1 Introduction

Rainfall is one of the most important parameters in the hydrologic cycle. Since Iran is located in an arid and semi-arid climatic region, the estimation of rainfall plays an important role to plan and manage water resources. Moreover, accurate estimation of rainfall is vital in the assessment of hydrological parameters such as runoff, flood, sediment, drought, agriculture, irrigation scheduling, groundwater, etc. Nevertheless, accurate estimation of rainfall is difficult due to its spatial and temporal variations (Kisi and Cimen 2012). Although the estimation of rainfall is not impossible, it faces great complexities due to its non-linear behavior.

Artificial neural networks (ANN) and gene expression programming (GEP) are considered as soft computing approaches which are extensively used to model hydrologic variables. The use of ANN is more common than GEP for estimating the rainfall. However, the GEP has widely been used in other engineering fields (Marti et al. 2013; Imani et al. 2014; Ebtehaj et al. 2015; Gocic et al. 2015; Yassin et al. 2016; Mehdizadeh et al. 2016, 2017; Shiri et al. 2017; Behmanesh and Mehdizadeh 2017). Ramirez et al. (2005) applied the ANN to forecast the rainfall in Sao Paulo, Brazil. The ANN, as a result, was superior to linear regression model. Partal and Kisi (2007) employed a conjunction model, namely wavelet-neuro fuzzy for prediction of daily precipitation in three stations in Eagean region, Turkey. The results showed that the proposed conjunction model performs better than the classical neuro-fuzzy model. Wu et al. (2010) predicted monthly and daily rainfall time series using modular ANN coupled with data-preprocessing methods in China and India. The results revealed that the benefits of modular ANN than other models are significant, especially in the prediction of daily rainfall. Moustris et al. (2011) predicted the precipitation using ANN at four meteorological stations in Greece. Their findings confirmed the satisfactory of precipitation forecasts by the ANN. Kashid and Maity (2012) estimated summer monsoon monthly rainfall on homogenous regions of India using genetic programming (GP). Mekanik et al. (2013) employed the ANN and multiple regression (MR) to forecast seasonal spring rainfall in Victoria, Australia. The outcomes revealed the superiority of the ANN over MR. Venkata Ramana et al. (2013) used wavelet neural networks to estimate monthly rainfall in Darjeeling rain gauge station. The wavelet neural networks models were found to perform better than the ANN models. Abbot and Marohasy (2014) used ANN to predict monthly rainfall as a continuous variable in Queensland, Australia. The results indicated the usefulness of climate indices for the estimation of rainfall. He et al. (2015) introduced hybrid wavelet neural network model with mutual information and particle swarm optimization to forecast the monthly rainfall in Australia. The results demonstrated that the proposed model increased the accuracy of the prediction of monthly rainfall compared to the reference models. Shenify et al. (2016) compared the performance of wavelet transform-support vector machine (WT-SVM) with GP and ANN for the estimation of monthly precipitation in Serbia. Consequently, the WT-SVM estimations were superior to the GP and ANN models.

Beside soft computing methods, there are several time series models to estimate the time series of hydrological variables such as rainfall. Chinchorkar et al. (2012) stated that the rainfall is often predicted using stochastic models due to the random properties of rainfall time series. Some of these models are autoregressive (AR), autoregressive moving average (ARMA), etc. Autoregressive conditional heteroscedasticity (ARCH) model which was presented by Engle (1982) performs as a non-linear model. The variance in this model is not considered as a constant overtime.

In fact, time series of hydrologic variables such as rainfall have composed of two components; deterministic and random parts. Literature review reveals that the previous conducted studies about the rainfall forecasting have focused only on the deterministic part of rainfall time series. This means that the random component of rainfall data has not been considered in previous studies. Herein, random component was considered in the modeling process. The present study attempts to create novel methods by combining the ARCH time series model with the GEP and ANN to enhance the estimation accuracy of the rainfall time series at different climatic regions in Iran. In the present study, the deterministic part of rainfall time series was estimated using the GEP and ANN models whereas the random part was determined by the ARCH model.

