Inferring time-derivatives, including cell growth rates, using Gaussian processes

Often the time-derivative of a measured variable is of as much interest as the variable itself. For a growing population of biological cells, for example, the population's growth rate is typically more important than its size. Here we introduce a non-parametric method to infer first and second time-derivatives as a function of time from time-series data. Our approach is based on established properties of Gaussian processes and therefore applies to a wide range of data. In tests, the method is at least as accurate as others, but has several advantages: it estimates errors both in the inference and in any summary statistics, such as lag times, allows interpolation with the corresponding error estimation, and can be applied to any number of experimental replicates. As illustrations, we infer growth rate from measurements of the optical density of populations of microbial cells and estimate the rate of in vitro assembly of an amyloid fibril and both the speed and acceleration of two separating spindle pole bodies in a single yeast cell. Being accessible through both a GUI and from scripts, our algorithm should have broad application across the sciences.