simplex {ExtremalDep}R Documentation

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 dd-dimensional simplex is defined by

S={(ω1,,ωd)R+d:i=1dωi=1}.S = \{ (\omega_1, \ldots, \omega_d) \in R^d_+: \sum_{i=1}^d \omega_i = 1 \}.

Here the function defines the simplex as

S={(ω1,,ωd)[a,b]d:i=1dωi=1}.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,b]=[0,1], a grid of points of the form {(ω1,ω2)[0,1]:ω1+ω2=1}\{ (\omega_1, \omega_2) \in [0,1]: \omega_1 + \omega_2 = 1 \}.

Value

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

id1=0n1id2=0nid1i1=1nid1i2i1\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, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, borisberanger@gmail.com https://www.borisberanger.com;

Examples


### 3-dimensional unit simplex

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


[Package ExtremalDep version 0.0.4-1 Index]