Principal component analysis using the alpha-transformation {Compositional} R Documentation

## Principal component analysis using the α-transformation

### Description

Principal component analysis using the α-transformation.

### Usage

```alfa.pca(x, a, center = TRUE, scale = TRUE, k = NULL, vectors = FALSE)
```

### Arguments

 `x` A matrix with the compositional data. Zero values are allowed. In that case "a" should be positive. `a` The value of α to use in the α-transformation. `center` Do you want your data centered? TRUE or FALSE. `scale` Do you want each of your variables scaled, i.e. to have unit variance? TRUE or FALSE. `k` If you want a specific number of eigenvalues and eigenvectors set it here, otherwise all eigenvalues (and eigenvectors if requested) will be returned. `vectors` Do you want the eigenvectors be returned? By dafault this is FALSE.

### Details

The α-transformation is applied to the compositional data and then PCA is performed. Note however, that the right multiplication by the Helmert sub-matrix is not applied in order to be in accordance with Aitchison (1983). When α=0, this results to the PCA proposed by Aitchison (1983).

### Value

A list including:

 `values` The eigenvalues. `vectors` The eigenvectors.

### Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

### References

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

Aitchison, J. (1983). Principal component analysis of compositional data. Biometrika, 70(1), 57-65.

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. http://arxiv.org/pdf/1106.1451.pdf

```logpca, alfa.pcr, kl.alfapcr ```
```x <- as.matrix(iris[, 1:4])