title: The conceptual framework of ontogenetic trajectories: parallel transport allows the recognition and visualization of pure deformation patterns.
abstract: Ontogeny is usually studied by analyzing a deformation series spanning over juvenile to adult shapes. In geometric morphometrics, this approach implies applying generalized Procrustes analysis coupled with principal component analysis on multiple individuals or multiple species datasets. The trouble with such a procedure is that it mixes intra- and inter-group variation. While MANCOVA models are relevant statistical/mathematical tools to draw inferences about the similarities of trajectories, if one wants to observe and interpret the morphological deformation alone by filtering inter-group variability, a particular tool, namely parallel transport, is necessary. In the context of ontogenetic trajectories, one should firstly perform separate multivariate regressions between shape and size, using regression predictions to estimate within-group deformations relative to the smallest individuals. These deformations are then applied to a common reference (the mean of per-group smallest individuals). The estimation of deformations can be performed on the Riemannian manifold by using sophisticated connection metrics. Nevertheless, parallel transport can be effectively achieved by estimating deformations in the Euclidean space via ordinary Procrustes analysis. This approach proved very useful in comparing ontogenetic trajectories of species presenting large morphological differences at early developmental stages.
authors.affiliation: Dipartimento di Scienze, Università Roma Tre, Rome, Italy.|Center for Evolutionary Ecology, Università Roma Tre, Rome, Italy.|Dipartimento di Ingegneria Strutturale e Geotecnica "Sapienza", Università di Roma, Rome, Italy.|Dipartimento di Scienze Cardiovascolari, Respiratorie, Nefrologiche, Anestesiologiche e Geriatriche, Sapienza, Università di Roma, Rome, Italy.
authors.forename: P
authors.initials: P
authors.lastname: Piras
authors.affiliation: Dipartimento di Matematica e Fisica, Università Roma Tre, Rome, Italy.
authors.forename: L
authors.initials: L
authors.lastname: Teresi
authors.affiliation: Dipartimento di Scienze, Università Roma Tre, Rome, Italy.
authors.forename: L
authors.initials: L
authors.lastname: Traversetti
authors.affiliation: Dipartimento di Architettura, Università Roma Tre, Rome, Italy.
authors.forename: V
authors.initials: V
authors.lastname: Varano
authors.affiliation: Dipartimento di Architettura, Università Roma Tre, Rome, Italy.
authors.forename: S
authors.initials: S
authors.lastname: Gabriele
authors.affiliation: Dipartimento di Scienze, Università Roma Tre, Rome, Italy.|Center for Evolutionary Ecology, Università Roma Tre, Rome, Italy.
authors.forename: T
authors.initials: T
authors.lastname: Kotsakis
authors.affiliation: Dipartimento di Scienze della Terra, dell'Ambiente e delle Risorse (DiSTAR), Università degli Studi di Napoli Federico II, Naples, Italy.
authors.forename: P
authors.initials: P
authors.lastname: Raia
authors.affiliation: Dipartimento di Scienze Cardiovascolari, Respiratorie, Nefrologiche, Anestesiologiche e Geriatriche, Sapienza, Università di Roma, Rome, Italy.
authors.forename: P E
authors.initials: PE
authors.lastname: Puddu
authors.affiliation: Dipartimento di Scienze, Università Roma Tre, Rome, Italy.
authors.forename: M
authors.initials: M
authors.lastname: Scalici
mesh_terms: D000818:Animals; D001699:Biometry; D008322:Mammals / Q000033:anatomy & histology* / Q000145:classification; D008954:Models, Biological; D015999:Multivariate Analysis; D025341:Principal Component Analysis; D012886:Skull / Q000033:anatomy & histology*