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JOSHUA MIKE ET AL.


FIGURE 5. k-means was performed with the MDS embedding of 100% resolution DCP, resulting in the figures shown. Left, k=2; center, k=3; right, k=4. These divisions help support the results of our aggregate clustering.


the pyriform and godoniform groups quite naturally. On the other hand, it appears that the vertical component in the plot captures the concavity between ambulacra. However, since the MDS algorithm does not embed the data set directly onto coordinate axes (as does the alternative method, PCA), these interpreta- tions are rather hard to prove. The final step in the analysis consists of


deploying AC on the dissimilarity matrices DCP and DAt at all five resolutions. Like MDS, this method also does not require the matrices to be of equal size. Dendrograms of these clusters can be seen (Fig. 6).Wecan also see the progression of variance as clusters are aggre- gated (Fig. 7). This progression of variance helps us to determine that there are three clusters present in our data. Figure 5, which depicts k-means as performed on our MDS embedding, also supports our clustering choice. k-means with k=3 splits our data into the same three groups as the AC does. A visual analysis of the AC outcome of


Figure 6 shows that the CP-distance dissim- ilarity matrix manages to clearly separate P. tulipiformis from the cluster containing P. fredericki and P. spicatus samples. This improves the result of the comparison study by Atwood and Sumrall (2012), in which the separation of P. tulipiformis and P. fredericki is incomplete, and a few P. fredericki samples are (incorrectly) categorized as P. tulipiformis. Furthermore, while both old and new methods separate P. pyriformis from the other species, CP distance–based analysis does a better job. Indeed, it separates the data into two distinct,


large clusters that correspond to the pyriform and godoniform groups; moreover, it is evident that this clustering happens at a coarser level than the separation of species. Finally, we see that as the resolution of scans decreases, the advantages of CP-distance method over the traditional techniques disap- pear: at 20% resolution, CP distance no longer separates P. tulipiformis and P. fredericki better than DP distance. However, even at 5% resolution, the larger clusters of the pyriform and godoniform groups still clearly separate, and some (coarse) information still can be obtained. Figure 7 depicts the results of our cross-


validation procedure. For each trial, Ward AC was used to cluster 90% of our data. These clusters were used to classify the remaining 10%, and these were compared to the bench- mark group. The details of the procedure can be seen in Table 2.


Discussion One of the most evident advantages of the


CP distance–based methodology over the DP- distance procedure is the speed at which samples can be analyzed. Although both analyses require specimens to be scanned, the proposed methodology nearly eliminates the tedious (and human error–prone) landmark- picking phase. An evident advantage of the traditional method in the blastoids example is that it can be deployed using only a portion of the surface scan, because the full set of land- marks lies entirely on one side of the specimens


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