fastmatrix (version 0.5-7721)

helmert: Helmert matrix

Description

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

Usage

helmert(n = 1)

Value

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

Arguments

n

order of the Helmert matrix.

Details

A Helmert matrix of order \(n\) is a square matrix defined as $$\bold{H}_n = \left[ {\begin{array}{ccccc} 1/\sqrt{n} & 1/\sqrt{n} & 1/\sqrt{n} & \dots & 1/\sqrt{n} \\ 1/\sqrt{2} & -1/\sqrt{2} & 0 & \dots & 0 \\ 1/\sqrt{6} & 1/\sqrt{6} & -2/\sqrt{6} & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \frac{1}{\sqrt{n(n-1)}} & \frac{1}{\sqrt{n(n-1)}} & \frac{1}{\sqrt{n(n-1)}} & \dots & -\frac{(n-1)}{\sqrt{n(n-1)}} \end{array}} \right].$$ Helmert matrix is orthogonal and is frequently used in the analysis of variance (ANOVA).

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)