Presentation and Interpretation of PCA Results

A plot of the samples on the PC axes is the primary output of PCA. This reduced space plot displays the relative positions of samples in multivariate space in fewer dimensions. Although a simple scatterplot of samples on the PC axes might provide some useful information on data structure - for example, whether samples are clustered together or occupy a gradient -additional information is usually included on the plot to assist interpretation. This can be illustrated by a PCA of triplefin (Pisces: Tripterygiidae) fish abundance at a range of sites in northeastern New Zealand. The data were collected from sites with different exposure and location characteristics and so graph symbols could be used to reflect these characteristics of the samples (Figure 2a). This contributes to the interpretation of patterns in the samples based on additional information and is an informal exploration that can identify hypotheses about causal processes. There appears to be a gradient in triplefin assemblages across exposure gradients from sheltered to exposed sites, but assemblages on offshore exposed and sheltered mainland sites are distinct from the semiexposed and exposed mainland sites (Figure 2a). Information about the value of individual variables can also be included in the reduced space plot to identify which variables are responsible for the observed patterns. If plot symbols are scaled to reflect the value of a single variable in the analysis we see that the triplefin Forsterygion varium was more abundant in semiexposed and exposed mainland sites, but relatively uncommon on sheltered mainland sites and practically absent from exposed offshore sites (Figure 2b). Presenting multiple bubble plots of species abundances is often not a suitable option due to the large number of graphs required; so a more compact and formal presentation of the dependent variables can be generated by plotting the eigenvectors into the reduced space plot (Figure 2c). This presentation is known as a biplot,

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Figure 2 Reduced space plots of a principal components analysis (PCA) of triplefin fish (Family: Tripterygiidae) abundances. The PCA was calculated using the covariance matrix of the square root of the proportional species abundance in a sample. Percent variance explained is derived from the eigenvalues. Note the equal scaling of the x and y axes - this ensures the ordination space is not distorted in the plot. (a) Information about wave exposure and location of sites is added by changing plot symbols: circle indicates sheltered mainland; diamond indicates semiexposed mainland; square indicates exposed mainland; downward triangle indicates exposed offshore. (b) Symbol size can be scaled in proportion to the value of the variables, in this case triplefin species abundance. Forsterygion varium is characteristic of exposed and semiexposed mainland sites, uncommon in sheltered sites, and practically absent from offshore exposed sites. (c) A joint presentation of the reduced space and eigenvectors forms a biplot. Distances between sites approximate the Euclidean distance of the transformed data and the eigenvectors are the projection of the original species axes into the space. Eigenvectors have been rescaled to half of their original value for clarity in the plot. The circle is the equilibrium contribution of the eigenvectors. Species outside this circle are influential in defining the ordination space. In this example, Notoclinops segmentatus is characteristic of exposed offshore and exposed mainland sites, F. varium is characteristic of exposed and semiexposed mainland sites, and F. lapillum is characteristic of sheltered sites.

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and follows from the mathematics of PCA in which the samples are projected into the space by premultiplication of the eigenvectors. In this example the importance of F. varium in characterizing mainland exposed/semiex-posed sites is clear from the length and direction of its eigenvector. Notoclinops segmentatus is characteristic of exposed sites, regardless of mainland or offshore status, and F. lapillum is characteristic of sheltered sites. Examination of bubble plots and species-frequency histograms at each site support this interpretation.

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