Statistical assessment of bilateral symmetry of shapes

This paper formulates the problem of measuring bilateral symmetry of object s analytically for landmark data, and develops various new testing procedur es and exploratory data analyses. The development is linked with the fundam ental biological problem of measuring directional asymmetry and fluctuating asymmetry. We distinguish two types of symmetry, object symmetry and match ing symmetry, and provide tests under assumptions of isotropic landmark var iability as well as non-isotropy. The tests require novel statistical and g eometrical analyses of the Procrustes shape manifold. For describing compon ents of symmetry and asymmetry within samples, an extension of principal co mponent analysis is introduced and illustrated by appropriate deformation t echniques. Various real examples are used to illustrate the practical relev ance of this work.