We study the problem of selection of regularization parameter in penalized Gaussian graphical models. When the goal is to obtain the model with good predicting power, cross validation is the gold standard. We present a new estimator of Kullback-Leibler loss in Gaussian Graphical model which provides a computationally fast alternative to cross-validation. The estimator is obtained by approximating leave-one-out-cross validation. Our approach is demonstrated on simulated data sets for various types of graphs. The proposed formula exhibits superior performance, especially in the typical small sample size scenario, compared to other available alternatives to cross validation, such as Akaike's information criterion and Generalized approximate cross validation. We also show that the estimator can be used to improve the performance of the BIC when the sample size is small.

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Graphical lasso is one of the most used estimators for inferring genetic networks. Despite its diffusion, there are several fields in applied research where the limits of detection of modern measurement technologies make the use of this estimator theoretically unfounded, even when the assumption of a multivariate Gaussian distribution is satisfied. Typical examples are data generated by polymerase chain reactions and flow cytometer. The combination of censoring and high-dimensionality make inference of the underlying genetic networks from these data very challenging. In this article, we propose an $\ell_1$-penalized Gaussian graphical model for censored data and derive two EM-like algorithms for inference. We evaluate the computational efficiency of the proposed algorithms by an extensive simulation study and show that, when censored data are available, our proposal is superior to existing competitors both in terms of network recovery and parameter estimation. We apply the proposed method to gene expression data generated by microfluidic Reverse Transcription quantitative Polymerase Chain Reaction technology in order to make inference on the regulatory mechanisms of blood development. A software implementation of our method is available on github (https://github.com/LuigiAugugliaro/cglasso).

Dynamic gene-regulatory networks are complex since the interaction patterns between their components mean that it is impossible to study parts of the network in separation. This holistic character of gene-regulatory networks poses a real challenge to any type of modelling. Graphical models are a class of models that connect the network with a conditional independence relationships between random variables. By interpreting these random variables as gene activities and the conditional independence relationships as functional non-relatedness, graphical models have been used to describe gene-regulatory networks. Whereas the literature has been focused on static networks, most time-course experiments are designed in order to tease out temporal changes in the underlying network. It is typically reasonable to assume that changes in genomic networks are few, because biological systems tend to be stable.We introduce a new model for estimating slow changes in dynamic gene-regulatory networks, which is suitable for high-dimensional data, e.g. time-course microarray data. Our aim is to estimate a dynamically changing genomic network based on temporal activity measurements of the genes in the network. Our method is based on the penalized likelihood with ℓ1-norm, that penalizes conditional dependencies between genes as well as differences between conditional independence elements across time points. We also present a heuristic search strategy to find optimal tuning parameters. We re-write the penalized maximum likelihood problem into a standard convex optimization problem subject to linear equality constraints. We show that our method performs well in simulation studies. Finally, we apply the proposed model to a time-course T-cell dataset.

Italian wine is increasingly appreciated in new world consumer countries and, in particular, in Russia where consumers associate its consumption with an Italian lifestyle. In this paper, market value for wine search attributes is measured through the estimation of a hedonic price model using online data from a Wine Searcher website and the information contained in the labels of wines marketed in Russia. Results show a premium price for wines from Piedmont and Tuscany, and in particular for non-native varieties and for Indicazione Geografica Tipica and Protected Geographical Indication wines. Additionally, vintage and higher alcohol content have a significant positive impact on the prices that consumers are willing to pay for wines. The knowledge of the implicit price of wine quality attributes has important implication for producers to aid in the adoption of effective marketing strategies, and for decision-makers for the adoption of effective measures to improve regional competitiveness in the world market

Dynamic networks models describe a growing number of important scientific processes, from cell biology and epidemiology to sociology and finance. There are many aspects of dynamical networks that require statistical considerations. In this paper we focus on determining network structure. Estimating dynamic networks is a difficult task since the number of components involved in the system is very large. As a result, the number of parameters to be estimated is bigger than the number of observations. However, a characteristic of many networks is that they are sparse. For example, the molecular structure of genes make interactions with other components a highly-structured and therefore sparse process.Penalized Gaussian graphical models have been used to estimate sparse networks. However, the literature has focussed on static networks, which lack specific temporal constraints. We propose a structured Gaussian dynamical graphical model, where structures can consist of specific time dynamics, known presence or absence of links and block equality constraints on the parameters. Thus, the number of parameters to be estimated is reduced and accuracy of the estimates, including the identification of the network, can be tuned up. Here, we show that the constrained optimization problem can be solved by taking advantage of an efficient solver, logdetPPA, developed in convex optimization. Moreover, model selection methods for checking the sensitivity of the inferred networks are described. Finally, synthetic and real data illustrate the proposed methodologies.

Factorial Gaussian graphical Models (fGGMs) have recently been proposed for inferring dynamic gene regulatory networks from genomic high-throughput data. In the search for true regulatory relationships amongst the vast space of possible networks, these models allow the imposition of certain restrictions on the dynamic nature of these relationships, such as Markov dependencies of low order - some entries of the precision matrix are a priori zeros - or equal dependency strengths across time lags - some entries of the precision matrix are assumed to be equal. The precision matrix is then estimated by l1-penalized maximum likelihood, imposing a further constraint on the absolute value of its entries, which results in sparse networks. Selecting the optimal sparsity level is a major challenge for this type of approaches. In this paper, we evaluate the performance of a number of model selection criteria for fGGMs by means of two simulated regulatory networks from realistic biological processes. The analysis reveals a good performance of fGGMs in comparison with other methods for inferring dynamic networks and of the KLCV criterion in particular for model selection. Finally, we present an application on a high-resolution time-course microarray data from the Neisseria meningitidis bacterium, a causative agent of life-threatening infections such as meningitis. The methodology described in this paper is implemented in the R package sglasso, freely available at CRAN, http://CRAN.R-project.org/package=sglasso.

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