Hi Corrado,

As I said before, PCA has major advantages, given the statistical assumptions are met and the limitations are known. In terms of summarizing environmental variables and getting uncorrelated PCs it is, I think, the best solution. That is because not only the PCs are uncorrelated, but also and mainly because you can use the loadings of the PCs and the original variables to do for example forecasting, by calculating the values of PCs posthoc, which you will need to derive a map. In certain cases you can do a PCA, run a model selection selection procedure, which extracts for you the significant predictors (PCs) of your dependent variable, and then in that set of selected variables you can predicted the value of your dependent variable from the model for areas, where you only know the independent environmental conditions. I usually (for reasons of how I started to do stats) check the SAS online help, especially the examples and chapter overview are very helpful (http://support.sas.com/91doc/docMainpage.jsp go to SAS/STAT and then SAS/STAT User's Guide).


good luck,

Kamran


That's what SAS has to say about it:

The purpose of principal component analysis (Rao 1964) is to derive a small number of linear combinations (principal components) of a set of variables that retain as much of the information in the original variables as possible. Often a small number of principal components can be used in place of the original variables for plotting, regression, clustering, and so on. Principal component analysis can also be viewed as an attempt to uncover approximate linear dependencies among variables. 
Given a data set with p numeric variables, you can compute p principal components. Each principal component is a linear combination of the original variables, with coefficients equal to the eigenvectors of the correlation or covariance matrix. The eigenvectors are customarily taken with unit length. The principal components are sorted by descending order of the eigenvalues, which are equal to the variances of the components.


Principal components have a variety of useful properties (Rao 1964; Kshirsagar 1972):

    * The eigenvectors are orthogonal, so the principal components represent jointly perpendicular directions through the space of the original variables.
    * The principal component scores are jointly uncorrelated. Note that this property is quite distinct from the previous one.
    * The first principal component has the largest variance of any unit-length linear combination of the observed variables. The jth principal component has the largest variance of any unit-length linear combination orthogonal to the first j-1 principal components. The last principal component has the smallest variance of any linear combination of the original variables.

    * The scores on the first j principal components have the highest possible generalized variance of any set of unit-length linear combinations of the original variables.
    * The first j principal components provide a least-squares solution to the model Y = XB + E where Y is an n 莥 matrix of the centered observed variables; X is the n 莌 matrix of scores on the first j principal components; B is the j 莥 matrix of eigenvectors; E is an n 莥 matrix of residuals; and you want to minimize trace(E'E), the sum of all the squared elements in E. In other words, the first j principal components are the best linear predictors of the original variables among all possible sets of j variables, although any nonsingular linear transformation of the first j principal components would provide equally good prediction. The same result is obtained if you want to minimize the determinant or the Euclidean (Schur, Frobenious) norm of E'E rather than the trace.
    * In geometric terms, the j-dimensional linear subspace spanned by the first j principal components provides the best possible fit to the data points as measured by the sum of squared perpendicular distances from each data point to the subspace. This is in contrast to the geometric interpretation of least squares regression, which minimizes the sum of squared vertical distances. For example, suppose you have two variables. Then, the first principal component minimizes the sum of squared perpendicular distances from the points to the first principal axis. This is in contrast to least squares, which would minimize the sum of squared vertical distances from the points to the fitted line. 

------------------------
Kamran Safi

Postdoctoral Research Fellow
Institute of Zoology
Zoological Society of London
Regent's Park
London NW1 4RY

http://www.zoo.cam.ac.uk/ioz/people/safi.htm

http://spatialr.googlepages.com
http://asapi.wetpaint.com



-----Original Message-----
From: Corrado [mailto:ct529@york.ac.uk]
Sent: Wed 17.12.2008 11:25
To: St廧hane Dray; Kamran Safi
Cc: Edzer Pebesma; Ashton Shortridge; r-sig-geo@stat.math.ethz.ch
Subject: Re: [R-sig-Geo] Principal Component Analysis - Selecting	components? + right choice?
 
Dear R-sig-geo friends,

I have been testing some of the suggested approaches, and they seem too 
confirm each others conclusion.

I think one of the nice things about PC is that it does actually generate 
variables that are a mixture of the original variables (in this case a 
combination of precipitation and temperature), and, in a sense, it seems to 
account for interaction between variables. Do you think this statement is 
true?

Best,

-- 
Corrado Topi

Global Climate Change & Biodiversity Indicators
Area 18,Department of Biology
University of York, York, YO10 5YW, UK
Phone: + 44 (0) 1904 328645, E-mail: ct529@york.ac.uk


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