# How to interpret variation explained by principal coordinates?

I have recently seen a couple of Principal Coordinates Analysis (PCoA) projection plots which show "percentage variation explained" by the respective principal coordinates. Given that the analysis is not done on a co-variance matrix (it is usually done on some ecological metric) it seems wrong to imply that the magnitude of the eigenvalues is in any way related to the variance explained.

My question is, what would the best interpretation of the eigenvalues be?

• Principal coordinate analysis, or Torgerson's metric MDS, treats the input distances as (noised) euclidean ones. The larger the noise the more pronounced will be the geometrical inconvergence among the distances and the greater will be negative eigenvalues. Positive eigenvalues describe the "undisturbed" portion of the euclidean scatter. – ttnphns Aug 29 '13 at 15:31
• Thanks you for the comment. Let me see if i got this right. You're basically saying that the eigenvalues are a reflection of the magnitude of the vector in the "undisturbed" euclidean scatter. Thus in no way related to the variation between points. Is that about right? – Paul Igor Costea Aug 29 '13 at 15:48
• Why? "Scatter" and "variation" are about synonyms. – ttnphns Aug 29 '13 at 16:15
• Fair enough, scatter and variation are rather the same things. Then , how does one handle the negative values to get the percentages? Ignore them (make them zero) or simply make them positive? – Paul Igor Costea Aug 31 '13 at 15:22
• Zero off negative eigenvalues. Then rescale the positive ones to sum up to the original sum of all eigenvalues. Compute percentages of "variance explained". – ttnphns Aug 31 '13 at 15:47

In preparing a workshop on ordination techniques, I realized I was having the same difficulty in interpreting the eigenvalues of principal coordinate analysis for the same reasons that have puzzled you (@Paul Igor Costea), so I started digging around for some answers.

I have a few books on multivariate statistics that are not for the statistically faint of heart, and occasionally certain explanations get lost in some heavy matrix algebra (not the best for an 101 on ordinations). The best answer I found was actually in an overview of ordination methods for non-experts by Lengendre & Birks 2012 in a chapter of "Tracking Environmental Change using Lake Sediments".

The eigenvectors are typically much easier to interpret as they are essentially the coordinates (in reduced space) of a given object along a given axis. The eigenvalues, however, represent:

"the variance (not divided by degrees of freedom) of the objects along that axis." (Lengendre & Birks 2012)

This is the most concise and precise interpretation I have found. While it is true that PCoA is not computed on a covariance matrix but on a distance matrix, PCoA and PCA are very similar, and the following simple example (from the same chapter) explains the mathematical relationship between the eigenvalues computed by each technique:

"From an object-by-variable data matrix Y, compute matrix D of Euclidean distances among the objects. Run PCA using matrix Y and PCoA using matrix D. The eigenvalues of the PCoA of matrix D are proportional to the PCA eigenvalues computed for matrix Y (they differ by the factor (n – 1) [i.e.the degrees of freedom]), while the eigenvectors of the PCoA of D are identical to matrix F [i.e. the matrix of eigenvectors] of the PCA of Y. Normally, one would not compute PCoA on a matrix of Euclidean distances since PCA is a faster method to obtain an ordination of the objects in Y that preserves the Euclidean distance among the objects.This was presented here simply as a way of understanding the relationship between PCA and PCoA in the Euclidean distance case. The real interest of PCoA is to obtain an ordination of the objects from some other form of distance matrix more appropriate to the data at hand — for example, a Steinhaus/Odum/Bray-Curtis distance matrix in the case of assemblage composition data."

• "Given that the analysis is not done on a covariance matrix," why does this interpretation apply at all? What is the connection? – whuber Feb 27 '15 at 0:19
• @whuber, indeed, sorry for the confusion, hope the above edits clarify... – Xavier GB Feb 27 '15 at 0:55