The alpha-IT transformation {Compositional}

The \alpha-IT transformation

Description

The \alpha-IT transformation.

Usage

ait(x, a, h = TRUE)

Arguments

x

A matrix with the compositional data.

a

The value of the power transformation, it has to be between -1 and 1. If zero values are present it has to be greater than 0. If \alpha=0 the isometric log-ratio transformation is applied.

h

A boolean variable. If is TRUE (default value) the multiplication with the Helmert sub-matrix will take place. When \alpha=0 and h = FALSE, the result is the centred log-ratio transformation (Aitchison, 1986). In general, when h = FALSE the resulting transformation maps the data onto a singualr space. The sum of the vectors is equal to 0. Hence, from the simplex constraint the data go to another constraint.

Details

The \alpha-IT transformation is applied to the compositional data.

Value

A matrix with the \alpha-IT transformed data.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris [e-mail].

References

Clarotto L., Allard D. and Menafoglio A. (2022). A new class of \alpha-transformations for the spatial analysis of Compositional Data. Spatial Statistics, 47.

See Also

aitdist, ait.knn, alfa, green, alr

Examples

library(MASS)
x <- as.matrix(fgl[, 2:9])
x <- x / rowSums(x)
y1 <- ait(x, 0.2)
y2 <- ait(x, 1)
rbind( colMeans(y1), colMeans(y2) )