NONLANDMARK CLASSIFICATION IN PALEOBIOLOGY
classification of species. Specifically,DP distance finds the minimum value that the sum of all distances between corresponding points within the two landmark sets achieves as the config- urations are “aligned” in space(seeAtwood and Sumrall [2012] for details). In contrast, the CP-distance method developed in Lipman et al. (2013) is an extension of the DP distance. It incorporates full geometric information (read- ily available in 3D scans) such as curvature into the analysis. DP distance is computed via minimizing a sum of distances over a discrete setoflandmarks,whereas CP distance is computed viaminimizing integrals of geometric quantities. Both methods are scale and rigid- motion invariant. This is to say, the outcomes of the calculations are independent of the size and position/orientation of the scans and resulting landmarks. An overview of the CP-distance algorithm appears in the Supplementary Mate- rials. For specific details, see Lipman et al. (2013). In this study we use DP distance and CP
Atwood and Sumrall (2012) to verify the
distance to generate dissimilarity matrices associated to each data set. Each matrix’s i–j entry corresponds to the distance (or dissim- ilarity) between specimen i and specimen j. These matrices are symmetric, nonnegative,
and have only zeros along the diagonal. We first use the DP distance on pairs of landmark configurations fromAtwood and Sumrall (2012) to generate the matrix DAt, in an attempt to reproduce the results of Atwood and Sumrall (2012) in the dissimilarity-matrix language of this study. We then use the CP-distance algorithm on pairs of new 3D scans to generate the
dissimilaritymatrixDCP.Finally,wedevisea methodology for comparing these matrices and determine whether they carry the same “clustering information” within them. Our specimen scans were performed at the
(medium) resolution of 4400 points per square inch. At this resolution, a mesh of about 5000 points represents a typical specimen (Fig. 2). To determine the breaking point of our method, we reran the whole matrix- comparison process using 3D scans artificially downsampled to resolutions of 50%, 20%, 10%, and 5% of the original scans. Downsampling was carried out using quadric edge collapse in MeshLab.
699 In this work we deploy three methods for
comparing dissimilarity matrices and their cluster information:
1. Mantel’s test (performed on two dissimilar- ity matrices of equal size) is used to check for correlations between each distance’s interspecimen measurements.
2. A multidimensional scaling algorithm is used for visualizing cluster information.
3. Aggregate clustering is applied to under- stand groupings at different scales.
1. Mantel’s Test Methodology.—Mantel’s test
is a standard statistical tool used to compare distance matrices, often used in a biological setting with actual distances (Sokal and Rohlf, 2012). Here we use Mantel’s test to compare the distance matrices DCP and DAt. It is important to note that both matrices represent mathematical metrics on the same kinds of objects,making this comparison a classical and direct use of Mantel’s test. The dimensions of these matrices must be equal for Mantel’s test to be used. Because we were not able to obtain full 3D scans of all the specimens of the Atwood and Sumrall (2012) study, the dissimilarity matrices DAt and DCP are of different sizes. To overcome this and at the same time use the most information possible, we proceed as follows. We randomly choose 20 specimens from within Atwood’s landmark data (52 specimens total) and produce a “submatrix” of DAt corresponding to the intersample DP distances between these 20 specimens; moreover, we make sure that the number of specimens from each species matches the amount in DCP. This sampling method means that the results of our test will address how similar the metrics are when considering the comparison of different species. We then carry out Mantel’s test between DCP and the randomly sampled submatrix of DAt. The sampling process is repeated 5000 times, 1000 times for each one of the four possible resolutions: 100%, 50%, 20%, and 10%, and once to cross-compare DAt with itself to see the effect of the sampling method on correlation values. In addition, Mantel’s test was also per-
formed pairwise between different resolutions ofDCP. This process was repeated 10,000 times,
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