PhD progress talk

Event status:

You are welcome to attend the following Statistics and Stochastic colloquium (part of the Colloquium Series of the Department of Mathematics and Statistics) at La Trobe University.

Thursday 19 November 2020 02:00 pm until Thursday 19 November 2020 03:00 pm (Add to calendar)
Andriy Olenko
Presented by:
Ravindi Nanayakkara
Type of Event:

Title: Stochastic Modelling and Statistical Analysis of Dependent Data


First, we discuss the obtained results about the analysis of spherical monofractal and multifractal random fields with cosmological applications. The Rényi function plays an important role in the analysis of multifractal random fields. For random fields on the sphere, there are three models in the literature where the Rényi function is known explicitly [1]. The main statistical model used to describe CMB data in the literature is isotropic Gaussian fields. We present some new theoretical models, numerical multifractality studies and methodology based on simulating random fields, computing the Rényi function and the multifractal spectrum for different scenarios and actual CMB data. The results suggest that there may exist a very minor multifractality of the CMB data [2].

Next, we discuss the obtained results about the asymptotic normality of simultaneous estimators of cyclic long-memory processes. Spectral singularities at non-zero frequencies play an important role in investigating cyclic or seasonal time series. The publication [3] introduced the generalized filtered method-of-moments approach to simultaneously estimate singularity location and long-memory parameters. This study [4] continues investigations of these simultaneous estimators. The results about asymptotic normality of several statistics are obtained. The methodology includes wavelet transformations as a particular case. The theoretical findings are illustrated by numerical results including Meyer, Shannon father wavelets and Mexican hat wavelets.

Finally, we discuss multifractionality of spherical random fields with cosmological applications. The Hölder exponent is used to measure the roughness in a rigorous mathematical way [5]. In this study, one dimensional and two dimensional pointwise Hölder exponent values are computed for the CMB data using the HEALPix ring ordering and nested ordering visualisations. The results suggest that there exist a considerable multifractionality in CMB data.


  1. Leonenko, N. & Shieh, N.R. (2013). Rényi function for multifractal random fields. Fractals, 21(2), 1350009.
  2. Leonenko, N., Nanayakkara, R., & Olenko, A. (2020). Analysis of Spherical Monofractal and Multifractal Random Fields. Stochastic Environmental Research and Risk Assessment Journal.
  3. Alomari, H. M., Ayache, A., Fradon, M. & Olenko, A. (2020). Estimation of cyclic long-memory parameters. Scandinavian Journal of Statistics, 47(1) 104-133.
  4. Ayache, A., Fradon, M., Nanayakkara, R., & Olenko, A. (2020). Asymptotic normality of simultaneous estimators of cyclic long-memory processes. Submitted.
  5. Ayache, A., & Véhel, J. L. (2004). On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion. Stochastic Processes and their Applications, 111(1), 119–56.




19th Oct 2021 2:32pm

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