ticEmp {satdad} | R Documentation |
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)
Arguments
sample |
A |
ind |
A character string among "with.singletons" and "all" (without singletons), or an integer in |
k |
An integer smaller or equal to |
sobol |
A boolean. 'FALSE' (the default). If 'TRUE': the index is normalized by the empirical global variance. |
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 thesample
, eachX_s
is a d-dimensional vectorX_s^{(t)}
fort=1,...,d
, -
R^{(t)}_s
denotes the rank ofX^{(t)}_s
amongX^{(t)}_1, ..., X^{(t)}_n
, and
\overline{R}^{(t)}_s = \min((n- R^{(t)}_s+1)/k,1)
.
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 cardinalityind
. Whenind
is the list, it corresponds tosubsets
.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 whensobol = "TRUE"
.
Author(s)
Cécile Mercadier (mercadier@math.univ-lyon1.fr
)
References
Mercadier, C. and Roustant, O. (2019) The tail dependograph. Extremes, 22, 343–372.
See Also
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)