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Sequential Barycentric Interpolation

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mODa 9 – Advances in Model-Oriented Design and Analysis

Part of the book series: Contributions to Statistics ((CONTRIB.STAT.))

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Abstract

Polynomial interpolators may exhibit oscillating behaviour which often makes them inadequate for modelling functions. A well-known correction to this problem is to use Chebyshev design points. However, in a sequential strategy it is not very clear how to add points, while still improving polynomial interpolation. We present a sequential design alternative by allocating an extra observation where the difference between consecutive interpolators is largest. Our proposal is independent of the response and does not require distributional assumptions. In simulated examples, we show the good interpolation performance of our proposal and its asymptotical convergence to the Chebyshev distribution.

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References

  • Berrut, J.-P. and L. N. Trefethen (2004). Barycentric Lagrange interpolation. SIAM Review 46, 501–517.

    Article  MATH  MathSciNet  Google Scholar 

  • Boyd, J. P. and J. R. Ong (2009). Exponentially-convergent strategies for defeating the Runge phenomenon for the approximation of non-periodic functions. I. Single-interval schemes. Communications in Computational Physics 5, 484–497.

    MathSciNet  Google Scholar 

  • Boyd, J. P. and F. Xu (2009). Divergence (Runge phenomenon) for least-squares polynomial approximation on an equispaced grid and Mock-Chebyshev subset interpolation. Applied Mathematics and Computation 210, 158–168.

    Article  MATH  MathSciNet  Google Scholar 

  • Bratley, P. and B. L. Fox (1988). ALGORITHM 659 Implementing Sobol’s quasirandom sequence generator. ACM Transactions on Mathematical Software 14, 88–100.

    Article  MATH  Google Scholar 

  • Dette, H. (1993a). A note on E-optimal designs for weighted polynomial regression. Annals of Statistics 21, 767–771.

    Article  MATH  MathSciNet  Google Scholar 

  • Dette, H. (1993b). On a mixture of the D- and D1-optimality criterion in polynomial regression. Journal of Statistical Planning and Inference 35, 233–249.

    Article  MATH  MathSciNet  Google Scholar 

  • Epperson, J. F. (1987). On the Runge example. American Mathematical Monthly 94, 329–341.

    Article  MATH  MathSciNet  Google Scholar 

  • Fedorov, V. V. (1972). Theory of Optimal Experiments. New York: Academic Press.

    Google Scholar 

  • Fedorov, V. V. and W. G. Müller (2007). Optimum design for correlated fields via covariance kernel expansions. In J. Lopez-Fidalgo, J. M. Rodriguez-Diaz, and B. Torsney (Eds.), mODa 8–-Advances in Model-oriented Design and Analysis, pp. 57–66. Heidelberg: Physica-Verlag.

    Chapter  Google Scholar 

  • Heiligers, B. (1998). E-optimal designs for polynomial spline regression. Journal of Statistical Planning and Inference 75, 159–172.

    Article  MATH  MathSciNet  Google Scholar 

  • Karlin, S. and W. J. Studden (1966). Optimal experimental designs. Annals of Mathematical Statistics 37, 783–815.

    Article  MATH  MathSciNet  Google Scholar 

  • Müller, W. G. (2001). Collecting Spatial Data, rev. edn. Heidelberg: Physica-Verlag.

    Google Scholar 

  • O’Hagan, A. (2006). Bayesian analysis of computer code outputs: a tutorial. Reliability Engineering & System Safety 37, 1290–1300.

    Article  Google Scholar 

  • Pázman, A. (1986). Foundations of Optimum Experimental Design. Dordrecht: Reidel.

    MATH  Google Scholar 

  • Platte, R. B. and T. A. Driscoll (2005). Polynomials and potential theory for Gaussian radial basis function interpolation. SIAM Journal on Numerical Analysis 43, 750–766.

    Article  MATH  MathSciNet  Google Scholar 

  • Pukelsheim, F. (1993). Optimal Design of Experiments. New York: Wiley.

    MATH  Google Scholar 

  • Pukelsheim, F. and B. Torsney (1991). Optimal weights for experimental designs on linearly independent support points. Annals of Statistics 19, 1614–1625.

    Article  MATH  MathSciNet  Google Scholar 

  • Studden, W. J. (1968). Optimal designs on Tchebycheff points. Annals of Mathematical Statistics 39, 1435–1447.

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, UK

    Hugo Maruri-Aguilar

  2. Departamento de Estadística e Investigación Operativa, Universidad de Valladolid, Prado de la Magdalena s/n, 47005, Valladolid, Spain

    Paula Camelia Trandafir

Authors
  1. Hugo Maruri-Aguilar
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  2. Paula Camelia Trandafir
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Corresponding author

Correspondence to Hugo Maruri-Aguilar .

Editor information

Editors and Affiliations

  1. Dipt. di Scienze Statistiche, University of Bologna, Via Belle Arti 41, Bologna, 40126, Italy

    Alessandra Giovagnoli

  2. and Political Science, Department of Statistics, London School Economics, Houghton Street, London, WC2A 2AE, United Kingdom

    Anthony C. Atkinson

  3. Department of Statistics, University of Glasgow, University Gardens 15, Glasgow, G12 8QW, United Kingdom

    Bernard Torsney

  4. , SEMEQ, Università del Piemonte Orientale, via Perrone 18, Novara, 28100, Italy

    Caterina May

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Maruri-Aguilar, H., Trandafir, P.C. (2010). Sequential Barycentric Interpolation. In: Giovagnoli, A., Atkinson, A., Torsney, B., May, C. (eds) mODa 9 – Advances in Model-Oriented Design and Analysis. Contributions to Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2410-0_16

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