Dj@GbdZddlmZmZddlZddlmZddlm Z GddZ Gdd eZ y) z Which Archimedean is Best? Extreme Value copulas formulas are based on Genest 2009 References ---------- Genest, C., 2009. Rank-based inference for bivariate extreme-value copulas. The Annals of Statistics, 37(5), pp.2990-3022. )ABCabstractmethodN)stats)utilsc8eZdZdZddZd dZd dZd dZd dZy) CopulaDistributionaMultivariate copula distribution Parameters ---------- copula : :class:`Copula` instance An instance of :class:`Copula`, e.g. :class:`GaussianCopula`, :class:`FrankCopula`, etc. marginals : list of distribution instances Marginal distributions. copargs : tuple Parameters for copula Notes ----- Status: experimental, argument handling may still change cN||_||_||_t||_yN)copula marginalscop_argslenk_vars)selfr r r s lC:\Crop_Prediction\Backend\crop-ai-system\venv\Lib\site-packages\statsmodels/distributions/copula/copulas.py__init__zCopulaDistribution.__init__'s$ #  )n Nc| |j}|dg|jz}|jj|||}t |j D]2\}}|j dd|dd|fdz zzg|||dd|f<4|S)a6Draw `n` in the half-open interval ``[0, 1)``. Sample the joint distribution. Parameters ---------- nobs : int, optional Number of samples to generate in the parameter space. Default is 1. cop_args : tuple Copula parameters. If None, then the copula parameters will be taken from the ``cop_args`` attribute created when initiializing the instance. marg_args : list of tuples Parameters for the marginal distributions. It can be None if none of the marginal distributions have parameters, otherwise it needs to be a list of tuples with the same length has the number of marginal distributions. The list can contain empty tuples for marginal distributions that do not take parameter arguments. random_state : {None, int, numpy.random.Generator}, optional If `seed` is None then the legacy singleton NumPy generator. This will change after 0.13 to use a fresh NumPy ``Generator``, so you should explicitly pass a seeded ``Generator`` if you need reproducible results. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` instance then that instance is used. Returns ------- sample : array_like (n, d) Sample from the joint distribution. Notes ----- The random samples are generated by creating a sample with uniform margins from the copula, and using ``ppf`` to convert uniform margins to the one specified by the marginal distribution. See Also -------- statsmodels.tools.rng_qrng.check_random_state N)nobsargs random_stateg?gA?)r rr rvs enumerater ppf)rrr marg_argsrsampleidists rrzCopulaDistribution.rvs0sZ  }}H  t{{*Id.:!<!0 3GAt#488C919K*L$L3%.q\3F1a4L 3 rctj|}| |j}|dg|jdz}g}t |j D]9}|j |j|j|d|fg||;tj|}|jdk(r|j}|jj||S)aCDF of copula distribution. Parameters ---------- y : array_like Values of random variable at which to evaluate cdf. If 2-dimensional, then components of multivariate random variable need to be in columns cop_args : tuple Copula parameters. If None, then the copula parameters will be taken from the ``cop_args`` attribute created when initiializing the instance. marg_args : list of tuples Parameters for the marginal distributions. It can be None if none of the marginal distributions have parameters, otherwise it needs to be a list of tuples with the same length has the number of marginal distributions. The list can contain empty tuples for marginal distributions that do not take parameter arguments. Returns ------- cdf values r.) npasarrayr shaperangerappendr cdf column_stackndimsqueezer )ryr rcdf_margrus rr(zCopulaDistribution.cdfjs2 JJqM  }}H  qwwr{*It{{# MA OO1DNN1-11!CF)KilK L M OOH % 66Q; A{{q(++rcPtj|j|||S)aPDF of copula distribution. Parameters ---------- y : array_like Values of random variable at which to evaluate cdf. If 2-dimensional, then components of multivariate random variable need to be in columns cop_args : tuple Copula parameters. If None, then the copula parameters will be taken from the ``cop_args`` attribute created when initiializing the instance. marg_args : list of tuples Parameters for the marginal distributions. It can be None if none of the marginal distributions have parameters, otherwise it needs to be a list of tuples with the same length has the number of marginal distributions. The list can contain empty tuples for marginal distributions that do not take parameter arguments. Returns ------- pdf values )r r)r#explogpdf)rr,r rs