ó šßÈ[c@`sdZddlmZmZmZmZddlmZddlZ ddl m Z ddl m Z mZdd lmZd d d d dgZde fd„ƒYZde fd„ƒYZde fd„ƒYZde fd„ƒYZde fd„ƒYZdS(u Power law model variants i(tabsolute_importtunicode_literalstdivisiontprint_function(t OrderedDictNi(tFittable1DModel(t ParametertInputParameterErrori(tQuantityu PowerLaw1DuBrokenPowerLaw1DuSmoothlyBrokenPowerLaw1DuExponentialCutoffPowerLaw1Du LogParabola1Dt PowerLaw1DcB`sqeZdZeddƒZeddƒZeddƒZed„ƒZed„ƒZ e d„ƒZ d„Z RS(uÛ 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} tdefaulticC`s||}||| S(u(One dimensional power law model function((txt amplitudetx_0talphatxx((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pytevaluate3s cC`sL||}|| }||||}| |tj|ƒ}|||gS(u?One dimensional power law derivative with respect to parameters(tnptlog(R R R RRt d_amplitudetd_x_0td_alpha((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyt fit_deriv9s   cC`s+|jjdkrdSi|jjd6SdS(Nux(R tunittNone(tself((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyt input_unitsEscC`s$td|dfd|dfgƒS(Nux_0uxu amplitudeuy(R(Rt inputs_unitt outputs_unit((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyt_parameter_units_for_data_unitsLs( t__name__t __module__t__doc__RR R Rt staticmethodRRtpropertyRR(((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyR s tBrokenPowerLaw1DcB`s€eZdZeddƒZeddƒZeddƒZeddƒZed„ƒZ ed„ƒZ e d„ƒZ d„Z RS(uV One dimensional power law model with a break. Parameters ---------- amplitude : float Model amplitude at the break point. x_break : float Break point. alpha_1 : float Power law index for x < x_break. alpha_2 : float Power law index for x > x_break. See Also -------- PowerLaw1D, ExponentialCutoffPowerLaw1D, LogParabola1D Notes ----- Model formula (with :math:`A` for ``amplitude`` and :math:`\alpha_1` for ``alpha_1`` and :math:`\alpha_2` for ``alpha_2``): .. math:: f(x) = \left \{ \begin{array}{ll} A (x / x_{break}) ^ {-\alpha_1} & : x < x_{break} \\ A (x / x_{break}) ^ {-\alpha_2} & : x > x_{break} \\ \end{array} \right. R icC`s2tj||k||ƒ}||}||| S(u/One dimensional broken power law model function(Rtwhere(R R tx_breaktalpha_1talpha_2RR((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyRxs c C`s tj||k||ƒ}||}|| }||||}| |tj|ƒ} tj||k| dƒ} tj||k| dƒ} ||| | gS(uFOne dimensional broken power law derivative with respect to parametersi(RR$R( R R R%R&R'RRRt d_x_breakRt d_alpha_1t d_alpha_2((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyR€s  cC`s+|jjdkrdSi|jjd6SdS(Nux(R%RR(R((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyRscC`s$td|dfd|dfgƒS(Nux_breakuxu amplitudeuy(R(RRR((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyR–s(RRR RR R%R&R'R!RRR"RR(((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyR#Qs tSmoothlyBrokenPowerLaw1DcB`s¿eZdZeddddƒZeddƒZeddƒZeddƒZeddddƒZej d„ƒZej d „ƒZe d „ƒZ e d „ƒZ e d „ƒZd „ZRS(u+ 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() R itminiiþÿÿÿigü©ñÒMbP?cC`s(tj|dkƒr$tdƒ‚ndS(Niuamplitude parameter must be > 0(RtanyR(Rtvalue((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyR þscC`s(tj|dkƒr$tdƒ‚ndS(Ngü©ñÒMbP?u delta parameter must be >= 0.001(RR-R(RR.((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pytdeltascC`sw||}tj|dtƒ}t|tƒrC|j}|j}nd}tj|ƒ|} d} | | k} | j ƒr¤||| | d||||| ((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyRBsH   ' 1'  '1' '*1 GcC`s+|jjdkrdSi|jjd6SdS(Nux(R%RR(R((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyR{scC`s$td|dfd|dfgƒS(Nux_breakuxu amplitudeuy(R(RRR((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyR‚s(RRR RR R%R&R'R/t validatorR!RRR"RR(((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyR+›s[89tExponentialCutoffPowerLaw1DcB`s€eZdZeddƒZeddƒZeddƒZeddƒZed„ƒZ ed„ƒZ e d„ƒZ d„Z RS(u 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}) R icC`s)||}||| tj| |ƒS(u;One dimensional exponential cutoff power law model function(RR7(R R R Rtx_cutoffR((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyR§s c C`s}||}||}|| tj| ƒ}||||}| |tj|ƒ} ||||d} ||| | gS(uROne dimensional exponential cutoff power law derivative with respect to parametersi(RR7R( R R R RRBRtxcRRRt d_x_cutoff((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyR®s  cC`s+|jjdkrdSi|jjd6SdS(Nux(R RR(R((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyR¼scC`s1td|dfd|dfd|dfgƒS(Nux_0uxux_cutoffu amplitudeuy(R(RRR((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyRÃs (RRR RR R RRBR!RRR"RR(((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyRA‡st LogParabola1DcB`s€eZdZeddƒZeddƒZeddƒZeddƒZed„ƒZ ed„ƒZ e d„ƒZ d„Z RS(ut 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 )}} R iicC`s.||}| |tj|ƒ}|||S(u+One dimensional log parabola model function(RR(R R R RtbetaRtexponent((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyRés c C`s‚||}tj|ƒ}| ||}||}| ||d} |||||||} | ||} || | | gS(uBOne dimensional log parabola derivative with respect to parametersi(RR( R R R RRFRtlog_xxRGRtd_betaRR((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyRñs  cC`s+|jjdkrdSi|jjd6SdS(Nux(R RR(R((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyRÿscC`s$td|dfd|dfgƒS(Nux_0uxu amplitudeuy(R(RRR((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyRs(RRR RR R RRFR!RRR"RR(((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyREÉs(R t __future__RRRRt collectionsRtnumpyRtcoreRt parametersRRtunitsRt__all__R R#R+RARE(((s9lib/python2.7/site-packages/astropy/modeling/powerlaws.pyts"   ;JìB