class: center, middle, inverse, title-slide .title[ # 10: Symmetry and Asymmetry ] .author[ ### ] --- <style type="text/css"> pre { max-width: 100%; overflow-x: scroll; } .scrollable { height: 80%; overflow-y: auto; } </style> ### Symmetry and Asymmetry + Many objects are built symmetrically + Paired structures and serially homologous body parts are organized in a symmetric way -- + Bilateral symmetry: structures arranged in either side of the body mid-line (axis of symmetry) + Symmetry poses some challenges for morphometric data analysis --- ### Types of Bilateral Symmetry - Two main types of bilateral symmetry: - **Matching symmetry**: pairs of structures are found across the mid-line of the object (e.g. insect wings, tetrapod limbs etc) - **Object symmetry**: single structure with internal structural symmetry (e.g. structures in vertebrate skull) <img src="LectureData/10.asymmetry/SymmTypes.png" width="100%" style="display: block; margin: auto;" /> --- ### The Problem with Symmetrical Objects + Landmark positions are not independent in symmetric objects + Lack of statistical independence for paired landmarks + Some dimensions of shape space have little variance + SSCP matrices become singular (or nearly so) + This causes issues with parametric statistical hypothesis-testing methods ('divide' by zero or nearly so) <img src="LectureData/10.asymmetry/SymmLizard.png" width="40%" style="display: block; margin: auto;" /> -- + The problem becomes more acute as objects become more symmetrical! --- ### The Problem with Symmetrical Objects (Cont.) + The data in reality have fewer dimensions, so use less data. How? + Digitize only half structure + Reflect structure and take the average: make a symmetrical structure + Using only `\(\frac{1}{2}\)` structure has challenges: + Does not represent entire shape + Can introduce assymetries because the midline is not 'anchored' during the analysis + Solution: use **entire structure** and **RRPP** for statistical evaluation! ###### As we've seen this week, RRPP, combined with robust summary test statistics, does not require inverting (nearly) singular covariance matrices, so model evaluation is not compromised by this issue. --- ### Symmetric Objects: Example + Note the large number of shape dimensions with no variation. Much of this is due to bilateral symmetry! .scrollable[ <img src="10-Asymmetry_files/figure-html/unnamed-chunk-4-1.png" width="40%" style="display: block; margin: auto;" /> ``` ## Importance of components: ## PC1 PC2 PC3 PC4 PC5 PC6 ## Standard deviation 0.02167 0.01764 0.01354 0.01115 0.009989 0.008932 ## Proportion of Variance 0.27485 0.18198 0.10735 0.07272 0.058380 0.046680 ## Cumulative Proportion 0.27485 0.45683 0.56418 0.63690 0.695290 0.741960 ## PC7 PC8 PC9 PC10 PC11 PC12 ## Standard deviation 0.00823 0.007665 0.006262 0.005993 0.00541 0.005021 ## Proportion of Variance 0.03964 0.034380 0.022940 0.021010 0.01713 0.014750 ## Cumulative Proportion 0.78160 0.815980 0.838920 0.859940 0.87706 0.891810 ## PC13 PC14 PC15 PC16 PC17 PC18 ## Standard deviation 0.004777 0.004682 0.004486 0.00432 0.004139 0.003805 ## Proportion of Variance 0.013350 0.012830 0.011780 0.01092 0.010020 0.008470 ## Cumulative Proportion 0.905170 0.918000 0.929770 0.94069 0.950710 0.959180 ## PC19 PC20 PC21 PC22 PC23 PC24 ## Standard deviation 0.00354 0.003408 0.003317 0.003162 0.003013 0.002664 ## Proportion of Variance 0.00733 0.006800 0.006440 0.005850 0.005310 0.004150 ## Cumulative Proportion 0.96652 0.973310 0.979750 0.985600 0.990910 0.995070 ## PC25 PC26 PC27 PC28 PC29 PC30 ## Standard deviation 0.00242 0.001605 3.621e-15 3.489e-15 3.463e-15 3.294e-15 ## Proportion of Variance 0.00343 0.001510 0.000e+00 0.000e+00 0.000e+00 0.000e+00 ## Cumulative Proportion 0.99849 1.000000 1.000e+00 1.000e+00 1.000e+00 1.000e+00 ## PC31 PC32 PC33 PC34 PC35 ## Standard deviation 3.232e-15 