# Investigating Shape Variation Using Generalized Procrustes Analysis and Machine Learning
## 2 Materials and Methods  
To evaluate the research hypotheses of this study, two main strategies were implemented. Initially, the learned features were visualized and manually inspected. For this inspection, the existing biological knowledge represented by the proposed landmark positions was used. The biological explainability of all used models was investigated. For the purpose of this study, we defined the biological explainability as the ability of models to explain the reasoning process based on meaningful biological information. To this end, we used the explainability of generalized Procrustes analysis, relying on landmarks placed on specimens as the reference for biological explainability. In addition to this visual inspection, the features and landmarks were investigated using hypothesis tests. Initially we used [PERSON]'s rank correlation tests to investigate the relation between the features as well as landmarks to the known population clusters. However, these tests do not compare the capability of the machine learning models and landmark-based approaches to identify visible information which is useful to unveil the actual population clusters. In this study, we aimed to measure the unambiguity of known population clusters relying on processed landmarks or features obtained by machine learning models. The performance of the unsupervised machine learning models in contrast to the landmark-based approaches was evaluated by comparing the unveiled cluster unambiguity. To investigate this unambiguity of the known clusters in the learned feature space as well as in the extracted landmarks, we proposed a novel and non-parametric test. For this test, we created several population cluster hypotheses and used a novel Bayesian extension of consensus clustering [33] as well as the principle of self-similarity for the formulation of a Bayesian hypothesis test for population discriminability. The technical core problem of this study was the quantification of a model's capability to unveil morphological structure. However, this quantification is a complex task due toambiguity. Different models and optimization strategies maybe result in biological meaningful morphological clusters using the same dataset [29]. Similarly, a model may unveil subpopulations, but fail to quantify obvious differences in water bodies and vice versa. This effect is visualized in Figure 1. On the left side, two dimensional representations of specimens (e.g., two principal coordinates of generalized Procrustes analysis features) are shown as blue points. The right side shows the visual representation of two models (red and green) with different global parameters \(\phi\) and local parameters \(\psi\). Using different optimization strategies, two valid cluster hypotheses, namely four clusters (red model) or two clusters (green model) in the two-dimensional space, maybe occur. Both cluster models unveil biological interpretable structures and may differ as a result of different mathematical formulations or optimization strategies. In order to be able to quantify methods, and inspired by infinite mixture models [34] as well as the idea of cluster ensemble [33; 35], this study combined multiple population structure hypotheses unveiled in landmark and machine learning-based visual diversity data. These visual diversity data were generated relying on generalized Procrustes analysis and unsupervised machine learning models. We aimed to fuse the information of all population structure hypotheses. The combined results, as well as known clusters, were used to compare the landmark-based approaches with the machine learning approaches. Our developed processing pipeline is shown in Figure 2.  
The remaining part of this chapter introduces the used data, landmark processing methods, machine learning models and morphological diversity investigation.