ExtremalDep (version 0.0.4-4)

simplex: Definition of a multivariate simplex

Description

Generation of grid points over the multivariate simplex

Usage

simplex(d, n=50, a=0, b=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 $$\sum _{i_{d-1}=0}^{n-1}\sum _{i_{d-2}=0}^{n-i_{d-1}}\cdots \sum _{i_1=1}^{n-i_{d-1}-\cdots -i_2}{i}_1$$

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.

Author

Simone Padoan, , https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, https://www.borisberanger.com;

Details

A $d$-dimensional simplex is defined by $$S=\{({\omega }_1,\dots ,{\omega }_d)\in R_{+}^d:\sum _{i=1}^d{\omega }_i=1\}.$$ Here the function defines the simplex as $$S=\{({\omega }_1,\dots ,{\omega }_d)\in [a,b{]}^d:\sum _{i=1}^d{\omega }_i=1\}.$$ When d=2 and $[a,b]=[0,1]$, a grid of points of the form $\{({\omega }_1,{\omega }_2)\in [0,1]:{\omega }_1+{\omega }_2=1\}$.

Examples

### 3-dimensional unit simplex

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