3.173e-15 3.109e-15 3.042e-15 3.02e-15 ## Proportion of Variance 0.000e+00 0.000e+00 0.000e+00 0.000e+00 0.00e+00 ## Cumulative Proportion 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.00e+00 ## PC36 PC37 PC38 PC39 PC40 ## Standard deviation 2.989e-15 2.919e-15 2.872e-15 2.796e-15 2.748e-15 ## Proportion of Variance 0.000e+00 0.000e+00 0.000e+00 0.000e+00 0.000e+00 ## Cumulative Proportion 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00 ## PC41 PC42 PC43 PC44 PC45 ## Standard deviation 2.723e-15 2.697e-15 2.64e-15 2.616e-15 2.565e-15 ## Proportion of Variance 0.000e+00 0.000e+00 0.00e+00 0.000e+00 0.000e+00 ## Cumulative Proportion 1.000e+00 1.000e+00 1.00e+00 1.000e+00 1.000e+00 ## PC46 PC47 PC48 PC49 PC50 ## Standard deviation 2.55e-15 2.493e-15 2.418e-15 2.387e-15 2.282e-15 ## Proportion of Variance 0.00e+00 0.000e+00 0.000e+00 0.000e+00 0.000e+00 ## Cumulative Proportion 1.00e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00 ## PC51 PC52 PC53 PC54 PC55 ## Standard deviation 2.205e-15 2.163e-15 1.437e-15 1.364e-15 1.936e-16 ## Proportion of Variance 0.000e+00 0.000e+00 0.000e+00 0.000e+00 0.000e+00 ## Cumulative Proportion 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00 ## PC56 ## Standard deviation 1.743e-17 ## Proportion of Variance 0.000e+00 ## Cumulative Proportion 1.000e+00 ``` ] --- ### The Biology of Asymmetry + Different types of asymmetry are thought to have a biological meaning + **Directional asymmetry**: consistent difference skewed towards one of the sides (at the population level); thought to reflect difference in use, e.g. fiddler crabs feeding vs. fighting displays <img src="LectureData/10.asymmetry/FiddlerCrab.png" width="40%" style="display: block; margin: auto;" /> -- + **Fluctuating asymmetry**: thought to reflect instability during development (genetic, environmental etc) + Long debate about the meaning of FA --- ### Types of Symmetry + Perfect symmetry is rare in biology; objects nearly always have some degree of asymmetry + Three main types of asymmetry have been described, based on the statistical distribution of asymmetry in the population -- + **1: Directional**: consistent `\(\small{(R-L)}\)` differences with one side always larger than the other: `\(\small{\overline{(R-L)}\neq0}\)` + **2: Fluctuating**: small random `\(\small{(R-L)}\)` deviations with `\(\small{\overline{(R-L)}=0}\)` + **3: Antisymmetry**: consistent `\(\small{(R-L)}\)` differences, but larger side random <img src="LectureData/10.asymmetry/SymmTypesHist.png" width="70%" style="display: block; margin: auto;" /> --- ### Analysis of Symmetry: General Procedure + The presence of various types of asymmetry may be evaluated using one of several factorial ANOVA designs + Traits need to be quantified multiple times, so that measurement error may be quantified + This allows one to evaluate if the (normally small) differences between sides are actually “real” or due to measurement error (ME) -- + ANOVA then often set up as: <img src="LectureData/10.asymmetry/AnovaDesign.png" width="70%" style="display: block; margin: auto;" /> --- ### Extensions to GM Shape Data <img src="LectureData/10.asymmetry/ReflectObjects.png" width="70%" style="display: block; margin: auto;" /> + **Procedure** + 1: Digitize right and left structures (or R and L sides), possibly multiple times for ME + 2: Reflect one side to match the other (and relabel landmarks for object symmetry) + 3: GPA + projection to tangent space + 4: Factorial ANOVA – SS for Individual, SS Side (DA), SS Side x Individual (FA) + 5: Assess model factors via permutation approaches ###### Klingenberg & McIntyre (1998). *Evolution.*; Klingenberg et al. (2002). *Evolution.