2 Materials and Methods

2.1 Study Area and Data Used

Iran, with an area of over 1,648,000 km2, is located in the southwest of Asia. About 94.8% of Iran has arid or semi-arid climate with low amounts of rainfall and high evapotranspiration (Khalili et al. 2016). In the present study, five regions with various climatic conditions were chosen. It should be noted that the aridity index defined by United Nations Environment Program (UNEP 1992) was employed to classify the selected stations in terms of climatic aspect (Ashraf et al. 2014). Table 1 presents the geographical and climatic information of the studied stations. Also, Fig. 1 demonstrates the spatial distribution of the considered stations in Iran.

Table 1 Geographical and climatic information of the studied stations
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Fig. 1
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The location of studied stations in Iran

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The required data, i.e. the monthly rainfall of the studied stations was obtained from the Islamic Republic of Iran Meteorological Organization (IRIMO) during 1981 to 2012. Here, the whole data was divided into two parts; training and testing datasets. Around 75% (1981–2004, i.e. 288 months) and 25% (2005–2012, i.e. 96 months) of the data were utilized in training and testing stages, respectively. Figure 2 depicts the time series of the monthly rainfall for the studied stations during 1981 to 2012. Descriptive statistics of the monthly rainfall data for the study regions are presented in Table 2. As seen, humid and hyper-arid stations; i.e. Rasht and Zahedan have the maximum and minimum values of standard deviation, respectively. On the other hand, the lowest and highest skewness are related to Gorgan and Zahedan, respectively.

Fig. 2
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Time series of the monthly rainfall for the studied stations during 1981–2012

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Table 2 Statistical properties of the monthly rainfall time series
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2.2 Artificial Neural Networks

Artificial neural networks (ANN) were created on the basis of the extensive internal communications like the human brain and nervous system. The ANN is presented as a structure like the biological one in the human brain. In these models, at first, the data is introduced to the model. Then, the training process will be done and the obtained (trained) model will be used for the next steps. The general structure of the ANN is composed of three layers;

  1. 1-

    Input layer: In this layer, the input data is introduced to the model.

  2. 2-

    Hidden layer: In the hidden layer, the input data is processed.

  3. 3-

    Output layer: The results of the model are produced in the output layer.

Each layer consists of some neurons. Each neuron receives information from a series of input data, and after processing, it delivers the results to the output. Finally, the output of the mentioned neuron is used as the next layer input. The number of neurons in the input and output layers are determined in accordance with the nature of the studied phenomenon. In this study, the lagged monthly rainfall data were considered as input variables, while the output of the ANN was defined as the rainfall of current month. Moreover, the number of neurons in the hidden layer is specified by user using trial and error procedure in order to reduce the error rate of the network. In the present study, the optimum number of neurons in the hidden layer changes between 1 and 6 for the different applied scenarios and considered stations. The neurons of each layer are connected to the neurons of next layer by weights. Beside weights, the biases are the elements of the hidden and output layers. During the training process, the weights and biases are continuously changed until the best structure of the network is found.

Furthermore, transfer functions are used to create a relationship between input and output parameters. In this study, tangent-sigmoid and linear transfer functions were used in the hidden and output layers, respectively. More details about the ANN can be found in Haykin (1998).

The ANN program was written in MATLAB programming language. In the present study, three-layer feed forward neural networks were used to estimate the monthly rainfall time series in the studied area. Moreover, Levenberg-Marquardt (LM) algorithm was used as the learning algorithm. Zanetti et al. (2007) stated that the LM is better than the back propagation learning algorithm due to a faster convergence in the network learning. Detailed information about the LM can be seen in Mehdizadeh et al. (2016).

