simplex {ExtremalDep}

Definition of a multivariate simplex

Description

Generation of grid points over the multivariate simplex

Usage

simplex(d, n=50, a=0, b=1)

Arguments

d

A positive integer indicating the dimension of the simplex.

n

A positive integer indicating the number of grid points to be generated on the univariate components of the simplex.

a, b

Two numeric values indicating the lower and upper bound of the simplex. By default a=0 and b=0, indicating the unit-simplex.

Details

A d-dimensional simplex is defined by

S = \{ (\omega_1, \ldots, \omega_d) \in R^d_+: \sum_{i=1}^d \omega_i = 1 \}.

Here the function defines the simplex as

S = \{ (\omega_1, \ldots, \omega_d) \in [a,b]^d: \sum_{i=1}^d \omega_i = 1 \}.

When d=2 and [a,b]=[0,1], a grid of points of the form \{ (\omega_1, \omega_2) \in [0,1]: \omega_1 + \omega_2 = 1 \}.

Value

Returns a matrix with d columns. When d=2, the number of rows is n. When d>2, the number of rows is equal to

\sum_{i_{d-1}=0}^{n-1} \sum_{i_{d-2}=0}^{n-i_{d-1}} \cdots \sum_{i_{1}=1}^{n-i_{d-1}-\cdots-i_{2}} i_{1}

Author(s)

Simone Padoan, [e-mail], https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, [e-mail] https://www.borisberanger.com;

Examples

### 3-dimensional unit simplex

W <- simplex(d=3, n=10)
plot(W[,-3], pch=16)