Lecture 6 Ordination Ordination contains a number of techniques to classify data according to predefined standards. The simplest ordination technique is cluster analysis. An easy but powerful technique is principal component analysis (PCA).

Factor analysis Is it possible to group the variables according to their values for the countries? T (Jan)T (July)Mean TDiff T GDP GDP/C Elev Factor 1 Factor 2 Factor 3 Correlations The task is to find coefficients of correlation etween the original variables and the exctracted factors from the analysis of the coefficiencts of correlation between the original variables.

Because the f values are also Z-transformed we have Eigenvalue

How to compute the factor loadings? The dot product of orthonormal matrices gives the unity matrix Fundamental theorem of factor analysis

F1F2 f 11 f 21 f 31 f 41 f 51 f 61 f 12 f 22 f 32 f 42 f 52 f 62 Z-trans- formed Factor values b Cases n Factors F Factors are new variables. They have factor values (independent of loadings) for each case. These factors can now be used in further analysis, for instance in regression analysis.

We are looking for a new x,y system were the data are closest to the longest axis. PCA in fact rotates the original data set to find a solution where the data are closest to the axes. PCA leaves the number of axes unchanged. Only a few of these rotated axes can be interpreted from the distances to the original axes. We interpret the new axis on the basis of their distance (measured by their angle) to the original axes. The new axes are the principal axes (eigenvectors) of the dispersion matrix obtained from raw data. X1 Y1 X’1 Y’1 PCA is an eigenvector method Principal axes are eigenvectors.

The programs differ in the direction of eigenvectors. This does not change the results but might pose problems with the interpretation of factors according to the original variables.

Pincipal coordinate analysis PCoA uses different metrics to generate the dispersion matrix

Using PCA or PCoA to group cases v A factor might be interpreted if more than two variables have loadings higher than 0.7. A factor might be interpreted if more than four variables have loadings higher than 0.6. A factor might be interpreted if more than 10 variables have loadings higher than 0.4.

Correspondence analysis (reciprocal averaging, seriation, contingency table analysis) Correspondence analysis ordinates rows and columns of matrices simultaneously according their principal axes. It uses the  2-distances instead of correlations coefficients or Euclidean distances.  distances Contingency table

We take the transposed raw data matrix and calculate eigenvectors in the same way Correspondence analyis is row and column ordination. Joint plot

The plots are similar but differ numerically and in orientation. The orientation problem comes again from the way Ecxel calculates eigenvalues. Row and column eigenvectors differ in scale. For a joint plot the vectors have to be rescaled.

Reciprocal averaging Sorting according to row/column eigenvalues rearranges the matrix in a way where the largest values are near the matrix diagonal.

=los() =(B85*B$97+C85*C$97+D85*D$97+E85*E$97)/$F85 =(H85-H$94)/H$95 Seriation using reciprocal averaging Repeat until scores become stable Weighed mean Z-transformed weighed means