Abstract. Gaussian graphical models are widely utilized to infer and visualize networks of dependencies between continuous variables. However, inferring the graph is difficult when the sample size is small compared to the number of variables. To reduce the number of parameters to estimate in the model, we propose a non-asymptotic model selection procedure supported by strong theoretical guarantees based on an oracle type inequality and a minimax lower bound. The covariance matrix of the model is approximated by a block-diagonal matrix. The structure of this matrix is detected by thresholding the sample covariance matrix, where the threshold is selected using the slope heuristic. Based on the block-diagonal structure of the covariance matrix, the estimation problem is divided into several independent problems: subsequently, the network of dependencies between variables is inferred using the graphical lasso algorithm in each block. The performance of the procedure is illustrated on simulated data. An application to a real gene expression dataset with a limited sample size is also presented: the dimension reduction allows attention to be objectively focused on interactions among smaller subsets of genes, leading to a more parsimonious and interpretable modular network.
We introduce in this survey the major concepts, models, and algorithms proposed so far to infer causal relations from observational time series, a task usually referred to as causal discovery in time series. To do so, after a description of the underlying concepts and modelling assumptions, we present different methods according to the family of approaches they belong to: Granger causality, constraint-based approaches, noise-based approaches, score-based approaches, logic-based approaches, topology-based approaches, and difference-based approaches. We then evaluate several representative methods to illustrate the behaviour of different families of approaches. This illustration is conducted on both artificial and real datasets, with different characteristics. The main conclusions one can draw from this survey is that causal discovery in times series is an active research field in which new methods (in every family of approaches) are regularly proposed, and that no family or method stands out in all situations. Indeed, they all rely on assumptions that may or may not be appropriate for a particular dataset.
We report large-scale molecular dynamics simulations of the glass formation from the liquid phase and homogeneous nucleation phenomena of pure zirconium. For this purpose, we have built a modified embedded atom model potential, in order to reproduce relevant structural, dynamic and thermodynamic properties from ab initio and experimental data near the melting point. By means of liquid-solid interface simulations, we show that this potential provides a thermodynamic melting temperature and densities of the solid and liquid state in good agreement with experiments. Using melt-quenching simulations with one million atoms, we determine the glass transition from the temperature evolution of the inherent structure energy as well as the nose of the time-temperature-transformation curve located in the deep undercooling regime. We identify the local structural origin of the glass forming ability as a competition between bcc and five-fold polytetrahedral structures that may represent an impediment of rapid homogeneous nucleation at such high undercoolings. This suggests the ability of single elemental zirconium to form a glass from the melt with cooling rates of at least 10 12 K/s compatible with modern experiments.
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