Approximating Heavy-Tailed Distributions with a Mixture of Bernstein Phase-Type and Hyperexponential Models
Résumé
Heavy-tailed distributions, prevalent in a lot of real-world applications such as finance, telecommunications, queuing theory, and natural language processing, are challenging to model accurately owing to their slow tail decay. Bernstein phase-type (BPH) distributions, through their analytical tractability and good approximations in the nontail region, can present a good solution, but they suffer from an inability to reproduce these heavy-tailed behaviors exactly, thus leading to inadequate performance in important tail areas. On the contrary, while highly adaptable to heavy-tailed distributions, hyperexponential (HE) models struggle in the body part of the distribution. Additionally, they are highly sensitive to initial parameter selection, significantly affecting their precision. To solve these issues, we propose a novel hybrid model of BPH and HE distributions, borrowing the most desirable features from each for enhanced approximation quality. Specifically, we leverage an optimization to set initial parameters for the HE component, significantly enhancing its robustness and reducing the possibility that the associated procedure results in an invalid HE model. Experimental validation demonstrates that the novel hybrid approach is more performant than individual application of BPH or HE models. More precisely, it can capture both the body and the tail of heavy-tailed distributions, with a considerable enhancement in matching parameters such as mean and coefficient of variation. Additional validation through experiments utilizing queuing theory proves the practical usefulness, accuracy, and precision of our hybrid approach.
Bernstein Phase-Type (BPH) distributions [3,4], a class of distributions defined in terms of Markov chains, offer fast parameter estimation and precise approximation of the body of a distribution, but fall short in capturing heavy tails. Hyperexponential (HE) distributions, on the other hand, can better approximate slowly decaying tails when used with proper parameters [5] but (i) they are highly sensitive to initialization, which can hinder performance, and (ii) they form an inflexible family of distributions (their probability density function can only be monotone decreasing).
To address these limitations, we propose a hybrid model combining BPH and HE distributions. This approach leverages the strengths of both: the approximation efficiency of BPH distributions and the tail-fitting flexibility of HE distributions. To improve the robustness of the HE component, we employ the Adam optimizer to enhance parameter initialization. This paper presents the hybrid BPH HE model and the optimization method, evaluates performance, and validates results through queuing theory applications.
A similar approach was presented in [6], where the authors proposed combining acyclic phase-type (APH) distributions with HE distributions. The improvements over that work are as follows: (i) instead of APH, we apply BPH distributions, which allow for much faster and more precise approximation of the body, and (ii) we automate the parameter initialization of the HE distributions that capture the tail, whereas in [5] and [6], this process is left to a non-trivial manual setting.
The remainder of the paper is organized as follows: Section 2 reviews related work, Section 3 gives background, Section 4 details the hybrid model and the optimization strategy, Section 5 covers experiments, Section 6 application results, and Section 7 concludes the study.
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