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Alignment-independent bilinear multivariate modeling (AIBIMM) for global analyses of 16S rRNA gene phylogeny, by K. Rudi, M. Zimonja and T. Næs

International Journal of Systematic and Evolutionary Microbiology vol. 56, part 7, pp. 1565 - 1575

Supplementary material

Appendix 1. Computer program for conducting AIBIMM analyses

We have developed the computer program PhyloMode for conducting AIBIMM analyses. The program can be downloaded free of charge from http://www.matforsk.no/web/sampro.nsf/downloadE/Microbial_community. PhyloMode was written in the Microsoft Visual Studio .net programming environment using C#. We used .net charting libraries from ZedGraph (http://sourceforge.net/projects/zedgraph) and multivariate statistical .net libraries from CenterSpace Software (http://www.centerspace.net).

The PhyloMode program contains two basic modules. The first module transforms DNA sequences into multimer data (n = 1 to n = 6). The input is a file in FASTA format (sequences begin with a single-line description which is distinguished by '>'). The output from the module can be exported in tab-delimited text for advanced multivariate statistical analyses by software packages such as The Unscrambler (CAMO Inc.; http://www.camo.com). The PhyloMode software also includes a module for principal component analyses (PCA) and 2D visualization of both the score and loading plots. Finally, the program has an option for creating dendrograms based on the principal component data using single, centroid or complete linkage. The linkage data are exported in a format compatible with the free-of-charge software TreeView (http://taxonomy.zoology.gla.ac.uk/rod/treeview.html), which is a software package for drawing phylogenetic trees.

Appendix 2. Step-by-step details of bilinear PCA modelling

PCA is a method for extracting/computing a set of components that explain as much of the variability of a dataset (here denoted by X) as possible. The method goes as follows:

1. Find the mean of the dataset and subtract it from X. This corresponds to moving the origin of the vector space to the mean.
   
   
2. Find the direction (defined by the unit vector p₁) in the vector space spanned by the rows of X that accounts for as much of the variance as possible. The first principal component score, t₁, is then obtained by projecting the data onto the direction defined by p₁ (i.e. t₁ = Xp₁). The elements of the vector p₁ are called the loadings. The elements of the loading vector tell us how much the different variables 'load' in the computation of the score.
   
   
3. After the first component is computed in this way, the second component is found in the same way under the restriction that it is orthogonal (perpendicular) to the first. An equivalent requirement is that the second score variable is uncorrelated with the scores from component 1.
   
   
4. The procedure continues until the desired number of components have been extracted (this is called A). The significant number of components can be found by cross-validation.

It can be shown that, if we organize the scores and loadings for the A first components in matrices T and P, the original data can be modelled as

X = TP^T + E

where E denotes noise, i.e. that part of the dataset which is not considered further.

The scores and loadings for the first couple of components are usually plotted in scatter plots in order to reveal information about relationships among objects and variables, respectively.

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