* --- ### Decomposition of Asymmetry Component + One can approach the problem more theoretically (rather than 'procedurally') + ANOVA effects represent Sums of Squares, which correspond to various asymmetry components + Therefore, decomposing SST (total shape variation) obtained via `\(\small{D}_{Proc}\)` provides variance components for both DA and FA `$$\small{SST}=\sum^n_1{D}^2_{(X_i,Y_i)}=nD^2_{(\overline{X},\overline{Y})}+\sum^n_1{D}^2_{(X_i-\overline{X},Y_i-\overline{Y})}$$` -- + where: `\(\small{nD}^2_{(\overline{X},\overline{Y})}\)` represents the **Directional Asymmetry** component + and: `\(\small\sum^n_1{D}^2_{(X_i-\overline{X},Y_i-\overline{Y})}\)` represents the **Fluctuating Asymmetry** component ###### Mardia et al. (2000). *Biometrika*. --- ### Matching Symmetry: Example <img src="LectureData/10.asymmetry/MatchExWings.png" width="80%" style="display: block; margin: auto;" /> --- ### Matching Symmetry: Example 2 .scrollable[ ```r data(mosquito) Y.gpa <- gpagen(mosquito$wingshape, print.progress = FALSE) plot(Y.gpa) ``` <img src="10-Asymmetry_files/figure-html/unnamed-chunk-10-1.png" width="80%" style="display: block; margin: auto;" /> ] --- ### Matching Symmetry: Example 2 (Cont.) .scrollable[ ```r mosquito.sym <- bilat.symmetry(A = Y.gpa, ind = mosquito$ind, side=mosquito$side, object.sym = FALSE, print.progress = FALSE) summary(mosquito.sym) ``` ``` ## ## Call: ## bilat.symmetry(A = Y.gpa, ind = mosquito$ind, side = mosquito$side, ## object.sym = FALSE, print.progress = FALSE) ## ## ## Symmetry (data) type: Matching ## ## Type I (Sequential) Sums of Squares and Cross-products ## Randomized Residual Permutation Procedure Used ## 1000 Permutations ## ## Shape ANOVA ## Df SS MS Rsq F Z Pr(>F) ## ind 9 0.104888 0.0116542 0.45533 2.7646 4.7037 0.001 ** ## side 1 0.003221 0.0032209 0.01398 0.7641 -0.3558 0.635 ## ind:side 29 0.122249 0.0042155 0.53069 ## Total 39 0.230358 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## ## Centroid Size ANOVA ## Df SS MS Rsq F Z Pr(>F) ## ind 9 4.1497e-09 4.6107e-10 0.18555 0.7484 -0.42684 0.665 ## side 1 3.4740e-10 3.4738e-10 0.01553 0.5638 0.15779 0.471 ## ind:side 29 1.7867e-08 6.1609e-10 0.79891 ## Total 39 2.2364e-08 ``` ] --- ### Matching Symmetry: Example 2 (Cont.) .scrollable[ ```r plot(mosquito.sym, warpgrids = TRUE) ``` <img src="10-Asymmetry_files/figure-html/unnamed-chunk-12-1.png" width="80%" style="display: block; margin: auto;" /> ] --- ### Object Symmetry: Example <img src="LectureData/10.asymmetry/ObjExLizards.png" width="80%" style="display: block; margin: auto;" /> --- ### Object Symmetry: Example 2 .scrollable[ ```r data('lizards') Y.gpa <- gpagen(lizards$coords, print.progress = FALSE) plot(Y.gpa) ``` <img src="10-Asymmetry_files/figure-html/unnamed-chunk-14-1.png" width="80%" style="display: block; margin: auto;" /> ] --- ### Object Symmetry: Example 2 (Cont.) .scrollable[ ```r lizard.sym <- bilat.symmetry(A = Y.gpa, ind = lizards$ind, replicate = lizards$rep, object.sym = TRUE, land.pairs = lizards$lm.pairs, print.progress = FALSE) summary(lizard.sym) ``` ``` ## ## Call: ## bilat.symmetry(A = Y.gpa, ind = lizards$ind, replicate = lizards$rep, ## object.sym = TRUE, land.pairs = lizards$lm.pairs, print.progress = FALSE) ## ## ## ## Symmetry (data) type: Object ## ## Type I (Sequential) Sums of Squares and Cross-products ## Randomized Residual Permutation Procedure Used ## 1000 Permutations ## ## Shape ANOVA ## Df SS MS Rsq F Z Pr(>F) ## ind 48 0.236788 0.0049331 0.83194 7.3721 -0.1011 0.536 ## side 1 0.009432 0.0094317 0.03314 14.0951 3.7540 0.001 ** ## ind:side 48 0.032119 0.0006692 0.11285 10.4367 19.7078 0.001 ** ## ind:side:replicate 98 0.006283 0.0000641 0.02208 ## Total 195 0.284622 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` ] --- ### Object Symmetry: Example 2 (Cont.) .scrollable[ ```r plot(lizard.sym, warpgrids = TRUE) ``` <img src="10-Asymmetry_files/figure-html/unnamed-chunk-16-1.png" width="80%" style="display: block; margin: auto;" /> ] --- ### Other Types of Asymmetry + Many