The first step after importing a file with raw data is almost always a new Procrustes fit. Some procedures in MorphoJ are automatically followed by a new Procrustes fit, for example, after excluding outliers or choosing a subset of landmarks. Moreover, a new Procrustes fit can also be used to change the alignment of the landmark configurations in the graphocal output windows (mostly for esthetical reasons).
MorphoJ uses a full Procrustes fit and projects the data to the tangent space by orthogonal projection (Dryden and Mardia 1998). For most applications, there is very little difference between the different methods for Procrustes fitting (full versus partial Procrustes fits), but for datasets with unusually large variation, the full Procrustes fit puts less weight on observations that are far from the average shape and will therefore be somewhat more robust against the influence of outliers.
The implementation of the Procrustes fit in MorphoJ includes reflection if that improves the fit. This is not the 'standard' for implementations of the Procrustes fit, although for most applications it will provide the same result. Strictly speaking, MorphoJ extracts the 'reflection shape' rather than shape (Dryden and Mardia 1998).
Reflection shape is standard (and indispensable) in shape analyses of asymmetry: first reflecting all configurations from one side and then using a standard Procrustes fit (e.g. Klingenberg and McIntyre 1998). If reflection were not included, there would be a huge shape difference between all the left and all the right sides, which would simply reflect the fact that mirror-image configurations give an extremely bad Procrustes fit. This difference would obscure any other shape differences in the analysis, including the biologically informative differences between the reflection shapes of the left and right sides.
For analyses that do not involve mirror images, the analyses using shape and reflection shape are identical. This is the typical type of study using one structure from a bilateral pair per individual, all from the same body side: e.g. left scapulae, right wings, etc.
The reason why the Procrustes fit with reflection is implemented in MorphoJ is that such studies often encounter the problem that the structure of interest is not intact or present on the body side preferred by the investigator. To avoid having to exclude a specimen, the structure from the other body side is therefore often included.But it is necessary to reflect the corresponding landmark configuration to make it compatible with the other specimens in the sample (which amounts, again, to using reflection shape). MorphoJ does this step automatically, if necessary, and includes the following line in the Reports window:
Reflections were used in the Procrustes fit.
When does it matter? Normally, the automatic reflection does not make a diffrence. In the vast majority of cases when it does make a difference (asymmetry studies and other analyses involving mirror images), it is helpful or even required to produce meaningful results. There are cases, however, when the reflection is undesirable:
To invoke a new Procrustes fit for a dataset, click on this dataset in the Project Tree window and select New Procrustes Fit from the Preliminaries menu.
The procedure for the Procrustes fit involves several steps, including the alignment of the specimens relative to the coordinate system for graphical output and, for data with object symmetry, the pairing of landmarks. MorphoJ will guide users through these steps as erquired for the dataset under study.
First, the following dialog box will appear:
This dialog box requests information on how the configurations are to be aligned. To be more precise, what is chosen here how the Procrustes mean is to be aligned relative to the coordinate axes -- the individual observations will follow this orientation more or less, depending on the shape variation in the dataset. This choice of the coordinate system is primarily of interest for the graphical output of the subsequent analyses. Note that the choice of the alignment does not affect the statistical outcomes of subsequent analyses, but may change the appearance of graphical output.
There are three alternative alignment methods: aligning with the first specimen in the dataset, aligning by the principal axis of the mean configuration and aligning with specific landmarks entered by the user.
Aligning with the first specimen is only advised in relatively rare cases when the first specimen is aligned in a specific way by the user. In particular, when there is object symmetry, this alignment method does not establish any specific reference to the axis or plane of symmetry.
Aligning by the principal axes of the mean configuration is the default method. The first axis will be oriented parallel to the long axis of the configuration, which usually provides a reasonable orientation in graphical outputs. If the dataset has object symmetry, this will automatically establish a relation to the axis or plane of symmetry. For 2D data, one of the resulting axes is parallel and the other is perpendicular to the axis of symmetry. For 3D data, two of the axes will be parallel to and one will be perpendicular to the plane of symmetry.
Aliging using specific landmarks provides the most control to the user in choosing the exact alignment, for example in the Frankfurt plane. This feature is particularly useful if the main anatomical axes are arranged at an oblique angle to the main axes of the landmark confguration (e.g. this often happens with facial landmarks).
Clicking on the button labeled Align using specific landmarks will activate the text fields below it. These text fields are for specifying two points on the left and right sides of the first axis (normally shown as the horizontal in graphs) and, for 3D data, a third points that will be used with the first two to define the plane of the first and second axes.
The points for specifying the alignment can either be landmarks or weighted means of the positions of different landmarks. Landmarks are specified by typing "L" and the number of the landmark. For instance, "L1" is the first landmark and "L15" is the 15th landmark. Combinations of landmarks are written as sums, for example "L1 + L15", "0.5L1 + 0.5L15" and "0.5*L1 + 0.5*L15" are three ways to specify the point midway between landmarks 1 and 15 (both landmarks are given equal weight). It is also possible to give unequal weights to the landmarks: "0.3L1 + 0.7L15" is a point 70% towards L15 on the line connecting landmarks 1 and 15, and "2L1 + L15" is a point two-thirds toward landmark 1. Negative weights can also be assigned, but the plus sign still must be used, for example "L1 + -0.2*L15". Finally, it is possible to use more than two landmark to define a point.
