satdad (version 1.1)

ticEmp: Empirical tail importance coefficients.

Description

Computes on a sample the tail importance coefficients (tic) associated with threshold k. The value may be renormalized by the empirical global variance (Sobol version).

Usage

ticEmp(sample, ind = 2, k, 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 the 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

sample

A (n times d) matrix.

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.

k

An integer smaller or equal to n.

sobol

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

Author

Cécile Mercadier ([e-mail])

Details

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 variance contribution is thus \(D_I(\ell)\) and the theoretical tail sobol index is \(S_I(\ell)=\dfrac{D_I(\ell)}{D(\ell)}\).

Here, the function ticEmp evaluates \(\hat{D}_{I,k,n}\) the empirical counterpart of \(D_I(\ell)\) under the option sobol = FALSE, and \(\hat{S}_{I,k,n}\) the empirical counterpart of \(S_I(\ell)\) under the option sobol = TRUE.

Proposition 1 and Theorem 2 of Mercadier and Roustant (2019) furnish their rank-based expressions. For the subset of components \(I\),

\(\hat{D}_{I,k,n}=\frac{1}{k^2}\sum_{s=1}^n\sum_{s^\prime=1}^n \prod_{t\in I}(\min(\overline{R}^{(t)}_s,\overline{R}^{(t)}_{s^\prime})-\overline{R}^{(t)}_{s}\overline{R}^{(t)}_{s^\prime}) \prod_{t\notin I} \overline{R}^{(t)}_s\overline{R}^{(t)}_{s^\prime}\)

\(\hat{D}_{k,n}=\frac{1}{k^2}\sum_{s=1}^n\sum_{s^\prime=1}^n \prod_{t\in I}\min(\overline{R}^{(t)}_s,\overline{R}^{(t)}_{s^\prime})- \prod_{t\in I}\overline{R}^{(t)}_{s}\overline{R}^{(t)}_{s^\prime}\)

and \(\hat{S}_{I,k,n}=\dfrac{\hat{D}_{I,k,n}}{\hat{D}_{k,n}}\)

where

• \(k\) is the threshold parameter,

• \(n\) is the sample size,

• \(X_1,...,X_n\) describes the sample, each \(X_s\) is a d-dimensional vector \(X_s^{(t)}\) for \(t=1,...,d\),

• \(R^{(t)}_s\) denotes the rank of \(X^{(t)}_s\) among \(X^{(t)}_1, ..., X^{(t)}_n\),

• and \(\overline{R}^{(t)}_s = \min((n- R^{(t)}_s+1)/k,1)\).

References

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

See Also

tic and tsicEmp

Examples

## Fix a 5-dimensional asymmetric tail dependence structure
(ds5 <- gen.ds(d = 5))

## Generate a 1000-sample of Mevlog random vectors associated with ds5
sample5 <- rMevlog(n = 1000, ds = ds5)

## Compute empirical tic values according cardinality
res2 <- ticEmp(sample5, ind = 2, k = 100, sobol = TRUE)
res3 <- ticEmp(sample5, ind = 3, k = 100, sobol = TRUE)
res4 <- ticEmp(sample5, ind = 4, k = 100, sobol = TRUE)

## Represent the empirical indices associated with pairs
barplot(res2$tic ~ as.character(res2$subsets), las = 2,
     xlab = "", ylab = "", main = "Tail Sobol Indices (cardinality 2)")

## Represent the empirical indices associated with triplets
barplot(res3$tic ~ as.character(res3$subsets), las = 2,
     xlab = "", ylab = "", main = "Tail Sobol Indices (cardinality 3)")

## Represent the empirical indices associated with quadriplets
barplot(res4$tic ~ as.character(res4$subsets), las = 2,
     xlab = "", ylab ="", main = "Tail Sobol Indices (cardinality 4)")

## Check the sum-to-one constraint of empirical tail Sobol indices
sum(ticEmp(sample5, ind = "with.singletons", k = 100,  sobol = TRUE)$tic)