On Convergence Properties of the EM Algorithm for Gaussian Mixtures

"Expectation-Maximization'' (EM) algorithm and gradient-based approaches for maximum likelihood learning of finite Gaussian mixtures. We show that the EM step in parameter space is obtained from the gradient via a projection matrix $P$, and we provide an explicit expression for the matrix. We then analyze the convergence of EM in terms of special properties of $P$ and provide new results analyzing the effect that $P$ has on the likelihood surface. Based on these mathematical results, we present a comparative discussion of the advantages and disadvantages of EM and other algorithms for the learning of Gaussian mixture models.

On the radius of centrality in evolving communication networks

In this article, we investigate how the choice of the attenuation factor in an extended version of Katz centrality influences the centrality of the nodes in evolving communication networks. For given snapshots of a network, observed over a period of time, recently developed communicability indices aim to identify the best broadcasters and listeners (receivers) in the network. Here we explore the attenuation factor constraint, in relation to the spectral radius (the largest eigenvalue) of the network at any point in time and its computation in the case of large networks. We compare three different communicability measures: standard, exponential, and relaxed (where the spectral radius bound on the attenuation factor is relaxed and the adjacency matrix is normalised, in order to maintain the convergence of the measure). Furthermore, using a vitality-based measure of both standard and relaxed communicability indices, we look at the ways of establishing the most important individuals for broadcasting and receiving of messages related to community bridging roles. We compare those measures with the scores produced by an iterative version of the PageRank algorithm and illustrate our findings with two examples of real-life evolving networks: the MIT reality mining data set, consisting of daily communications between 106 individuals over the period of one year, a UK Twitter mentions network, constructed from the direct \emph{tweets} between 12.4k individuals during one week, and a subset the Enron email data set.

Chain motifs: The tails and handles of complex networks

A great part of the interest in complex networks has been motivated by the presence of structured, frequently nonuniform, connectivity. Because diverse connectivity patterns tend to result in distinct network dynamics, and also because they provide the means to identify and classify several types of complex network, it becomes important to obtain meaningful measurements of the local network topology. In addition to traditional features such as the node degree, clustering coefficient, and shortest path, motifs have been introduced in the literature in order to provide complementary descriptions of the network connectivity. The current work proposes a different type of motif, namely, chains of nodes, that is, sequences of connected nodes with degree 2. These chains have been subdivided into cords, tails, rings, and handles, depending on the type of their extremities (e.g., open or connected). A theoretical analysis of the density of such motifs in random and scale-free networks is described, and an algorithm for identifying these motifs in general networks is presented. The potential of considering chains for network characterization has been illustrated with respect to five categories of real-world networks including 16 cases. Several interesting findings were obtained, including the fact that several chains were observed in real-world networks, especially the world wide web, books, and the power grid. The possibility of chains resulting from incompletely sampled networks is also investigated.

The impact of measurement errors in the identification of regulatory networks

Background: There are several studies in the literature depicting measurement error in gene expression data and also, several others about regulatory network models. However, only a little fraction describes a combination of measurement error in mathematical regulatory networks and shows how to identify these networks under different rates of noise. Results: This article investigates the effects of measurement error on the estimation of the parameters in regulatory networks. Simulation studies indicate that, in both time series (dependent) and non-time series (independent) data, the measurement error strongly affects the estimated parameters of the regulatory network models, biasing them as predicted by the theory. Moreover, when testing the parameters of the regulatory network models, p-values computed by ignoring the measurement error are not reliable, since the rate of false positives are not controlled under the null hypothesis. In order to overcome these problems, we present an improved version of the Ordinary Least Square estimator in independent (regression models) and dependent (autoregressive models) data when the variables are subject to noises. Moreover, measurement error estimation procedures for microarrays are also described. Simulation results also show that both corrected methods perform better than the standard ones (i.e., ignoring measurement error). The proposed methodologies are illustrated using microarray data from lung cancer patients and mouse liver time series data. Conclusions: Measurement error dangerously affects the identification of regulatory network models, thus, they must be reduced or taken into account in order to avoid erroneous conclusions. This could be one of the reasons for high biological false positive rates identified in actual regulatory network models.

