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Dynamics, Games and Science I pp 545–571Cite as

Dynamical Gene-Environment Networks Under Ellipsoidal Uncertainty: Set-Theoretic Regression Analysis Based on Ellipsoidal OR

Part of the Springer Proceedings in Mathematics book series (PROM,volume 1)

Abstract

We consider dynamical gene-environment networks under ellipsoidal uncertainty and discuss the corresponding set-theoretic regression models. Clustering techniques are applied for an identification of functionally related groups of genes and environmental factors. Clusters can partially overlap as single genes possibly regulate multiple groups of data items. The uncertain states of cluster elements are represented in terms of ellipsoids referring to stochastic dependencies between the multivariate data variables. The time-dependent behaviour of the system variables and clusters is determined by a regulatory system with (affine-) linear coupling rules. Explicit representations of the uncertain multivariate future states of the system are calculated by ellipsoidal calculus. Various set-theoretic regression models are introduced in order to estimate the unknown system parameters. Hereby, we extend our Ellipsoidal Operations Research previously introduced for gene-environment networks of strictly disjoint clusters to possibly overlapping clusters. We analyze the corresponding optimization problems, in particular in view of their solvability by interior point methods and semidefinite programming and we conclude with a discussion of structural frontiers and future research challenges.

Keywords

  • Genetic Cluster
  • Regression Problem
  • Interior Point Method
  • Environmental Cluster
  • Ellipsoidal Approximation

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Acknowledgements

The authors express their cordial gratitude to Professor Alberto Pinto for inviting us to contribute with our chapter to this distinguished book in honour of Prof. Dr. Mauricio Peixoto and Prof. Dr. David Rand.

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Authors and Affiliations

  1. Institute for Theoretical Computer Science, Mathematics and Operations Research, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577, Neubiberg, München, Germany

    Erik Kropat

  2. Institute of Applied Mathematics, Middle East Technical University, 06531, Ankara, Turkey

    Gerhard-Wilhelm Weber

  3. Faculty of Economics, Business and Law, University of Siegen, Siegen, Germany

    Gerhard-Wilhelm Weber

  4. Center for Research on Optimization and Control, University of Aveiro, Aveiro, Portugal

    Gerhard-Wilhelm Weber

  5. Faculty of Science, Universiti Teknologi Malaysia (UTM), Skudai, Malaysia

    Gerhard-Wilhelm Weber

  6. CAG University, Yenice-Tarsus, 33800, Mersin, Turkey

    Selma Belen

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  1. Erik Kropat
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Correspondence to Erik Kropat .

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Editors and Affiliations

  1. Pura e Aplicada (IMPA), Instituto de Matemática, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Rio de Janeiro, Brazil

    Mauricio Matos Peixoto

  2. Faculdade de Ciências, Departamento de Matemática, Universidade do Porto, Rua do Campo Alegre 687, Porto, 4169-007, Portugal

    Alberto Adrego Pinto

  3. Warwick Systems Biology, University of Warwick, Coventry House, Coventry, CV4 7AL, United Kingdom

    David A. Rand

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Kropat, E., Weber, GW., Belen, S. (2011). Dynamical Gene-Environment Networks Under Ellipsoidal Uncertainty: Set-Theoretic Regression Analysis Based on Ellipsoidal OR. In: Peixoto, M., Pinto, A., Rand, D. (eds) Dynamics, Games and Science I. Springer Proceedings in Mathematics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11456-4_35

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