A major problem for biological time series is that often experiments (such as gene expression measurements using microarrays or RNA-seq) require the organism or cells to be destroyed. This means that a particular time series is often a series of measurements of different organisms (or batches of cells) at different times. Biological replicates normally consist of a separate biological sample measured at the same time. With the advent of single cell expression experiments, where it is not currently conceivable to make genome-wide gene expression measurements without destroying the cell, we expect such set ups to be sustained.

Many existing approaches to modelling transcriptional data postulate a differential equation model for continuous-time expression profiles from which the repeated observations arise. Two ways of modelling repeat experiments would be either to handle repeated observations as being from a shared profile, or from completely independent profiles. The former approach assumes that gene expression profile for each experiment does not vary, whilst the latter approach assumes no relationship between the gene expression profiles. For many experimental set ups we might expect something in between these two extremes where, whilst each individual measurement comes from a different collection of cells or a different organism, the experimental set up is broadly the same. We therefore expect some shared affects and some independent affects for the experiments.

In this work we propose an integrated Gaussian process framework for analysis of such experiments. In our approach, independent aspects of the experiments are modelled as independent Gaussian process draws, while the common profile across the experiments is modelled by a separate Gaussian process. The method adds power through sharing of replicates for the common profile while being robust to outliers from individual rogue experiments.