Conditional orthonormal basis

Source: R/utils.R

Compute orthonormal ilr bases adapted to row-wise conditioning patterns.

conditional_obasis(X, scheme = c("zero", "zero_na"))

Arguments

X

A numeric matrix or data frame with one observation or conditioning pattern per row and one part per column.

scheme

Character string indicating the conditioning scheme. Possible values are `"zero"` and `"zero_na"`. Default is `"zero"`.

Value

A three-dimensional array of dimension `(D - 1, D, nrow(X))`, where `D` is the number of parts. Each slice contains one orthonormal ilr basis.

Details

Each row of `X` defines one conditioning pattern on the parts of a composition. According to `scheme`, the parts are split into ordered blocks:

  • `"zero"`: parts equal to `0` and parts with strictly positive values,

  • `"zero_na"`: missing values (`NA`), zeros, and strictly positive values.

For each row, the function constructs an orthonormal basis of the clr-plane preserving the block structure induced by the selected scheme.

Under `scheme = "zero"`, if a row contains `nz` zeros, then:

  • the first `nz - 1` coordinates describe the internal log-ratio structure of the zero block,

  • the coordinate `nz` describes the balance between the zero block and the positive block,

  • the remaining coordinates describe the internal log-ratio structure of the positive block.

Under `scheme = "zero_na"`, the blocks are ordered as:

  • missing values (`NA`),

  • zeros,

  • strictly positive values.

In this case:

  • the first coordinates describe the internal structure of the `NA` block,

  • the next coordinate contrasts the `NA` block with the positive block,

  • the following coordinates describe the internal structure of the zero block,

  • the next coordinate contrasts the zero block with the positive block,

  • the remaining coordinates describe the internal structure of the positive block.

Examples

C <- rbind(
  c(0, 0, 1, 1, 0),
  c(0, 1, 0, 1, 0)
)

conditional_obasis(C)
#> , , 1
#> 
#>           [,1]       [,2]       [,3]       [,4]       [,5]
#> [1,] 0.7071068 -0.7071068  0.0000000  0.0000000  0.0000000
#> [2,] 0.4082483  0.4082483  0.0000000  0.0000000 -0.8164966
#> [3,] 0.3651484  0.3651484 -0.5477226 -0.5477226  0.3651484
#> [4,] 0.0000000  0.0000000  0.7071068 -0.7071068  0.0000000
#> 
#> , , 2
#> 
#>           [,1]       [,2]       [,3]       [,4]       [,5]
#> [1,] 0.7071068  0.0000000 -0.7071068  0.0000000  0.0000000
#> [2,] 0.4082483  0.0000000  0.4082483  0.0000000 -0.8164966
#> [3,] 0.3651484 -0.5477226  0.3651484 -0.5477226  0.3651484
#> [4,] 0.0000000  0.7071068  0.0000000 -0.7071068  0.0000000
#> 

X <- rbind(
  c(1, NA, 0, 2),
  c(NA, 3, 0, 4),
  c(1, 2, 3, 4)
)

conditional_obasis(X, scheme = "zero_na")
#> , , 1
#> 
#>            [,1]      [,2]      [,3]       [,4]
#> [1,] -0.4082483 0.8164966 0.0000000 -0.4082483
#> [2,] -0.4082483 0.0000000 0.8164966 -0.4082483
#> [3,]  0.7071068 0.0000000 0.0000000 -0.7071068
#> 
#> , , 2
#> 
#>           [,1]       [,2]      [,3]       [,4]
#> [1,] 0.8164966 -0.4082483 0.0000000 -0.4082483
#> [2,] 0.0000000 -0.4082483 0.8164966 -0.4082483
#> [3,] 0.0000000  0.7071068 0.0000000 -0.7071068
#> 
#> , , 3
#> 
#>           [,1]       [,2]       [,3]       [,4]
#> [1,] 0.7071068 -0.7071068  0.0000000  0.0000000
#> [2,] 0.4082483  0.4082483 -0.8164966  0.0000000
#> [3,] 0.2886751  0.2886751  0.2886751 -0.8660254
#>