other types of (a)symmetry are present in biological data + How can we quantify asymmetry for these structures? <img src="LectureData/10.asymmetry/HigherSymm.png" width="70%" style="display: block; margin: auto;" /> --- ### Symmetry Groups + One can approach the problem by defining **Symmetry Groups** + Symmetry groups: transformations that leave the data invariant + e.g.: bilateral symmetry = reflection across the midline <img src="LectureData/10.asymmetry/SymmGroups.png" width="40%" style="display: block; margin: auto;" /> + Symmetry groups define transformations such that there are invariances in those symmetric 'dimensions' + These groups describe the ways in which symmetry can be defined, and thus quantified for more complex structures (e.g., radial symmetry) ###### Savriama and Klingenberg. (2011). *BMC Evol. Biol.* --- ### Symmetry: Extended Protocol .pull-left[ + For bilateral asymmetry: transformation = reflection + For other types, decompose asymmetry to the components defining the symmetry type based on the appropriate symmetry groups + Example: Rotational Symmetry + Digitize all p components + GPA + projection + Decompose shape variation into symmetry components, quantify and evaluate ] .pull-right[ <img src="LectureData/10.asymmetry/RadialSymm.png" width="50%" style="display: block; margin: auto;" /> ] ###### Savriama and Klingenberg. (2011). *BMC Evol. Biol.* --- ### Complex Symmetry: Example + Symmetry Analysis in corals <img src="LectureData/10.asymmetry/CoralSymmetry.png" width="70%" style="display: block; margin: auto;" /> --- ### A Note on Geometric Transformations + Any captured shape difference that follows a geometric rule can be modelled using the Procrustes paradigm + We can account for undesired effects + We can model shapes and other effects separately -- + Articulations frequently cause this kind of effects + Shape effects + positional effects <img src="LectureData/10.asymmetry/ArticConcept.png" width="80%" style="display: block; margin: auto;" /> --- ### Geometric Transformations: Articulations + For articulated structures, several solutions exist + Fixing the angle in all specimens through a mathematical transformation + Separating the subsets to analyse separately, etc. <img src="LectureData/10.asymmetry/ArticMath.png" width="80%" style="display: block; margin: auto;" /> ###### Adams. (1999). *Evol. Ecol. Res.* --- # Articulation Standardization: Example + Standardize some data for relative jaw position ```r jaw.fixed <- fixed.angle(gpa.rand, art.pt=1, angle.pts.1 = 5, angle.pts.2 = 6, rot.pts = c(2,3,4,5)) gpa.fixed <- gpagen(jaw.fixed, print.progress = FALSE)$coords ``` <img src="10-Asymmetry_files/figure-html/unnamed-chunk-25-1.png" width="40%" style="display: block; margin: auto;" /> --- ### Another Note on Geometric Transformations + Considering 'fixing' articulation angles in reverse results in a different pattern **motion analysis** + For shape devoid of movement, one fixes the angle. + For quantifying movement, one compares shapes across the motion path interested + The latter forms a *trajectory* that describes the motion path + It is represented by a sequence of shapes (a trajectory) in shape space <img src="LectureData/10.asymmetry/MotionAnal.png" width="50%" style="display: block; margin: auto;" /> ###### NOTE: see Phenotypic Trajectory Analysis lecture for analytical details --- ### Symmetry: Summary + Symmetry causes redundancy in shape data + Can account for symmetry effects analytically + Asymmetry: biological significance + FA: developmental instability + DA: adaptation? + Use ANOVA procedures to separate variation within (asymmetry) and among individuals + Extensions to different types of symmetry + Geometric transformations in general: any effect that follows a geometric rule can be taken into account, partialed out, or targeted in the Procrustes framework