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JOSHUA MIKE ET AL.
1000 times for each possible pairing. We note that at 5% resolution, three of the samples could not be processed by the CP distance due to lack of mesh points. (We observed a thres- hold of about 80 mesh points for the algorithm towork.) Because of this, our Mantel’s test with 5% resolution always involves sampling of P. tulipiformis. 2. Multidimensional Scaling (MDS) Algorithm.—
MDS is a tool used for dimensional reduction or presentation of complicated high-dimensional data (Cox and Cox 2001). We use this procedure to obtain a “visual realization” of the cluster structures that lie within the dissimilarity matrices. The MDS methodology differs little from the principal components analysis method (PCA)usedinAtwood andSumrall
(2012)
but presents some graphical advantages. For example, it allows us to remove the requirement of a global landmark alignment. For our aggregate clustering, MDS was used with six dimensions. This number is chosen based on a drop in the eigenvalues of the resulting covariance matrix, in a manner similar to what is done for PCA (Fig. 3). The dimension is fixed on all runs to obtain consistency across the clustering procedure. 3. Aggregate Clustering (AC).—AC is a well-
established method (Tan et al. 2005). It is particularly useful for determining clusters within data sets. It produces dendrograms,which convey grouping information at different scales
(Fig. 6). Within AC, we use the complete algorithm for all resolutions of DCP, because it is appropriate for high-dimensional data sets but does not assume our clusters have any particular shape (Ferreira and Hitchcock 2009). The Ward algorithm was used forDAt, because it is the best general agglomerative method,we expect elliptical clusters from our MDS graph (Ferreira and Hitchcock 2009), and the Ward algorithm yields resultsmost similar to those in Atwood and Sumrall (2012). To test for robustness, we performed a cross-
validation. The Ward AC was repeated on random samples of 90% of the (DCP or DAt) specimens. The resulting clustering is given labels and used to classify the remaining 10% by the cluster of each specimen’s nearest neighbor. This classification is then compared with the results of Atwood and Sumrall (2012) for benchmarking purposes; we aim to repro- duce the number of correct identifications. For this process, the data was always split into three clusters, with labels of (1) P. fredericki/ P. spicatus,(2) P. tulipiformis,and (3) P. pyriformis. The same labels are used to describe the benchmark group, although P. spicatus is not present in these data.
Results For each level of resolution, we performed
Mantel’s test with 1000 iterations over 1000 random samples of submatrices of DAt. Our results (Table 1, left) show that the correlation between the submatrices of DAt and DCP is consistently above 70%, with a very low p-value (p-value<10-6) at all levels of resolu- tion. Considering all possible multiple-way comparisons, we can express an overall p-value of at most 0.00072. Mantel’s test was also performed between
FIGURE 3. Diagram depicting the second through tenth eigenvalues of the MDS procedure for each dissimilarity matrix. For most matrices, there is a noticeable decline between eigenvalues 6 and 7, and so six eigenvalues were used for all clustering algorithms at all resolutions for consistency. The first eigenvalue is always much larger than the rest, and so is omitted for scale purposes.
different resolutions of CP distance. Our results (Table 1, right) show correlation between 100% and 50%resolution is above 97%. Similarly, 10% and 5% resolution are highly correlated. The correlations between higher and lower resolu- tions are lower, suggesting a qualitative change as resolution drops. The next step in the comparison process was
to implement the MDS algorithm on the dissimilarity matrices DCP and DAt at all five
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