Introduction
Theory, Examples & Exercises

 Constrained ordination



Principal component analysis (PCA) is a linear unconstrained ordination method. It is implicitly based on Euclidean distances among samples (see the algorithm below), and as such, it is not suitable for heterogeneous compositional datasets with many zeros (so common in case of ecological datasets with many species missing in many samples). It is suitable on quantitative variables (could be negative), and also presenceabsence data; it cannot handle qualitative variables.
(a) Use the matrix of samples x species (or, generally, samples x descriptors), and display each sample into the multidimensional space where each dimension is defined by an abundance of one species (or descriptor). In this way, the samples will produce a cloud located in the multidimensional space.
(b) Calculate the centroid of the cloud.
(c) Move the centres of axes to this centroid.
(d) Rotate the axes in such a way that the first axis goes through the cloud in the direction of the highest variance, the second is perpendicular to the first and goes in the direction of the second highest variance, and so on. The position of samples on resulting rotated axes are sample scores on ordination axes.
Fig. 1 (from Legendre & Lengendre 1998) illustrates this algorithm on a very simple case with only two species (descriptors) and five samples. Fig. 2 illustrates the same logic on the data cloud in threedimensional space (three species/descriptors).
When considering ecological data, PCA has three main applications:
1) Describe correlation structure between different variables, e.g. environmental variables measured for each sample, or species characteristics (traits) measured for individual species. In this case, the variables need to be standardized to zero mean and unit standard deviation, otherwise, the variable with higher absolute values or variance would be more important in the analysis. Resulting PCA ordination can show the main dimensions of variation in the data. This information can be further processed in several ways:
2) Analysis of relatively homogeneous species composition data. “Relatively homogeneous” means that in these data, we assume that species response along the (hypothetical) environmental gradient can be described by a linear relationship. Such data should contain few zeros, thus lowering the issue of the double zero problem, to which Euclidean distance is sensitive (see Ecological resemblance > Distance indices > Euclidean distance). If applied on heterogeneous dataset with many zeros, the result often shows strong horseshoe artefact, when sites with no species in common appear very close to each other in the ordination diagram.
3) Relatively recently was suggested that PCA applied on pretransformed species composition data (e.g. by Hellinger) can solve the problem of Euclidean distances in PCA and double zeros. In case of Hellinger transformation, Euclidean distance (implicit in PCA) applied on Hellingertransformed raw species composition data results in PCA representing Hellinger distances between samples, and Hellinger distances are known to be not influenced by double zero problem. This method is called transformationbased PCA (tbPCA) and is described in a separate section. Note, however, that not everybody agrees that this is a good idea (see ESA 2010 presentation of Peter Minchin & Lauren Rennie on this topic).
PCA axes are sorted in descending order according to their eigenvalues, i.e. the amount of variance they represent. There are several options how to decide which axes are important and representative (e.g. for visualizing data onto ordination diagrams). Two of these options are the following ( Borcard et al. 2011):
There is no single way how to display sites and variables (species) in the same biplot diagram (i.e. diagram showing two types of results, here sites and variables), that's why there are two ways of scaling results^{1)}:
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The circle sometimes projected onto ordination diagram to estimate the importance of individual species/descriptors/variables. The radius is calculated as √(d/p), where d is the number of displayed PCA axes (usually d = 2) and p is the number of variables (columns in the dataset). The arrow of the same length as the circle radius contributes equally to all axes in PCA; arrows longer than circle radius make a higher contribution than average and can be interpreted with confidence (in the context of given number of ordination axes, here two, Fig. 5).