rpdfzCopulaDistribution.pdfs"0vvdkk!h)kLMMrc.tj|}| |j}|tdg|jdz}d}g}t |j D]d}||j|j|d|fg||z }|j|j|j|d|fg||ftj|}|jdk(r|j}||jj||z }|S)aLog-pdf of copula distribution. Parameters ---------- y : array_like Values of random variable at which to evaluate cdf. If 2-dimensional, then components of multivariate random variable need to be in columns cop_args : tuple Copula parameters. If None, then the copula parameters will be taken from the ``cop_args`` attribute creating when initiializing the instance. marg_args : list of tuples Parameters for the marginal distributions. It can be None if none of the marginal distributions have parameters, otherwise it needs to be a list of tuples with the same length has the number of marginal distributions. The list can contain empty tuples for marginal distributions that do not take parameter arguments. Returns ------- log-pdf values rr!g.r")r#r$r tupler%r&rr r1r'r(r)r*r+r )rr,r rlpdfr-rr.s rr1zCopulaDistribution.logpdfs2 JJqM  }}H  rdQWWR[01It{{# MA ,DNN1%,,QsAvYF1F FD OO1DNN1-11!CF)KilK L M OOH % 66Q; A  ""1h// rr)r"NNN)NN) __name__ __module__ __qualname____doc__rrr(r2r1rrrrrs#"%8t&,PN4*rrcpeZdZdZd dZddZeddZddZeddZ ddZ dd Z dd Z d Z d Zy)Copulau_A generic Copula class meant for subclassing. Notes ----- A function :math:`\phi` on :math:`[0, \infty]` is the Laplace-Stieltjes transform of a distribution function if and only if :math:`\phi` is completely monotone and :math:`\phi(0) = 1` [2]_. The following algorithm for sampling a ``d``-dimensional exchangeable Archimedean copula with generator :math:`\phi` is due to Marshall, Olkin (1988) [1]_, where :math:`LS^{−1}(\phi)` denotes the inverse Laplace-Stieltjes transform of :math:`\phi`. From a mixture representation with respect to :math:`F`, the following algorithm may be derived for sampling Archimedean copulas, see [1]_. 1. Sample :math:`V \sim F = LS^{−1}(\phi)`. 2. Sample i.i.d. :math:`X_i \sim U[0,1], i \in \{1,...,d\}`. 3. Return:math:`(U_1,..., U_d)`, where :math:`U_i = \phi(−\log(X_i)/V), i \in \{1, ...,d\}`. Detailed properties of each copula can be found in [3]_. Instances of the class can access the attributes: ``rng`` for the random number generator (used for the ``seed``). **Subclassing** When subclassing `Copula` to create a new copula, ``__init__`` and ``random`` must be redefined. * ``__init__(theta)``: If the copula does not take advantage of a ``theta``, this parameter can be omitted. * ``random(n, random_state)``: draw ``n`` from the copula. * ``pdf(x)``: PDF from the copula. * ``cdf(x)``: CDF from the copula. References ---------- .. [1] Marshall AW, Olkin I. “Families of Multivariate Distributions”, Journal of the American Statistical Association, 83, 834–841, 1988. .. [2] Marius Hofert. "Sampling Archimedean copulas", Universität Ulm, 2008. .. rvs[3] Harry Joe. "Dependence Modeling with Copulas", Monographs on Statistics and Applied Probability 134, 2015. c||_yr )k_dim)rr>s rrzCopula.__init__ s  rNct)aEDraw `n` in the half-open interval ``[0, 1)``. Marginals are uniformly distributed. Parameters ---------- nobs : int, optional Number of samples to generate from the copula. Default is 1. args : tuple Arguments for copula parameters. The number of arguments depends on the copula. random_state : {None, int, numpy.random.Generator}, optional If `seed` is None then the legacy singleton NumPy generator. This will change after 0.13 to use a fresh NumPy ``Generator``, so you should explicitly pass a seeded ``Generator`` if you need reproducible results. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` instance then that instance is used. Returns ------- sample : array_like (nobs, d) Sample from the copula. See Also -------- statsmodels.tools.rng_qrng.check_random_state NotImplementedError)rrrrs rrz Copula.rvs s >"!rcy)a[Probability density function of copula. Parameters ---------- u : array_like, 2-D Points of random variables in unit hypercube at which method is evaluated. The second (or last) dimension should be the same as the dimension of the random variable, e.g. 2 for bivariate copula. args : tuple Arguments for copula parameters. The number of arguments depends on the copula. Returns ------- pdf : ndarray, (nobs, k_dim) Copula pdf evaluated at points ``u``. Nrrr.rs rr2z Copula.pdf.rcNtj|j|g|S)aYLog of copula