# alfa: The alpha-transformation In Compositional: Compositional Data Analysis

 The alpha-transformation R Documentation

## The α-transformation

### Description

The α-transformation.

### Usage

```alfa(x, a, h = TRUE)
alef(x, a)
```

### 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 α=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 α=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 α-transformation is applied to the compositional data. The command "alef" is the same as "alfa(x, a, h = FALSE)", but reurns a different element as well and is necessary for the functions `a.est`, `a.mle` and `alpha.mle`.

### Value

A list including:

 `sa` The logarithm of the Jacobian determinant of the α-transformation. This is used in the "profile" function to speed up the computations. `sk` If the "alef" was called, this will return the sum of the α-power transformed data, prior to being normalised to sum to 1. If α=0, this will not be returned. `aff` The α-transformed data.

### Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

### References

Tsagris Michail and Stewart Connie (2020). A folded model for compositional data analysis. Australian and New Zealand Journal of Statistics, 62(2): 249-277. https://arxiv.org/pdf/1802.07330.pdf

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

```alfainv, pivot, alfa.profile, alfa.tune a.est, alpha.mle, alr, bc, fp, green ```

### Examples

```library(MASS)
x <- as.matrix(fgl[, 2:9])
x <- x / rowSums(x)
y1 <- alfa(x, 0.2)\$aff
y2 <- alfa(x, 1)\$aff
rbind( colMeans(y1), colMeans(y2) )
y3 <- alfa(x, 0.2)\$aff
dim(y1)  ;  dim(y3)
rowSums(y1)
rowSums(y3)
```

Compositional documentation built on July 8, 2022, 1:06 a.m.