Detecting (non)parallel evolution in multidimensional spaces: angles, correlations and eigenanalysis.

Area of Science:

• Evolutionary Biology
• Quantitative Genetics
• Geometric Statistics

Background:

• Parallelism in evolutionary trajectories suggests repeatable phenotypic evolution.
• Angles between trajectories are crucial for analyzing evolutionary parallelism.
• Geometric and statistical properties of angles in multidimensional spaces are underappreciated in biology.

Purpose of the Study:

• To provide an overview of geometric and statistical aspects of angles in multidimensional spaces.
• To establish statistical baselines for analyzing evolutionary trajectory parallelism.
• To equip biologists with methods for statistically justified inferences on (non)parallel evolution.

Main Methods:

• Review of geometric and statistical properties of angles in multidimensional trait spaces.
• Exploration of probability distributions (related to t- and beta distributions) for angles between trajectory vectors under a null hypothesis of random directions.
• Connection of eigenanalysis of vector correlation matrices to tests for vector concentration.
• Application of directional statistics tools like the Rayleigh test.

Main Results:

• Under the null hypothesis, the angle between two independent evolutionary trajectory vectors concentrates around 90 degrees, especially in higher dimensions.
• This angular distribution relates to established statistical distributions (t and beta), enabling hypothesis testing.
• Methods like eigenanalysis and directional statistics offer frameworks for testing vector concentration and correlation among multiple trajectories.

Conclusions:

• Understanding multidimensional angles is essential for robust analysis of evolutionary parallelism.
• Statistical frameworks presented provide necessary baselines for inferring repeatability in phenotypic evolution.
• These tools facilitate statistically sound conclusions regarding parallel evolutionary trajectories.