est.tpdm {ExtrPatt}

Estimation of TPDM

Description

Estimation of tail pairwise dependence matrix (TPDM)

Sub-Routine of est.row.tpdm. Calculates one element of the TPDM

Usage

est.tpdm(X, Y = NULL, anz_cores = 1, clust = NULL, q = 0.98)

est.row.tpdm(x, Y, clust = NULL, q = 0.98)

est.element.tpdm(x, y, clust = NULL, q = 0.98)

Arguments

X

A t x n dimensional, numeric data-matrix with t: Number of time steps and n: Number of grid points/stations

Y

A t x n dimensional, numeric Data-matrix with t: Number of time steps and n: Number of grid points/stations

anz_cores

Number of cores for parallel computing (default:1); Be careful not to overload your computer!

clust

Optional: If clust = NULL, no declustering is performed. Else, declustering according to cluster-length 'clust'.

q

Threshold for computation of TPDM. Only data above the 'q'-quantile will be used for estimation. Choose such that 0<q<1.

x

Array of length t, where t is the number of time steps

y

Same as x

Details

Given a random vector X with components x_{t,i}, x_{t,j} with i,j = 1, \ldots, n and it's radial component r_{t,ij} = \sqrt{x_{t,i}^2 + x_{t,j}^2} and angular components w_{t,i} = x_{t,i}/r_{t,ij} and w_{t,j} = x_{t,j}/r_{t,ij}, the i'th,j'th element of the TPDM is estimated as:

\hat{\sigma}_{ij} = 2 n_{ij,exc}^{-1} \sum_{t=1}^{n} w_{t,i} w_{t,j} |_{(r_{t,ij} > r_{0,ij})}

. Given two random vectors X and Y with components x_{t,i}, y_{t,j} with i,j = 1, \ldots, n, and it's radial component r_{t,ij} = \sqrt{x_{t,i}^2 + y_{t,j}^2} and angular components w_{t,i}^x = \frac{x_{t,i}}{r_{t,ij}} ; w_{t,j}^y = \frac{y_{t,j}}{r_{t,ij}}, the i'th,j'th element of the cross-TPDM is estimated as:

\hat{\sigma}_{ij} = 2 n^{-1}_{exc} \sum_{t=1}^{n} w^x_{t,i} w^y_{t,j} |_{(r_{t,ij} > r_{0,ij})}

.

Value

An n x n matrix, containing the estimate of the TPDM

Array containing the estimate of one row of the TPDM.

Value containing the estimate of one element of the TPDM.

References

Jiang & Cooley (2020) <doi:10.1175/JCLI-D-19-0413.1>; Szemkus & Friederichs (2023)

Examples

data    <- precipGER

data.alpha2       <- to.alpha.2(data$pr)
Sigma   <- est.tpdm(data.alpha2,anz_cores =1)