Orthonormal contrast matrix

matrix_k() returns the $(n - 1) \times n$ matrix $\mathbf{K}_n$ whose rows are orthonormal and orthogonal to $\mathbf{1}_n$ (Helmert contrasts). bayes_ammi() writes the main effects as $\boldsymbol{\tau} = \mathbf{K}_G^\top\mathbf{t}$, which keeps them summing to zero.

Usage

matrix_k(n)

# Default S3 method
matrix_k(n)

Arguments

n: number of columns (at least 2).

Value

The $(n - 1) \times n$ matrix $\mathbf{K}_n$.

Details

Row $r$ ($r = 1, \ldots, n - 1$) has $0$ in its first $r - 1$ positions, $(n - r)/c_r$ in position $r$ and $-1/c_r$ after it, with $c_r = \sqrt{(n - r)(n - r + 1)}$, so that $$\mathbf{K}_n\mathbf{K}_n^\top = \mathbf{I}_{n-1}, \qquad \mathbf{K}_n\mathbf{1}_n = \mathbf{0}, \qquad \mathbf{K}_n^\top\mathbf{K}_n = \mathbf{I}_n - \frac{1}{n}\mathbf{J}_n.$$

References

Crossa, J., Perez-Elizalde, S., Jarquin, D., Cotes, J. M., Viele, K., Liu, G. and Cornelius, P. L. (2011). Bayesian estimation of the additive main effects and multiplicative interaction model. \emphCrop Science, \strong51(4), 1458–1469. \doi10.2135/cropsci2010.06.0343

Author

Muhammad Yaseen (myaseen208@gmail.com)

Examples

K <-
  matrix_k(
    n = 4
  )
K
#>           [,1]       [,2]       [,3]       [,4]
#> [1,] 0.8660254 -0.2886751 -0.2886751 -0.2886751
#> [2,] 0.0000000  0.8164966 -0.4082483 -0.4082483
#> [3,] 0.0000000  0.0000000  0.7071068 -0.7071068
round(x = K %*% t(K), digits = 10)
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1