helmert

Helmert matrix

Description

This function returns the Helmert matrix of order \(n\) .

Usage

helmert(n = 1)

Arguments

Argument	Description
n	order of the Helmert matrix.

Details

A Helmert matrix of order \(n\) is a square matrix defined as

\[\mathbf{H}_n = \left[{\begin{array}{*{20}{c}}1/\sqrt{n} & 1/\sqrt{n} & 1/\sqrt{n} & \dots & 1/\sqrt{n} \\1/\sqrt{2} & -1/\sqrt{2} & 0 & \dots & 0 \\1/\sqrt{6} & 1/\sqrt{6} & -2/\sqrt{6} & \dots & 0 \\\vdots & \vdots & \vdots & \ddots & \vdots \\\frac{1}{\sqrt{n(n-1)}} & \frac{1}{\sqrt{n(n-1)}} & \frac{1}{\sqrt{n(n-1)}} & \dots & -\frac{(n-1)}{\sqrt{n(n-1)}}\end{array}}\right].\]

Helmert matrix is orthogonal and is frequently used in the analysis of variance (ANOVA).

Value

Returns an \(n\) by \(n\) matrix.

References

Lancaster, H.O. (1965). The Helmert matrices. The American Mathematical Monthly 72 , 4-12.

Gentle, J.E. (2007). Matrix Algebra: Theory, Computations, and Applications in Statistics . Springer, New York.

Examples

n <- 1000
set.seed(149)
x <- rnorm(n)

H <- helmert(n)
object.size(H) # 7.63 Mb of storage
K <- H[2:n,]
z <- c(K %*% x)
sum(z^2) # 933.1736

# same that
(n - 1) * var(x)