Package dev.nm.stat.evt.evd.bivariate
Class BivariateEVDBilogistic
java.lang.Object
dev.nm.stat.distribution.multivariate.AbstractBivariateProbabilityDistribution
dev.nm.stat.evt.evd.bivariate.AbstractBivariateEVD
dev.nm.stat.evt.evd.bivariate.BivariateEVDBilogistic
- All Implemented Interfaces:
BivariateProbabilityDistribution
,MultivariateProbabilityDistribution
,BivariateEVD
,RandomVectorGenerator
,Seedable
The bilogistic 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 bilogistic model is
\[
G(z_1,z_2) = \exp\left\{-y_1 q^{(1-\alpha)} -y_2 (1-q)^{(1-\beta)}\right\}
\]
where \(q = q(y_1,y_2;\alpha,\beta)\) is the root of the equation
\[
(1-\alpha) y_1 (1-q)^\beta - (1-\beta) y_2 q^\alpha = 0,
\]
and \(0 < \alpha,\beta < 1\) are the dependence parameters. When \(\alpha=\beta\), the
bilogistic model is equivalent to the logistic model with dependence parameter
\(r=\alpha=\beta\). Complete dependence is obtained in the limit as \(\alpha=\beta\) approaches
zero. Independence is obtained as \(\alpha=\beta\) approaches one, and when one of \(\alpha\),
\(\beta\) is fixed and the other approaches one. Different limits occur when one of \(\alpha\),
\(\beta\) is fixed and the other approaches zero.
The R equivalent functions are
evd::dbvbilog
, evd::pbvbilog
, evd::rbvbilog
, evd::hbvbilog
,
evd::abvbilog
, evd::ccbvevd
.-
Constructor Summary
ConstructorsConstructorDescriptionBivariateEVDBilogistic
(double alpha, double beta) BivariateEVDBilogistic
(double alpha, double beta, GeneralizedEVD marginal) BivariateEVDBilogistic
(double alpha, double beta, GeneralizedEVD marginal1, GeneralizedEVD marginal2) -
Method Summary
Modifier and TypeMethodDescriptiondouble
cdf
(double x1, double x2) The joint distribution function \(F_{X_1,X_2}(x_1,x_2) = Pr(X_1 \le x_1, X_2 \le x_2)\).double
conditionalCopula
(double x1, double x2) The conditional copula function conditioning on either margin.double
density
(double x1, double x2) The joint distribution density \(f_{X_1,X_2}(x_1,x_2)\).double
dependence
(double x) The dependence function \(A\) for the parametric bivariate extreme value model.double[]
Gets the next random vector.void
seed
(long... seeds) Seed the random number/vector/scenario generator to produce repeatable experiments.double
spectralDensity
(double x) The density \(h\) of the spectral measure \(H\) on the interval (0,1).Methods inherited from class dev.nm.stat.evt.evd.bivariate.AbstractBivariateEVD
covariance, entropy, mean, mode, moment
Methods inherited from class dev.nm.stat.distribution.multivariate.AbstractBivariateProbabilityDistribution
cdf, density
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
Methods inherited from interface dev.nm.stat.distribution.multivariate.MultivariateProbabilityDistribution
cdf, density
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Constructor Details
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BivariateEVDBilogistic
public BivariateEVDBilogistic(double alpha, double beta) -
BivariateEVDBilogistic
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BivariateEVDBilogistic
public BivariateEVDBilogistic(double alpha, double beta, GeneralizedEVD marginal1, GeneralizedEVD marginal2)
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Method Details
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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 \(X_2\)
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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 \(X_2\)
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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:
- \(h(x)\)
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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:
- \(A(x)\)
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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 \(P(U_1 < x_1|U_2 = x_2)\)
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nextVector
public double[] nextVector()Description copied from interface:RandomVectorGenerator
Gets the next random vector.- Returns:
- the next random vector
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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
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