## Morphometrics with RThis book aims to explain how to use R to perform morphometrics. Morpho- tric analysis is the study of shape and size variations and covariations and their covariations with other variables. Morphometrics is thus deeply rooted within stat- tical sciences. While most applications concern biology, morphometrics is becoming common tools used in archeological, palaeontological, geographical, or medicine disciplines. Since the recent formalizations of some of the ideas of predecessors, such as D’arcy Thompson, and thanks to the development of computer techno- gies and new ways for appraising shape changes and variation, morphometrics have undergone, and are still undergoing, a revolution. Most techniques dealing with s- tistical shape analysis have been developed in the last three decades, and the number of publications using morphometrics is increasing rapidly. However, the majority of these methods cannot be implemented in available software and therefore prosp- tive students often need to acquire detailed knowledge in informatics and statistics before applying them to their data. With acceleration in the accumulation of me- ods accompanying the emerging science of statistical shape analysis, it is becoming important to use tools that allow some autonomy. R easily helps ful?ll this need. Risalanguage andenvironment forstatisticalcomputingandgraphics. Although there is an increasing number of computer applications that perform morphometrics, using R has several advantages that confer to users considerable power and possible new horizons in a world that requires rapid adaptability. |

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### Contents

1 | |

3 | |

13 An R Approach to Morphometrics | 5 |

14 Starting with R | 9 |

142 Objects | 10 |

143 Functions | 17 |

144 Operators | 21 |

146 Loops | 23 |

42 Superimposition Methods | 138 |

421 Removing the Size Effect | 139 |

422 Baseline Registration and Bookstein Coordinates | 141 |

423 Procrustes Methods and Kendall Coordinates | 148 |

424 The Kendall Shape Space and the Tangent Euclidean Shape Space | 166 |

425 Resistantﬁt Superimposition | 170 |

43 ThinPlate Splines | 181 |

44 Form and Euclidean Distance Matrix Analysis | 189 |

Problems | 24 |

Acquiring and Manipulating Morphometric Data | 25 |

212 Organizing Data | 27 |

22 Data Acquisition with R | 31 |

222 Entering Data by Hand | 32 |

224 Reading and Converting Image Files | 33 |

225 Graphical Visualization | 35 |

226 Image Analysis and Morphometric Data Acquisition with R | 41 |

23 Manipulating and Creating Data with R | 48 |

231 Obtaining Distance from Coordinates of Points | 49 |

232 Calculating an Angle from Two Interlandmark Vectors | 50 |

233 Regularly Spaced Pseudolandmarks | 51 |

234 Outline Smoothing | 54 |

24 Saving and Converting Data | 56 |

25 Missing Data | 60 |

252 Estimating Missing Landmarks on Symmetrical Structures | 61 |

26 Measurement Error | 63 |

262 Protocols for Estimating Measurement Error | 65 |

Problems | 66 |

Traditional Statistics for Morphometrics | 68 |

311 Visualizing and Testing the Distribution | 70 |

312 When Data are Organization in Several Groups | 72 |

32 Bivariate Analyses | 80 |

322 Analyzing the Relationship Between two Distance Measurements Regression | 81 |

323 Analyzing the Relationship Between Two Distance Measurements in Different Groups | 84 |

324 A Short Excursion to Generalized Linear Models | 89 |

325 Interspecific Measurements and Phylogenetic Data | 92 |

326 Allometry and Isometry | 95 |

A Problem of Deﬁnition | 98 |

34 Multivariate Morphometrics | 105 |

342 Principal Component Analysis | 106 |

343 Analyzing Several Groups with Several Variables | 111 |

344 Analyzing Relationships Between Different Sets of Variables | 124 |

345 Comparing Covariation or Dissimilarity Patterns Between Two Groups | 128 |

Problems | 129 |

Modern Morphometrics Based on Conﬁgurations of Landmarks | 133 |

45 Anglebased Approaches for the Study of Shape Variation | 198 |

Problems | 203 |

Statistical Analysis of Outlines | 205 |

51 Open Outlines | 206 |

512 Splines | 207 |

513 Bezier Polynomials | 209 |

52 Fourier Analysis | 212 |

521 Fourier Analysis Applied to Radii Variation of Closed Outlines | 213 |

522 Fourier Analysis applied to the Tangent Angle | 217 |

523 Elliptic Fourier Analysis | 221 |

53 Eigenshape Analysis and Other Methods | 229 |

Problems | 232 |

Statistical Analysis of Shape using Modern Morphometrics | 233 |

611 Landmark Data | 234 |

62 Discriminant and Multivariate Analysis of Variance | 248 |

622 Procrustes Data | 251 |

63 Clustering | 254 |

64 Morphometrics and Phylogenies | 257 |

65 Comparing Covariation Patterns | 262 |

66 Analyzing Developmental Patterns with Modern Morphometrics | 267 |

662 Developmental Stability | 272 |

663 Developmental Integration | 276 |

Problems | 279 |

Going Further with R | 280 |

72 Writing Functions and Implementing Methods | 287 |

Contour Acquisition Revisited | 289 |

73 Interfacing and Hybridizing R | 293 |

Using ImageMagick to Display High Resolution Images | 296 |

74 Conclusion | 297 |

Problems | 298 |

Functions Developed in this Text | 299 |

Packages Used in this Text | 301 |

References | 303 |

311 | |

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### Common terms and phrases

3D data aligned allometry analyzing angle ANOVA argument array asymmetry baseline Bookstein calculate centered preshape centroid clustering coefﬁcients columns compute conﬁdence configuration conﬁguration matrix coordinates correlation corresponds covariance curve D’Arcy Thompson dataset deﬁned deformation grids degrees of freedom digitized display eigenvectors ellipse elliptic Fourier equally spaced estimate Euclidean distance matrix example f f f factor ﬁnd ﬁrst ﬁt ﬁtting Fourier analysis graph graphical groups harmonic image ﬁles individual interlandmark distances Iris setosa isometry Mahalanobis distance Mantel test mean shape measurement error median methods morphological morphometrics multivariate number of landmarks object observations obtain outline package parameters perform pixel plot points Procrustes analysis Procrustes distance Procrustes superimposition pseudolandmarks reﬂection regression relationship Required functions residual rotation rotation matrix sample scale shape change shape space shape variation signiﬁcance simulated singular-value decomposition species statistical sum of squares superimposed conﬁgurations tangent tion transformation triangle values variables variance variance-covariance matrix vector