Skip to main content
SpringerLink
Log in

Multivariate data modelling through Piecewise generalized HDMR method

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

This work aims to develop a new High Dimensional Model Representation (HDMR) based method which can construct an analytical structure for a given multivariate data modelling problem. Modelling multivariate data through a divide-and-conquer method stands for multivariate data partitioning process in which we deal with a number of less variate data sets instead of a single N dimensional problem. Generalized HDMR is one of these methods used to model a multivariate data set which has a number of scattered nodes with associated function values. However, Generalized HDMR includes a linear equation system with huge number of unknowns and equations to be solved. This equation sometimes has linearly dependent equations in it and this is an undesirable situation. This work offers a new method named Piecewise Generalized HDMR method which bypasses this disadvantage as well as reducing the mathematical complexity and CPU time needed to complete the algorithm of the previous method. Our new method splits the given problem domain into subdomains, applies the Generalized HDMR philosophy to each subdomain and superpositions the information coming from these subdomains. The algorithm of this new method and a number of numerical implementations are given in this paper.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Sobol I.M.: Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp. (MMCE) 1, 407–414 (1993)

    Google Scholar 

  2. Rabitz H., Alış Ö.F., Shorter J., Shim K.: Efficient input-output model representations. Comput. Phys. Commun. 117, 11–20 (1999)

    Article  CAS  Google Scholar 

  3. Rabitz H., AlışÖ.: General foundations of high dimensional model representations. J. Math. Chem. 25, 197–233 (1999)

    Article  CAS  Google Scholar 

  4. Li G., Wang S.-W., Rabitz H.: Practical approaches to construct RS-HDMR component functions. J. Phys. Chem. A 106, 8721–8733 (2002)

    Article  CAS  Google Scholar 

  5. Demiralp M.: High dimensional model representation and its application varieties. Math. Res. 9, 146–159 (2003)

    Google Scholar 

  6. Tunga B., Demiralp M.: Hybrid high dimensional model representation approximants and their utilization in applications. Math. Res. 9, 438–446 (2003)

    Google Scholar 

  7. Demiralp M.: Illustrative implementations to show how logarithm based high dimensional model representation works for various function structures. WSEAS Trans. Comput. 5, 1339–1344 (2006)

    Google Scholar 

  8. Yaman  İ., Demiralp M.: A new rational approximation technique based on transformational high dimensional model representation. Numer. Algorithms 52, 385–407 (2009)

    Article  Google Scholar 

  9. Tunga B., Demiralp M.: Constancy maximization based weight optimization in high dimensional model representation. Numer. Algorithms 52, 435–459 (2009)

    Article  Google Scholar 

  10. Tunga M.A., Demiralp M.: A factorized high dimensional model representation on the partitioned random discrete data. Appl. Numer. Anal. Comp. Math. 1, 231–241 (2004)

    Article  Google Scholar 

  11. Ziehn T., Tomlin A.S.: A global sensitivity study of sulfur chemistry in a premixed methane flame model using HDMR. Int. J. Chem. Kinet. 40, 742–753 (2008)

    Article  CAS  Google Scholar 

  12. Ziehn T., Tomlin A.S.: GUI-HDMR—a software tool for global sensitivity analysis of complex models. Environ. Model. Softw. 24, 775–785 (2009)

    Article  Google Scholar 

  13. Sridharan J., Chen T.: Modeling multiple input switching of CMOS gates in DSM technology using HDMR. Proc. Des. Autom. Test Eur. 1(3), 624–629 (2006)

    Google Scholar 

  14. Rao B.N., Chowdhury R.: Probabilistic analysis using high dimensional model representation and fast fourier transform. Int. J. Comput. Methods Eng. Sci. Mech. 9, 342–357 (2008)

    Article  Google Scholar 

  15. Chowdhury R., Rao B.N.: Hybrid high dimensional model representation for reliability analysis. Comput. Methods Appl. Mech. Eng. 198, 753–765 (2009)

    Article  Google Scholar 

  16. Gomez M.C., Tchijov V., Leon F., Aguilar A.: A tool to improve the execution time of air quality models. Environ. Model. Softw. 23, 27–34 (2008)

    Article  Google Scholar 

  17. Banerjee I., Ierapetritou M.G.: Design optimization under parameter uncertainty for general black-box models. Ind. Eng. Chem. Res. 41, 6687–6697 (2002)

    Article  CAS  Google Scholar 

  18. Banerjee I., Ierapetritou M.G.: Parametric process synthesis for general nonlinear models. Comput. Chem. Eng. 27, 1499–1512 (2003)

    Article  CAS  Google Scholar 

  19. Banerjee I., Ierapetritou M.G.: Model independent parametric decision making. Ann. Oper. Res. 132, 135–155 (2004)

    Article  Google Scholar 

  20. Shan S., Wang G.G.: Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions. Struct. Multidiscip. Optim. 41, 219–241 (2010)

    Article  Google Scholar 

  21. Tunga M.A., Demiralp M.: A new approach for data partitioning through high dimensional model representation. Int. J. Comput. Math. 85, 1779–1792 (2008)

    Article  Google Scholar 

  22. Tunga M.A., Demiralp M.: Data partitioning via generalized high dimensional model representation (GHDMR) and multivariate interpolative applications. Math. Res. 9, 447–462 (2003)

    Google Scholar 

  23. M.A. Tunga, M. Demiralp, Data Partitioning Through Piecewise Based Generalized GHDMR: Univariate Case, the 2nd International Symposium on Computing in Science and Engineering (ISCSE 2011), June 1–4, 2011 (Kuşadas  Aydın, Turkey, 2011), pp. 128–133

  24. Tunga M.A.: An approximation method to model multivariate interpolation problems: indexing HDMR. Math. Comput. Model. 53(9–10), 1970–1982 (2011)

    Article  Google Scholar 

  25. Tunga M.A.: A matrix based indexing HDMR method for multivariate data modelling. J. Math. Chem. 49(5), 1092–1114 (2011)

    Article  CAS  Google Scholar 

  26. Oevel W., Postel F., Wehmeier S., Gerhard J.: The MuPAD Tutorial. Springer, New York (2000)

    Google Scholar 

  27. Deitel H.M., Deitel P.J., Nieto T.R., McPhie D.C.: How to Program Perl. Prentice Hall, New Jersey (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Software Engineering Department, Faculty of Engineering, Bahçeşehir University, Beşiktaş, 34349, Istanbul, Turkey

    M. Alper Tunga

  2. Informatics Institute, Computational Science and Engineering Program, İstanbul Technical University, Maslak, 34469, Istanbul, Turkey

    Metin Demiralp

Authors
  1. M. Alper Tunga
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Metin Demiralp
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Alper Tunga.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tunga, M.A., Demiralp, M. Multivariate data modelling through Piecewise generalized HDMR method. J Math Chem 50, 1711–1726 (2012). https://doi.org/10.1007/s10910-012-0001-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-012-0001-2

Keywords

Search

Navigation