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Multiresolution analysis of point processes and statistical thresholding for Haar wavelet-based intensity estimation

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  • Youssef Taleb

    (Imperial College London)

  • Edward A. K. Cohen

    (Imperial College London)

Abstract

We take a wavelet-based approach to the analysis of point processes and the estimation of the first-order intensity under a continuous-time setting. A Haar wavelet multiresolution analysis is formulated which motivates the definition of homogeneity at different scales of resolution, termed J-th level homogeneity. Further to this, the activity in a point process’ first-order behaviour at different scales of resolution is also defined and termed L-th level innovation. Likelihood ratio tests for both these properties are proposed with asymptotic distributions provided, even when only a single realization is observed. The test for L-th level innovation forms the basis for a collection of statistical strategies for thresholding coefficients in a wavelet-based estimator of the intensity function. These thresholding strategies outperform the existing local hard thresholding strategy on a range of simulation scenarios. This methodology is applied to NetFlow data, characterizing multiscale behaviour on computer networks.

Suggested Citation

  • Youssef Taleb & Edward A. K. Cohen, 2021. "Multiresolution analysis of point processes and statistical thresholding for Haar wavelet-based intensity estimation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(2), pages 395-423, April.
    • Handle: RePEc:spr:aistmt:v:73:y:2021:i:2:d:10.1007_s10463-020-00753-4
    DOI: 10.1007/s10463-020-00753-4
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    References listed on IDEAS

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    1. Antoniadis, Anestis & Bigot, Jeremie & Sapatinas, Theofanis, 2001. "Wavelet Estimators in Nonparametric Regression: A Comparative Simulation Study," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 6(i06).
    2. José Miranda & Pedro Morettin, 2011. "Estimation of the intensity of non-homogeneous point processes via wavelets," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(6), pages 1221-1246, December.
    3. Roelof Helmers & Ričardas Zitikis, 1999. "On Estimation of Poisson Intensity Functions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 51(2), pages 265-280, June.
    4. Ogden, Todd & Parzen, Emanuel, 1996. "Data dependent wavelet thresholding in nonparametric regression with change-point applications," Computational Statistics & Data Analysis, Elsevier, vol. 22(1), pages 53-70, June.
    5. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504, October.
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