Skip to main content
SpringerLink
Log in

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

  • Chapter
  • First Online:
Dynamics, Games and Science I

Part of the book series: Springer Proceedings in Mathematics ((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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alparslan Gök, S.Z.: Cooperative interval games. PhD Thesis, Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey (2009)

    Google Scholar 

  2. Alparslan Gök, S.Z., Branzei, R., Tijs, S.: Convex interval games. Preprint at IAM, Middle East Technical University, Ankara, Turkey, and Center for Economic Research, Tilburg University, The Netherlands (2008)

    Google Scholar 

  3. Alparslan Gök, S.Z., Branzei, R., Tijs, S.: Airport interval games and their Shapley value. Operations Research and Decisions, 2, 9–18 (2010)

    Google Scholar 

  4. Alparslan Gök, S.Z., Miquel, S., Tijs, S.: Cooperation under interval uncertainty. Math. Methods Oper. Res. 69, 99–109 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Alparslan Gök, S.Z., Weber, G.-W.: Cooperative games under ellipsoidal uncertainty. In: The Proceedings of PCO 2010, 3rd Global Conference on Power Control and Optimization, Gold Coast, Queensland, Australia, Feb 2–4, 2010

    Google Scholar 

  6. Alter, O., Brown, P.O., Botstein, D.: Singular value decomposition for genome-wide expression data processing and modeling. PNAS. 97(18), 10101–10106 (2000)

    Article  Google Scholar 

  7. Aster, A., Borchers, B., Thurber, C.: Parameter Estimation and Inverse Problems. Academic, Amsterdam (2004)

    Google Scholar 

  8. Bagirov, A.M., Yearwood, J.: A new nonsmooth optimization algorithm for minimum sum-of-squares clustering problems. Eur. J. Oper. Res. 170(2), 578–596 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Barzily, Z., Volkovich, Z.V., Akteke-Öztürk, B., Weber, G.-W.: Cluster stability using minimal spanning tree. ISI Proceedings of 20th Mini-EURO Conference, Continuous Optimization and Knowledge-Based Technologies, pp. 248–252. Neringa, Lithuania, 20–23 May 2008

    Google Scholar 

  10. Benedetti, R.: Real algebraic and semi-algebraic sets. Hermann, Ed. des Sciences et des Arts, Paris (1990)

    Google Scholar 

  11. Ben-Tal, A.: Conic and robust optimization. Lecture notes (2002)Available athttp://iew3.technion.ac.il/Home/Users/morbt.phtml.

  12. Bochnak, J., Coste, M., Roy, M.-F.: Real algebraic geometry. Springer, New York (1998)

    MATH  Google Scholar 

  13. Borenstein, E., Feldman, M.W.: Topological signatures of species interactions in metabolic networks. J. Comput. Biol. 16(2), 191–200 (2009). doi: 10.1089/cmb.2008.06TT

    Article  Google Scholar 

  14. Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  15. Brown, M.P.S., Grundy, W.N., Lin, D., Cristianini, N., Sugnet, C.W., Furey, T.S., Ares, M., Haussler, D.: Knowledge-based analysis of microarray gene expression data by using support vector machines. PNAS 97(1), 262–267 (2000)

    Article  Google Scholar 

  16. Büyükbebeci, E.: Comparison of MARS, CMARS and CART in predicting default probabilities for emerging markets. MSc. Term Project Report/Thesis in Financial Mathematics, at IAM, METU, Ankara, August 2009

    Google Scholar 

  17. Chipman, H., Tibshirani, R.: Hybrid hierarchical clustering with applications to microarray data. Biostatistics 7(2), 286–301 (2006)

    Article  MATH  Google Scholar 

  18. Mol, C. De, Mosci, S., Traskine, M., Verri, A.: A Regularized Method for selecting nested groups of relevant genes from microarray data. J. Comput. Biol. 16(5), 677–690 (2009)

    Article  MathSciNet  Google Scholar 

  19. Defterli, Ö., Fügenschuh, A., Weber, G.-W.: New discretization and optimization techniques with results in the dynamics of gene-environment networks. In: The proceedings of PCO 2010, 3rd Global Conference on Power Control and Optimization, Gold Coast, Queensland, Australia, Feb 2–4, 2010 (ISBN: 978-983-44483-1-8)

