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We describe a general approach to transforming molecular models between different levels of resolution, based on machine learning methods. The approach uses a matched set of models at both levels of resolution for training, but requires only the coordinates of their particles and no side information (e.g., templates for substructures, defined mappings, or molecular mechanics force fields). Once trained, the approach can transform further molecular models of the system between the two levels of resolution in either direction with equal facility.

Network data are becoming increasingly available, and so there is a need to develop suitable methodology for statistical analysis. Networks can be represented as graph Laplacian matrices, which are a type of manifold-valued data. Our main objective is to estimate a regression curve from a sample of graph Laplacian matrices conditional on a set of Euclidean covariates, for example in dynamic networks where the covariate is time. We develop an adapted Nadaraya-Watson estimator which has uniform weak consistency for estimation using Euclidean and power Euclidean metrics. We apply the methodology to the Enron email corpus to model smooth trends in monthly networks and highlight anomalous networks. Another motivating application is given in corpus linguistics, which explores trends in an author's writing style over time based on word co-occurrence networks., Comment: 15 pages, 4 figures

Mathematical modelling can play a key role in understanding as well as quantifying uncertainties surrounding the presence and fate of emerging pollutants in wastewater treatment processes (WWTPs). This paper presents for the first time a simplified emerging pollutant pathway in the WWTP that incorporates two potential pathways to sequestration. It develops de-terministic and stochastic ordinary differential equations to gain insight into the fate and behaviour of a case study pharmaceutical, with particular focus on sorption to the solid phase, as well as the nature of the experimentally measured solid parent compound. Statistical estimation and inferential procedures are developed and via a proof-of-concept examination, the study explores the transformation pathways of the bioactive chemicals (BACs) in the bioreactor, which is the heart of the WWTP. With a focus on the case study pharmaceutical 17??ethinlyestradiol (EE2), the simulation results show good agreement with the EE2 data. In addition, the results suggest that the experimentally measured solid EE2-parent concentration is very similar to the model-based sequestered EE2-parent concentration.

Solid dispersions can be a successful way to enhance the bioavailability of poorly soluble drugs. Here 60 solid dispersion formulations were produced using ten chemically diverse, neutral, poorly soluble drugs, three commonly used polymers, and two manufacturing techniques, spray drying and melt extrusion. Each formulation underwent a six-month stability study at accelerated conditions, 40 °C and 75% relative humidity (RH). Significant differences in times to crystallisation (onset of crystallisation) were observed between both the different polymers and the two processing methods. Stability from zero days to over one year was observed. The extensive experimental dataset obtained from this stability study was used to build multiple linear regression models to correlate physicochemical properties of the active pharmaceutical ingredients (API) with the stability data. The purpose of these models is to indicate which combination of processing method and polymer carrier is most likely to give a stable solid dispersion. Six quantitative mathematical multiple linear regression-based models were produced based on selection of the most influential independent physical and chemical parameters from a set of 33 possible factors, one model for each combination of polymer and processing method, with good predictability of stability. Three general rules are proposed from these models for the formulation development of suitably stable solid dispersions. Namely, increased stability is correlated with increased glass transition temperature (Tg) of solid dispersions, as well as decreased number of H-bond donors and increased molecular flexibility (such as rotatable bonds and ring count) of the drug molecule.

A method is introduced for variable selection and prediction in linear regression problems where the number of predictors can be much larger than the number of observations. The methodology involves minimising a penalised Euclidean distance, where the penalty is the geometric mean of the $\ell_1$ and $\ell_2$ norms of the regression coefficients. This particular formulation exhibits a grouping effect, which is useful for model selection in high dimensional problems. Also, an important result is a model consistency theorem, which does not require an estimate of the noise standard deviation. An algorithm for estimation is described, which involves thresholding to obtain a sparse solution. Practical performances of variable selection and prediction are evaluated through simulation studies and the analysis of real datasets.

We introduce methodology for analysing the mean size-and-shape and covariance matrix of landmark data that are collected over time. Motivated by a study of DNA damage, we study some permutation based tests for investigating significant differences in the structure of the mean and the variability/covariance of size-and-shape of point sets which evolve over time. The covariance matrix tests make use of some recently introduced metrics for comparing covariance matrices. We demonstrate that the tests have the correct significance level in various simulation studies, and we also investigate the relative power of the tests. Finally we apply the procedures to the DNA datasets, providing practical insights into different types of DNA damage.

Contributed discussion and rejoinder to "Geodesic Monte Carlo on Embedded Manifolds" (arXiv:1301.6064), Comment: Discussion of arXiv:1301.6064. To appear in the Scandinavian Journal of Statistics. 18 pages