dens {ExtremalDep} | R Documentation |
Evaluates the angular density or calculates the likelihood function of the Pairwise Beta, Husler-Reiss, Dirichlet, Extremal-$t$, Extremal Skew-$t$ and Asymmetric Logistic models at one or more locations on the unit simplex.
dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Pairwise", par=c(2,2,2,4), c, log=FALSE, vectorial=TRUE)
x |
A (n x d) matrix of angular components, where the rows represent n independent points
in the d-dimensional unit simplex. See Details.
The default is |
model |
A string with the name of the parametric model to be estimated. Models are
Pairwise Beta ( |
par |
A vector containing the parameters of the model. See Details. |
c |
A real value in [0,1], providing the decision rule to allocate a data point to a subset of the simplex. Only required for the Extremal-t, Extremal Skew-t and Asymmetric Logistic models. |
log |
Logical; if |
vectorial |
Logical; if |
The Extremal-$t$ and Asymmetric Logistic models are available up to 3 dimensions; mass on all the subsets of the simplex is included.
For the Pairwise Beta model, the parameter vector is decomposed as:
A vector of size choose(d,2)
. Controls the dependence between
pairs. The default is b=c(2,2,2)
.
A positive real that controls the general dependence between all the variables. The default is 4.
For the Husler-Reiss model, the parameter vector is of size choose(d,2)
.
For the Dirichlet model, the parameter vector is decomposed a vector of size d which controls the dependence between pairs.
For the Extremal-$t$ model, the parameter vector is decomposed as:
A vector of size choose(d,2)
representing the corrleation parameters.
A positive integer, μ ≥q 1, representing the degree of freedom.
For the Extremal Skew-$t$ model, the parameter vector is decomposed as:
A vector of size choose(d,2)
representing the corrleation parameters.
A vector of size d
representing the shape parameters.
A positive integer, μ ≥q 1, representing the degree of freedom.
For the Asymmetric Logistic model, the parameter vector is decomposed as:
A vector of size 1 or 4 depending on whether d=2 or 3.
A vector of size 2 or 9 depending on whether d=2 or 3.
If log=TRUE
and vectorial=FALSE
then the log-likelihood function is
calculated.
Returns a n-dimensional vector if vectorial=TRUE
or a single value
if vectorial=FALSE
.
Simone Padoan, simone.padoan@unibocconi.it, http://faculty.unibocconi.it/simonepadoan; Boris Beranger, borisberanger@gmail.com
Cooley, D.,Davis, R. A., and Naveau, P. (2010). The pairwise beta distribution: a flexible parametric multivariate model for extremes. Journal of Multivariate Analysis, 101, 2103–2117.
Husler, J. and Reiss, R.-D. (1989), Maxima of normal random vectors: between independence and complete dependence, Statistics and Probability Letters, 7, 283–286.
Engelke, S., Malinowski, A., Kabluchko, Z., and Schlather, M. (2015), Estimation of Husler-Reiss distributions and Brown-Resnick processes, Journal of the Royal Statistical Society, Series B (Methodological), 77, 239–265.
Coles, S. G., and Tawn, J. A. (1991), Modelling Extreme Multivariate Events, Journal of the Royal Statistical Society, Series B (Methodological), 53, 377–392.
Nikoloulopoulos, A. K., Joe, H., and Li, H. (2009) Extreme value properties of t copulas. Extremes, 12, 129–148.
Opitz, T. (2013) Extremal t processes: Elliptical domain of attraction and a spectral representation. Jounal of Multivariate Analysis, 122, 409–413.
Beranger, B. and Padoan, S. A. (2015). Extreme dependence models, chapater of the book Extreme Value Modeling and Risk Analysis: Methods and Applications, Chapman Hall/CRC.
Beranger, B., Padoan, S. A. and Sisson, S. A. (2017). Models for extremal dependence derived from skew-symmetric families. Scandinavian Journal of Statistics, 44(1), 21-45.
Tawn, J. A. (1990), Modelling Multivariate Extreme Value Distributions, Biometrika, 77, 245–253.
