On generating random Gaussian graphical models

Córdoba, Irene; Varando, Gherardo; Bielza, Concha; Larrañaga, Pedro

doi:10.1016/j.ijar.2020.07.007

Cited by 3 publications

(2 citation statements)

References 27 publications

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“…In particular, Chapter 4 is based on the conference article Córdoba et al (2018b), with Section 4.4.2 and the proof of Proposition 4.2.1 being novel material first appearing in this thesis. Such chapter serves as an introduction to the sampling methodology that is later applied to Gaussian graphical models in Chapter 5, largely based on both the conference article Córdoba et al (2018c) and the journal article Córdoba et al (2020c). Proposition 5.1.1 and Algorithm 6 are novel content, and serve to further illustrate sampling problems in acyclic directed models, which were not covered in the aforementioned papers but are nevertheless relevant to the thesis content.…”

Section: Discussion Of the Resultsmentioning

confidence: 99%

“…Proposition 5.1.1 and Algorithm 6 are novel content, and serve to further illustrate sampling problems in acyclic directed models, which were not covered in the aforementioned papers but are nevertheless relevant to the thesis content. Some figures also differ because of such unification, for example, Figures 5.10 and 5.11 merge the respective versions of Córdoba et al (2018c) and Córdoba et al (2020c), for better clarity and to avoid duplication.…”

Section: Discussion Of the Resultsmentioning

confidence: 99%

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Unifying methodologies for graphical models with Gaussian parametrization

Sánchez¹

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Graphical models represent conditional independences of a multivariate distribution by absent edges in a graph, which typically is directed, undirected or mixed. This compact modelling allows to decompose statistical inference into efficient computations over the associated graph. As such, graphical models originated mainly at the interface between statistics and artificial intelligence, with Markov networks (undirected graph) and Bayesian networks (acyclic digraph) being the classic representatives. Nowadays graphical models are widely applied and a significant amount of research is devoted to them.Gaussian Markov networks and Gaussian Bayesian networks, although not being equivalent models, share a common intersection consisting of chordal graphs (or acyclic digraphs with no v-structures). A typical approach for model selection in both model classes is hypothesis testing, which amounts to selecting the graph that parametrizes the model. Absent edges in both models are represented by a zero pattern in the inverse covariance or partial correlation matrix (Gaussian Markov networks) or in its Cholesky decomposition (Gaussian Bayesian networks). Afterwards, their parameters are estimated by maximum likelihood. Alternatively, there exist state-of-the-art regularisation methods for both model classes, which simultaneously perform model selection and estimation.A popular method for model selection via hypothesis testing is the PC algorithm, which can be applied for both Gaussian Markov networks and Gaussian Bayesian networks. This method mainly depends on two parameters: the statistical test type and the significance level at which the hypotheses are tested. However, the usual approach in the literature is to use a Gaussian test for a transformation of the partial correlation, and a grid search for its significance level. By contrast, when using an automatic procedure for parameter tuning, such as Bayesian optimization, it is shown how model selection performance is significantly improved when employing an uncommonly used test in the literature. Furthermore, these automatic parameter tuning procedures allow to select a significance level optimized for each test type.Validation of methodologies for Gaussian graphical model selection is also deeply affected, apart from how parameter tuning is performed, by how synthetic test models are simulated. It can be shown that state-of-the-art methodologies addressing this task are biased towards certain regions, thereby significantly influencing validation results. It would be therefore desirable to have a uniform sampling procedure for Gaussian graphical models. In particular, both Gaussian Bayesian networks and Gaussian Markov networks are intimately related with the partial correlation matrix, thereby uniform sampling methods for such set, called elliptope, can be a departing point. A novel Metropolis uniform sampling from the elliptope is proposed, which can be straightforwardly extended to chordal Gaussian graphical models. However, in the general case, a partial orthogonali...

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Section: Discussion Of the Resultsmentioning

confidence: 99%

Section: Discussion Of the Resultsmentioning

confidence: 99%

Unifying methodologies for graphical models with Gaussian parametrization

Sánchez¹

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Sparse Cholesky Covariance Parametrization for Recovering Latent Structure in Ordered Data

et al. 2020

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The sparse Cholesky parametrization of the inverse covariance matrix is directly related to Gaussian Bayesian networks. Its counterpart, the covariance Cholesky factorization model, has a natural interpretation as a hidden variable model for ordered signal data. Despite this, it has received little attention so far, with few notable exceptions. To fill this gap, in this paper we focus on arbitrary zero patterns in the Cholesky factor of a covariance matrix. We discuss how these models can also be extended, in analogy with Gaussian Bayesian networks, to data where no apparent order is available. For the ordered scenario, we propose a novel estimation method that is based on matrix loss penalization, as opposed to the existing regression-based approaches. The performance of this sparse model for the Cholesky factor, together with our novel estimator, is assessed in a simulation setting , as well as over spatial and temporal real data where a natural ordering arises among the variables. We give guidelines, based on the empirical results, about which of the methods analysed is more appropriate for each setting.

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Control Co-Design Optimization of Spacecraft Trajectory and System for Interplanetary Missions

Harris,

He,

Abdelkhalik

2024

Journal of Spacecraft and Rockets

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This paper develops a control co-design (CCD) framework to simultaneously optimize the spacecraft’s trajectory and onboard system (rocket engine) and quantify its benefit. An open-loop optimal control problem (two-finite burn Mars missions) is used as the benchmark, and the engine design considers the combustion equilibrium and nozzle geometry. The objective function is the fuel burn. The design variables are the trajectory control parameters (such as burn times, burn directions, and time of flight), initial fuel mass, and engine design parameters (such as throat area, mixture ratio, and chamber pressure). The constraints include the final velocities and positions of spacecraft. Single-point optimizations are conducted for three departure dates in May, July, and September 2020. A multipoint optimization is also performed to balance the engine performance for these dates with 49 design variables and 20 constraints. It is found that the CCD optimizations exhibit 22–28% more fuel burn reduction than the trajectory-only optimization with fixed engine parameters and 16–20% more fuel burn reduction than the decoupled trajectory-engine optimization. The proposed CCD optimization framework can be extended to more spacecraft trajectory control parameters and onboard systems and has the potential to design more efficient spacecraft missions.

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On generating random Gaussian graphical models

Cited by 3 publications

References 27 publications

Unifying methodologies for graphical models with Gaussian parametrization

Unifying methodologies for graphical models with Gaussian parametrization

Sparse Cholesky Covariance Parametrization for Recovering Latent Structure in Ordered Data

Control Co-Design Optimization of Spacecraft Trajectory and System for Interplanetary Missions

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