2.3 Gene Expression Programming

Gene expression programming (GEP) was first presented by Ferreira (2001). The GEP has been developed on the basis of Darwinian Evolution Theory. The GEP acts like the genetic algorithm (GA) and genetic programming (GP). The sample of individuals is used in the GEP. The individuals are selected on the basis of their merits. Genetic variations are exerted using genetic operators. The main difference among GA, GP and GEP is the nature of their individuals. In the GA, the individuals are linear strings of fixed length (chromosomes), while the individuals in the GP are non-linear entities of different sizes and shapes (parse trees). Nevertheless in the GEP, the individuals are encoded as linear strings of fixed-length (similar to the GA), and then, they are expressed in the form of non-linear entities of different sizes and shapes (similar to the GP).

In the GEP, each gene is encoded in the form of expression tree which represents a solution for the problem. All expression trees are connected to each other using a linking function in the case of multi-gene chromosomes. Readers can refer to Ferreira (2001) to see more information about the GEP.

In this study, GeneXpro Tools program was used. The estimation process of the monthly rainfall using the GEP is as follows:

The first stage is to determine a suitable fitness function. In the present research, root mean square error (RMSE) was used as the fitness function. Terminals and functions sets are selected in the second stage. In fact, terminals set are independent variables including different parameters which may affect the estimation of rainfall. In this study, one to five previous monthly rainfall in the form of lagged data were considered as the terminals set. Also, four arithmetic operators (+,  − ,  × , ÷) along with mathematical functions (\( {x}^2,{x}^3,{e}^x, Lnx,\sqrt{x},\sqrt[3]{x}, Sinx, Cosx,A\mathrm{tanx} \)) were used as the functions set. The third stage is to select the structure of the chromosomes including head size and the number of genes and chromosomes. The head size, the number of genes and chromosomes were selected 8, 3 and 30, respectively. The selection of linking function is the fourth stage. In this study, an addition function was employed to link the selected three genes (or expression trees). The final stage is to choose the genetic operators and their rates. In this process, a combination of all operators including three types of transposition, three types of recombination, as well as mutation and inversion were utilized.

The values of used parameters in the GEP are given in Table 3.

Table 3 The values of parameters used for estimation of the monthly rainfall using GEP
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2.4 Developing the GEP and ANN Models

In this study, one to five previous monthly rainfall in the form of lagged data were used as inputs to develop the different GEP and ANN scenarios. These scenarios are named as below,

  • GEP1, ANN1: Pt-1

  • GEP2, ANN2: Pt-1, Pt-2

  • GEP3, ANN3: Pt-1, Pt-2, Pt-3

  • GEP4, ANN4: Pt-1, Pt-2, Pt-3, Pt-4

  • GEP5, ANN5: Pt-1, Pt-2, Pt-3, Pt-4, Pt-5

For example, GEP1 and ANN1 are scenarios in which the monthly rainfall of previous month was used to estimate the rainfall of current month.

2.5 Introducing New GEP-ARCH and ANN-ARCH Hybrid Models

One of the main problems in modeling of hydrological time series such as rainfall is to determine the random component of the time series. In this direction, entering a series of random data may not affect the estimated values. In this study, the random component of the monthly rainfall time series data is modeled using the ARCH model. In the most hydrological studies, the main attention is focused on the mean of data. However, little attention has been paid to the variance changes over the time. Regarding the progress of conducted studies on the field of risk and uncertainty in water resources engineering, it is essential to develop modeling techniques for considering the variance changes with respect to the time. The present study attempts to create novel methods by combining the ARCH time series model with the GEP and ANN to enhance the estimation accuracy of the rainfall time series.

The ARCH model which was first presented by Engle (1982) is a non-linear time series model for modeling the variance changes over the time as below,

$$ {\sigma}_t^2={a}_0+\sum_{i=1}^m{b}_i{\varepsilon}_{t-i}^2 $$
(1)
$$ {\varepsilon}_t={\sigma}_t{z}_t $$
(2)

where, \( {\sigma}_t^2 \)is the conditional variance, ε t is the error phrase or model residual, a 0 and b i are the model parameters, m is the order of model and z t ~N(0, 1) is the normalized and standardized dataset.

