Graphical continuous Lyapunov models

The linear Lyapunov equation of a covariance matrix parametrizes the equilibrium covariance matrix of a stochastic process. This parametrization can be interpreted as a new graphical model class, and we show how the model class behaves under marginalization and introduce a method for structure learning via $\ell_1$-penalized loss minimization. Our proposed method is demonstrated to outperform alternative structure learning algorithms in a simulation study, and we illustrate its application for protein phosphorylation network reconstruction.Comment: 10 pages, 5 figure

Foundational principles for large scale inference: Illustrations through correlation mining

When can reliable inference be drawn in the "Big Data" context? This paper presents a framework for answering this fundamental question in the context of correlation mining, with implications for general large scale inference. In large scale data applications like genomics, connectomics, and eco-informatics the dataset is often variable-rich but sample-starved: a regime where the number $n$ of acquired samples (statistical replicates) is far fewer than the number $p$ of observed variables (genes, neurons, voxels, or chemical constituents). Much of recent work has focused on understanding the computational complexity of proposed methods for "Big Data." Sample complexity however has received relatively less attention, especially in the setting when the sample size $n$ is fixed, and the dimension $p$ grows without bound. To address this gap, we develop a unified statistical framework that explicitly quantifies the sample complexity of various inferential tasks. Sampling regimes can be divided into several categories: 1) the classical asymptotic regime where the variable dimension is fixed and the sample size goes to infinity; 2) the mixed asymptotic regime where both variable dimension and sample size go to infinity at comparable rates; 3) the purely high dimensional asymptotic regime where the variable dimension goes to infinity and the sample size is fixed. Each regime has its niche but only the latter regime applies to exa-scale data dimension. We illustrate this high dimensional framework for the problem of correlation mining, where it is the matrix of pairwise and partial correlations among the variables that are of interest. We demonstrate various regimes of correlation mining based on the unifying perspective of high dimensional learning rates and sample complexity for different structured covariance models and different inference tasks

mgm: Estimating Time-Varying Mixed Graphical Models in High-Dimensional Data

We present the R-package mgm for the estimation of k-order Mixed Graphical Models (MGMs) and mixed Vector Autoregressive (mVAR) models in high-dimensional data. These are a useful extensions of graphical models for only one variable type, since data sets consisting of mixed types of variables (continuous, count, categorical) are ubiquitous. In addition, we allow to relax the stationarity assumption of both models by introducing time-varying versions MGMs and mVAR models based on a kernel weighting approach. Time-varying models offer a rich description of temporally evolving systems and allow to identify external influences on the model structure such as the impact of interventions. We provide the background of all implemented methods and provide fully reproducible examples that illustrate how to use the package

Selection and Estimation for Mixed Graphical Models

We consider the problem of estimating the parameters in a pairwise graphical model in which the distribution of each node, conditioned on the others, may have a different parametric form. In particular, we assume that each node's conditional distribution is in the exponential family. We identify restrictions on the parameter space required for the existence of a well-defined joint density, and establish the consistency of the neighbourhood selection approach for graph reconstruction in high dimensions when the true underlying graph is sparse. Motivated by our theoretical results, we investigate the selection of edges between nodes whose conditional distributions take different parametric forms, and show that efficiency can be gained if edge estimates obtained from the regressions of particular nodes are used to reconstruct the graph. These results are illustrated with examples of Gaussian, Bernoulli, Poisson and exponential distributions. Our theoretical findings are corroborated by evidence from simulation studies