    Google Scholar 

  20. Durieu, P., Walter, É., Polyak, B.: Multi-input multi-output ellipsoidal state bounding. J. Optim. Theory Appl. 111(2), 273–303 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gebert, J., Lätsch, M., Quek, E.M.P., Weber, G.-W.: Analyzing and optimizing genetic network structure via path-finding. J. Comput. Technol. 9(3), 3–12 (2004)

    MATH  Google Scholar 

  22. Gökmen, A., Kayalgil, S., Weber, G.-W., Gökmen, I., Ecevit, M., Sürmeli, A., Bali, T., Ecevit, Y., Gökmen, H., DeTombe, D.J.: Balaban valley project: improving the quality of life in rural area in Turkey. Int. Sci. J. Methods Models Complex. 7(1) (2004)

    Google Scholar 

  23. Harris, J.R., Nystad, W., Magnus, P.: Using genes and environments to define asthma and related phenotypes: applications to multivariate data. Clin. Exp. Allergy 28(1), 43–45 (1998)

    Article  Google Scholar 

  24. Hastie, T., Tibshirani, R.: Discriminant adaptive nearest neighbor classification. IEEE Trans. Pattern Anal. Mach. Intell. 18(6), 607–616 (1996)

    Article  Google Scholar 

  25. Hastie, T., Tibshirani, R., Friedman, J.: The elements of statistical learning. Springer, New York (2001)

    MATH  Google Scholar 

  26. Herrero, J., Valencia, A., Dopazo, J.: A hierarchical unsupervised growing neural network for clustering gene expression patterns. Bioinformatics 17(2), 126–136 (2001)

    Article  Google Scholar 

  27. Hooper, S.D., Boué, S., Krause, R., Jensen, L.J., Mason, C.E., Ghanim, M., White, K.P., Furlong, E.E.M., Bork, P.: Identification of tightly regulated groups of genes during Drosophila melanogaster embryogenesis. Mol. Syst. Biol. 3, 72 (2007)

    Article  Google Scholar 

  28. Işcanoğlu, A., Weber, G.-W., Taylan, P.: Predicting default probabilities with generalized additive models for emerging markets. Invited lecture, Graduate Summer School on New Advances in Statistics, METU (2007)

    Google Scholar 

  29. Kropat, E., Pickl, S., Rössler, A., Weber, G.-W.: On theoretical and practical relations between discrete optimization and nonlinear optimization. In: Special issue Colloquy Optimization – Structure and Stability of Dynamical Systems (at the occasion of the colloquy with the same name, Cologne, October 2000) of Journal of Computational Technologies, vol. 7, pp. 27–62 (2002)

    Google Scholar 

  30. Kropat, E., Weber, G.-W., Akteke-Öztürk, B.: Eco-finance networks under uncertainty. In: Herskovits, J., Canelas, A., Cortes, H., Aroztegui, M. (eds.) Proceedings of the International Conference on Engineering Optimization (ISBN 978857650156-5, CD), EngOpt 2008, Rio de Janeiro, Brazil (2008)

    Google Scholar 

  31. Kropat, E., Weber, G.-W., Rückmann, J.-J.: Regression analysis for clusters in gene-environment networks based on ellipsoidal calculus and optimization. Preprint 157 at IAM, METU, Ankara, Turkey (2009). Submitted to Dynamics of Continuous, Discrete and Impulsive Systems

    Google Scholar 

  32. Kropat, E., Weber, G.-W., Pedamallu, C.S.: Regulatory networks under ellipsoidal uncertainty – optimization theory and dynamical systems. Preprint at IAM, METU, Ankara, Turkey, 2009. Submitted to SIAM Journal on Optimization

    Google Scholar 

  33. Kurzhanski, A.B., Vályi, I.: Ellipsoidal Calculus for Estimation and Control. Birkhäuser, Boston (1997)

    MATH  Google Scholar 

  34. Kurzhanski, A.A., Varaiya, P.: Ellipsoidal Toolbox Manual. EECS Department, University of California, Berkeley (2008)

    Google Scholar 

  35. Lee, Y., Lin, Y., Wahba, G.: Multicategory support vector machines: theory and application to the classification of microarray data and satellite radiance data. J. Am. Stat. Assoc. 99, 67–81 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  36. Mahony, S., McInerney, J.O., Smith, T.J., Golden, A.: Gene prediction using the self-organizing map: automatic generation of multiple gene models. BMC Bioinform 5, 23 (2004). doi:10.1186/1471-2105-5-23