The user must provide a valid specification in each of the active text fields. For 2D data, only the first two text fields are activated because only two points are to be specified. For 3D data, all three text fields are activated.
If the dataset has object symmetry and is in 2D, the median axis of symmetry can be set to be horizontal by specifying two landmarks on the median axis as the left and right points (or alternatively, midpoints between paired landmarks). The median axis can be set to be vertical by specifying two paired landmarks as the left and right points. If the dataset has object symmetry and is in 3D, it is recommended to use three points in the median plane. The first two points should be chosen to provide the horizontal direction for the first axis, and the third point should be chosen to be above the other two (it provides the information of what is "up" and "down" for the second axis).
Clicking the button Perform Procrustes Fit will invoke the next step of the procedure. Clicking Cancel will abort the Procrustes fit.
If the configuration does not have object symmetry, the dialog box will disappear, the Procrustes fit will be performed and the landmark coordinates after the Procrustes fit will appear as a new tab inside the Graphics tab, similar to the example shown below. If there is object symmetry, the pairing of landmarks must be done before that.
If the landmark configuration has object symmetry, an interface similar to the following will appear in the Graphics output tab:
This interface presents an initial guess of the pairing of landmarks and the items required to retify any problems. The paired landmarks are represented by blue dots with light blue lines connecting pairs. The median unpaired landmarks are represented by green dots. The user can invoke a pop-up menu for this graph to change the axes that are displayed (for 3D data) and to rotate or flip the diagram.
For many datasets this initial guess of the pairing of landmarks is correct and the user can choose to go on to the actual Procrustes fit by clicking the Accept button.
If there are inconsistencies with the pairing of landmarks, the landmarks affected by these inconsistencies will be represented by red dots. Moreover, a warning will be displayed above the Accept button, and the button itself will be deactivated.
In order to carry out the Procrustes fit, the user first must rectify the problem with the landmark pairing. The two drop-down menus in the upper-right corner of the interface can be used to select landmark pairs, and clicking the Pair button will establish the two selected landmarks as a pair. To specify a landmark as unpaired, select it in both drop-down menus and click the Pair button. Once every inconsistency in the pairing is eliminated (if landmark a is declared as paired with landmark b, landmark b must be declared as paired with landmark a), the Warning message disappears and the Accept button is activated and can be used. Nevertheless, the user should still check whether the pairing is indeed correct.
Once the pairing is correct, clicking the Accept button will start the Procrustes fit. If the user clicks Cancel, the whole procedure will be stopped and the Procrustes fit will not be performed.
The final output, here shown for a 3D dataset with object symmetry, may look as follows:
The graphics tab with the output of the Procrustes fit has the name of the dataset (here "St Bernard dogs") as its title. It contains one or more graphs as separate tabs, depending on whether the dataset is in 2D or 3D and whether there is object symmetry.
If the dataset has oject symmetry, separate graphs are shown for the symmetric and asymmetry components (abbreviations "Symm" and "Asym", respectively). If the dataset does not have object symmetry, the plots are shown for the different axes. The number of plots of each category depends on whether the data is in 2D or 3D.
The blue dots represent the mean landmark positions and the small black dots represent the landmark positions for individual configurations in the sample. For the asymmetry component, the symmetric overall mean of the configuration is added to the coordinates, so that it is clear to which landmark the scatters of points belong, and this symmetric mean configuration is shown in light blue (but may be hidden is average asymmetry is small). The numbers of the landmark are indicated by red numbers.
This procedure modifies the dataset for which the Procrustes fit was performed by adding the landmark positions after Procrustes superimosition (or the symmetric and asymmetry components, if there is object symmetry) and centroid size.
The raw data are kept as part of the dataset. The new data items are added to, but do not replace the raw data (indeed, swapping landmarks as part of the Find Outliers procedure, is the only maneuver that actually changes the raw data).
The data matrix Procrustes coordinates is produced if the data do not have object symmetry, and contains the shape data for all observations. If there is object symmetry, this data matrix is not produced, but instead two separate matrices Symmetric component and Asymmetry component (Klingenberg et al. 2002). These two components of variation are normally used in different contexts: the symmetric component represents shape variation among individuals in what might be considered a left-right average, and which will be of most interest in the majority of studies, whereas asymmetry is usually the focus of more specialized studies of asymmetry and developmental integration.
Finally, the Procrustes fit procedure adds a data matrix with the Centroid size to the dataset. This dataset contains two variables: centroid size and log centroid size (the natural logarithm of centroid size). These are useful, for example, in studies of allometry etc.
Dryden, I. L., and K. V. Mardia. 1998. Statistical shape analysis. Wiley, Chichester.
Klingenberg, C. P., and G. S. McIntyre. 1998. Geometric morphometrics of developmental instability: analyzing patterns of fluctuating asymmetry with Procrustes methods. Evolution 52:1363–1375.
Klingenberg, C. P., M. Barluenga, and A. Meyer. 2002. Shape analysis of symmetric structures: quantifying variation among individuals and asymmetry. Evolution 56:1909–1920.