Fitting steps of the ARCH model are as follows,

  1. 1-

    The best AR (Eq. 3) or ARMA (Eq. 4) is fitted on the stationary data series (z t ),

$$ {z}_t(p)=\sum_{i=1}^p\left({\phi}_i.{\mathrm{z}}_{t-i}\right)+{\varepsilon}_t $$
(3)
$$ {z}_t\left(p,q\right)=\sum_{i=1}^p\left({\phi}_i.{\mathrm{z}}_{t-i}\right)-\sum_{i=1}^q\left({\theta}_i.{\varepsilon}_{t-i}\right)+{\varepsilon}_t $$
(4)

where, p is the order of AR model, q is the order of MA model, ϕ i and θ i are the coefficients of the models.

  1. 2-

    The error rates or residuals (ε t ) are calculated by Eqs. 3 or 4.

  2. 3-

    \( {\varepsilon}_t^2 \)is calculated using step 2,

  3. 4-

    ARCH model is fitted on the obtained \( {\varepsilon}_t^2 \)values and then Eqs. 1 and 2 are used to obtain the random component of rainfall data (ε t ).

  4. 5-

    The final step is the combination of ARCH time series model with the GEP and ANN models.

Since the rainfall time series consist of the deterministic and random parts, the results of the GEP and ANN models are combined with the results of the ARCH model as below,

$$ {P}_t={D}_t+{R}_t $$
(5)

where, D t is the generated deterministic part of the rainfall time series using GEP and ANN models, R t (or ε t ) is the obtained random part of the rainfall time series using ARCH model and P t is the final value of the rainfall time series. Both D t and R t are the normalized and standardized data of the monthly rainfall time series.

Equations 6 (Delleur and Karamouz 1982) and 7 were used to normalize and standardize the monthly rainfall time series, respectively,

$$ {N}_t= Ln\left({X}_t+c\right) $$
(6)
$$ {Z}_t=\frac{N_t-\mu }{\sigma } $$
(7)

where, N t is the normalized values of the rainfall data, X t is the actual values of the rainfall data, c is a constant, Z t is the standardized values of the rainfall data. Also, μand σ are the mean and standard deviation of the normalized monthly rainfall data, respectively, in monthly scale. It should be noted that the c coefficient in Eq. 6 is obtained using trial and error procedure to achieve the lowest value for the skewness coefficient.

2.6 Performance Evaluation Criteria

To investigate the performance of the GEP and ANN models, as well as the proposed GEP-ARCH and ANN-ARCH hybrid models in estimating the monthly rainfall, two statistical indicators including root mean square error (RMSE) and coefficient of determination (R2) were used as below,

$$ RMSE=\sqrt{\frac{\sum_{i=1}^n{\left({E}_i-{O}_i\right)}^2}{n}} $$
(8)
$$ {R}^2={\left(\frac{\sum_{i=1}^n\left({E}_i-\overline{E_i}\right)\left({O}_i-\overline{O_i}\right)}{\sqrt{\sum_{i=1}^n{\left({E}_i-\overline{E_i}\right)}^2{\sum}_{i=1}^n{\left({O}_i-\overline{O_i}\right)}^2}}\right)}^2 $$
(9)

where, E i is the ith estimated monthly rainfall using GEP, ANN, GEP-ARCH and ANN-ARCH models; O i is the ith observed monthly rainfall; \( \overline{E_i} \)is the average of the estimated monthly rainfall data; \( \overline{O_i} \)is the average of the observed monthly rainfall data and n is the number of observations. RMSE is measured in mm in the present study.

3 Results and Discussion

In the present study, one to five previous monthly rainfall in the form of lagged data were used to estimate the monthly rainfall of current month. At first, the accuracy of GEP and ANN models was investigated at various climatic regions in Iran. Then, the performance of proposed novel models including GEP-ARCH and ANN-ARCH was evaluated and then compared to the GEP and ANN models.