### Pairwise Beta : # Examples on the 3-dimensional simplex # Returns the bivariate angular density at two locations dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Pairwise", par=c(2,2,2,4), log=FALSE, vectorial=TRUE) # returns the likelihood function at two locations dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Pairwise", par=c(2,2,2,4), log=FALSE, vectorial=FALSE) # returns the log-likelihood function dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Pairwise", par=c(2,2,2,4), log=TRUE, vectorial=FALSE) # Examples on the 4-dimensional simplex # returns the bivariate angular density at two locations dens(x=rbind(c(0.1,0.3,0.3,0.3),c(0.1,0.2,0.3,0.4)), model="Pairwise", par=c(2,2,2,1,0.5,3,4), log=FALSE, vectorial=TRUE) # returns the likelihood function at two locations dens(x=rbind(c(0.1,0.3,0.3,0.3),c(0.1,0.2,0.3,0.4)), model="Pairwise", par=c(2,2,2,1,0.5,3,4), log=FALSE, vectorial=FALSE) # returns the log-likelihood function dens(x=rbind(c(0.1,0.3,0.3,0.3),c(0.1,0.2,0.3,0.4)), model="Pairwise", par=c(2,2,2,1,0.5,3,4), log=TRUE, vectorial=FALSE) ### Husler-Reiss # Example on the 2-dimensional simplex # returns the log-likelihood at two locations dens(x=rbind(c(0.1,0.9),c(0.3,0.7)), model="Husler", par=1.7, log=TRUE, vectorial=FALSE) # Example on the 3-dimensional simplex # returns the likelihood function at two locations dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Husler", par=c(1.7,0.7,1.1), log=FALSE, vectorial=FALSE) # Example on the 4-dimensional simplex # returns the bivariate angular density at two locations dens(x=rbind(c(0.1,0.1,0.4,0.4),c(0.1,0.2,0.3,0.4)), model="Husler", par=rep(1,6), log=FALSE, vectorial=TRUE) ### Dirichlet # Example on the 2-dimensional simplex # returns the log-likelihood at two points dens(x=rbind(c(0.1,0.9),c(0.3,0.7)), model="Dirichlet", par=c(1.7,0.7), log=TRUE, vectorial=FALSE) # Example on the 3-dimensional simplex # returns the likelihood function at three locations dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Dirichlet", par=c(1.7,0.7,1.1), log=FALSE, vectorial=FALSE) # Example on the 4-dimensional simplex # returns the bivariate angular density at two locations dens(x=rbind(c(0.1,0.1,0.4,0.4),c(0.1,0.2,0.3,0.4)), model="Dirichlet", par=c(1.7,0.7,1.1,0.1), log=FALSE, vectorial=TRUE) ### Extremal-t # Example on the 2-dimensional simplex # Returns the log-likelihood dens(x=rbind(c(0.4,0.6),c(0.3,0.7)), model="Extremalt", par=c(0.7,2), c=0.1, log=TRUE, vectorial=FALSE) # Density in the corner dens(x=c(0.08,0.92), model="Extremalt", par=c(0.7,2), c=0.1, log=FALSE, vectorial=FALSE) # Example on the 3-dimensional simplex # Returns the log-likelihood dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Extremalt", par=c(rep(0.1,3),2), c=0.03, log=FALSE, vectorial=FALSE) # Returns the evalutaion of the angular density at three locations: # The first one is set to be on the edge linking the second and third components # The second one is set to be on the interior of the simplex # The third one is set to be on the corner near the third component dens(x=rbind(c(0.001,0.3,0.699),c(0.1,0.2,0.7),c(0.001,0.001,0.998)), model="Extremalt", par=c(rep(0.1,3),2), c=0.01, log=FALSE, vectorial=TRUE) ### Extremal Skew-t # Example on the 2-dimensional simplex # Returns the log-likelihood dens(x=rbind(c(0.4,0.6),c(0.3,0.7)), model="Skewt", par=c(0.7,0,0,2), c=0.1, log=TRUE, vectorial=FALSE) dens(x=rbind(c(0.4,0.6),c(0.3,0.7)), model="Skewt", par=c(0.7,2,-1,2), c=0.1, log=TRUE, vectorial=FALSE) # Density in the corner dens(x=c(0.08,0.92), model="Skewt", par=c(0.7,0,0,2), c=0.1, log=FALSE, vectorial=FALSE) dens(x=c(0.08,0.92), model="Skewt", par=c(0.7,-1,2,2), c=0.1, log=FALSE, vectorial=FALSE) # Example on the 3-dimensional simplex # Returns the log-likelihood dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Skewt", par=c(rep(0.1,3),rep(0,3),2), c=0.03, log=FALSE, vectorial=FALSE) # Returns the evalutaion of the angular density at three locations: # The first one is set to be on the edge linking the second and third components # The second one is set to be on the interior of the simplex # The third one is set to be on the corner near the third component dens(x=rbind(c(0.001,0.3,0.699),c(0.1,0.2,0.7),c(0.001,0.001,0.998)), model="Skewt", par=c(rep(0.1,3),rep(0,3),2), c=0.01, log=FALSE, vectorial=TRUE) ### Asymmetric Logistic # Example on the 3-dimensional simplex # Returns the angular density at three points: # The first one is set to be on the edge linking the second and third components # The second one is set to be on the interior of the simplex # The third one is set to be on the corner near the third component dens(x=rbind(c(0.001,0.3,0.699),c(0.1,0.2,0.7),c(0.001,0.001,0.998)), c=0.05, model="Asymmetric", par=c(1.2,1.8,4,2,rep(0.3,9)), log=FALSE, vectorial=TRUE)