| helmert {fastmatrix} | R Documentation |
Helmert matrix
Description
This function returns the Helmert matrix of order n.
Usage
helmert(n = 1)
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).
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)