Phylogenetics - IB 200B 15 Feb Morphometrics

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Egbert Armstrong
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1 Morphometrics Morphometrics is the branch of mathematics studying the metrical and statistical properties of shapes and shape changes of geometric objects like molecules, fossils, brains, bird wings, ancient handcraft, modern cars, etc. On Growth and Form Sir D'Arcy W. Thompson, The mathematization of natural history. Thompson comes out punching with an array of arguments and stresses the importance of understanding the natural world quantitatively, but is limited philosophically to descriptive and classificatory methods (although embryology had already embarked on experimental manipulation). Bivariant data plots (X,Y) -- Correlations between morphological characters. Data exploration -- should I plot everything? Two morphological values can be combined into a single variable or ratio, Ratios are excellent for removing size, weight, etc. from a term, but what happens to the variance? Is it still normally distributed in the new term?

2 Multivariant Morphometrics Multivariant Methods Principal component analysis (PCA) has been a commonly used mathematical procedure that transforms a number of (possibly) correlated variables into a (smaller) number of uncorrelated variables called principal components. The first principal component accounts for as much of the variability in the data as possible (typically size), and each succeeding component accounts for as much of the remaining variability as possible.

Phylogenetics - IB 200B 15 Feb 2011 Principal component analysis is performed on covariance (scaled sums of squares and cross products), or, correlation (sums of squares and cross products

3 Phylogenetics - IB 200B 15 Feb 2011 Principal component analysis is performed on covariance (scaled sums of squares and cross products), or, correlation (sums of squares and cross products from standardized data) matrices. A correlation matrix is used when the variances of individual variates substantially differ or the units of measurement of the individual variates differ. Trus

4 Truss techniques Uses distances between landmarks to form a truss that represents the configuration of the organism. Homologous landmarks characters are determined and distances between these landmarks measured. The truss measurements can be treated as Cartesian coordinates and can be averaged and standardized and analyzed using principal Procrustes analysis -- The shape of an object can be described by the coordinates of a set of well defined points or landmarks. Coordinate data from similar points across a group of individuals can be used to compare and contrast their shapes provided they have been superimposed (i.e., translated, rotated and scaled) in a common coordinate system In this coordinate system differences in the relative positions of the corresponding points on different configurations are directly reflected by the differences In their coordinates. Point selection.

5 Figure 3. Full Procrustes fit of the bivalve mollusk Mercenaria campechiensis. Data are 13 landmarks for 52 shells (Bush, A. 1999, unpub. M.S thesis, Geological Sciences, VT) Geometric Morphometrics: Geometric morphometrics is a collection of approaches for the multivariate statistical analysis of Cartesian coordinate data, usually (but not always) limited to landmark point locations. The "geometry" referred to by the word "geometric" is the geometry of Kendall's shape space: the estimation of mean shapes and the description of sample variation of shape using the geometry of Procrustes distance. The multivariate part of geometric morphometrics is usually carried out in a linear tangent space to the non- Euclidean shape space. More generally, it is the class of morphometric methods that preserve complete information about the relative spatial arrangements of the data throughout an analysis. As such, these methods allow for the visualization of group and individual differences, sample variation, and other results in the space of the original specimens. (Slice et al. AND

6 Patellogastropoda Sorbeoconcha A B A-D A-D Q 2Q 3Q 1 4q 1Q 2Q 3Q 4q Original Original C D A-D 1Q 2Q Dummy 3Q 4q 1 A-D 1Q 2Q Dummy 4q 3Q 1 cell number E F Original Dummy

Scaphopoda Patellogastropoda Vetigastropoda Neritopsina Sorbeoconcha

06 Opist Sorb C 0.08 0.06 Veti Arch Val 0.04 Veti 0.04 0.02 0.00-0.02-0.

04 Pat HTU2 Nerit HTU3 Pul HTU6-7 HTU4-5 Opist Sorb B -0.06-0.08-0.15-0.10-0.

08 Opist Sorb Arch Pul Nerit -0.10-0.12-0.10-0.08-0.06-0.04-0.02 0.00 0.02 0.04 0.

10-0.08-0.06-0.04-0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.08 Veti 0.06 0.04 0.02 0.00 Pat -0.

7 Scaphopoda Patellogastropoda Vetigastropoda Neritopsina Sorbeoconcha Architaenoglossa Valvatoidea Opisthobranchia Pulmonata A Opist Sorb C Veti Arch Val 0.04 Veti Pul Val Scaph Nerit Pat Pat HTU2 Nerit HTU3 Pul HTU6-7 HTU4-5 Opist Sorb B Opist Sorb Arch Pul Nerit Val PCA 1 Arch Scaph Veti Pat Poly D Scaph HTU Veti Pat Arch Val HTU2 HTU3 Pul HTU6-7 HTU4-5 Opist Nerit Sorb Scaph HTU PCA 1

8 Aligned landmark data - In late 2010 Catalano et al. proposed a new method for the use of landmark data in phylogenetics. Their method uses Farris optimization to estimate ancestral values that minimize the distance between ancestor descendant morphologies through the tree. They contend that existing superimposition methods of to align landmarks (finding the best fit to the data through rotation, translation and size) are satisfactory for comparing shape in pairwise comparisons, but insufficient for evaluating shape changes along a phylogeny because shape change is determined from the changes in the relative position of each individual landmark. The method they propose explicitly evaluates change between observed taxa and hypothetical ancestors and is parsimony based. The choice of alignment algorithms is important and one that minimizes the sum of the linear distances is preferred. The final section of the paper provides a good overview of the supposed incompatibility between morphometrics and cladistics, which the authors argue is not the case. Figg.1.Comparisonbetween(a)Farrisoptimizationand(b)3D landmarkoptimization.infarrisoptimization,thesumofthe numericaldifferencesbetweenstatesisminimized.inlandmark optimization,thedistancesbetweenlandmarkpositionsareminimized.

9 Fig.4.Ahypotheticalexampleshowingancestrallandmarkpositionscalculatedwith spatialoptimization,showingthedifferencebetweenalignmentsproducedwith(a) resistantfittheta rhoanalysisor(b)generalizedleast squaresprocrustesalignment.the changesin(b)occurattheexpectednode,butarespreadoverallthelandmarksinstead ofonlythelandmarksrepresentingthethumb. Fig.5.ThesamealignmentasFig. 4a,butwithancestrallandmark positionscalculatedusingleastsquares parsimony insteadof spatialoptimization.thechanges arespreadoveralltreenodes.

Phylogenetics - IB 200A Morphometrics

Phylogenetics - IB 200A Morphometrics Phylogenetics - IB 200A 2012 Morphometrics Morphometrics is the branch of mathematics studying the metrical and statistical properties of shapes and shape changes of geometric objects like molecules, fossils,