THE UNIVERSITY OF CALGARY Eric Snively A ... - Ohio University

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$Introduction to Numerical Math and Matlab ... - Ohio University$

Different variables in a PCA will have diering contributions to a principal component axis. These contributions are described in terms of numerical loadings, which can be either positive or negative. The absolute value of the loading is directly proportional to the variable's contribution to the principal component. If the loadings of neariy al1 or al1 variables are positive for that principal component, the component describes size variation for the sample. If many loadings are negative, the principal component probably describes shape variation (Pimente1 1979). Careful examination of loadings can detemine the type of shape variation that a given principal wmponent reveals. The relative importance of a variable to a principal component is determined by its correlation with that component. A correlation is simply the correlation coefficient between a variable and its Ioading on the PC. A high absolute value for a correlation indicates a strong association between the loading and the original variable. If the correlation is low, the variable is less important in explaining the varianœ along that PC. Regardless of the relative importance of a variable to a PC, the sign of the mrrelation indicates a shape or size contribution congruently with the sign of the variable's loading. For example, if a variable has both a negative correlation and a negative loading, its importance for shape variation if strongly validated. PCA therefore determines which Principal Components are responsible for given amounts of variation (Pimentel 1979). Principal Component 1 or PC1 (the largest eigenvector), is responsible for the most variation, PC2 for the second
highest degree, and so on. Specimens that are similar for a given kind of variation will ciuster together, perpendicular to a principal camporient axis. White it is difïtcult to visualize dustering of specimens in a multidimensional space, PCA can show where specimens reside relative to each other along whatever principal component axes we choose. The rnultidimensional matrix can be rotated mathernatically so that PC1 and PC2 (or other combinations) became axes on a two dimensional graph. This process of infotmatively projecting rnultidimensional coordinates ont0 a two dimensional plot is called ordination (Pielou 1984). The last five figures of Chapter 2 depict examples of ordination.
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THE UNIVERSITY OF CALGARY Functiona
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Many figures in this thesis use col
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ACKNOWLEûGEM ENTS Many friends and
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Approval page Acknowledgements Tabi
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CONCLUDING COMMENTS Figures for Cha
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CHAPTER 5: Mechanical and phylogene
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LIST OF FIGURES Figure 1 .l. Phylog
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Figure 3.6. Osteological correlates
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AMNH GI HMN IVPP MOR NMC OMNH PIN P
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and constructional morphology revea
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1985, Bryant and Seymour 1990). If
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surfaces of a Tyrannosaurus rex spe
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3. Troodontidae (Figure 1.2): Trwdo
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Hypothesized functions of the tyran
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1. Atomization (Chapters 2 and 3).
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Figure 1.1. P hylogenetic diagram o
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Figure 1.2. Phylogenetic diagram of
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Figure 1 .S. Skeletal restorations
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did not quantify the degree of prox
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qualitative debate over arctometata
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This hypothesis focuses on tyrannos
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Table 2.1. Theropod and Plafeosauru
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specimens in the figures reflects a
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were not attempted. While this limi
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Chapter 3). 1 now address variation
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- Ç6'P 1 68'€tr 1 L'OC 99'66 L'9
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Table 2.3: Log-transformed values f
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experiences a sharp laterally indin
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ecause it can be cursorily dassifie
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arctometatarsalian specimens, the l
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) Shaf€ and region of proximal ar
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a) Ginglymus. Figure 2.14 shows the
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) Shaft and region of proximal arti
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measurements of width pV2S%TU-DE an
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Table 2.4b Variancelmeasurement eac
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the largest sum of linear measureme
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Table 2.5: Specimen statistics Herr
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and Sinraptor dongi specimens. With
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In addition, a hook like proximal a
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oth physical narrowing and elongati
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1s the MT 111 shape segregation fun
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tyrannosaurids, ornithomimid, and t
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proximal constriction, white EImisa
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Figure 2.2. Tyrannosaurid, trwdonti
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Figure 2.4. Dromaeosaurid and carno
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- Anterior (flexor) surface medial
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Figure 2.7. Left MT III of Tyrannos
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Figure 2.8. Left MT III of Gorgosau
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Figure 2.9. Left MT III of an omith
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Figure 2.10. Left metatarsus of Tmd
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Figure 2.1 1. Left MT III of Elmisa
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Figure 2.12. Right MT III of Deinon
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Figure 2.13. Left MT III of Allosau
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Figure 2.14. Left metatarsus of Sin
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Figure 2.15. Plots of MT Ill specir
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Figure 2.16. Plot of third metatars
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tv . ... . Ri. . . .
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Figure 2.19. Plot of third metatars
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CHAPTER 3: Tensile keystone model o
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dynamically transmitted Iommotor fo
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Table 3.1 . Metatarsi examined in t
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interrnetatarsal dynamics in these
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compelled analysis of movement evid
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Once prepared, TMP 94.1 2.602 and i
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Daspletosaurus tomsus (MOR 590), an
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2. Freedom of movement inferred fro
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Table 3.2. Surface areas of intemet
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phylogenetic bracketing Wtmer 1995)
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4) Forces from anterior displacerne
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e expected for tyrannosaurid interm
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The preceding discussion derives fr
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weapons sparring), and the arctomet
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Figure 3.1. Schematic representatio
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Figure 3.3. CT reconstruction of ri
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Figure 3.5. MT II (left element) an
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Figure 3.1 1 ae. The metatarsal rec
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Figure 3.13. Ligament contribution
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Figure 3.15. Anatorny of the equid
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Figure 3.16. Cornparison of loading
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(nodes), and solving stresslstrain
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The practicality of the finite elem
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collective density of trabeculae (K
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A FORCE INPUTS AND MATERIAL PROPERT
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Alexander et al. (1 979) rewrded a
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The preceding loading regime entait
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detemined by using the photographic
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2. Preparation of data for contour
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. - Elastic modulus (GP4 Ex EY Ez P
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conversion. Volume-to-nuages facili
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maintained. An angle of 2 degrees p
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of MT III (Figure 4.4, pinpointed b
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anterodorsal rotation of the distal
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allowed to stretch slightly under t
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Figure 4.1. Initial loads and bound
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Figure 4.3. Finite element mesh of
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Figure 4.5. Strain distribution in
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CHAPTER 5: Mechanical and evolution
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y distal intermetatarsal ligaments,
Page 240 and 241: Probable fundion of the tyrannosaur
Page 242 and 243: Ornitholestes, and camosaurs) incre
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Page 248 and 249: face of the skull met part of the m
Page 250 and 251: Alternately, a morphological novelt
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Page 258 and 259: dromaeosaurids], and a dade compris
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Page 262 and 263: Proximal intermetatarsal ligaments
Page 264 and 265: witti a surfeit of potentially supp
Page 267: Figure 5.2. Phylogeny of the Therop
Page 278 and 279: Bock W.J. and von Wahlert G. Evolut
Page 280 and 281: Cume P.J. 1997. Theropoda. In: Fari
Page 282 and 283: Grenard S. 1991. Handbook of Alliga
Page 284 and 285: Liem K.F. and Osse J.W.M. 1975. Bio
Page 286 and 287: Richtsmeier J.T. and Cheverud J.M.
Page 288 and 289: Woo S., Maynard J., Butler O. 1987.
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THE UNIVERSITY OF CALGARY Eric Snively A ... - Ohio University

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