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Chronological clustering of fossil stickleback fish In this section, we show how chronological clustering can be used to detect discontinuities in a multivariate time series data set of fossil stickleback fish (Gasterosteus doryssus). The data were taken from Table 1 in Bell and Legendre (1987), a paper which is downloadable from Piere Legendre's website. We strongly advise to download their paper, as results produced here are merely used to demonstrate chronological clustering. A detailed description of the data and ecological interpretation of the results are given in the Bell and Legendre paper. The data in Table 1 in Bell and Legendre (1987) are morphotypes of Gasterosteus doryssus at (approximately) 5000 year time intervals. There are six morphotypes, denoted by A, B, C, D, E and F, see Figure 1 in Bell and Legendre (1987). Chronological clustering is explained in detail in Legendre and Legendre (1998), and Legendre et al. (1985). A short introduction is given in Bell and Legendre (1987), the Brodgar manual, and on the Brodgar website. Figure 1 shows a time series plot of the 6 morphotypes. One can clearly detect a change in patterns from time point 20 onwards. Results of chronological clustering are presented in Figure 2. The figure shows results for different values of alpha, which is a clustering intensity parameter. For small values of alpha, the major breakpoints in the time series are shown, whereas larger values of alpha visualise smaller scale variation.
Figure 1. Time series plot of 6 morphotypes.
Figure 2. Results of chronological clustering. Small values of alpha show the most important breakpoints. Note that the results show that the most important breakpoint is at sample number 21 (the start of a new group). This means that relationships between the variables (as measured by Whittaker's index of association in this case) has changed from time point 21 onwards. For alpha=0.05, we obtain two breakpoints; at sample numbers 15 and 22. Lower values of alpha show more small scale variation during the first 15 time points. To aid interpretation, principal coordinate analysis can be applied on the same distance matrix (obtained by Whittaker's index of association), see Figure 3. Both PCO ordination plots are the same, except that labeling of the upper graph is based on the grouping obtained by alpha=0.05 and the lower graph on alpha=0.1. Note that there is a clear distinction between the groups, and that one group is rather different from the others. The lower graph suggest that the two points on the right hand side (labeled as 6) are rather different from the three samples that were in the same group at the alpha=0.05 level (labeled by a 3 in the upper graph).
Brodgar can also perform an a posteriori test. The numerical output for alpha=0.1 is given by: GROUP EXPANSION TESTS
The group expansion test (above) will show whether breakpoints are abrupt, or whether the groups could me expanded on either side. Look for H values larger than 0.1 (alpha). In most cases, the groups can be further expanded. The following output shows whether non-adjacent groups are similar. H values larger than 0.1 (alpha) indicate that two groups are similar. These are the groups that are connected by arrows (alpha=0.1) in Figure 2 in Bell and Legendre (1987). TESTS AMONG GROUPS [ 5 .. 7] against [ 8 .. 11] H: 0.02857
[ 12 .. 14] against [ 15 .. 20] H: 0.01190 [ 15 .. 20] against [ 21 .. 22] H: 0.03571 [ 21 .. 22] against [ 23 .. 26] H: 0.06667
References. Bell, M.A. and Legendre, P. (1987). Multicharacter Chronological Clustering in a Sequence of Fossil Sticklebacks. Systematic Zoology, 36: 52-61. Legendre, P., Dallot, S. and Legendre, L. (1985). Succession of species within a community: Chronological clustering, with application to marine and freshwater zooplankton. Am. Nat. 125: 257-288. Legendre, P., and L. Legendre. 1998. Numerical ecology. Second edition. Elsevier, Amsterdam, The Netherlands. The first two papers can be downloaded from: http://www.fas.umontreal.ca/biol/legendre/reprints/ |
