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## Introduction to Geometric Morphometrics

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**Introduction to Geometric Morphometrics**François Gould, Ph.D.**What is geometric morphometrics?**• A increasingly common buzzword**What is geometric morphometrics?**• A toolkit of methods for the numerical analysis of 2D and 3D shape variation. • Several different approaches!**What does geometric morphometrics examine?**• Form: aspects of geometry invariant to rotation, translation, reflection • Most geometric morphometric approaches also scale: leave “pure shape”. • Size can be examined separately with a metric**About Size and Shape**• Key concepts in understanding the morphology of organisms. • Size: absolute difference in magnitude between objects • Shape: relative differences in geometry between organisms • These concepts are tricky!**Allometry and Scaling**• The allometric relation is a power relation: • y=m*xb or ln(y)=b*ln(x)+ln(m)**Where did geometric morphometrics come from?**• Result of a synthesis of two trends (Bookstein, 1991)**Quantitative representation of shape I**• Role of coordinate points: the landmark concept 3 0 5 -5 -5 7**Quantitative representation of shape II**• Mathematical theory of shape space • A space where each point defines a single configuration of landmarks • Classical shape space non-euclidean: projection**Getting into shape space: the Procrustes transform**• Translate, rotate, scale. Least squares fit • Creates Procrustescoordinates**Analysis of Procrustes coordinates**• Project the shapes into a tangent space passing through the mean shape • Calculate the variance-covariance matrix of the projected procrustes coordinates • These can either be analysed directly (Principal components) or using the Thin Plate Spline (Partial and Relative warps)**The Procrustes transform: problems**• Assumptions about variance: equal distribution • Iterative algorithm without true solution: data dependent • May be statistically problematic: requires estimation of nuisance parameters**Other approaches**• Bookstein coordinates, Resistant fit: different variance assumptions • EDMA: Euclidean distance matrix analysis • Calculates all pairwise distances and compares them as ratios • Does not require estimation of nuissance parameters • Eigenshape approaches: Phi function (angle change). Ideal for outlines.**On landmarks**• Pivotal in geometric morphometrics**Criteria for landmark selection**• Landmark homology • Classical three-tier formulation (Bookstein 1991) • Type I: meeting of tissue types (“true” landmarks) • Type II: maxima of curvature (orientation independent) • Type III: extremal points**Limitations of the Bookstein paradigm**• Many structures cannot be reduced to type I landmarks**Methods for the analysis of curves and surfaces**• Semilandmarks approaches (Bookstein, 1997) • Fourier transform • Eigenshape approaches (Macleod and Rose, 1993, Macleod 1999)**Limitations of the Bookstein paradigm**• Cannot deal with novel structures. PERISSODACTYL ARTIODACTYL**What is landmark homology?**• Individual landmarks are not biologically homologous. • Moving towards a recognition of importance of homology of the underlying biological structure. • Even Bookstein now agrees! (Gunz et al., 2005) • Think about the BIOLOGY, not the theory**Uses of Geometric Morphometrics**• Data exploration • Exploration of distribution of data (ordination) • Exploration of coordinated shape change (visualisation) • Source of hypothesis • Hypothesis testing: • Development studies (fluctuating asymmetry, integration) • Evolutionary (modularity, morphological evolution) • Ecomorphology**Choose the best tool**• What is your biological question? • Data exploration • Hypothesis testing • Type of data: • 2D or 3D? • Landmark? Outline? Surface? • Sample size?**Collecting your data**• From specimens? • Microscribe • From photographs • ImageJ • Be VERY careful about parallax • From 3D models • Laser scans • CT scans**Measurement Error**• Morphometric data can be assessed for error • Global measurement error • Error associated with landmarks • Need to assess each stage of data collection protocol for error • Error less of a problem in cross-taxonomic studies**Transforming your data into shape coordinates: WISYWIG**software • Written by researchers, increasingly powerful and easy to use • TPS suite • MorphoJ • WinEDMA • Can be found at SUNY morphometrics website • http://life.bio.sunysb.edu/morph/ • REFLECT BIASES OF AUTHORS!**Transforming your data into shape coordinates: the hard way**• Can code analysis in Matlab, Mathematica and R. • Full geometric morphometrics R package: Geomorph(Adams, 2012) • Often necessary if working with analyses outside what other researchers do. • Get on the morphmet listserv: active community.**Analysis**• Exploratory analysis • Ordination (PCA or Relative warps) • Shape change visualisation • Discrimination • CVA • Discriminant function • Hypothesis testing • MANOVA • Regression • 2 Block Partial least squares**Exploring your shape space**• All methods allow visualisations of changes in shape. • HOWEVER, need to know if you are in a shape space or not: different approaches to modelling in shape space (e.g. PCA) versus non-shape space (e.g. CVA). • Do not overinterpret your shapes: do not extrapolate beyond data**Example: Ecomorphological pattern in distal femoral**variation