Optimization of the AZO dyes decoloration process through neural networks: Determination of the H2O2 addition critical point

The concentration of hydrogen peroxide is an important parameter in the azo dyes decoloration process through the utilization of advanced oxidizing processes, particularly by oxidizing via UV/H2O2. It is pointed out that, from a specific concentration, the hydrogen peroxide works as a hydroxyl radical self-consumer and thus a decrease of the system`s oxidizing power happens. The determination of the process critical point (maximum amount of hydrogen peroxide to be added) was performed through a ""thorough mapping"" or discretization of the target region, founded on the maximization of an objective function objective (constant of reaction kinetics of pseudo-first order). The discretization of the operational region occurred through a feedforward backpropagation neural model. The neural model obtained presented remarkable coefficient of correlation between real and predicted values for the absorbance variable, above 0.98. In the present work, the neural model had, as phenomenological basis the Acid Brown 75 dye decoloration process. The hydrogen peroxide addition critical point, represented by a value of mass relation (F) between the hydrogen peroxide mass and the dye mass, was established in the interval 50 < F < 60. (C) 2007 Elsevier B.V. All rights reserved.

The choice of the best among the shortest routes in transparent optical networks

This work introduces the problem of the best choice among M combinations of the shortest paths for dynamic provisioning of lightpaths in all-optical networks. To solve this problem in an optimized way (shortest path and load balance), a new fixed routing algorithm, named Best among the Shortest Routes (BSR), is proposed. The BSR`s performance is compared in terms of blocking probability and network utilization with Dijkstra`s shortest path algorithm and others algorithms proposed in the literature. The evaluated scenarios include several representative topologies for all-optical networking and different wavelength conversion architectures. For all studied scenarios, BSR achieved superior performance. (C) 2010 Elsevier B.V. All rights reserved.

On the convergence of successive Galerkin approximation for nonlinear output feedback H(infinity) control

Although the formulation of the nonlinear theory of H(infinity) control has been well developed, solving the Hamilton-Jacobi-Isaacs equation remains a challenge and is the major bottleneck for practical application of the theory. Several numerical methods have been proposed for its solution. In this paper, results on convergence and stability for a successive Galerkin approximation approach for nonlinear H(infinity) control via output feedback are presented. An example is presented illustrating the application of the algorithm.

Optimization of the Topology of Electric Energy Distribution Networks by Using Algorithm Inspired on Ant Behaviors

An algorithm inspired on ant behavior is developed in order to find out the topology of an electric energy distribution network with minimum power loss. The algorithm performance is investigated in hypothetical and actual circuits. When applied in an actual distribution system of a region of the State of Sao Paulo (Brazil), the solution found by the algorithm presents loss lower than the topology built by the concessionary company.

Exponential Rates of Convergence in the Ergodic Theorem: A Constructive Approach

We prove that, once an algorithm of perfect simulation for a stationary and ergodic random field F taking values in S(Zd), S a bounded subset of R(n), is provided, the speed of convergence in the mean ergodic theorem occurs exponentially fast for F. Applications from (non-equilibrium) statistical mechanics and interacting particle systems are presented.

Characterization of dimethacrylate polymeric networks: A study of the crosslinked structure formed by monomers used in dental composites

The resin phase of dental composites is mainly composed of combinations of dimethacrylate comonomers, with final polymeric network structure defined by monomer type/reactivity and degree of conversion. This fundamental study evaluates how increasing concentrations of the flexible triethylene glycol dimethacrylate (TEGDMA) influences void formation in bisphenol A diglycidyl dimethacrylate (BisGMA) co-polymerizations and correlates this aspect of network structure with reaction kinetic parameters and macroscopic volumetric shrinkage. Photopolymerization kinetics was followed in real-time by a near-infrared (NIR) spectroscopic technique, viscosity was assessed with a viscometer, volumetric shrinkage was followed with a linometer, free volume formation was determined by positron annihilation lifetime spectroscopy (PALS) and the sol-gel composition was determined by extraction with dichloromethane followed by (1)H NMR analysis. Results show that, as expected, volumetric shrinkage increases with TEGDMA concentration and monomer conversion. Extraction/(1)H NMR studies show increasing participation of the more flexible TEGDMA towards the limiting stages of conversion/crosslinking development. As the conversion progresses, either based on longer irradiation times or greater TEGDMA concentrations, the network becomes more dense, which is evidenced by the decrease in free volume and weight loss after extraction in these situations. For the same composition (BisGMA/TEGDMA 60-40 mol%) light-cured for increasing periods of time (from 10 to 600 s), free volume decreased and volumetric shrinkage increased, in a linear relationship with conversion. However, the correlation between free volume and macroscopic volumetric shrinkage was shown to be rather complex for variable compositions exposed for the same time (600 s). The addition of TEGDMA decreases free-volume up to 40 mol% (due to increased conversion), but above that concentration, in spite of the increase in conversion/crosslinking, free volume pore size increases due to the high concentration of the more flexible monomer. In those cases, the increase in volumetric shrinkage was due to higher functional group concentration, in spite of the greater free volume. Therefore, through the application of the PALS model, this study elucidates the network formation in dimethacrylates commonly used in dental materials. (C) 2010 Elsevier Ltd. All rights reserved.

