Bayesian learning of Gaussian mixture model for calculating debris flow exceedance probability

ABSTRACT

Probabilistic modelling of debris flow data provides useful information for quantitative risk assessment, such as exceedance probabilities (EPs) of debris flow quantities. This task can defy many classical statistical models because debris flow data are frequently collected over years or even decades, and the nonuniformity of the nature and the complex physical mechanism of debris flows lead to multimodal distribution characteristics of observational data. This paper proposes a Bayesian framework for learning Gaussian mixture model (GMM) of debris flow quantities (e.g. total discharge Q_total and maximum impact pressure P_max) and calculating their EPs for risk-informed decision making. GMM provides great flexibility to fit observation data, but are intrinsically unidentifiable due to the label switching. These computational difficulties are addressed using Random Gibbs Sampling and Bridge Sampling in the proposed framework, allowing incorporating the statistical uncertainty in GMM parameters into EP estimation. Equations are derived for the proposed approach and are illustrated using Q_total and P_max data at Jiangjia Ravine, China. Results show that the proposed approach identifies a bivariate GMM of Q_total and P_max reflecting the multimodal characteristics of the observed data and quantifies the statistical uncertainty of GMM parameters. Incorporating the statistical uncertainty into EP estimation provides robust estimates.

KEYWORDS:

• Bayesian learning
• Gaussian mixture model
• debris flow
• exceedance probability
• identification uncertainty

Acknowledgements

Authors would like to acknowledge Prof. Sylvia Frühwirth-Schnatter at Vienna University of Economics and Business (WU) and her colleagues for great works on finite mixture model and developing the open source MATLAB package bayesf. The codes implemented in this study are modified based on bayesf, which is available via the link https://statmath.wu.ac.at/~fruehwirth/monographie/.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work described in this paper was supported by grants from National Natural Science Foundation of China [Project numbers 51879205, 51779189] and the Fundamental Research Funds for the Central Universities [Project number 2042020kf0193]. The financial support is gratefully acknowledged.