrrerr rV)r*rphi_starrdrfactorrrrrzs zSchechter1D.evaluatec Cs|||}dtd||dt| }||}|ddtd|dtd||}|t|} ||| gS)z^ Schechter luminosity function derivative with respect to parameters. rfrbrrg) r*rrhrdrriZ d_phi_starfuncZd_m_starr#rrrr$s &zSchechter1D.fit_derivcCs"|jjdkrdS|jd|jjiSr%)rdr'r(r)rrrr+s zSchechter1D.input_unitscCs||jd||jddS)Nr)rdrhr-r/rrrr0s  z+Schechter1D._parameter_units_for_data_unitsN)r1r2r3r4r rhrdrr5rerr$r6r+r0rrrrrsG   )r4numpyrZ astropy.unitsrrrrrcorer parametersr r __all__r r r rrrrrrrs$  <KtEF