ddirimix function

Angular density/likelihood function in the Dirichlet Mixture model.

Likelihood function (spectral density on the simplex) and angular data sampler in the Dirichlet mixture model.

ddirimix(
  x = c(0.1, 0.2, 0.7),
  par,
  wei = par$wei,
  Mu = par$Mu,
  lnu = par$lnu,
  log = FALSE,
  vectorial = FALSE
)

rdirimix(
  n = 10,
  par = get("dm.expar.D3k3"),
  wei = par$wei,
  Mu = par$Mu,
  lnu = par$lnu
)

Arguments

• x: An angular data set which may be reduced to a single point: A $n*p$ matrix or a vector of length p, where $p$ is the dimension of the sample space and $n$ is the sample size. Each row is a point on the simplex, so that each row sum to one. The error tolerance is set to 1e-8

  in this package.

• par: The parameter list for the Dirichlet mixture model.

• wei: Optional. If present, overrides the value of par$wei.

• Mu: Optional. If present, overrides the value of par$Mu.

• lnu: Optional. If present, overrides the value of par$lnu.

• log: Logical: should the density or the likelihood be returned on the log-scale ?

• vectorial: Logical: Should a vector of size $n$ or a single value be returned ?

• n: The number of angular points to be generated

Returns

ddirimix returns the likelihood as a single number if vectorial ==FALSE, or as a vector of size nrow(x) containing the likelihood of each angular data point. If log == TRUE, the log-likelihood is returned instead. rdirimix returns a matrix with n points and p=nrow(Mu) columns.

Details

The spectral probability measure defined on the simplex characterizes the dependence structure of multivariate extreme value models. The parameter list for a mixture with $k$ components, is made of

• Mu: The density kernel centers $\mu[1:p,1:k]$ : A $p*k$ matrix, which columns sum to one, and such that Mu %*% wei=1, for the moments constraint to be satisfied. Each column is a Dirichlet kernel center.
• wei: The weights vector for the kernel densities: A vector of $k$ positive numbers summing to one.
• lnu: The logarithms of the shape parameters $\nu[1:k]$ for the density kernels: a vector of size $k$.

The moments constraint imposes that the barycenter of the columns in Mu, with weights wei, be the center of the simplex.