Introduction

Precipitation plays an important role in hydrological research and meteorological disaster warning. Due to the complexity and diversity of climatic conditions, the process of rainfall has a lot of ambiguity, uncertainty and randomness. The effect of varying precipitation in both timing and amount can cause serious natural disasters, while atmospheric and water circulation are boosted in the cycles of global hydrological variation \cite{Oki_2006,Sun_2018,Sun_2016}. For example, in Hong Kong and Macau, heavy rain in the summer is typically associated with the southwest monsoon, monsoon troughs and tropical cyclones. Though less frequently, excessive rainfall can also occur during the cool season due to land-sea breeze convergence and the passage of frontal systems. Persistent heavy rain can also lead to landslips because both of them have a hilly topography \cite{Li_2004}. Therefore, accurate and reliable precipitation forecasting is of great significance for weather forecasting, climate analysis, monitoring and warning of natural disasters such as floods, landslides, and mudslides.
Traditionally, the precipitation forecasting models can be classified into two main types: the physical prediction models and the statistical prediction models \cite{Nourani_2011}. Technically, physical-based models usually consider multi-factors including atmospheric circulation, pressure belt, ocean current, and Monsoon, etc., while statistical prediction models usually take advantage of the implicit information contained in the historical precipitation time series to forecast the future precipitation \cite{Yang_2016}. Physically-based models can suffer from the drawbacks that physical relationships between different impact features are complex, and sometimes the lack of multi-factors data makes it difficult to complete physical modeling \cite{Napolitano_2011}.  
The main technologies of the statistical simulating for precipitation forecasting can be classified into three types: the linear methods, the nonlinear methods and the hybrid methods \cite{Jiao_2016}. The linear methods, such as the Auto Regressive (AR) model, the Auto Regressive Moving Average (ARMA) model and the Auto Regressive Integrated Moving Average (ARIMA) model, are reliable and fast \cite{Burlando_1993}. \cite{Wang_2013} investigated the prediction performance of some linear and non-linear seasonal models. These models have satisfactory performance \cite{Valipour_2013}. The nonlinear methods, such as Support Vector Machine (SVM), Support Vector Regression (SVR), Grey Forecasting (GF), Artificial Neural Network (ANN) and so on, can always deal with the nonlinear and complex precipitation series accurately \cite{Yu_2000}, and compared with statistical approaches, they can achieve better forecasting results. The traditional ANN models for rainfall forecasting usually include Feed-forward Neural Network (FNN), Back Propagation Neural Network (BPNN), Radial Basis Function Neural Network (RBFNN) and so on. The benefits of these models have been investigated in some cases. \cite{Abbot_2014} use ANN to forecasts rainfall for sites in three geographically distinct regions in Queensland. \cite{2007} designed the optimal BP neural network for downscaling of monthly precipitation forecast. \cite{Farajzadeh_2014} forecast rainfall in the Urmia basin of northern Iran by using a feed-forward neural network. The proposed model had satisfactory performance in the nonlinear prediction performance. \cite{Kisi_2012} showed that their proposed SVM model was more accurate than the ANN models in precipitation forecasting. The hybrid methods are the combinations of some mainstream models and algorithms. Due to the good abilities of the hybrid models for the feature-extracting, these models have been widely applied in the precipitation time series forecasting.
Recently, many hybrid prediction models in precipitation forecasting have been proposed and investigated, which mainly contain the data preprocessing and the forecast modeling. The main technologies for these two modeling types are the signal decomposition algorithms and the prediction algorithms \cite{Tao_2017,Sulaiman_2017}.
The decomposition algorithms can effectively improve the prediction performance of the built models through decomposing the precipitation time series into several more stationary sub-layers \cite{Sun_2018}. Among the decomposition algorithms, the Singular Spectrum Analysis (SSA), Wavelet Decomposition (WD), the Empirical Mode Decomposition (EMD) and the Ensemble Empirical Mode Decomposition (EEMD) are widely recognized and used in rainfall prediction. \cite{Kalteh_2017} provided a prediction model combined with SSA and ANN (Artificial Neural Network). \cite{Ding_2015} presented the hybrid forecasting model using the WD, phase space reconstruction (PSR) and the ANN. \cite{Ouyang_2017} compared the performance of different decomposition algorithms such as WD, SSA and EEMD. Ordinarily, the WD algorithms have excellent ability of time-frequency analysis, and the EMD and EEMD algorithms have good self-adaptive ability of removing the stochastic volatility. However, the aforementioned decomposition algorithms have some disadvantages: (1) the performance of the WD algorithm depends on the wavelet basis and decomposition levels highly; (2) the EMD and EEMD algorithms lack the strict mathematical theory \cite{Liu_2015}. To overcome these drawbacks, \cite{Gilles_2013} proposed the Empirical Wavelet Transform (EWT), it can extract a series of modes of a signal by using an appropriate wavelet, so it is very effective in the non-stationary signal processing \cite{Shaari_2018}. In the study, the EWT is adopted to decompose the original precipitation time series.
