# Multivariate Statistics For Wildlife And Ecology Research Papers

The term "biplot" originates in Ruben Gabriel's *Biometrika* paper in 1971:

Gabriel, K.R. (1971). The biplot graphic display of matrices with application to principal component analysis. *Biometrika* 58, 453-467.

This paper, which at the time of writing has 1008 citations on Google Scholar and 682 on the Science Citation Index (ISI Web of Knowledge), is widely regarded as the origin of the idea.

A less cited paper by Ruben Gabriel, but nevertheless one of my favourite ones on the biplot, appeared the following year in the *Journal of Applied Meteorology* (Ruben was also well-known for his work as a statistician in weather modification projects):

Gabriel, K.R. (1972). Analysis of meteorological data by means of canonical decompositions and biplots. *Journal of Applied Meteorology* 11, 1071-1077.

Another gem is by Dan Bradu and Ruben Gabriel in Technometrics in 1978:

Bradu, D. and Gabriel, K.R. (1972). The biplot as a diagnostic tool for models of two-way tables. ** Technometrics** 20, 47-68.

Other authors also had the idea of adding variables to an existing configuration of points to make joint displays, although they did not call them biplots. For example, Doug Carroll's vector model for preferences is a biplot:

Carroll, J.D. (1972). Individual differences and multidimensional scaling. In R.N. Shepard, A.K. Romney, and S.B. Nerlove, eds, *Multidimensional Scaling: Theory and Applications in the Behavioral Sciences (Vol. 1)*, 105-155. Seminar Press, New York.

Only one book exists to date specifically on the topic of biplots, by John Gower and David Hand:

Gower, J.C. and Hand, D.J (1996). *Biplots*. Chapman & Hall, London, UK.

This book is very complete, both on linear and nonlinear biplots, giving a rigorous theoretical treatment of the subject. Another book by John Gower is with coauthors Sugnet Gardner-Lubbe and Niel le Roux:

Gower, J.C., Gardner-Lubbe, S. and le Roux, N. (2010). *Understanding Biplots*. Wiley, Chichester, UK.

As far as the vast literature on the singular value decomposition (SVD) is concerned, I mention only two sources, by the author of one of the landmark algorithms for the SVD, Gene Golub in 1971, which seems to be an important year for the biplot:

Golub, G.H. and Reinsch, C. (1971). The singular value decomposition and least squares solutions. In: J.H. Wilkinson and C. Reinsch, eds, *Handbook for Automatic Computation*, 134-151. Springer-Verlag, Berlin.

And the other a classic book by Paul Green and Doug Carroll, originally published in 1976, which was the first time I saw the geometric interpretation of the SVD (called "basic structure" by these authors)-this book is invaluable as a practical introduction to matrix and vector geometry in multivariate analysis:

Green, P.E. and Carroll, J.D. (1997). *Mathematical Tools for Applied Multivariate Analysis, Revised Edition*. Academic Press, New York.

Most books or articles that treat the methods presented in this book will have a section or chapter on biplots and their interpretation in the context of that method. This is just a tiny selection of some of the literature that can be consulted, and by no means the primary references:

**Principal component analysis**

- Joliffe, I.T. (2002).
*Principal Component Analysis*(2nd edition). Springer, New York.

**Log-ratio analysis (unweighted form) **

- Aitchison, J. and Greenacre, M. (2002). Biplots of compositional data.
*Applied Statistics*51, 375-392.

**Log-ratio analysis (weighted form)**

- Greenacre, M. and Lewi, P.J. (2009). Distributional equivalence and subcompositional coherence in the analysis of compositional data, contingency tables and ratio scale measurements.
*Journal of Classification*26, 29-54.

**Correspondence analysis**

**Multiple correspondence analysis**

- Greenacre, M. and Blasius, J., eds (2006). (eds),
*Multiple Correspondence Analysis and Related Methods*, Chapman & Hall/CRC Press, London.

- Michalidis, G. and de Leeuw, J. (1998). The Gifi system for descriptive multivariate analysis.
*Statistical Science*13, 307-336.

**Discriminant analysis/centroid biplots**

- Hastie, T., Tibshirani, R. and Friedman, J. (2009).
*The Elements of Statistical Learning*(2nd edition). Springer, New York. This book may be freely downloaded at www-stat.stanford.edu

**Constrained biplots**

- Legendre, P. and Legendre, L. (1998).
*Numerical Ecology*(2nd edition). Elsevier, Amsterdam.

Finally we give some resources on the internet, on R packages relevant to this book, *Biplots in Practice* (in alphabetic order of package names).

- Thioulouse, J. and Dray, S. (2007). Interactive multivariate data analysis in R with the ade4 and ade4TkGUI packages.
*Journal of Statistical Software*. Download from http://www.jstatsoft.org/v22/i05/paper - De Leeuw, J. and Mair, P. (2009). Simple and canonical correspondence analysis using the R package anacor.
*Journal of Statistical Software*. Download from http://www.jstatsoft.org/v31/i05/paper - De Leeuw, J. and Mair, P. (2009). Gifi methods for optimal scaling in R: the package homals.
*Journal of Statistical Software*. Download from http://www.jstatsoft.org/v31/i04/paper - La Grange, A., le Roux, N. and Gardner-Lubbe, S. (2000). BiplotGUI: Interactive biplots in R.
*Journal of Statistical Software*. Download from http://www.jstatsoft.org/v30/i12/paper - Nenadic, O. and Greenacre, M. (2007). Correspondence analysis in R, with two- and three-dimensional graphics: The ca package.
*Journal of Statistical Software*. Download from http://www.jstatsoft.org/v20/a03/paper - Markos, A. (2010). caGUI: a Tcl/Tk GUI for the functions in the ca package. Download from http://cran.r-project.org/web/packages/caGUI/index.html
- Graffelman, J. (2010). calibrate: Calibration of scatterplot and biplot axes. Download from http://cran.r-project.org/web/packages/calibrate/index.html
- Oksanen, J. (2010). vegan: Community Ecology Package. Download from http://cran.r-project.org/web/packages/vegan/index.html

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