| helmert {fastmatrix} | R Documentation |
This function returns the Helmert matrix of order n.
helmert(n = 1)
n |
order of the Helmert matrix. |
A Helmert matrix of order n is a square matrix defined as
\bold{H}_n = ≤ft[{\begin{array}{*{20}{c}} 1/√{n} & 1/√{n} & 1/√{n} & … & 1/√{n} \\ 1/√{2} & -1/√{2} & 0 & … & 0 \\ 1/√{6} & 1/√{6} & -2/√{6} & … & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \frac{1}{√{n(n-1)}} & \frac{1}{√{n(n-1)}} & \frac{1}{√{n(n-1)}} & … & -\frac{(n-1)}{√{n(n-1)}} \end{array}}\right].
Helmert matrix is orthogonal and is frequently used in the analysis of variance (ANOVA).
Returns an n by n matrix.
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.
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)