RESUMO
Keerthi and Shevade (2007) proposed an efficient algorithm for constructing an approximate least angle regression least absolute shrinkage and selection operator solution path for logistic regression as a function of the regularization parameter. In this brief, their approach is extended to multinomial regression. We show that a brute-force approach leads to a multivariate approximation problem resulting in an infeasible path tracking algorithm. Instead, we introduce a noncanonical link function thereby: 1) repeatedly reusing the univariate approximation of Keerthi and Shevade, and 2) producing an optimization objective with a block-diagonal Hessian. We carry out an empirical study that shows the computational efficiency of the proposed technique. A MATLAB implementation is available from the author upon request.
RESUMO
Probabilistic Boolean networks (PBNs) have recently been introduced as a promising class of models of genetic regulatory networks. The dynamic behaviour of PBNs can be analysed in the context of Markov chains. A key goal is the determination of the steady-state (long-run) behaviour of a PBN by analysing the corresponding Markov chain. This allows one to compute the long-term influence of a gene on another gene or determine the long-term joint probabilistic behaviour of a few selected genes. Because matrix-based methods quickly become prohibitive for large sizes of networks, we propose the use of Monte Carlo methods. However, the rate of convergence to the stationary distribution becomes a central issue. We discuss several approaches for determining the number of iterations necessary to achieve convergence of the Markov chain corresponding to a PBN. Using a recently introduced method based on the theory of two-state Markov chains, we illustrate the approach on a sub-network designed from human glioma gene expression data and determine the joint steadystate probabilities for several groups of genes.
