Package dev.nm.stat.evt.evd.bivariate

Class BivariateEVDColesTawn

  • All Implemented Interfaces:
    BivariateProbabilityDistribution, MultivariateProbabilityDistribution, BivariateEVD, RandomVectorGenerator, Seedable
    public class BivariateEVDColesTawn
extends AbstractBivariateEVD
    The Coles-Tawn model. Define \[ y_i = y_i(z_i) = \left[1+\frac{\xi_i(z_i-\mu_i)}{\sigma_i}\right]^{-1/\xi_i} \] for \(1+\xi_i(z_i-\mu_i)/\sigma_i > 0\) and \(i=1,2\) where the marginal univariate GEV parameters are given by \((\mu_i,\sigma_i,\xi_i)\), and \(G_i(z_i) = \exp(-y_i)\). The distribution function of the Coles-Tawn model is \[ G(z_1,z_2) = \exp\left\{-y_1 [1 - B(q;\alpha+1,\beta)] - y_2 B(q;\alpha,\beta+1)\right\} \] where \(\alpha,\beta > 0\) are the dependence parameters, \(q=\alpha y_2/(\alpha y_2+\beta y_1)\) and \(B(q;\alpha,\beta)\) is the beta distribution function evaluated at \(q\) with shape parameters \(\alpha\) and \(\beta\). Complete dependence is obtained in the limit as \(\alpha=\beta\) tends to infinity. Independence is obtained as \(\alpha=\beta\) approaches zero, and when one of \(\alpha\), \(\beta\) is fixed and the other approaches zero. Different limits occur when one of \(\alpha\), \(\beta\) is fixed and the other tends to infinity.

    The R equivalent functions are evd::dbvct, evd::pbvct, evd::rbvct, evd::hbvct, evd::abvct, evd::ccbvevd.

    • Constructor Detail

      • BivariateEVDColesTawn

        public BivariateEVDColesTawn​(double alpha,
                             double beta)

      • BivariateEVDColesTawn

        public BivariateEVDColesTawn​(double alpha,
                             double beta,
                             GeneralizedEVD marginal)

    • Method Detail

      • density

        public double density​(double x1,
                      double x2)
        Description copied from interface: BivariateProbabilityDistribution
        The joint distribution density \(f_{X_1,X_2}(x_1,x_2)\).
        Parameters:
        x1 - the value drawn from \(X_1\)
        x2 - the value drawn from \(X_2\)
        Returns:
        the joint density of \(X_1\) and \(X_2\)

      • cdf

        public double cdf​(double x1,
                  double x2)
        Description copied from interface: BivariateProbabilityDistribution
        The joint distribution function \(F_{X_1,X_2}(x_1,x_2) = Pr(X_1 \le x_1, X_2 \le x_2)\).
        Parameters:
        x1 - the value drawn from \(X_1\)
        x2 - the value drawn from \(X_2\)
        Returns:
        the joint distribution of \(X_1\) and \(X_2\)

      • spectralDensity

        public double spectralDensity​(double x)
        Description copied from interface: BivariateEVD
        The density \(h\) of the spectral measure \(H\) on the interval (0,1). Any bivariate extreme value distribution can be written as \[ G(z_1,z_2) = \exp\left\{-\int_0^1 \max(w y_1, (1-w) y_2) H(dw)\right\} \] where \(y_i=(1+\xi_i(z_i-\mu_i)/\sigma_i)^{(-1/\xi_i)}\), and \(\mu_i\), \(\sigma_i\), \(\xi_i\) are the location, scale and shape parameters.

        For some function \(H()\) defined on [0,1], satisfying \[ \int_0^1 w H(dw) = \int_0^1 (1-w) H(dw) = 1. \] \(H()\) is called the spectral measure, with density \(h\) on the interval (0,1).

        For differentiable models, \(H\) may have up to two point masses: at zero and one. Assuming that the model parameters are in the interior of the parameter space, we have the following. For the asymmetric logistic and asymmetric negative logistic models the point masses are of size \((1-t_1)\) and \((1-t_2)\) respectively. For the asymmetric mixed model they are of size \((1-\alpha-\beta)\) and \((1-\alpha-2*\beta)\) respectively. For all other models the point masses are zero.

        At independence, \(H\) has point masses of size one at both zero and one. At complete dependence [a non-differentiable model] \(H\) has a single point mass of size two at 1/2. In either case, \(h\) is zero everywhere.

        Parameters:
        x - x
        Returns:
        \(h(x)\)

      • dependence

        public double dependence​(double x)
        Description copied from interface: BivariateEVD
        The dependence function \(A\) for the parametric bivariate extreme value model. Any bivariate extreme value distribution can be written as \[ G(z_1,z_2) = \exp\left\{-(y_1+y_2)A\left[y_1/(y_1+y_2)\right]\right\} \] for some function \(A()\) defined on [0,1], where \(y_i=(1+\xi_i(z_i-\mu_i)/\sigma_i)^{(-1/\xi_i)}\), and \(\mu_i\), \(\sigma_i\), \(\xi_i\) are the location, scale and shape parameters.

        It follows that \(A(0)=A(1)=1\), and that \(A()\) is a convex function with \(\max(x,1-x) \le A(x) \le 1\) for all \(0 \le x \le 1\).

        The lower and upper limits of \(A\) are obtained under complete dependence and independence respectively. \(A()\) does not depend on the marginal parameters.

        Parameters:
        x - x
        Returns:
        \(A(x)\)

      • conditionalCopula

        public double conditionalCopula​(double x1,
                                double x2)
        Description copied from interface: BivariateEVD
        The conditional copula function conditioning on either margin. The function calculates \(P(U_1 < x_1|U_2 = x_2)\), where \((U_1,U_2)\) is a random vector with Uniform(0,1) margins and with a dependence structure given by the specified parametric model.
        Parameters:
        x1 - an observation from \(U_1\)
        x2 - an observation from \(U_2\)
        Returns:
        the conditional copula \(P(U_1 < x_1|U_2 = x_2)\)

      • nextVector

        public double[] nextVector()
        Description copied from interface: RandomVectorGenerator
        Gets the next random vector.
        Returns:
        the next random vector

      • seed

        public void seed​(long... seeds)
        Description copied from interface: Seedable
        Seed the random number/vector/scenario generator to produce repeatable experiments.
        Parameters:
        seeds - the seeds