satdad (version 1.1)

tic: Tail importance coefficients for Mevlog models.

Description

Computes the tail importance coefficients (tic) on a Mevlog model which is a multivariate extreme value (symmetric or asymmetric) logistic model, descibed here by its dependence structure.

Usage

tic(ds, ind = 2, n.MC = 1000, sobol = FALSE)

Value

The function returns a list of two elements:

• subsets A list of subsets from \(\{1,...,d\}\).

  When ind is given as an integer, subsets is the list of subsets from \(\{1,...,d\}\) with cardinality ind.

  When ind is a list, it corresponds to subsets.

  When ind = "with.singletons" subsets is the list of all non empty subsets in \(\{1,...,d\}\).

  When ind = "all" subsets is the list of all subsets in \(\{1,...,d\}\) with cardinality larger or equal to 2.

• tic A vector of tail importance coefficients, or their Sobol versions when sobol = "TRUE".

Arguments

ds

An object of class ds.

ind

A character string among "with.singletons" and "all" (without singletons), or an integer in \(\{2,...,d\}\) or a list of subsets from \(\{1,...,d\}\). The default is ind = 2, all pairwise coefficients are computed.

n.MC

Monte Carlo sample size. Default value is 1000. See details in tsic.

sobol

A boolean. `FALSE` (the default). If `TRUE`: the index is normalized by the theoretical global variance.

Author

Cécile Mercadier ([e-mail])

Details

The tail dependence structure is specified using a ds object, which corresponds to the stable tail dependence function \(\ell\). The process for deducing the stable tail dependence function \(\ell\) from ds is explained in the Details section of gen.ds.

The theoretical functional decomposition of the variance of the stdf \(\ell\) consists in writing \(D(\ell) = \sum_{I \subseteq \{1,...,d\}} D_I(\ell) \) where \(D_I(\ell)\) measures the variance of \(\ell_I(U_I)\) the term associated with subset \(I\) in the Hoeffding-Sobol decomposition of \(\ell\) ; note that \(U_I\) represents a random vector with independent standard uniform entries. The theoretical tail importance coefficient (tic) is thus \(D_I(\ell)\) and its sobol version is \(S_I(\ell)=\dfrac{D_I(\ell)}{D(\ell)}\).

The function tic uses the Mobius inversion formula, see Formula (8) in Liu and Owen (2006), to derive the tic from the tsic. The latter are the tail superset importance coefficients obtained by the function tsic.

References

Liu, R. and Owen, A. B. (2006) Estimating mean dimensionality of analysis of variance decompositions. J. Amer. Statist. Assoc., 101(474):712--721.

Mercadier, C. and Roustant, O. (2019) The tail dependograph. Extremes, 22, 343--372.

See Also

tsic, ticEmp

ticEmp and tsic

Examples

## Fix a 4-dimensional asymmetric tail dependence structure
ds4 <- gen.ds(d = 4, sub = list(1:2,3:4,1:3))

## Compute all tic values
res4 <- tic(ds4, ind = "with.singletons", sobol = TRUE)

## Check the sum-to-one constraint of tail Sobol indices
sum(res4$tic)