Exploring variation in fitness surfaces over time or space

As the number of studies estimating selection on multiple traits has increased in recent years, fitness surfaces have become a fundamental tool for understanding multivariate selection and evolution. However, rigorous statistical comparisons of multivariate selection surfaces over time or space have been limited to parametric analyses of selection coefficients estimated using a quadratic regression model. Although parametric comparisons are useful when selection is approximately linear or quadratic in nature, they are limited when confronting the complex nature of rugged fitness surfaces. Here, I present a novel solution to comparing non-parametric fitness surfaces over time or space. Using a Tucker3 tensor decomposition, which is essentially a higher-order principal components analysis, I show how major features of fitness surfaces can be compared statistically. Combined with a bootstrap algorithm, I develop three statistical tests that identify 1) Differences in the shape of non-parametric fitness surfaces, 2) Differences in the contribution of each surface to variation in fitness across time or space, and 3) Specific areas of the surfaces (trait combinations) that vary significantly over time or space. I illustrate the tensor decomposition and statistical analyses using idealized fitness surfaces.