Principal component analysis and redundancy analysis

Abstract

In the previous chapter, Bray-Curtis ordination was explained, and more re-cently developed multivariate techniques were mentioned. Principal component analysis (PCA), correspondence analysis (CA), discriminant analysis (DA) and non-metric multidimensional scaling (NMDS) can be used to analyse data without explanatory variables, whereas canonical correspondence analysis (CCA) and re-dundancy analysis (RDA) use both response and explanatory variables. In this chapter, we present PCA and RDA, and in the next chapter CA and CCA are dis-cussed. The chapter is mainly based on Ter Braak and Prentice (1988), Ter Braak (1994), Legendre and Legendre (1998) and Jolliffe (2002). More easy readings are chapters in Kent and Coker (1992), McCune and Grace (2002), Quinn and Keough (2002), Everitt (2005) and especially Manly (2004) 12.1 The underlying principle of PCA Let Yy be the value of variable7 (j = 1, ..^V) for observation i (i = 1, ..,M). Most ordination techniques create linear combinations of the variables: Zu = c n Y u + C] 2 Y i2 + ... + c 1N Y iN (12.1) Assume you have a spreadsheet where each column is a variable. The linear combination can be imagined as multiplying all elements in a column with a par-ticular value, followed by a summation over the columns. The idea of calculating a linear combination of variables is perhaps difficult to grasp at first. However, think of a diversity index like the total abundance. This is the sum of all variables (all c t jS are one), and summarise a large number of variables with a single diversity index. The linear combination, Zi = (Z n ,...,Z M1 y, is a vector of length M, and is called a principal component, gradient or axis. The underlying idea is that the most im-portant features in the N variables are caught by the new variable Z b Obviously one component cannot represent all features of the N variables and a second com-ponent may be extracted: Z i2 = c 21 Y n + c 22 Y i2 + ... + c 2N Y iN (12.2)