    Article  Google Scholar 

  37. Marvanova, M., Toronen, P., Storvik, M., Lakso, M., Castren, E., Wong, G.: Synexpression analysis of ESTs in the rat brain reveals distinct patterns and potential drug targets. Mol. Brain Res. 104(2), 176–183 (2002)

    Article  Google Scholar 

  38. Mattes, W.B., Pettit, S.D., Sansone, S.A., Bushel, P.R., Waters, M.D.: Database development in toxicogenomics: issues and efforts. Environ. Health Perspect. 112(4), 495–505 (2004)

    Article  Google Scholar 

  39. Nemirovski, A.: Five lectures on modern convex optimization. C.O.R.E. Summer School on Modern Convex Optimization (2002). Available at http://iew3.technion.ac.il/Labs/Opt/opt/LN/Final.pdf

  40. Nemirovski, A.: Lectures on modern convex optimization, Israel Institute of Technology (2002). Available at http://iew3.technion.ac.il/Labs/Opt/opt/LN/Final.pdf

  41. Nemirovski, A.: Interior point polynomial time algorithms in convex programming. Lecture Notes (2004). Available at https://itweb.isye.gatech.edu

    Google Scholar 

  42. Nemirovski, A.: Modern convex optimization. Lecture at PASCAL Workshop, Thurnau, Germany, March 16–18 (2005)

    Google Scholar 

  43. Nesterov, Y.E., Nemirovskii, A.S.: Interior Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994)

    MATH  Google Scholar 

  44. Nikkila, J., Törönen, P., Kaski, S., Venna, J., Castrén, E., Wong, G.: Analysis and visualization of gene expression data using self-organizing maps. Neural Netw. 15(8–9), 953–966 (2002)

    Article  Google Scholar 

  45. Partner, M., Kashtan, N., Alon, U.: Environmental variability and modularity of bacterial metabolic network. BMC Evol. Biol. 7, 169 (2007). doi:10.1186/1471-2148-7-169

    Article  Google Scholar 

  46. Pickl, S.: Der τ-value als Kontrollparameter – Modellierung und Analyse eines Joint-Implementation Programmes mithilfe der dynamischen kooperativen Spieltheorie und der diskreten Optimierung. Thesis, Darmstadt University of Technology, Department of Mathematics (1998)

    Google Scholar 

  47. Quackenbush, J.: Computational analysis of microarray data. Nat. Rev. Genet. 2, 418–427 (2001)

    Article  Google Scholar 

  48. Rivolta, M.N., Halsall, A., Johnson, C.M., Tones, M.A., Holley, M.C.: Transcript profiling of functionally related groups of genes during conditional differentiation of a mammalian cochlear hair cell line. Genome Res. 12, 1091–1099 (2002)

    Article  Google Scholar 

  49. Ros, L., Sabater, A., Thomas, F.: An ellipsoidal calculus based on propagation and fusion. IEEE Trans. Syst. Man Cybern. B Cybern. 32(4), 430–442 (2002)

    Article  Google Scholar 

  50. She, Y.: Sparse regression with exact clustering. PhD Thesis, Department of Statistics, Stanford University, USA (2008)

    Google Scholar 

  51. Shapiro, A., Dentcheva, D., Ruszczy\acute{{ n}}ski, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009, in press)

    Google Scholar 

  52. Tamayo, P., Slonim, D., Mesirov, J., Zhu, Q., Kitareewan, S., Dmitrovsky, E., Lander, E.S., Golub, T.R.: Interpreting patterns of gene expression with self-organizing maps: methods and application to hematopoietic differentiation. PNAS 96, (6), 2907–2912 (1999)

    Google Scholar 

  53. Taştan, M.: Analysis and prediction of gene expression patterns by dynamical systems, and by a combinatorial algorithm. MSc Thesis, Institute of Applied Mathematics, METU, Turkey (2005)

    Google Scholar 

  54. Taştan, M., Ergenç, T., Pickl, S.W., Weber, G.-W.: Stability analysis of gene expression patterns by dynamical systems and a combinatorial algorithm. In: HIBIT – Proceedings of International Symposium on Health Informatics and Bioinformatics, Turkey ’05, pp. 67–75. Antalya, Turkey, 2005

