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Non-negative matrix factorization is a useful tool for reducing the dimension of large datasets. This work considers simultaneous non-negative matrix factorization of multiple sources of data. In particular, we perform the first study that involves more than two datasets. We discuss the algorithmic issues required to convert the approach into a practical computational tool and apply the technique to new gene expression data quantifying the molecular changes in four tissue types due to different dosages of an experimental panPPAR agonist in mouse. This study is of interest in toxicology because, whilst PPARs form potential therapeutic targets for diabetes, it is known that they can induce serious side-effects. Our results show that the practical simultaneous non-negative matrix factorization developed here can add value to the data analysis. In particular, we find that factorizing the data as a single object allows us to distinguish between the four tissue types, but does not correctly reproduce the known dosage level groups. Applying our new approach, which treats the four tissue types as providing distinct, but related, datasets, we find that the dosage level groups are respected. The new algorithm then provides separate gene list orderings that can be studied for each tissue type, and compared with the ordering arising from the single factorization. We find that many of our conclusions can be corroborated with known biological behaviour, and others offer new insights into the toxicological effects. Overall, the algorithm shows promise for early detection of toxicity in the drug discovery process.

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When one considers a Poincaré return map on a general unbounded (n−1)-dimensional section in a vector field in R n there are typically points where the flow is tangent to the section. The only notable exception is when the system is (equivalent to) a periodically forced system. The tangencies cause bifurcations of the Poincaré return map when the section is moved when there are no bifurcations in the underlying vector field. The interaction of invariant manifolds and the tangency loci on the surface gives rise to discontinuities of the Poincaré map and there can be open regions where the map is not defined. We study the case of the four-dimensional phase space R 4 . Specifically, we make use of tools from singularity theory and flowbox theory to present normal forms of the codimension-one tangency bifurcations in the neighbourhood of a tangency point.AMS classification scheme numbers: 37C10, 37G05 37G25, 58K50.