34 research outputs found
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
-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
On generating random Gaussian graphical models
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. In this work we investigate different methods to generate
random symmetric positive definite matrices with undirected graphical
constraints. We show that if the graph is chordal it is possible to sample
uniformly from the set of correlation matrices compatible with the graph, while
for general undirected graphs we rely on a partial orthogonalization method.Comment: Improved figures, algorithm descriptions and text exposition. arXiv
admin note: substantial text overlap with arXiv:1807.0309
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
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
The R Package stagedtrees for Structural Learning of Stratified Staged Trees
stagedtrees is an R package which includes several algorithms for learning the structure of staged trees and chain event graphs from data. Score-based and clustering-based algorithms are implemented, as well as various functionalities to plot the models and perform inference. The capabilities of stagedtrees are illustrated using mainly two datasets both included in the package or bundled in R
Expressive power of binary relevance and chain classifiers based on Bayesian Networks for multi-label classification
Bayesian network classifiers are widely used in machine learning because they intuitively represent causal relations. Multi-label classification problems require each instance to be assigned a subset of a defined set of h labels. This problem is equivalent to finding a multi-valued decision function that predicts a vector of h binary classes. In this paper we obtain the decision boundaries of two widely used Bayesian network approaches for building multi-label classifiers: Multi-label Bayesian network classifiers built using the binary relevance method and Bayesian network chain classifiers. We extend our previous single-label results to multi-label chain classifiers, and we prove that, as expected, chain classifiers provide a more expressive model than the binary relevance method