Source code for copul.family.other.asymmetric_xi_rho_si_copula

# file: copul/families/other/biv_counterexample_band_copula.py
import sympy as sp
import numpy as np

from copul.family.core.biv_copula import BivCopula
from copul.wrapper.sympy_wrapper import SymPyFuncWrapper




class AsymmetricSICopulaWithXiEqualsRho(BivCopula):
    r"""
    Singular “band” copula defined implicitly by

    .. math::

       h_v(t) := \partial_1 C(t,v)
       \;=\; \mathbf{1}_{\{\,t < v/2\,\}}
             \;+\; v\,\mathbf{1}_{(\,v/2,\,(v+1)/2\,)}(t),

    and explicitly by

    .. math::

       C(u,v) \;=\;
       \begin{cases}
         u, & 0 \le u \le v/2,\\[0.6ex]
         \dfrac{v}{2} + v\!\left(u - \dfrac{v}{2}\right)
         \;=\; v\,u + \dfrac{v(1-v)}{2}, & v/2 < u \le (v+1)/2,\\[1.0ex]
         v, & (v+1)/2 < u \le 1.
       \end{cases}

    This copula is valid (non-symmetric) with an absolutely continuous part
    of density :math:`1` on the strip :math:`\{(u,v): 2u-1 < v < 2u\}` and a
    singular part supported on the boundary line segments
    :math:`v=2u` for :math:`u\in[0,\tfrac12]` and
    :math:`v=2u-1` for :math:`u\in[\tfrac12,1]`.

    **Basic functionals**

    - Chatterjee's :math:`\xi(C)=\tfrac12`.
    - Spearman's footrule :math:`\psi(C)=\tfrac12`.

    **Notes**

    - This class is parameter-free.
    - The returned PDF corresponds to the absolutely continuous part only.
    """

    # No parameters
    params = []
    intervals = {}

    # convenience symbols
    u, v = sp.symbols("u v", real=True, nonnegative=True)

    # -------- Symbolic CDF -------- #
    @property
    def _cdf_expr(self):
        u, v = self.u, self.v
        v_over_2 = v / 2
        vp1_over_2 = (v + 1) / 2
        middle = v_over_2 + v * (u - v_over_2)  # = v*u + v*(1-v)/2
        return sp.Piecewise(
            (u, u <= v_over_2),
            (middle, u <= vp1_over_2),
            (v, True),
        )

    # -------- Symbolic PDF (a.c. part only) -------- #
    def _pdf_expr(self):
        u, v = self.u, self.v
        # density = 1 on the open strip (2u-1, 2u), 0 elsewhere (ignoring singular lines)
        expr = sp.Piecewise(
            (1, sp.And(v > 2 * u - 1, v < 2 * u)),
            (0, True),
        )
        return SymPyFuncWrapper(expr)

    # -------- Vectorized CDF -------- #


    def cdf_vectorized(self, u, v):
        """
        Fast numpy implementation of the piecewise CDF.
        """
        u = np.asarray(u)
        v = np.asarray(v)

        a = v / 2.0
        b = (v + 1.0) / 2.0
        middle = a + v * (u - a)  # = v*u + v*(1-v)/2

        return np.select([u <= a, u <= b], [u, middle], default=v)


    # -------- Properties / metadata -------- #
    @property
    def is_absolutely_continuous(self) -> bool:
        # There is a singular component on two line segments.
        return False

    @property
    def is_symmetric(self) -> bool:
        # C(u,v) ≠ C(v,u) in general.
        return False

    # -------- Measures requested -------- #


    def chatterjees_xi(self):
        r"""Chatterjee's rank correlation :math:`\xi(C)=\tfrac12`."""
        return sp.Rational(1, 2)




    def spearmans_footrule(self):
        r"""Spearman's footrule :math:`\psi(C)=\tfrac12`."""
        return sp.Rational(1, 2)




if __name__ == "__main__":
    cop = AsymmetricSICopulaWithXiEqualsRho()
    # Quick sanity checks
    cop.plot_cdf()
    cop.plot_cond_distr_1()
    cop.plot_cond_distr_2()