Arc-cosine kernel matrix

Formula: K(x_i, x_j) = (1/pi) * ||x_i|| * ||x_j|| * [sin(theta) + (pi - theta) cos(theta)], theta = angle(x_i, x_j)

Usage

Arckern(X, Z = NULL, depth = 1L)

Arguments

X

One record per row.

Z

Second set of records; None gives the square Gram matrix of X.

depth

Number of times the kernel is composed with itself.

Value

List with K, depth, n, m.

References

Montesinos Lopez, Montesinos Lopez and Crossa (2022), Multivariate Statistical Machine Learning Methods for Genomic Prediction, Springer, doi:10.1007/978-3-030-89010-0. Chapter 8, Sect. 8.2.2, the arc-cosine kernel named on p. 252 and developed later in the chapter; the implementation delegates to the chapter-8 arc-cosine kernel verified against the book.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Arckern(V)
#> $K
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,]    1    2    3    4    5    6    7    8
#> [2,]    2    4    6    8   10   12   14   16
#> [3,]    3    6    9   12   15   18   21   24
#> [4,]    4    8   12   16   20   24   28   32
#> [5,]    5   10   15   20   25   30   35   40
#> [6,]    6   12   18   24   30   36   42   48
#> [7,]    7   14   21   28   35   42   49   56
#> [8,]    8   16   24   32   40   48   56   64
#> 
#> $depth
#> [1] 1
#> 
#> $n
#> [1] 8
#> 
#> $m
#> [1] 8
#> 
#> $method
#> [1] "Arc-cosine kernel, MVSML Chap. 8"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"