M/Ph9EpdZddlmZmZddlZddlmZddlm Z GddZ Gdd eZ dS) 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)utilsc:eZdZdZd dZd dZd dZd dZd d ZdS) 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 cX||_||_||_t||_dSN)copula marginalscop_argslenk_vars)selfr r rs h/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/statsmodels/distributions/copula/copulas.py__init__zCopulaDistribution.__init__'s* #  )nn Nc||j}| dg|jz}|j|||}t |jD]3\}}|jdd|dd|fdz zzg||R|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 Nr )nobsargs random_stateg?gA?)rrr rvs enumerater ppf)rrr marg_argsrsampleidists rrzCopulaDistribution.rvs0sZ  }H  t{*Id.:!<<!00 3 3GAt#48C919K*L$L3%.q\333F111a4LL rctj|}||j}|dg|jdz}g}t |jD]<}||j|j|d|fg||R=tj |}|j dkr| }|j ||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 Nr .r) npasarrayrshaperangerappendr cdf column_stackndimsqueezer )ryrrcdf_margrus rr(zCopulaDistribution.cdfjs2 JqMM  }H  qwr{*It{## M MA OO1DN1-1!CF)KilKKK L L L L OH % % 6Q;; A{q(+++rcVtj||||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 )rr)r#explogpdf)rr,rrs rpdfzCopulaDistribution.pdfs&0vdkk!h)kLLMMMrc tj|}||j}|tdg|jdz}d}g}t |jD]f}||j|j|d|fg||Rz }| |j|j |d|fg||Rgtj |}|j dkr| }||j||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 Nr r"g.r)r#r$rtupler%r&rr r1r'r(r)r*r+r )rr,rrlpdfr-rr.s rr1zCopulaDistribution.logpdfs2 JqMM  }H  rdQWR[011It{## M MA ,DN1%,QsAvYF1FFF FD OO1DN1-1!CF)KilKKK L L L L OH % % 6Q;; A  ""1h/// rr )rNNN)NN) __name__ __module__ __qualname____doc__rrr(r2r1r rrrrs"%%%%8888t&,&,&,&,PNNNN4******rrc~eZdZdZddZddZeddZdd Zedd Z dd Z ddZ ddZ dZ dZdS)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||_dSr )k_dim)rr?s rrzCopula.__init__ s  rrr Nct)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 >"!rcdS)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``. Nr rr.rs rr2z Copula.pdf.rcBtj|j|g|RS)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#logr2rDs rr1z Copula.logpdfCs(&vhdhq(4((()))rcdS)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``. Nr rDs rr(z Copula.cdfXrErcF|jdkrtd||||}tj|\}}||dddf|dddf|d|d||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 r=z#Can only plot 2-dimensional Copula.N)rrrrr.v)r? ValueErrorrr create_mpl_axscatter set_xlabel set_ylabel)rrrraxfigs r plot_scatterzCopula.plot_scattermsF :??BCC C >XX4lXCCF%b))R 6!!!Q$<1... c cF{r c .ddlm}|jdkrddl}|dd}d}t jt j|d|z |t j|d|z |\}}t j| | gj } | | j |j } t j| d } t j| d } tj|\} }t j| | | }| | g}|||| |d |d|d }|d|d|dd|dd|d|||}|d| | S)aPlot the PDF. Parameters ---------- ticks_nbr : int, optional Number of color isolines for the PDF. Default is 10. ax : AxesSubplot, optional If given, this subplot is used to plot in instead of a new figure being created. Returns ------- fig : Figure If `ax` is None, the created figure. Otherwise the figure to which `ax` is connected. r)pyplotr=NzPlotting 2-dimensional Copula.dg-C6?r_)numT) antialiasedvminvmaxr.rKequal)ticksp) matplotlibrVr?warningswarnr#meshgridlinspacevstackravelTr2reshaper% nanpercentilerrMcontourfrOrPset_xlimset_ylim set_aspectcolorbar set_label tight_layout)r ticks_nbrrQpltrb n_samplesepsuuvvpointsdatamin_max_rRvticks range_cbarcscbars rplot_pdfzCopula.plot_pdfs$ -,,,,, :?? OOO MM: ; ; ; R[a#gyAA[a#gyAACCBBHHJJ 3446xx!))"(33a((b))%b))RT4Y777D\ [[Rv%) 1 (m-- c c Aq Aq g||Bf|-- s  rc|||}tj|dddf|dddfdS)zKendall's tau based on simulated samples. Returns ------- tau : float Kendall's tau. )rNrr)rr kendalltau)rrrxs r tau_simulatedzCopula.tau_simulatedsI HHT H 5 5!!!Q$111a411!44rcHtj|jddkr0tjdddfdddfd}n7|jfdt D}tj|}||S)aCopula correlation parameter using Kendall's tau of sample data. Parameters ---------- data : array_like Sample data used to fit `theta` using Kendall's tau. Returns ------- corr_param : float Correlation parameter of the copula, ``theta`` in Archimedean and pearson correlation in elliptical. If k_dim > 2, then average tau is used. rr=Nrc g|]C}t|dzD]-}tjd|fd|fd.DS)r.r)r&rr).0rjkrs r z)Copula.fit_corr_param..sk>>>uQqS!}}>>*+$QsAvY#q& ::1=>>>>r) r#r$r%rrr?r&mean _arg_from_tau)rrytautausrrs @@rfit_corr_paramzCopula.fit_corr_params Jt   71:??"1QQQT7AaaadG44Q7CC A>>>>>"1XX>>>D'$--C!!#&&&rct)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. rA)rrs rrzCopula._arg_from_taus "!r)r=)rr Nr6)NrINN)rTN)rN)r7r8r9r:rrrr2r1r(rSrrrrr rrr<r<s..`""""B   ^ (*****   ^ (....`3333j 5 5 5 5'''4"""""rr<) r:abcrrnumpyr#scipyrstatsmodels.graphicsrrr<r rrrs  $#######&&&&&&AAAAAAAAHn"n"n"n"n"Sn"n"n"n"n"r