Autoregressive Asymmetric Linear Gaussian Hidden Markov Models

In a real life process evolving over time, the relationship between its relevant variables may change. Therefore, it is advantageous to have different inference models for each state of the process. Asymmetric hidden Markov models fulfil this dynamical requirement and provide a framework where the trend of the process can be expressed as a latent variable. In this paper, we modify these recent asymmetric hidden Markov models to have an asymmetric autoregressive component, allowing the model to choose the order of autoregression that maximizes its penalized likelihood for a given training set. Additionally, we show how inference, hidden states decoding and parameter learning must be adapted to fit the proposed model. Finally, we run experiments with synthetic and real data to show the capabilities of this new model.Comment: 34 pages, 16 figures, intended to be published in IEEE Transactions on Pattern Analysis and Machine Intelligenc

A partial orthogonalization method for simulating covariance and concentration graph matrices

Structure learning methods for covariance and concentration graphs are often validated on synthetic models, usually obtained by randomly generating: (i) an undirected graph, and (ii) a compatible symmetric positive definite (SPD) matrix. In order to ensure positive definiteness in (ii), a dominant diagonal is usually imposed. However, the link strengths in the resulting graphical model, determined by off-diagonal entries in the SPD matrix, are in many scenarios extremely weak. Recovering the structure of the undirected graph thus becomes a challenge, and algorithm validation is notably affected. In this paper, we propose an alternative method which overcomes such problem yet yielding a compatible SPD matrix. We generate a partially row-wise-orthogonal matrix factor, where pairwise orthogonal rows correspond to missing edges in the undirected graph. In numerical experiments ranging from moderately dense to sparse scenarios, we obtain that, as the dimension increases, the link strength we simulate is stable with respect to the structure sparsity. Importantly, we show in a real validation setting how structure recovery is greatly improved for all learning algorithms when using our proposed method, thereby producing a more realistic comparison framework.Comment: 12 pages, 5 figures, conferenc

Learning an L1-regularized Gaussian Bayesian Network in the Equivalence Class Space

Learning the structure of a graphical model from data is a common task in a wide range of practical applications. In this paper, we focus on Gaussian Bayesian networks, i.e., on continuous data and directed acyclic graphs with a joint probability density of all variables given by a Gaussian. We propose to work in an equivalence class search space, specifically using the k-greedy equivalence search algorithm. This, combined with regularization techniques to guide the structure search, can learn sparse networks close to the one that generated the data. We provide results on some synthetic networks and on modeling the gene network of the two biological pathways regulating the biosynthesis of isoprenoids for the Arabidopsis thaliana plan

Forward Stagewise Naive Bayes

The naïve Bayes approach is a simple but often satisfactory method for supervised classification. In this paper, we focus on the naïve Bayes model and propose the application of regularization techniques to learn a naïve Bayes classifier. The main contribution of the paper is a stagewise version of the selective naïve Bayes, which can be considered a regularized version of the naïve Bayes model. We call it forward stagewise naïve Bayes. For comparison’s sake, we also introduce an explicitly regularized formulation of the naïve Bayes model, where conditional independence (absence of arcs) is promoted via an L 1/L 2-group penalty on the parameters that define the conditional probability distributions. Although already published in the literature, this idea has only been applied for continuous predictors. We extend this formulation to discrete predictors and propose a modification that yields an adaptive penalization. We show that, whereas the L 1/L 2 group penalty formulation only discards irrelevant predictors, the forward stagewise naïve Bayes can discard both irrelevant and redundant predictors, which are known to be harmful for the naïve Bayes classifier. Both approaches, however, usually improve the classical naïve Bayes model’s accuracy

Multi-objective Estimation of Distribution Algorithm Based on Joint Modeling of Objectives and Variables

This paper proposes a new multi-objective estimation of distribution algorithm (EDA) based on joint modeling of objectives and variables. This EDA uses the multi-dimensional Bayesian network as its probabilistic model. In this way it can capture the dependencies between objectives, variables and objectives, as well as the dependencies learnt between variables in other Bayesian network-based EDAs. This model leads to a problem decomposition that helps the proposed algorithm to find better trade-off solutions to the multi-objective problem. In addition to Pareto set approximation, the algorithm is also able to estimate the structure of the multi-objective problem. To apply the algorithm to many-objective problems, the algorithm includes four different ranking methods proposed in the literature for this purpose. The algorithm is applied to the set of walking fish group (WFG) problems, and its optimization performance is compared with an evolutionary algorithm and another multi-objective EDA. The experimental results show that the proposed algorithm performs significantly better on many of the problems and for different objective space dimensions, and achieves comparable results on some compared with the other algorithms

Sparse Cholesky covariance parametrization for recovering latent structure in ordered data

The sparse Cholesky parametrization of the inverse covariance matrix can be interpreted as a Gaussian Bayesian network; however its counterpart, the covariance Cholesky factor, has received, with few notable exceptions, little attention so far, despite having a natural interpretation as a hidden variable model for ordered signal data. 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.Comment: 24 pages, 12 figure

Conductance interaction identification by means of Boltzmann distribution and mutual information analysis in conductance-based neuron models

Conductance interaction identification by means of Boltzmann distribution and mutual information analysis in conductance-based neuron models