    Google Scholar 

  55. Taştan, M., Pickl, S.W., Weber, G.-W.: Mathematical modeling and stability analysis of gene-expression patterns in an extended space and with Runge-Kutta discretization. In: Proceedings of Operations Research 2005, pp. 443–450. Springer, Bremen, Sept 2005

    Google Scholar 

  56. Taylan, P., Weber, G.-W., Beck, A.: New approaches to regression by generalized additive models and continuous optimization for modern applications in finance, science and techology. In: Burachik, B., Yang, X. (guest eds.) The special issue in honour of Prof. Dr. Alexander Rubinov, Optimization, vol. 56, 5–6, 1–24 (2007)

    Google Scholar 

  57. Thomas, B., Raju, G., Sonam, W.: A modified fuzzy c-means algorithm for natural data exploration. World Acad. Sci. Eng. Technol. 49 (2009)

    Google Scholar 

  58. Uğur, Ö., Pickl, S.W., Weber, G.-W., Wünschiers, R.: Operational research meets biology: An algorithmic approach to analyze genetic networks and biological energy production. Preprint no. 50, Institute of Applied Mathematics, METU, 2006. Submitted for the special issue of Optimization at the occasion of the 5th Ballarat Workshop on Global and Non-Smooth Optimization: Theory, Methods, and Applications (2006)

    Google Scholar 

  59. Uğur, Ö., Pickl, S.W., Weber, G.-W., Wünschiers, R.: An algorithmic approach to analyze genetic networks and biological energy production: an introduction and contribution where OR meets biology. Optimization 58(1), 1–22 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  60. Uğur, Ö., Weber, G.-W.: Optimization and dynamics of gene-environment networks with intervals. In the special issue at the occasion of the 5th Ballarat Workshop on Global and Non-Smooth Optimization: Theory, Methods and Applications, Nov 28–30, 2006, of J. Ind. Manag. Optim., vol. 3(2), 357–379 (2007)

    Google Scholar 

  61. Vazhentsev, A.Y.: On internal ellipsoidal approximations for problems of control synthesis with bounded coordinates. J. Comput. Syst. Sci. Int. 39(3) (2000)

    Google Scholar 

  62. Vázques, F.G., Rückmann, J.-J., Stein, O., Still, G.: Generalized semi-infinite programming: a tutorial. J. Comput. Appl. Math. 217(2), 394–419 (2008)

    Article  MathSciNet  Google Scholar 

  63. Wall, M., Rechtsteiner, A., Rocha, L.: Singular Value Decomposition and Principal Component Analysis. In: Berrar, D.P., Dubitzky, W., Granzow, M. (eds.) A Practical Approach to Microarray Data Analysis, pp. 91–109, Kluwer, Norwell, MA (2003)

    Google Scholar 

  64. Weber, G.-W.: Charakterisierung struktureller Stabilität in der nichtlinearen Optimierung. In: Bock, H.H., Jongen, H.T., Plesken, W.: (eds.) Aachener Beiträge zur Mathematik 5. Augustinus publishing house (now: Mainz publishing house) Aachen (1992)

    Google Scholar 

  65. Weber, G.-W.: Minimization of a max-type function: Characterization of structural stability. In: Guddat, J., Jongen, H.Th., Kummer, B., No\check{{ z}}i\check{{ c}}ka, F. (eds.) Parametric Optimization and Related Topics III, pp. 519–538. Peter Lang publishing house, Frankfurt a.M., Bern, New York (1993)

    Google Scholar 

  66. Weber, G.-W.: Generalized semi-infinite optimization and related topics. In: Hofmannn, K.H., Wille, R. (eds.) Research and Exposition in Mathematics, vol. 29, Heldermann Publishing House, Lemgo (2003)

    Google Scholar 

  67. Weber, G.-W., Alparslan-Gök, S.-Z., Defterli, O., Kropat, E.: Modeling, Inference and Optimization of Regulatory Networks Based on Time Series Data. Preprint at Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey, submitted to European Journal of Operational Research (EJOR)

    Google Scholar 

  68. Weber, G.-W., Alparslan-Gök, S.Z., Dikmen, N.: Environmental and life sciences: gene-environment networks – optimization, games and control – a survey on recent achievements. In: DeTombe, D. (guest ed.) Invited paper, in the special issue of Journal of Organisational Transformation and Social Change, vol. 5(3), pp. 197–233 (2008)

