Prediction of rainfall state using a hidden state Markov model

Abstract

ainfall prediction is a challenging problem because of the various uncertainties involved in rainfall processes. Nevertheless, long-term prediction of rainfall state is of great significance for the planning and management of water resources. In the past, different heuristic and statistical approaches have been used to address this problem. However, there still remains a need to develop a reliable rainfall long-term state prediction model. In this study, the rainfall state prediction problem has been introduced and solved using a Hidden State Markov (HSM) model. This work is motivated by the desire to understand and describe the issues associated with rainfall state prediction and to investigate the usefulness of the HSM model for solving this problem.

After doing an extensive review of the literature survey on different techniques for prediction of hydrologic time series, it was noticed that there are few models available for prediction of long-term events, and most of the models developed proved to be effective for short-term predictions but failed with long-term predictions. In the literature, it was also observed that the previous applications that used Hidden Markov Models concentrated on the simulation of daily rainfall processes (Zucchini and Guttorp [1991], Hughes et al. [1999]). Thyer and Kuczera [2000] stated that the HMM modeling approaches developed for the simulation of daily rainfall processes are not suitable for simulation of long-term hydrologic time series. Recently, Thyer and Kuczera [2000] introduced a new HSM model to simulate hydroclimatic time series with long-term persistence. They used a Markov Chain Monte Carlo (MCMC) method known as the Gibbs sampler for model calibration and developed a methodology to analyze the calibrated results for identifying the true underlying persistence structure in the rainfall time series.

In the present work, the HSM model introduced by Thyer and Kuczera [2000] is extended to predict rainfall state one year in advance. The study includes the development of a two-state HSM prediction model, which assumes rainfall state in a particular year as either wet or dry; a three-state HSM prediction model, which assumes rainfall state in a particular year as wet, normal, or dry; and a multi-stage HSM prediction model, which assumes rainfall state in a particular year as either wet or non-wet in the first stage, then in the second stage, the non-wet year is classified as either dry or normal. All the models assume that each state has an independent distribution, assumed to be Gaussian.

The MCMC method, known as the Gibbs sampler, is used for model calibration. It is an iterative procedure, and in each iteration, the previous set of model parameters (each state mean, variance, transition probability matrix, and state sequence) is used to obtain the next set of model parameters (parameter vector). It generates a Markov chain, and after some iterations, the Markov chain induced by the Gibbs sampler reaches a stationary distribution. The convergence is achieved by monitoring the time series plots between each parameter and the number of iterations. After convergence is achieved, using the values of the parameter vector, the future year rainfall state is predicted.

Two prediction techniques, namely the Maximum Probable Event (MPE) prediction method and the Regression prediction method (Yapo et al. [1993]), are used to predict the future year rainfall state. All these models are applied to the annual and monsoon rainfall data of 50 rain gauge stations of Karnataka. Results are discussed in terms of a confusion matrix for each prediction method, only for a few selected stations. Using the elements of the confusion matrix, accuracies for predicting wet, normal, and dry states are computed. A comparative study between two-state, three-state, and multi-stage models is also performed.

The observed results show that the state prediction models based on the HSM approach can be reliably used for one-year-ahead rainfall state prediction. The proposed models performed reasonably well in predicting wet, normal, and dry states. The comparative study between all the developed HSM models shows that the two-state HSM prediction model is better than the other (three-state and multi-stage) HSM models in predicting future year rainfall state.

It is also observed that in the case of the two-state HSM prediction model, the Percentage Accuracy of Dry state (PAD) prediction is better than the Percentage Accuracy of Wet state (PAW) prediction for all the selected stations (in different meteorological subdivisions), particularly for monsoon prediction.