Geometric Morphometrics

The main goal of morphometrics is to study how shapes vary and their covariance with other variables. Even though morphometrics can be used to describe the form of any object it is mostly used in biology to describe organisms. Morphometrics is very important in biology because it allows quantitative descriptions of organisms. Quantitative approach allowed scientists to compare shapes of di erent organisms much better and they no longer had to rely on word descriptions that usually had the problem of being interpreted di erently by each scientist. This shift to quantitative descriptions was caused by advances in statistical analysis methods that allowed to interpret collected data. First method of morphometrics called Traditional morphometrics was done by measuring linear distances (such as length, width, and height) and multivariate statistical tools were used to describe patterns of shape variation within and among groups. This approach also sometimes used counts, ratios, areas and angles measures. The biggest advantage of this method was that it was very simple, however it had several di culties. The biggest problem was that linear distance measurements are usually highly correlated with size and this makes shape analysis di cult. Another problem was that measurements taken from two di erent shapes could produce equal results because the data did not include the location of where the measurements were taken relative to each other. And it was also not possible to reconstruct graphical representation of the shape from taken measurements. Figure 1 illustrates the problems of Traditional morphometrics. To overcome these problems a more sophisticated method called Geometric morphometrics was created. Landmark-based geometric morphometrics uses a set of landmarks to describe shape. Landmark is a twoor three-dimensional point described by a tightly de ned set of rules. The results that are generated by this method directly depend on the quality of landmarks. A lot of e orts have to be put to choose landmarks that would have high evolutionary signi cance. Each landmark also has to be present on every studied organism. If a landmark is not present on at least one of studied organisms it either has to be marked approximately or it can not be used at all. The number of landmarks selected should not exceed the number of specimen samples because extra landmarks will be redundant. Usually number of landmarks is approximately equal to the number of specimen samples. There are three types of landmarks that can be used. True landmarks that have some biological signi cance. Pseudo-landmarks are de ned by relative locations e.g. "the point of highest curvature of this bone". Semi-landmarks are de ned by a location relative to other landmarks e.g. "midway between landmarks X and Y". Landmarks can sometimes have weighted value in analysis according to their importance. Extracted landmark data has a lot of variations in position, orientation and scale between specimens. These non-shape variations have to be removed before further analysis. There are several methods used to superimpose landmarks each of them having di erent optimization criteria. The most simple one is two-point registration. This method translates, scales and rotates all landmarks such that two named landmarks are in the same place in all specimens. The biggest disadvantage of this method is that it removes all the data from those two landmarks. Another popular method is Generalized Procrustes analysis (GPA, also sometimes called Generalized least squares). This method rst calculates the centroid of landmark con gurations and translates it to the origin. This is done by taking k points in two dimensional space