3.1 Results of the GEP and ANN Models

The results of the GEP and ANN models including RMSE and R2 statistical indices in estimating monthly rainfall during the testing stage are presented in Tables 4 and 5, respectively. As seen, the highest and lowest values of the RMSE indicator are observed in Rasht and Zahedan stations, respectively. In Rasht station with a humid climate, the GEP models in all scenarios have higher precision than the corresponding ANN scenarios, except GEP1. In other stations, constant state is not observed, i.e. in some scenarios, the GEP performs better than the ANN and vice versa. The GEP is better than the ANN in three scenarios at Urmia and Kerman stations. However, the ANN had slightly better performance in four and three scenarios at Gorgan and Zahedan stations, respectively. As seen in Tables 4 and 5, the performance of scenarios are not necessarily improved by increasing the input variables in the GEP. For example, this condition is seen in Rasht and Urmia stations. In the mentioned stations, the GEP5 which uses full inputs has the lowest accuracy. In the ANN, the situation is different. The ANN5 in Urmia (RMSE = 23.95 mm), Kerman (RMSE = 12.77 mm) and Zahedan (RMSE = 11.65 mm) presents better performance than other scenarios with fewer inputs. Also, the ANN5 is the third superior model after ANN4 and ANN2, in Gorgan. The only exception in this field is Rasht station in which the ANN5 (similar to the GEP5) is the worst scenario compared to the others. The reason of this fact can be explained with attention to the type of climate of the mentioned station and the large differences between the values of the rainfall during various months.

Table 4 Calculated statistical indices for the different GEP and GEP-ARCH models in the testing stage
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Table 5 Calculated statistical indices for the different ANN and ANN-ARCH models in the testing stage
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3.2 Comparison of Novel GEP-ARCH and ANN-ARCH Hybrid Models with GEP and ANN

As mentioned before, the time series of hydrologic processes such as rainfall compose two parts; deterministic and random components. In this study, the deterministic part of the monthly rainfall time series was obtained by the GEP and ANN models. Moreover, the random component of rainfall time series was obtained using the ARCH model. Then, the obtained random component was combined and added with the results of the GEP and ANN models (according to Eq. 5). Results of the fitted ARCH model on time series of the monthly rainfall data are listed in Table 6. Also, calculated statistical indices for the proposed GEP-ARCH and ANN-ARCH hybrid models are given in the second part of the Tables 4 and 5, respectively. As seen, by combining the GEP and ANN with ARCH model, the performance of models is considerably improved for all of the stations and used scenarios in comparison with the GEP and ANN models. For example, the values of RMSE and R2 statistical indices in the GEP4 for Rasht station are 74.59 mm and 0.342, respectively. The mentioned statistics improve to 56.48 mm and 0.947, respectively in the GEP4-ARCH model. Similar trends were observed in other GEP-ARCH, as well as ANN-ARCH models for other stations. The performance comparison of the GEP-ARCH and ANN-ARCH models reveals that the GEP-ARCH scenarios are superior to the ANN-ARCH models in Gorgan. On the other hand, the ANN-ARCH scenarios in Rasht and Urmia stations present better results than the corresponding GEP-ARCH models (except for ANN2-ARCH in Rasht). Furthermore, the precision of the GEP-ARCH and ANN-ARCH models are slightly similar for the different scenarios in Kerman and Zahedan stations. The only exception is ANN3-ARCH in Kerman where the mentioned scenario has better performance than the GEP3-ARCH.

