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Identifying local smoothness for spatially inhomogeneous functions

Author

Listed:
  • Dongik Jang

    (The Korea Transport Institute)

  • Hee-Seok Oh

    () (Seoul National University)

  • Philippe Naveau

    (Laboratoire des Sciences du Climat et l’Environnement)

Abstract

Abstract We consider a problem of estimating local smoothness of a spatially inhomogeneous function from noisy data under the framework of smoothing splines. Most existing studies related to this problem deal with estimation induced by a single smoothing parameter or partially local smoothing parameters, which may not be efficient to characterize various degrees of smoothness of the underlying function when it is spatially varying. In this paper, we propose a new nonparametric method to estimate local smoothness of the function based on a moving local risk minimization coupled with spatially adaptive smoothing splines. The proposed method provides full information of the local smoothness at every location on the entire data domain, so that it is able to understand the degrees of spatial inhomogeneity of the function. A successful estimate of the local smoothness is useful for identifying abrupt changes of smoothness of the data, performing functional clustering and improving the uniformity of coverage of the confidence intervals of smoothing splines. We further consider a nontrivial extension of the local smoothness of inhomogeneous two-dimensional functions or spatial fields. Empirical performance of the proposed method is evaluated through numerical examples, which demonstrates promising results of the proposed method.

Suggested Citation

  • Dongik Jang & Hee-Seok Oh & Philippe Naveau, 2017. "Identifying local smoothness for spatially inhomogeneous functions," Computational Statistics, Springer, vol. 32(3), pages 1115-1138, September.
  • Handle: RePEc:spr:compst:v:32:y:2017:i:3:d:10.1007_s00180-016-0694-y
    DOI: 10.1007/s00180-016-0694-y
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    References listed on IDEAS

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    1. M. Giacofci & S. Lambert-Lacroix & G. Marot & F. Picard, 2013. "Wavelet-Based Clustering for Mixed-Effects Functional Models in High Dimension," Biometrics, The International Biometric Society, vol. 69(1), pages 31-40, March.
    2. Alexandre Pintore & Paul Speckman & Chris C. Holmes, 2006. "Spatially adaptive smoothing splines," Biometrika, Biometrika Trust, vol. 93(1), pages 113-125, March.
    3. Smith, Michael & Kohn, Robert, 1996. "Nonparametric regression using Bayesian variable selection," Journal of Econometrics, Elsevier, vol. 75(2), pages 317-343, December.
    4. Jeffrey S. Morris & Raymond J. Carroll, 2006. "Wavelet‐based functional mixed models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(2), pages 179-199, April.
    5. Jang, Dongik & Oh, Hee-Seok, 2011. "Enhancement of spatially adaptive smoothing splines via parameterization of smoothing parameters," Computational Statistics & Data Analysis, Elsevier, vol. 55(2), pages 1029-1040, February.
    6. Sain, Stephan R., 2002. "Multivariate locally adaptive density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 39(2), pages 165-186, April.
    7. Shubhankar Ray & Bani Mallick, 2006. "Functional clustering by Bayesian wavelet methods," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(2), pages 305-332, April.
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