Skip to main content
SpringerLink
Log in

Dimension-wise Decompositions

  • Chapter
  • First Online:
Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 77))

  • 1260 Accesses

Abstract

In this chapter, we introduce the classical ANOVA and the anchored-ANOVA decomposition of a multivariate function f. Based on these decompositions, we then define different notions of effective dimensions of f and derive error bounds for approximation and integration.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
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. Caflisch, R., Morokoff, W., Owen, A.: Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension. J. Comp. Finance 1(1), 27–46 (1997)

    Google Scholar 

  2. Dick, J., Sloan, I., Wang, X., Woźniakowski, H.: Liberating the weights. J. Complexity 20(5), 593–623 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Efron, B., Stein, C.: The jackknife estimate of variance. Annals of Statistics 9, 586–596 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  4. Gerstner, T., Griebel, M.: Dimension–adaptive tensor–product quadrature. Computing 71(1), 65–87 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Griebel, M.: Sparse grids and related approximation schemes for higher dimensional problems. In: L. Pardo, A. Pinkus, E. Suli, M. Todd (eds.) Foundations of Computational Mathematics (FoCM05), Santander, pp. 106–161. Cambridge University Press (2006)

    Google Scholar 

  6. Griebel, M., Holtz, M.: Dimension-wise integration of high-dimensional functions with applications to finance. J. Complexity (2010). Appeared online.

    Google Scholar 

  7. Griebel, M., Kuo, F., Sloan, I.: The smoothing effect of the ANOVA decomposition. J. Complexity (2010). Appeared online.

    Google Scholar 

  8. Griebel, M., Woźniakowski, H.: On the optimal convergence rate of universal and non-universal algorithms for multivariate integration and approximation. Mathematics of Computation 75(255), 1259–1286 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hegland, M.: Adaptive sparse grids. ANZIAM J. 44, C335–C353 (2003)

    MathSciNet  Google Scholar 

  10. Hickernell, F., Woźniakowski, H.: Integration and approximation in arbitrary dimensions. Adv. Comput. Math. 12, 25–58 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hoeffding, W.: A class of statistics with asympotically normal distributions. Annals of Math. Statist. 19, 293–325 (1948)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hooker, G.: Generalized functional ANOVA diagnostics for high dimensional functions of dependent variables. J. Comput. Graph. Statist. 16(3), 709–732 (2007)

    Article  MathSciNet  Google Scholar 

  13. Jinag, T., Owen, A.: Quasi-regression with shrinkage.Math. Comput. Simul. 62, 231–241 (2003)

    Article  Google Scholar 

  14. Kuo, F., Sloan, I., Wasilkowski, G., Woźniakowski, H.: On decompositions of multivariate functions. Math. of Comp. 79, 953–966 (2010)

    Article  MATH  Google Scholar 

  15. Lemieux, C.: Monte Carlo and Quasi-Monte Carlo Sampling. Springer Series in Statistics. Springer (2009)

    Google Scholar 

  16. Lemieux, C., Owen, A.: Quasi-regression and the relative importance of the ANOVA components of a function. In: K. Fang, F. Hickernell, H. Niederreiter (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2000, pp. 331–344. Springer (2002)

    Google Scholar 

  17. Liu, R., Owen, A.: Estimating mean dimensionality of analysis of variance decompositions. J. the American Statistical Association 101, 712–721 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Owen, A.: Dimension distribution and quadrature test functions. Statistica Sinica 13, 1–17 (2003)

    MATH  MathSciNet  Google Scholar 

  19. Rabitz, H., Alis, O.: General foundations of high-dimensional model representations. J. Mathematical Chemistry 25, 197–233 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. Sloan, I., Wang, X., Woźniakowski, H.: Finite-order weights imply tractability of multivariate integration. J. Complexity 20, 46–74 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Sobol, I.: Sensitivity estimates for nonlinear mathematical models. Mathematical Modeling and Computational Experiment 1, 407–414 (1993)

    MATH  MathSciNet  Google Scholar 

  22. Sobol, I.: Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simulation 55, 271–280 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Sobol, I.: Theorems and examples on high dimensional model representation. Reliability Engineering and System Safty 79, 187–193 (2003)

    Article  Google Scholar 

  24. Tezuka, S.: On the necessity of low-effective dimension. J. Complexity 21, 710–721 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Wahba, G.: Spline Models for Observational Data, vol. 59. CBMS-NSF Regional Conf. Ser. Appl. Math., SIAM, Philadelphia (1990)

    MATH  Google Scholar 

  26. Wang, X.: On the approximation error in high dimensional model representation. In: S. Mason et al. (ed.) Proceeding of the 2008 Winter Simulation Conference, pp. 453–462 (2008)

    Google Scholar 

  27. Wang, X., Fang, K.T.: The effective dimension and quasi-Monte Carlo integration. J. Complexity 19(2), 101 – 124 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Wang, X., Sloan, I.: Why are high-dimensional finance problems often of low effective dimension? SIAM J. Sci. Comput. 27(1), 159 – 183 (2005)

    MATH  MathSciNet  Google Scholar 

  29. Wasilkowski, G., Woźniakowski, H.: Finite-order weights imply tractability of linear multivariate problems. J. Approximation Theory 130(1), 57–77 (2004)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität Bonn, Institut für Numerische Simulation, Wegelerstr. 5, 53115, Bonn, Germany

    Markus Holtz

Authors
  1. Markus Holtz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Markus Holtz .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Berlin Heidelberg

About this chapter

Cite this chapter

Holtz, M. (2011). Dimension-wise Decompositions. In: Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance. Lecture Notes in Computational Science and Engineering, vol 77. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16004-2_2

Download citation

Publish with us

Policies and ethics

Search

Navigation