Table 6 Results of the fitted ARCH model on the monthly rainfall time series during 1981–2012
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To graphically illustrate the performance of proposed hybrid models compared to the GEP and ANN models, superior scenarios of the testing stage (with bolded statistical indices in Tables 4 and 5) in the hybrid models, as well as the corresponding scenarios in the GEP and ANN models were chosen. Scatter plots between the observed and estimated monthly rainfall for the best scenarios in the proposed GEP-ARCH and ANN-ARCH hybrid models, as well as the corresponding GEP and ANN models in the testing period are depicted in Figs. 3 and 4, respectively. It is clear from the scatter plots that the estimated monthly rainfall data by the proposed GEP-ARCH and ANN-ARCH hybrid models are closer to the observed data than those obtained by the GEP and ANN models. The highest correlations are observed in Gorgan, Rasht and Urmia stations (see also other scenarios in Tables 4 and 5). On the other hand, the lowest correlation is seen for Zahedan as a hyper-arid region. By assuming y = ax + b equation (y is the estimated rainfall and x is the observed rainfall) for fitted lines to the models, it is clear that a (slope) and b (intercept) coefficients in the GEP-ARCH and ANN-ARCH hybrid models are closer to 1 and 0, respectively in comparison with the GEP and ANN models.

Fig. 3
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Scatter plots between the observed and estimated rainfall for the best scenarios in the GEP-ARCH models and the corresponding GEP models during the testing stage

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Fig. 4
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Scatter plots between the observed and estimated rainfall for the best scenarios in the ANN-ARCH models and the corresponding ANN models during the testing stage

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Figures 5 and 6 demonstrate the time series of observed and estimated monthly rainfall for the superior scenarios in the hybrid models and the corresponding GEP and ANN scenarios during the testing period. It is clear from the Figures that a fixed trend is observed in the time series of estimated monthly rainfall data for all stations in the GEP and ANN models. In addition, the mentioned models are not able to estimate peak points of the monthly rainfall data. However, the proposed GEP-ARCH and ANN-ARCH models perform better than the GEP and ANN in estimating time series of the monthly rainfall, as well as peak values of the rainfall data. It is seen from the Figs. 5 and 6 that both proposed models overestimate peak values of the rainfall time series in Rasht, Gorgan and Urmia stations. Furthermore, the peak values of rainfall data are underestimated in Kerman and Zahedan stations as arid and hyper-arid climates, respectively. Similar to the obtained results, Kisi and Cimen (2012), Venkata Ramana et al. (2013), Feng et al. (2015) and Shenify et al. (2016) reported that peak points of the rainfall data were closely matched to the observed data using the hybrid models such as wavelet-support vector machine and wavelet-neural networks.

Fig. 5
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Time series of the observed and estimated rainfall for the best scenarios in the GEP-ARCH models and the corresponding GEP models during the testing stage

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Fig. 6
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Time series of the observed and estimated rainfall for the best scenarios in the ANN-ARCH models and the corresponding ANN models during the testing stage

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4 Conclusion

In the present study, at first, the capability of the GEP and ANN in estimating monthly rainfall time series was investigated at five stations with various climatic conditions in Iran. The results showed that both models presented slightly similar performances. Moreover, the models’ accuracies were not acceptable. In the second part, two novel models were introduced by combining the GEP and ANN intelligent models with the ARCH time series model. So, the GEP-ARCH and ANN-ARCH hybrid models were developed. The obtained results demonstrated that the proposed hybrid models yielded better performances compared to the GEP and ANN models. On the other hand, these hybrid models estimated the same trends of the observed rainfall data, as well as peak values of the rainfall time series with high accuracy compared to the GEP and ANN models. However, the proposed hybrid models are slightly over estimating the peak points of rainfall data in Rasht (Humid), Gorgan (sub-humid) and Urmia (semi-arid) stations. Moreover, the mentioned models had underestimation at selected stations in arid and hyper-arid climates. It is expected that it can be achieved better results by combining the intelligent models with time series models such as ARCH model in the hydrological time series modeling. Moreover, it is recommended that other types of ARCH model such as GARCH, EGARCH and Non-linear GARCH to be used in the next studied to model the random part of hydrological time series (e.g. rainfall). Further studies can also be accomplished to present other novel models in order to more accurate modeling of the rainfall time series, especially peak points of the rainfall data.