# Geometric Morphometrics

Geometric morphometrics is one of my areas of methodological expertise.

Most of my published papers use 2D or 3D geometric morphometric methods either alone or in conjunction with other quantitative approaches, including genomics and transcriptomics.

I have also published methodological geometric morphometric papers and  implemented some of these methods in easy-to-use software (R packages; please, see the Software section of the website).

Finally, every year I give a 5-day course in Geometric Morphometrics organized by Physalia Courses.

This page is intended to give an informal and intuitive idea of how these methods work.

## (Geometric) morphometrics

Morphometrics is the statistical analysis of shape.
It is a set of techniques which are widely used in biological sciences to
characterize, and quantitatively study, shapes of biological interest.

Geometric morphometrics is a younger
area of morphometrics.
It comprises a set of techniques which,
apart from allowing the statistical analysis of shape (as in “traditional
morphometrics"), retain geometric properties of objects
throughout the analysis.

In my research, geometric morphometrics is both a tool to answer biological questions and an object investigation in itself.

For instance, I have used geometric morphometrics either alone or in combination with other tools such as genomics to study phenotypic evolution in fish and other organisms. At the same time, I have developed new approaches to study the variation in modularity in organisms, and studied measurement error in geometric morphometric data.

## An intuitive explanation of geometric morphometrics

When studying a biological shape, such as fish body shape, with traditional morphometrics we could take a series of linear measurements between pairs of points, such as standard length (SL). From the moment we take these measurements onward, we loose all the geometric properties of the original shape and we have only a series of numbers representing distances.

When using geometric morphometrics, instead, we keep the geometric information throughout the analysis. For instance, in the case of landmark-based geometric morphometrics (which is the most widely used “version" of geometric morphometrics), we acquire coordinates of points capturing the shape we want to analyze and carry out a series of statistical analyses on them. Given that we retained geometric properties throughout the analysis, at the end of the analyses we can visualize results as shapes, deformations of the original shapes or deformation grids. There are numerous other advantages in using geometric morphometrics as opposed to traditional morphometrics. For instance, the information contained in traditional measurements is often redundant. Similarly, geometric morphometrics often allows capturing very subtle variation in shape, which would be otherwise lost using traditional methods. The advantages of geometric morphometrics over more traditional approaches have been widely documented in the literature.

A typical workflow for landmark-based geometric morphometrics is the following:

• Obtain coordinates of landmarks (homologous points)
• Obtain centroid size (which is a measure of size independent of shape which can be used in following analyses)
• Carry out a Procustes superimposition (which optimally superimposes configurations of points by removing the effects of translation, rotation and scale)
• Analyze the resulting coordinates (or their transformation) with statistical methods to answer biological questions (for instance, using discriminant analysis to discriminate a priori-defined groups or performing a multivariate regression of shape on centroid size to study allometric variation in shape)
• As the geometrical properties are not lost during the analysis, the results can be visualized as shapes, or deformation grids While landmark-based geometric morphometrics is probably the most used “version” of geometric morphometrics, there are many other methods that go under the “umbrella category” of “geometric morphometrics”. Among them, some of the most common are methods that analyze outlines by fitting mathematical functions to the coordinates of points of the outline (as in the case elliptic Fourier analysis) or by using points (called “semilandmarks”) that are not homologous but retain positional correspondence in a framework similar to the one provided by landmark-based geometric morphometrics.