Geometric Morphometrics

Transcription

1 Geometric Morphometrics First analysis J A K L E DBC F I G H Mathematica map of Marmot localities from Assignment 1

2 Tips for data manipulation in Mathematica Flatten[ ] - Takes a nested list and flattens it into a single list. Partition[ ] - Opposite of flatten: groups items into a nested list. Transpose[ ] - Switches columns to rows and rows to columns.

3 Taking rows and columns of data in Mathematica Example 1: take column 2 of row 5 Double square brackets after a variable allow you to select parts of a list or matrix. For these examples, data is a matrix with ten rows and seven columns. The first number in the brackets indicates rows, the second number indicates columns. Mathematica code: data[[5, 2]] Example 2: take all of row 4 The code for taking the highlighted data is shown beneath each example. Mathematica code: data[[4]]

4 Taking rows and columns (cont.) Example 3: take rows 3 to the last row Example 5: take rows 3 then 6 Mathematica code: data[[3;;]] Example 4: take rows 3 through 6 Mathematica code: data[[{3,6}]] Example 6: take column 1 Mathematica code: data[[3;;6]] Mathematica code: data[[1;;,1]]

5 Taking rows and columns (cont.) Example 7: take columns 3 through the end Example 9: take rows 4 through 8 in columns 3 through 5 Mathematica code: data[[1;;,3;;]] Example 8: take columns 3 through 5 Mathematica code: data[[1;;8,3;;5]] Example 10: take columns 3 then 6 Mathematica code: data[[1;;,3;;5]] Mathematica code: data[[1;;,{3,6}]]

6 Taking rows and columns (cont.) Example 11: take columns 6 then 3 Example 12: take rows 1-6 of columns 2, 4, and 6 Mathematica code: data[[1;;,{6,3}]] Mathematica code: data[[1;;6,{2,4,6}]]

7 Taking longitude and latitude from Assignment 1 For the map assignment you needed longitude and latitude from the table stored in data. Furthermore, you need longitude in your first column because it should be on the horizontal x-axis, whereas latitude should be in the second column so it will appear on the y-axis. You also need to discard the first row because it contains column labels instead of numbers. Mathematica code: data[[2;;, {9,8}]]

8 Map[] a function for repeating the same thing for every item in a list H* three ways to put your coordinates into the Point@D function *L coords Out[12]= , 90.45<, , <, , <, , <, 838., 73.<, , <, , <, , <, , <, , 7.934<, , <, , << H* table works by counting out each item in the list and putting it into the Point@D funtion *L Table@Point@coords@@xDDD, 8x, Length@coordsD<D Out[15]= 8Point@ , 90.45<D, Point@ , <D, Point@834.62, <D, Point@ , <D, Point@838., 73.<D, Point@ , <D, Point@ , <D, Point@ , <D, Point@ , <D, Point@ , 7.934<D, Point@ , <D, Point@ , <D< H* Map@D function replaces the apple with each element of coords. The & is placed at the end to indicate that the replacement should continue until all elements in coords have been used. *L Map@Point@appleD &, coordsd Out[16]= 8Point@ , 90.45<D, Point@ , <D, Point@834.62, <D, Point@ , <D, Point@838., 73.<D, Point@ , <D, Point@ , <D, Point@ , <D, Point@ , <D, Point@ , 7.934<D, Point@ , <D, Point@ , <D< Out[17]= H* shortcut for Map@D. Takes each element on the right of the êû and replaces apple on the left with it. & causes the replacement to repeat for all items in coords. This syntax requires much less typing than the Table@D format above. *L Point@appleD & êû coords 8Point@ , 90.45<D, Point@ , <D, Point@834.62, <D, Point@ , <D, Point@838., 73.<D, Point@ , <D, Point@ , <D, Point@ , <D, Point@ , <D, Point@ , 7.934<D, Point@ , <D, Point@ , <D<

9 Steps in a Geometric Morphometric Methods (GMM) Analysis 1. Collect landmark coordinates 2. Do a Procrustes superimposition Standardizes landmarks by rescaling them and rotating them to a common orientation using leastsquares fitting 3. Analyze similarity and difference of shape Analysis usually starts with a Principal Components Analysis, which (A) shows similarity and differences as simple scatter plots, and (B) provides new variables for further statistical analysis

10 Performing a GMM analysis in Mathematica GMM functions are in the Polly Morphometrics add-in package for Mathematica 1. Download the latest version of the package at (right click on link to save as file) 2. Open the file in Mathematica 3. From the File menu, choose Install 4. From Type of Item choose Package, from Source choose PollyMorphometrics8.x.m, under Install Name choose a short name for the package (e.g., PollyMorphometrics ) 5. Once installed, enter the command <<PollyMorphometrics` to use the functions For detailed information about the functions, see the Guide to Morphometrics for Mathematica available from the same web page.

