Evolutionary Inference for Function-valued Traits: Gaussian Process Regression on Phylogenies

In this paper we consider statistical inference for function-valued data which are correlated due to phylogenetic relationships. A schematic example is given in Figure 1A: in this case, given functional data observed at the tips of a phylogeny, the task is to perform inference on the (unobserved) functional data at the root of the phylogeny. Alternatively, if the phylogeny is uncertain we may wish to perform phylogenetic inference, or our interest may be inferring the dynamics of the evolutionary process which produced the data. The term'function-valued' is meant in the sense of [1], where a datum is a continuous functionf (x) of a variablex, such as time or temperature: an examples are therefore curves for ambient temperature versus growth rate for caterpillars, a heart rhythm time series [2], or a spectrogram of audio data. Our approach is to combine the theory of Gaussian processes with assumptions from phylogenetics, to obtain a flexible nonparametric model for such data.