pdf, loglikelihood. Parameters ---------- u : array_like, 2-D Points of random variables in unit hypercube at which method is evaluated. The second (or last) dimension should be the same as the dimension of the random variable, e.g. 2 for bivariate copula. args : tuple Arguments for copula parameters. The number of arguments depends on the copula. Returns ------- cdf : ndarray, (nobs, k_dim) Copula log-pdf evaluated at points ``u``. )r#logr2rCs rr1z Copula.logpdfCs#&vvhdhhq(4())rcy)akCumulative distribution function evaluated at points u. Parameters ---------- u : array_like, 2-D Points of random variables in unit hypercube at which method is evaluated. The second (or last) dimension should be the same as the dimension of the random variable, e.g. 2 for bivariate copula. args : tuple Arguments for copula parameters. The number of arguments depends on the copula. Returns ------- cdf : ndarray, (nobs, k_dim) Copula cdf evaluated at points ``u``. NrrCs rr(z Copula.cdfXrDrc|jdk7r td||j||}tj|\}}|j |dddf|dddf|j d|jd||fS) aSample the copula and plot. Parameters ---------- sample : array-like, optional The sample to plot. If not provided (the default), a sample is generated. nobs : int, optional Number of samples to generate from the copula. random_state : {None, int, numpy.random.Generator}, optional If `seed` is None then the legacy singleton NumPy generator. This will change after 0.13 to use a fresh NumPy ``Generator``, so you should explicitly pass a seeded ``Generator`` if you need reproducible results. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` instance then that instance is used. ax : AxesSubplot, optional If given, this subplot is used to plot in instead of a new figure being created. Returns ------- fig : Figure If `ax` is None, the created figure. Otherwise the figure to which `ax` is connected. sample : array_like (n, d) Sample from the copula. See Also -------- statsmodels.tools.rng_qrng.check_random_state z#Can only plot 2-dimensional Copula.N)rrrr"r.v)r> ValueErrorrr create_mpl_axscatter set_xlabel set_ylabel)rrrraxfigs r plot_scatterzCopula.plot_scattermsF ::?BC C >XX4lXCF%%b)R 6!Q$<1. c cF{rc ddlm}|jdk7rddl}|j dd}d}t j t j|d|z |t j|d|z |\}}t j|j|jgj} |j| jj|j} t j| d } t j| d } tj |\} }t j| | | }| | g}|j#||| |d |d|d }|j%d|j'd|j)dd|j+dd|j-d|j/||}|j1d| j3| S)aPlot the PDF. Parameters ---------- ticks_nbr : int, optional Number of color isolines for the PDF. Default is 10. ax : AxesSubplot, optional If given, this subplot is used to plot in instead of a new figure being created. Returns ------- fig : Figure If `ax` is None, the created figure. Otherwise the figure to which `ax` is connected. r)pyplotrINzPlotting 2-dimensional Copula.dg-C6?r"_)numT) antialiasedvminvmaxr.rJequal)ticksp) matplotlibrTr>warningswarnr#meshgridlinspacevstackravelTr2reshaper% nanpercentilerrLcontourfrNrOset_xlimset_ylim set_aspectcolorbar set_label tight_layout)r ticks_nbrrPpltr` n_samplesepsuuvvpointsdatamin_max_rQvticks range_cbarcscbars rplot_pdfzCopula.plot_pdfs$ - ::?  MM: ; R[[a#gyA[[a#gyACBBHHJ 3466xx!!))"((3a(b)%%b)RT4Y7D\ [[Rv%) 1 (m- c c Aq Aq g||Bf|- s  rcv|j||}tj|dddf|dddfdS)zKendall's tau based on simulated samples. Returns ------- tau : float Kendall's tau. )rNrr")rr kendalltau)rrrxs r tau_simulatedzCopula.tau_simulateds> HHT H 5!Q$1a41!44rc tj|}|jddk(r(tj|dddf|dddfd}np|j }t |Dcgc]9}t |dz|D]%}tj|d|f|d|fd';}}}tj|}|j|Scc}}w)aCopula correlation parameter using Kendall's tau of sample data. Parameters ---------- data : array_like Sample data used to fit `theta` using Kendall's tau. Returns ------- corr_param : float Correlation parameter of the copula, ``theta`` in Archimedean and pearson correlation in elliptical. If k_dim > 2, then average tau is used. r"rINr.) r#r$r%rrr>r&mean _arg_from_tau)rrwrtaukrjtauss rfit_corr_paramzCopula.fit_corr_params JJt  771:?""1QT7AadG4Q7C A"1X>uQqS!}>*+$$QsAvY#q& :1=>=>D>''$-C!!#&&>s*>Cct)a@Compute correlation parameter from tau. Parameters ---------- tau : float Kendall's tau. Returns ------- corr_param : float Correlation parameter of the copula, ``theta`` in Archimedean and pearson correlation in elliptical. r@)rrs rrzCopula._arg_from_taus "!r)rI)r"rNr6)NiNN) N)iN)r7r8r9r:rrrr2r1r(rRr~rrrrrrr<r<s[.`"B  (**  (.`3j 5'4"rr<) r:abcrrnumpyr#scipyrstatsmodels.graphicsrrr<rrrrs3 $&AAHn"Sn"r