    Google Scholar 

  69. Weber, G.-W., Alparslan-Gök, S.Z., Söyler, B.: A new mathematical approach in environmental and life sciences: gene-environment networks and their dynamics. Environ. Model. Assess. 14(2), 267-Ű288 (2009)

    Google Scholar 

  70. Weber, G.-W., Batmaz, I., Köksal, G., Taylan, P., Yerlikaya-Özkur, F.: CMARS: A new contribution to nonparametric regression with multivariate adaptive regression splines supported by continuous optimisation. Preprint at IAM, METU, Ankara

    Google Scholar 

  71. Weber, G.-W., Kropat, E., Akteke-Öztürk, B., Görgülü, Z.-K.: A survey on OR and mathematical methods applied on gene-environment networks. Special Issue on Innovative Approaches for Decision Analysis in Energy, Health, and Life Sciences of Central European Journal of Operations Research (CEJOR) at the occasion of EURO XXII 2007 (Prague, Czech Republic, July 8–11, 2007), vol. 17(3), 315–341 (2009)

    Google Scholar 

  72. Weber, G.-W., Kropat, E., Tezel, A., Belen, S.: Optimization applied on regulatory and eco-finance networks – survey and new developments. In: Fukushima, M. et al. (guest eds.) Pac. J. Optim., vol. 6(3), Special Issue in memory of Professor Alexander Rubinov (2011, in press)

    Google Scholar 

  73. Weber, G.-W., Özögür-Akyüz, S., Kropat, E.: A review on data mining and continuous optimization applications in computational biology and medicine. Embryo Today, Birth Defects Research (Part C), 87, 165–181 (2009)

    Google Scholar 

  74. Weber, G.-W., Taylan, P., Alparslan-Gök, S.-Z., Özöğür, S., Akteke-Öztürk, B.: Optimization of gene-environment networks in the presence of errors and uncertainty with Chebychev approximation. TOP, the Operational Research journal of SEIO (Spanish Statistics and Operations Research Society) vol. 16(2), 284–318 (2008)

    Google Scholar 

  75. Weber, G.-W., Tezel, A.: On generalized semi-infinite optimization of genetic networks. TOP 15(1), 65–77 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  76. Weber, G.-W., Tezel, A., Taylan, P., Soyler, A., Çetin, M.: Mathematical contributions to dynamics and optimization of gene-environment networks. In: Pallaschke, D., Stein, O. (guest eds.) Special Issue: In Celebration of Prof. Dr. Dr. Hubertus Th. Jongen’s 60th Birthday, of Optimization, vol. 57(2), pp. 353–377 (2008)

    Google Scholar 

  77. Weber, G.-W., Uğur, Ö., Taylan, P., Tezel, A.: On optimization, dynamics and uncertainty: a tutorial for gene-environment networks. In: the Special Issue Networks in Computational Biology of Discrete Appl. Math., vol. 157(10), pp. 2494–2513 (2009)

    MATH  Google Scholar 

  78. Yerlikaya, F.: A new contribution to nonlinear robust regression and classification with MARS and its applications to data mining for quality control in manufacturing. Thesis, Middle East Technical University, Ankara, Turkey (2008)

    Google Scholar 

  79. Yeung, K.Y., Ruzzo, W.L.: Principal component analysis for clustering gene expression data. Bioinformatics 17(9), 763–774 (2001)

    Article  Google Scholar 

  80. Yılmaz, F.B.: A mathematical modeling and approximation of gene expression patterns by linear and quadratic regulatory relations and analysis of gene networks. MSc Thesis, Institute of Applied Mathematics, METU, Ankara, Turkey (2004)

    Google Scholar 

  81. Yılmaz, F.B., Öktem, H., Weber, G.-W.: Mathematical modeling and approximation of gene expression patterns and gene networks. In: Fleuren, F., den Hertog, D., Kort, P. (eds.) Operations Research Proceedings, pp. 280–287 (2005)

    Google Scholar 

  82. Zhang, A.: Advanced Analysis of Gene Expression Microarray Data. World Scientific Pub. Co. Ltd., Singapore (2006)

    MATH  Google Scholar 

Download references

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.

Author information

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

Authors
  1. Erik Kropat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gerhard-Wilhelm Weber
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Selma Belen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Erik Kropat .

Editor information

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

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

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

Download citation

Publish with us

Policies and ethics

Search

Navigation