Hydrological Uncertainty Processor (HUP) with Estimation of the Marginal Distribution by a Gaussian Mixture Model

Abstract

Uncertainty assessments of hydrological prediction results can reflect additional hydrological information and reveal important hydrological characteristics of river basins, which is of great significance to disaster prevention and reduction. The hydrological uncertainty processor (HUP), which is a key part of the Bayesian forecasting system (BFS), has derived a variety of methods for hydrological uncertainty forecasting. The HUP allows for any form of marginal distributions of hydrological data and does not require a unified estimation structure for the marginal distribution function. The Gaussian mixture model (GMM) is a probability distribution estimation model that can approximate any probability distribution with arbitrary precision. In this paper, the GMM was used to estimate the marginal distribution of observed and modelled data, and this method is called HUP-GMM. The uncertainty of river discharge at the Yichang hydrological station on the main stem of the Yangtze River in China is predicted by the HUP-GMM. The Weibull and Gamma distributions, which are commonly used hydrological probability distributions, are compared to analyse the performance of the GMM. In June, when the measured flow h₃ is 13,850 m³/s and the GMM, Gamma and Weibull distributions are used, the prior probabilities are 1.63E-04, 1.05E-04 and 9.50E-05 and the posterior probabilities are 2.57E-04, 1.61E-04 and 1.38E-04, respectively. In September, when the measured flow h₃ is 35,400 m³/s and the GMM, Gamma and Weibull distributions are used, the prior probabilities are 5.98E-05, 2.21E-05 and 2.18E-05 and the posterior probabilities are 1.64E-04, 9.15E-05 and 8.43E-05, respectively. The results show that the performance of the uncertainty estimation of the prior and posterior probability distributions in the HUP-GMM has been improved.

Appendix

1.1 Expectation Maximum

Input: observation data y₁, y₂, ⋯, y_N, Gaussian mixture model;

Output: parameters of the Gaussian mixture model.

• (1) take the initial value of the parameter and start the iteration

• (2) E step: according to the current model parameters, the response of model k to observed data was calculated

$$ {\hat{\gamma}}_{jk}=\frac{\alpha_k\phi \left({y}_j|{\theta}_k\right)}{\sum \limits_{k=1}^K{\alpha}_k\phi \left({y}_j|{\theta}_k\right)},j=1,2,\cdots, N,k=1,2,\cdots, K $$

(A1)

• (3) M step: calculate the model parameters of the next iteration

$$ {\hat{\mu}}_k=\frac{\sum \limits_{j=1}^N{\hat{\gamma}}_{jk}{y}_j}{\sum \limits_{j=1}^N{\hat{\gamma}}_{jk}},k=1,2,\cdots, K $$

(A2)

$$ {\hat{\sigma}}_k^2=\frac{\sum \limits_{j=1}^N{\hat{\gamma}}_{jk}{\left({y}_j-{\mu}_k\right)}^2}{\sum \limits_{j=1}^N{\hat{\gamma}}_{jk}},k=1,2,\cdots, K $$

(A3)

$$ {\hat{\sigma}}_k=\frac{\sum \limits_{j=1}^N{\hat{\gamma}}_{jk}}{N},k=1,2,\cdots, K $$

(A4)

(4) repeat step (2) and (3) until they converge.