Ductile fabrics in the zone of active oblique convergence near the Alpine Fault, New Zealand: identifying the neotectonic overprint

The mid-crustal Alpine Schist in central Southern Alps, New Zealand has been exhumed during the past similar to3 m.y. on the hanging wall of the oblique-slip Alpine Fault. These rocks underwent ductile deformation during their passage through the similar to 150-km-wide Pacific-Australia plate boundary zone. Likely to be Cretaceous in age, peak metamorphism predates the largely Pliocene and younger oblique convergence that continues to uplift the Southern Alps today. Late Cenozoic ductile deformation constructively reinforced a pre-existing fabric that was well oriented to accommodate a dextral-transpressive overprint. Quartz microstructures below a recently exhumed brittle-ductile transition zone reflect a late Cenozoic increment of ductile strain that was distributed across deeper levels of the Pacific Plate. Deformation was transpressive, including a dextral-normal shear component that bends and rotates a delaminated panel of Pacific Plate crust onto the oblique footwall ramp of the Alpine Fault. Progressive ductile shear in mylonites at the base of the Pacific Plate overprints earlier fabrics in a dextral-reverse sense, a deformation that accompanies translation of the schists up the Alpine Fault. Ductile shear along that structure affects not only the 12-km-thick section of Alpine mylonites, but is distributed across several kilometres of overlying nonmylonitic rocks. (C) 2001 Elsevier Science Ltd. All rights reserved.

Sorting and convergence of primary olfactory axons are independent of the olfactory bulb

Primary olfactory axons expressing the same odorant receptor gene sort out and converge to fixed sites in the olfactory bulb. We examined the guidance of axons expressing the P2 odorant receptor when they were challenged with different cellular environments in vivo. In the mutant extratoes mouse, the olfactory bulb is lacking and is replaced by a fibrocellular mass. In these animals, primary olfactory axons form glomerular-like loci despite the absence of normal postsynaptic targets. P2 axons are able to sort out from other axons in this fibrocellular mass and converge to form loci of like axons. The sites of these loci along mediolateral and ventrodorsal axes were highly variable. Similar convergence was observed for larger subpopulations of axons expressing the same cell surface carbohydrates. The sorting out and convergence of like axons also occurred during regeneration following bulbectomy. Olfactory axon behaviour in these models demonstrates that sorting and convergence of axons are independent of the target, which instead provides distinct topographic cues for guidance. (C) 2003 Wiley-Liss, Inc.

The highly rearranged mitochondrial genome of the plague thrips, Thrips imaginis (Insecta : thysanoptera): Convergence of two novel gene boundaries and an extraordinary arrangement of rRNA genes

To help understand the mechanisms of gene rearrangement in the mitochondrial (mt) genomes of hemipteroid insects, we sequenced the mt genome of the plague thrips, Thrips imaginis (Thysanoptera). This genome is circular, 15,407 by long, and has many unusual features, including (1) rRNA genes inverted and distant from one another, (2) an extra gene for tRNA-Ser, (3) a tRNA-Val lacking a D-arm, (4) two pseudo-tRNA genes, (5) duplicate control regions, and (6) translocations and/or inversions of 24 of the 37 genes. The mechanism of rRNA gene transcription in T. imaginis may be different from that of other arthropods since the two rRNA genes have inverted and are distant from one another. Further, the rRNA genes are not adjacent or even close to either of the two control regions. Tandem duplication and deletion is a plausible model for the evolution of duplicate control regions and for the gene translocations, but intramitochondrial recombination may account for the gene inversions in T. imaginis. All the 18 genes between control regions #1 and #2 have translocated and/or inverted, whereas only six of the 20 genes outside this region have translocated and/or inverted. Moreover, the extra tRNA gene and the two pseudo-tRNA genes are either in this region or immediately adjacent to one of the control regions. These observations suggest that tandem duplication and deletion may be facilitated by the duplicate control regions and may have occurred a number of times in the lineage leading to T. imaginis. T. imaginis shares two novel gene boundaries with a lepidopsocid species from another order of hemipteroid insects, the Psocoptera. The evidence available suggests that these shared gene boundaries evolved by convergence and thus are not informative for the interordinal phylogeny of hemipteroid insects. We discuss the potential of hemipteroid insects as a model system for studies of the evolution of animal rut genomes and outline some fundamental questions that may be addressed with this system.