PCoA
1. PCoA finds the optimal solution by eigenvalue decomposition. The PCoA solution is unique.
2. In a PCoA, the dissimilarities are not distorted in the ordination solution.
3. PCoA can be used to find the first ordination axes for a given dissimilarity matrix.
These axes are those that maximize the variance of the observations.
4. PCoA produces all ordination axes corresponding to a given dissimilarity matrix.
The ordination solution is exact.
5. The full matrix of PCoA axes allows one to precisely reconstruct the distances
among objects. Reason: the matrix of Euclidean distances among objects computed from the PCoA axes is strictly equal to the dissimilarity matrix subjected to
PCoA, whatever the dissimilarity function that has been used. Proof of that
property is found in Gower (1966) and in Legendre and Legendre (2012).
6. Hence, the set {dissimilarity function, ordination method} produces a unique
transformation of the data, and the solution is reproduced exactly if one runs
PCoA again on the same data, irrespective of the user or program, except for
possible inversion of the signs along any one axis.
7. A pseudo-R
2 statistic is computed as the sum of the eigenvalues of the axes of
interest (for example the first 2 axes) divided by the sum of all eigenvalues. It
indicates the fraction of the variance of the data represented in the reduced-space
ordination. This is a useful statistic to assess the ordination result.
Applications to ecological analysis – (1) PCoA produces ordinations of the
objects in reduced 2-D or 3-D space. (2) PCoA can also act as a data transformation
after computation of an appropriately chosen dissimilarity measure. The coordinates
of the objects in full-dimensional PCoA space represent the transformed data. They
can be used as starting point for new analyses, for example db-RDA (Chap. 6) or kmeans partitioning (Chap. 4). Recommendation: use PCoA in most applications.
5.7 Hand-Written PCA Ordination Function
To conclude this chapter, let us dive into the bowels of an ordination method...
198
5 Unconstrained Ordination
1. PCoA finds the optimal solution by eigenvalue decomposition. The PCoA solution is unique.
2. In a PCoA, the dissimilarities are not distorted in the ordination solution.
3. PCoA can be used to find the first ordination axes for a given dissimilarity matrix.
These axes are those that maximize the variance of the observations.
4. PCoA produces all ordination axes corresponding to a given dissimilarity matrix.
The ordination solution is exact.
5. The full matrix of PCoA axes allows one to precisely reconstruct the distances
among objects. Reason: the matrix of Euclidean distances among objects computed from the PCoA axes is strictly equal to the dissimilarity matrix subjected to
PCoA, whatever the dissimilarity function that has been used. Proof of that
property is found in Gower (1966) and in Legendre and Legendre (2012).
6. Hence, the set {dissimilarity function, ordination method} produces a unique
transformation of the data, and the solution is reproduced exactly if one runs
PCoA again on the same data, irrespective of the user or program, except for
possible inversion of the signs along any one axis.
7. A pseudo-R
2 statistic is computed as the sum of the eigenvalues of the axes of
interest (for example the first 2 axes) divided by the sum of all eigenvalues. It
indicates the fraction of the variance of the data represented in the reduced-space
ordination. This is a useful statistic to assess the ordination result.
Applications to ecological analysis – (1) PCoA produces ordinations of the
objects in reduced 2-D or 3-D space. (2) PCoA can also act as a data transformation
after computation of an appropriately chosen dissimilarity measure. The coordinates
of the objects in full-dimensional PCoA space represent the transformed data. They
can be used as starting point for new analyses, for example db-RDA (Chap. 6) or kmeans partitioning (Chap. 4). Recommendation: use PCoA in most applications.
5.7 Hand-Written PCA Ordination Function
To conclude this chapter, let us dive into the bowels of an ordination method...
198
5 Unconstrained Ordination