11 Step 1: Collecting landmarks 1. Each image must have the same number of landmarks; 2. The landmarks on each image must be in the same order; 3. Landmarks are ordinarily placed on homologous points, points that can be replicated from object to object based on common morphology, common function, or common geometry Osteostracan head shield from Sansom, 2009

12 Step 1: Collecting landmarks (cont.) 1. use tpsdig and tpsutil programs from Jim Rohlf to produce a tps-format file ( 2. ImageJ program with the PointPicker plug-in along with a spreadsheet to produce a tps-format file ( ( 3. Use built in Get Coordinates tool in Mathematica See handout for detailed instructions on these three methods The tps files generated by methods 1 and 2 can be imported with the tpsimport[] function in the PollyMorphometrics package data = tpsimport["/users/pdavidpolly/osteostraci_tps.txt"];

13 Step 2: Procrustes superimposition Procrustes superimposition is the standardization step in GMM. Procrustes removes (1) size, (2) translation, (3) rotation from the original landmark data. In other words, it centers them all together, scales them to the same size, and rotates them into the same orientation. The Procrustes step removes statistical degrees of freedom from your data, which has implications for later statistical analyses. After landmarks have been superimposed, the similarities and differences in their shape can be analyzed.

14 Step 2: Procrustes (cont.) Once you have your landmarks, arrange them in a matrix where each row is a different object, and each column is a landmark coordinate. There should be no column labels Save that matrix in a variable called data then superimpose the landmarks using the Procrustes[] function: proc = Procrustes[data, 13, 2] where 13 is the number of landmarks and 2 indicates they are two-dimensional. HINT: Don t know how many landmarks? Count them by finding out the length of elements in one row (i.e., the number of columns) and divide by two: Length[data[[1]]] / 2

15 Step 3: Principal Components Analysis Principal Components Analysis (PCA) ordinates the objects in your analysis by arranging them in a shape space. Similarities and differences can easily be seen in a PCA plot. The axes of a PCA plot are Principal Components (PCs). The first PC of any analysis is, by definition, the one that shows the largest axis of variation in shape. The second PC shows the next largest axis of variation that is uncorrelated with the first, the third PC shows the third largest axis of variation, and so on. Each point on a PCA plot represents the shape of a single object from your analysis. The closer two objects are, the more similar they are in shape. 0.3 PCA Plot 0.2 Stensiopelta_pustulata Benneviaspis_lankesteri 0.1 Boreaspis_ceratops Dicranaspis_gracilis Parameteroaspis_gigis Zenaspis_salweyi PC Scolenaspis_signata Spatulaspis_costata Ateleaspis_tesselata Pattenaspis_acuminata Hirella_gracilis PC 1

16 Step 3: PCA (cont.) A PCA plot is often called a morphospace in GMM because each point on the plot represents a different shape or, more specifically, a different configuration of landmarks. An important part of understanding PCA results is to explore how shape varies in the PCA plot. The thin-plate spline grids in this PCA plot show how shape varies along PC1 and PC2 for a data set of osteostracan fish head shields (shown at right). For example, at the left of the plot landmark 1 is located far in front of 8 and 11, at the right it is very close to 8 and 11. The species at the left of the plot on the previous slide have shapes like the grids on the left in this plot. Exploring morphospace with these grids can help understand the meaning of the PCA. Morphospace 0.4 PCA Plot PC PC 1

17 Step 3: PCA (cont.) You can also explore the distribution of shape by referring back to your original photographs. Compare these shapes to the grids on the previous slide PCA Plot Benneviaspis_lankesteri Stensiopelta_pustulata Boreaspis_ceratops Dicranaspis_gracilis Parameteroaspis_gigis Zenaspis_salweyi PC Scolenaspis_signata Spatulaspis_costata Ateleaspis_tesselata Pattenaspis_acuminata Hirella_gracilis PC 1

18 Step 3: PCA (cont.) To do a PCA of shape in Mathematica: PrincipalComponentsOfShape[proc, {1,2}, labels] where proc is matrix of Procrustes superimposed coordinates from Step 2, the list {1,2} tells the function to plot the first and second principal components, and labels is a list of text labels for each object in proc. This function provides the following output: 1. A PCA plot showing the objects with labels 2. Text output explaining how much of the variation in shape is explained by each PC 3. A graphic representation of the mean shape in your data set, where each landmark is indicated by its number and a convex hull has been placed around the landmarks 4. A morphospace model showing thin-plate spline snapshots of shape variation in the PCA plot.

19 Step 3: PCA (cont.) Mean Shape

20 Assignment 2 - Faces project 1. Download the face photographs from Oncourse 2. Choose a landmark scheme for the faces, remembering that you must place the same number of landmarks on each face and they must always be placed in the same order. 3. Import the landmarks into Mathematica. 4. Superimpose them with the Procrustes[] function. 5. Do a Principal Components Analysis of Shape on your Procrustes superimposed landmarks. 6. Study the results to determine what the major axes of your PCA plot show, and to decide whether the results accurately pick up differences in people s facial features or whether other biases affect the outcome of the analysis. 7. Turn in Assignment 2. We will discuss results next week in class.