Entropy2013, 15(12), 5492-5509; doi:10.3390/e15125492 (doi registration under processing) - published online 12 December 2013 Show/Hide Abstract
Abstract: In this paper the combinations of maximum entropy method and Bayesian inference for reliability assessment of deteriorating system is proposed. Due to various uncertainties, less data and incomplete information, system parameters usually cannot be determined precisely. These uncertainty parameters can be modeled by fuzzy sets theory and the Bayesian inference which have been proved to be useful for deteriorating systems under small sample sizes. The maximum entropy approach can be used to calculate the maximum entropy density function of uncertainty parameters more accurately for it does not need any additional information and assumptions. Finally, two optimization models are presented which can be used to determine the lower and upper bounds of systems probability of failure under vague environment conditions. Two numerical examples are investigated to demonstrate the proposed method.
Entropy2013, 15(12), 5475-5491; doi:10.3390/e15125475 - published online 10 December 2013 Show/Hide Abstract
Abstract: Multi-objective particle swarm optimization (MOPSO) is a search algorithm based on social behavior. Most of the existing multi-objective particle swarm optimization schemes are based on Pareto optimality and aim to obtain a representative non-dominated Pareto front for a given problem. Several approaches have been proposed to study the convergence and performance of the algorithm, particularly by accessing the final results. In the present paper, a different approach is proposed, by using Shannon entropy to analyze the MOPSO dynamics along the algorithm execution. The results indicate that Shannon entropy can be used as an indicator of diversity and convergence for MOPSO problems.
Entropy2013, 15(12), 5464-5474; doi:10.3390/e15125464 - published online 9 December 2013 Show/Hide Abstract
Abstract: Complex systems are usually represented as an intricate set of relations between their components forming a complex graph or network. The understanding of their functioning and emergent properties are strongly related to their structural properties. The finding of structural patterns is of utmost importance to reduce the problem of understanding the structure–function relationships. Here we propose the analysis of similarity measures between nodes using hierarchical clustering methods. The discrete nature of the networks usually leads to a small set of different similarity values, making standard hierarchical clustering algorithms ambiguous. We propose the use of multidendrograms, an algorithm that computes agglomerative hierarchical clusterings implementing a variable-group technique that solves the non-uniqueness problem found in the standard pair-group algorithm. This problem arises when there are more than two clusters separated by the same maximum similarity (or minimum distance) during the agglomerative process. Forcing binary trees in this case means breaking ties in some way, thus giving rise to different output clusterings depending on the criterion used. Multidendrograms solves this problem by grouping more than two clusters at the same time when ties occur.
Entropy2013, 15(12), 5439-5463; doi:10.3390/e15125439 - published online 9 December 2013 Show/Hide Abstract
Abstract: We investigate the statistical properties of maximum entropy density estimation, both for the complete data case and the incomplete data case. We show that under certain assumptions, the generalization error can be bounded in terms of the complexity of the underlying feature functions. This allows us to establish the universal consistency of maximum entropy density estimation.
Entropy2013, 15(12), 5419-5438; doi:10.3390/e15125419 - published online 6 December 2013 Show/Hide Abstract
Abstract: The topic of community detection in social networks has attracted a lot of attention in recent years. Existing methods always depict the relationship of two nodes using the snapshot of the network, but these snapshots cannot reveal the real relationships, especially when the connection history among nodes is considered. The problem of detecting the stable community in mobile social networks has been studied in this paper. Community cores are considered as stable subsets of the network in previous work. Based on these observations, this paper divides all nodes into a few of communities due to the community cores. Meanwhile, communities can be tracked through incremental computing. Experimental results based on real-world social networks demonstrate that our proposed method performs better than the well-known static community detection algorithm in mobile social networks.
Entropy2013, 15(12), 5384-5418; doi:10.3390/e15125384 - published online 4 December 2013 Show/Hide Abstract
Abstract: Divergence functions are the non-symmetric “distance” on the manifold, Μθ, of parametric probability density functions over a measure space, (Χ,μ). Classical information geometry prescribes, on Μθ: (i) a Riemannian metric given by the Fisher information; (ii) a pair of dual connections (giving rise to the family of α-connections) that preserve the metric under parallel transport by their joint actions; and (iii) a family of divergence functions ( α-divergence) defined on Μθ x Μθ, which induce the metric and the dual connections. Here, we construct an extension of this differential geometric structure from Μθ (that of parametric probability density functions) to the manifold, Μ, of non-parametric functions on X, removing the positivity and normalization constraints. The generalized Fisher information and α-connections on M are induced by an α-parameterized family of divergence functions, reflecting the fundamental convex inequality associated with any smooth and strictly convex function. The infinite-dimensional manifold, M, has zero curvature for all these α-connections; hence, the generally non-zero curvature of M can be interpreted as arising from an embedding of Μθ into Μ. Furthermore, when a parametric model (after a monotonic scaling) forms an affine submanifold, its natural and expectation parameters form biorthogonal coordinates, and such a submanifold is dually flat for α = ± 1, generalizing the results of Amari’s α-embedding. The present analysis illuminates two different types of duality in information geometry, one concerning the referential status of a point (measurable function) expressed in the divergence function (“referential duality”) and the other concerning its representation under an arbitrary monotone scaling (“representational duality”).