The prediction algorithms are the core part of the precipitation forecasting. In recent years, some new prediction algorithms have been proposed, among these algorithms, deep learning methods, such as the Deep Belief Network (DBN), the Convolutional Neural Network (CNN) and the Recurrent Neural Network (RNN), have been developed rapidly. Deep learning is widely used in many fields such as bioinformatics \cite{Lan_2018}, speech recognition, visual object recognition, object detection and many other domains such as drug discovery and genomics \cite{LeCun_2015}. Compared with the shallow models, the deep learning models can extract the deep inherent features in data \cite{Najafabadi_2015}. Based on the following forecasting studies, deep learning methods can have good prediction performance in precipitation forecasting \cite{Liu_2017}\cite{Zhang_2017} built a DBN for the time series forecasting. \cite{Qiu_2017}  used the CNN for short-term rainfall forecasting tasks. Their results demonstrated that the proposed model was effective. \cite{Zhang_2018} forecasted rainfall using multi-layer perceptron (MLP) combined with dynamic regional, only short-term (3 hours) time series data was researched. \cite{Liu_2017} applied deep neural network (DNN) to weather datasets and showed it a potential tool for the feature fusion of time series problems. \cite{Grover_2015} presented a weather forecasting model based on deep hybrid model and forecasts made at 6, 12 and 24 hours. However they do not predict with the model more difficult weather data, such as rainfall dataset. \cite{shi2017deep} proposed a convolutional long short-term memory (ConvLSTM) model for precipitation nowcasting based on spatiotemporal sequence forecasting problem. \cite{Tao_2017a} improved performance of satellite rainfall retrievals by means of deep neural network approach, but rainfall times series data was not been researched. Therefore, the deep learning methods have not yet been widely used in the precipitation forecasting. By considering that the precipitation time series often have the long-term and short-term dependency, the Long Short Term Memory (LSTM) network \cite{Zhou_2016}, a special kind of RNN, is employed to predict the decomposed sub-layer in this study.
Based on forecasting algorithms, some other technologies are also used in forecast modeling, such as SVM model optimized by Particle Swam Optimization (PSO) algorithm, ANN connection weights optimized by social-based algorithm (SBA) \cite{Ramezani_2014} and mind evolutionary algorithm (MEA) \cite{Liu_2015a} and ANN model incorporated with Markov chains. In recent years, some prediction models are worked with Markov chains to get better predicting performance. \cite{Haidar_2017} proposed a hybrid Genetic Algorithm (GA) to select input features, network parameters to train neural network topologies in rainfall forecasting. \cite{Gui_2017} applied Markov chain to predict the rainfall and described the changing trend of precipitation easily. \cite{Aksoy_2018} proposed a new model combined with Markov chain to forecast monthly precipitation in arid regions. The proposed model had satisfactory performance. In the study, the Markov chain incorporated with LSTM network is employed as the prediction algorithm of precipitation forecasting model.
In this paper, a novel hybrid precipitation prediction model is proposed based on the EWT, Markov chain and LSTM network. The model is composed of three steps as: (a) the EWT is adopted to decompose the raw precipitation data into several sub-layers; (b) the LSTM network is incorporated  with Markov chain (MC) to predict the decompose sub-layers; (c) the prediction results of each sub-layer are summarized to obtain the final results for the original precipitation series. The provided EWT-MC-LSTM model, compared with LSTM models and BPNN model with and without EWT, MC technologies, have been designed and conducted to apply to monthly precipitation data of four regions to highlight the better performance. It has been demonstrated by the results of this study that the proposed hybrid model can provide an effective modeling approach to capture the nonlinear characteristics of monthly precipitation series, thus providing more satisfactory forecasting results.