Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

13.3. Principal Component Analysis for Compositional Data 347At first sight, it might seem that no real difficulty is implied by theconditionx i1 + x i2 + ···+ x ip =1, (13.3.1)which holds for each observation. If a PCA is done on x, there will be a PCwith zero eigenvalue identifying the constraint. This PC can be ignored becauseit is entirely predictable from the form of the data, and the remainingPCs can be interpreted as usual. A counter to this argument is that correlationsand covariances, and hence PCs, cannot be interpreted in the usualway when the constraint is present. In particular, the constraint (13.3.1)introduces a bias towards negative values among the correlations, so thata set of compositional variables that are ‘as independent as possible’ willnot all have zero correlations between them.One way of overcoming this problem is to do the PCA on a subset of(p−1) of the p compositional variables, but this idea has the unsatisfactoryfeature that the choice of which variable to leave out is arbitrary, anddifferent choices will lead to different PCs. For example, suppose that twovariables have much larger variances than the other (p − 2) variables. If aPCA is based on the covariance matrix for (p−1) of the variables, then theresult will vary considerably, depending on whether the omitted variablehas a large or small variance. Furthermore, there remains the restrictionthat the (p−1) chosen variables must sum to no more than unity, so that theinterpretation of correlations and covariances is still not straightforward.The alternative that is suggested by Aitchison (1983) is to replace x byv = log[x/g(x)], where g(x) =( ∏ pi=1 x i) 1 p is the geometric mean of theelements of x. Thus,thejth element of v isv j = log x j − 1 p∑log x i , j =1, 2,...,p. (13.3.2)pi=1A PCA is then done for v rather than x. There is one zero eigenvaluewhose eigenvector is the isometric vector with all elements equal; theremaining eigenvalues are positive and, because the corresponding eigenvectorsare orthogonal to the final eigenvector, they define contrasts (thatis, linear functions whose coefficients sum to zero) for the log x j .Aitchison (1983) also shows that these same functions can equivalentlybe found by basing a PCA on the non-symmetric set of variables v (j) , wherev (j) = log[x (j) /x j ] (13.3.3)and x (j) is the (p−1)-vector obtained by deleting the jth element x j from x.The idea of transforming to logarithms before doing the PCA can, of course,be used for data other than compositional data (see also Section 13.2).However, there are a number of particular advantages of the log-ratio transformation(13.3.2), or equivalently (13.3.3), for compositional data. Theseinclude the following, which are discussed further by Aitchison (1983)