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Thinning-Stable Point Processes as a Model for Spatial Burstiness

Sergei Zuyev

TL;DR

This work addresses spatial burstiness in telecommunications data, where classical Poisson models fail to capture highly irregular spatial activity. It introduces thinning-stable ($ ext{TαS}$) point processes, representing bursty data as Poisson clusters driven by a stable random mechanism, with a cluster structure expressed as $Φ ≡ ∑_{μ_i} Υ_i$ where $Υ_i ∼ Sib(α, μ_i)$ and $α ∈ (0,1]$. The authors develop inference methods to estimate the Sibuya cluster measure $μ_0$, the cluster-density $λ$, and the stability parameter $α$, using p.g.fl. and void-probability approaches, along with thinning-based regression to exploit stability properties. Simulations and a Paris festival dataset analysis yield $α$ values around $0.17$–$0.28$, illustrating pronounced burstiness and demonstrating the practical utility of $ ext{TαS}$ modeling for spatial traffic analysis and network planning. The framework offers a principled alternative to Poisson models and points toward future generalizations, such as branching-stable processes, to further enhance modeling flexibility for complex spatial burst patterns.

Abstract

In modern telecommunications, spatial burstiness of data traffic poses challenges to traditional Poisson-based models. This paper describes application of thinning-stable point processes, which provide a more appropriate framework for modeling bursty spatial data. We discuss their properties, representation, inference methods, and applications, demonstrating the advantages over classical approaches.

Thinning-Stable Point Processes as a Model for Spatial Burstiness

TL;DR

This work addresses spatial burstiness in telecommunications data, where classical Poisson models fail to capture highly irregular spatial activity. It introduces thinning-stable () point processes, representing bursty data as Poisson clusters driven by a stable random mechanism, with a cluster structure expressed as where and . The authors develop inference methods to estimate the Sibuya cluster measure , the cluster-density , and the stability parameter , using p.g.fl. and void-probability approaches, along with thinning-based regression to exploit stability properties. Simulations and a Paris festival dataset analysis yield values around –, illustrating pronounced burstiness and demonstrating the practical utility of modeling for spatial traffic analysis and network planning. The framework offers a principled alternative to Poisson models and points toward future generalizations, such as branching-stable processes, to further enhance modeling flexibility for complex spatial burst patterns.

Abstract

In modern telecommunications, spatial burstiness of data traffic poses challenges to traditional Poisson-based models. This paper describes application of thinning-stable point processes, which provide a more appropriate framework for modeling bursty spatial data. We discuss their properties, representation, inference methods, and applications, demonstrating the advantages over classical approaches.
Paper Structure (9 sections, 22 equations, 3 figures, 1 table)

This paper contains 9 sections, 22 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical Background
  3. Poisson Limit Theorem
  4. Statistical Inference for $\mathrm{T\alpha S}$ Processes
  5. Estimation of the Sibuya parameter measure
  6. Estimation of the Stability Parameter and the Cluster Density
  7. Estimation of the stability parameter from cluster sizes
  8. Applications and Simulations
  9. Conclusion

Figures (3)

  • Figure 1: Wireless communication activity during Fête de la Musique festival in Paris in 2008, courtesy of Orange.fr. The height of the torches represents the number of calls in progress at their locations.
  • Figure 2: $\lambda=0.4$, $\alpha=0.6$ and $\mu_0 \sim \mathcal{MVN}(0,0.5^2\mathrm{I})$
  • Figure 3: Realisation of a Gaussian cluster model with $\mu_0\sim\mathrm{MVN}(0,\mathrm{I})$, $\lambda=0.1, \alpha=0.7$ and the relative error of estimation of $G(1)$ for various values of $p$ in \ref{['eq:vp_est']}.