Elliptic Fourier shape analysis of fossil bivalves: some practical considerations

Abstract

Elliptic Fourier shape analysis is a powerful, though under‐utilized, biometric tool that is particularly suited for the description of fossils lacking many homologous landmarks, such as several common bivalve groups. The method is conceptually more parsimonious than more traditional biometric methods based on discrete linear and angular measurements. Most importantly, however, shape analysis captures a much higher proportion of the morphological information resident in any fossil than analyses based on discrete measurements. The number of harmonics required in an elliptic Fourier analysis can be estimated from a series of inverse Fourier reconstructions, or from the power spectrum. In most studies it is appropriate to normalize Fourier coefficients for size, although this information can be reincorporated at a later stage. The coefficients should probably not be standardized, unless there is evidence to suggest that high‐frequency information was genetically as important as low‐frequency information. Depending upon the aims of a particular study and the morphological disparity of the fossils in question, it might be appropriate to eliminate the first harmonic (‘best‐fitting’) ellipse from an analysis. Meaningful comparison of the left and right valves of bivalves requires the digitized coordinates of one or other to be mirrored prior to computation of the Fourier coefficients. □Biometric analysis, Bivalvia, elliptic Fourier analysis, morphometrics.

Phylogeny and ancient DNA of Sus provides insights into neolithic expansion in Island Southeast Asia and Oceania

“…The coefficients of the first harmonic, describing the best-fitting ellipse to the original outline, were used to standardize the size, orientation, and starting point of the molar outlines. These coefficients correspond to residuals after standardization and should not be included in following statistical analyses (37).…”

Otolith shape variation provides a marker of stock origin for north Atlantic bluefin tuna (Thunnus thynnus)

“…Elliptical Fourier harmonics were extracted from smoothed otolith outlines (using 200 smoothing iterations) using the momocs package in R (http://cran.r-project.org/web/packages/Momocs/ Momocs.pdf, accessed 12 August 2015. The Fourier power equation (Crampton 1995) showed that over 99.9% of the variability in shape was captured by the first 12 harmonics. Each harmonic is composed of four coefficients (a n , b n , c n and d n ).…”

Seasonal variation in molar outline of bank voles: An effect of wear?

“…The first ellipse (defined by the coefficients A 1 , B 1 , C 1 , and D 1 ) contains information about the size, position and rotation of the molar, and has been used to standardize the outline data. Hence, the first three coefficients were not retained for the statistical analysis, as they correspond to residuals after standardization (Crampton 1995;Renaud et al 1996). The fourth one, D 1 , corresponding to the minor axis of the ellipse, contains information about the elongation of the molar and was conserved in the analysis .…”