rmevspec {mev}

Random samples from spectral distributions of multivariate extreme value models.

Description

Generate from Q_i, the spectral measure of a given multivariate extreme value model based on the L1 norm.

Usage

rmevspec(
  n,
  d,
  param,
  sigma,
  model = c("log", "neglog", "bilog", "negbilog", "hr", "br", "xstud", "smith",
    "schlather", "ct", "sdir", "dirmix", "pairbeta", "pairexp", "wdirbs", "wexpbs",
    "maxlin"),
  weights = NULL,
  vario = NULL,
  coord = NULL,
  grid = FALSE,
  dist = NULL,
  ...
)

Arguments

n

number of observations

d

dimension of sample

param

parameter vector for the logistic, bilogistic, negative bilogistic and extremal Dirichlet (Coles and Tawn) model. Parameter matrix for the Dirichlet mixture. Degree of freedoms for extremal student model. See Details.

sigma

covariance matrix for Brown-Resnick and extremal Student-t distributions. Symmetric matrix of squared coefficients \lambda^2 for the Husler-Reiss model, with zero diagonal elements.

model

for multivariate extreme value distributions, users can choose between 1-parameter logistic and negative logistic, asymmetric logistic and negative logistic, bilogistic, Husler-Reiss, extremal Dirichlet model (Coles and Tawn) or the Dirichlet mixture. Spatial models include the Brown-Resnick, Smith, Schlather and extremal Student max-stable processes. Max linear models are also supported

weights

vector of length m for the m mixture components that sum to one. For the "maxlin" model, weights should be a matrix with d columns that represent the weight of the components and whose column sum to one (if provided, this argument overrides asy).

vario

semivariogram function whose first argument must be distance. Used only if provided in conjunction with coord and if sigma is missing

coord

d by k matrix of coordinates, used as input in the variogram vario or as parameter for the Smith model. If grid is TRUE, unique entries should be supplied.

grid

Logical. TRUE if the coordinates are two-dimensional grid points (spatial models).

dist

symmetric matrix of pairwise distances. Default to NULL.

...

additional arguments for the vario function

Details

The vector param differs depending on the model

• log: one dimensional parameter greater than 1

• neglog: one dimensional positive parameter

• bilog: d-dimensional vector of parameters in [0,1]

• negbilog: d-dimensional vector of negative parameters

• ct, dir, negdir: d-dimensional vector of positive (a)symmetry parameters. Alternatively, a d+1 vector consisting of the d Dirichlet parameters and the last entry is an index of regular variation in (0, 1] treated as scale

• xstud: one dimensional parameter corresponding to degrees of freedom alpha

• dirmix: d by m-dimensional matrix of positive (a)symmetry parameters

• pairbeta, pairexp: d(d-1)/2+1 vector of parameters, containing the concentration parameter and the coefficients of the pairwise beta, in lexicographical order e.g., \beta_{1,2}, \beta_{1,3}, \ldots

• wdirbs, wexpbs: 2d vector of d concentration parameters followed by the d Dirichlet parameters

Value

an n by d exact sample from the corresponding multivariate extreme value model

Note

This functionality can be useful to generate for example Pareto processes with marginal exceedances.

Author(s)

Leo Belzile

References

Dombry, Engelke and Oesting (2016). Exact simulation of max-stable processes, Biometrika, 103(2), 303–317.

Boldi (2009). A note on the representation of parametric models for multivariate extremes. Extremes 12, 211–218.

Examples

set.seed(1)
rmevspec(n=100, d=3, param=2.5, model='log')
rmevspec(n=100, d=3, param=2.5, model='neglog')
rmevspec(n=100, d=4, param=c(0.2,0.1,0.9,0.5), model='bilog')
rmevspec(n=100, d=2, param=c(0.8,1.2), model='ct') #Dirichlet model
rmevspec(n=100, d=2, param=c(0.8,1.2,0.5), model='sdir') #with additional scale parameter
#Variogram gamma(h) = scale*||h||^alpha
#NEW: Variogram must take distance as argument
vario <- function(x, scale=0.5, alpha=0.8){ scale*x^alpha }
#grid specification
grid.coord <- as.matrix(expand.grid(runif(4), runif(4)))
rmevspec(n=100, vario=vario,coord=grid.coord, model='br')
## Example with Dirichlet mixture
alpha.mat <- cbind(c(2,1,1),c(1,2,1),c(1,1,2))
rmevspec(n=100, param=alpha.mat, weights=rep(1/3,3), model='dirmix')