1. Introduction

Drought is a climate feature that occurs occasionally. This phenomenon, which affects more people than any other hazard, is considered by many to be the most complex but least understood of all the natural vulnerabilities (^{Mishra and Desai, 2005}). In Iran, arid climate extends to an area of 573 884 km² (35.54% of the territory). The Maharlu-Bakhtegan basin is located in this area; therefore, precipitation and water shortage in this region are very problematic. Meteorological drought occurs when the precipitation average is less than the precipitation average during the long-term period. In Iran, we are confronting challenges in many areas that have arid and semi-arid climates and suffer drought events, so it is necessary to pay more attention to precipitation. Pre-knowledge of the possible amount of precipitation is important in planning water recourses, management of agriculture and droughts, etc. Previous studies show that large-scale climate modes (e.g., North Atlantic Oscillation [NAO], South Oscillation Index [SOI]) have an influence on climate and precipitation in different parts of the world (^{Nazemosadat and Cordey, 2000}; ^{Karabörk et al., 2005}; ^{Gaughan and Waylen, 2012}; ^{Berg et al., 2013}; ^{Choubin et al., 2014b}).

In this study, we used large-scale climate indices for predicting the standard precipitation index (SPI). Among the several proposed drought monitoring indices, SPI has widespread application for describing and comparing droughts among different time periods and regions with different climatic conditions (^{Cancelliere et al., 2007}). SPI prediction is a critical issue that has attracted much attention in recent decades all over the world in order to carry out hydrological modeling in arid and semi-arid regions (^{Rezaeian-Zadeh et al., 2012}). Today, more non-linear models are applied to prediction. In previous studies, ^{Dahamsheh and Aksoy (2009)}, ^{Azadi and Sepaskhah (2012)}, and ^{Rezaeian-Zadeh et al. (2012)} used artificial neural networks (ANNs), and ^{El-Shafie et al. (2011)}, ^{Sanikhani and Kisi (2012)}, ^{Jeong et al. (2012)}, and ^{Choubin et al. (2014a)} successfully applied the adaptive neuro-fuzzy inference system (ANFIS) to predict precipitation. In eastern Australia, ^{Deo and Sahin (2015)} investigated the application of the ANN model for the prediction of monthly SPIs using hydrometeorological parameters and climate indices. The results showed that the ANN model is a useful data-driven tool for forecasting monthly SPIs.

In the Awash River Basin (Ethiopia), ^{Belayneh et al. (2014)} forecasted the long term SPI drought using wavelet neural networks. The forecasted results indicated that the coupled wavelet neural network (WA-ANN) models were better than all the other models in this study for forecasting SPI 12 and SPI 24 values. ^{Ruigar and Golian (2016)} predicted the precipitation in the Golestan dam watershed using climate indices: their results indicated that the MLP model is capable of accurately predicting monthly maximum precipitation.

In this study we compared the performances of three modeling techniques for predicting drought in a 43-yr period (1967-2009) in the Maharlu-Bakhtegan basin of Iran. We used the M5P model tree in addition to ANFIS and the multilayer perceptron (MLP) network to predict the SPI using large-scale climate indices as input data, over lead times of 1 to 12 months.

2. Methodology

2.1 Study area

The Maharlu-Bakhtegan basin spreads over 31 000 km². This area located in southwestern Iran (29° 00' to 31° 14' N, 51° 42' to 54° 31' W), with annual precipitation of270 mm, is one of the most important agricultural centers of Iran (Fig. 1). In this paper, precipitation data were collected from the Iranian Water Resource Management Company (TAMAB) for four meteorological stations: Shiraz synoptic station, Dashtbal, Ali Abad Khatr and Dehkade Shahid. First, station data were analyzed and missing data were reconstructed by using the correlation method; then homogeneity and independence of data were evaluated using the run-test method. Homogeneity and dependence were accepted at a high level. We used Thiessen polygons between stations to calculate the average of watershed precipitation.

Fig. 1 Study area. 

2.4 MLP network

ANNs are simplified versions of a human brain and consist of input, hidden and output layers (^{Gunaydin, 2009}). MLP is the most common neural network model (^{Zurada, 1992}; ^{Hagan et al., 1996}).

In this paper, we used the Levenberg-Marquardt (LM) training algorithm to obtain the weight of the MLP network. LM can be thought as a combination of steepest descent and the Newton method. The MLP network consists of an input layer of source neurons, at least one middle or hidden layer of computational neurons, and an output layer of computational neurons. The output of an artificial neuron can be expressed as follows:

(1)

where n is the total number of inputs, x ₁,x ₂,...,x_n are the inputs, w₁, w₂,...,w_n are corresponding weights for the inputs.

In this study, the optimum number of hidden neurons and transfer functions was obtained by experiments or by trial and error. Logsig and Purelin transfer functions were used in the hidden and output layers, respectively.

2.7 Data normalization and evaluation criteria

Climate data in a semi-arid region are sparse and irregular in distribution; the best way to improve the robustness of climate information would be data normalization. The best range for data normalization is 0.05-0.95 (^{Hsu et al., 1955}), as follows:

(2)

where, x_norm and x_r are the normalized and the original inputs, and x_min and x_max are the minimum and maximum input ranges, respectively.

Some of the common parameters, including root mean square error (RMSE), mean error (ME) and percent bias (PBIAS) were used to check the performance of the applied models. These indices are valuable because they disclose errors in the units (or squared units) of the constituent of interest, which aids in the analysis of results (^{Moriasi et al., 2007}). The PBIAS measures the average tendency of the simulated data to be larger or smaller than their observed counterparts; the optimal value is 0.0. Positive values indicate a model bias toward underestimation, whereas negative values indicate a bias toward over-estimation (^{Gupta et al., 1999}). These parameters were calculated as follows:

(3)

(4)

(5)

where N is the number of data points considered, and O_i and P_i are the observed and predicted values, respectively.

3. Results

Factor analysis showed that the Kaiser-Meyer-Olkin (KMO) statistic equals to 0.69, so the input variables are suitable for factor analysis (^{Shrestha and Kazama, 2007}). Eight components had eigenvalues greater than 1 and contained 81% of the total variance. Thus, eight climate signals (AMO, AMM, BEST, NINO3.4, NINO4, NTA, SOI, TNA) were selected as most effective after a Varimax rotation with factor loadings of 0.904, 0.826, 0.952, 0.918, 0.855, 0.908, -0.849, and 0.927, respectively.

Table I shows the performance of the ANFIS, M5P and MLP models in predicting SPI time series 12 months in advance. Regarding data testing, the best performance of ANFIS was found for eight-months in advance prediction, with RMSE, ME and PBIAS values of 1.032, 0.011, and 3.55, respectively (Table I). For the M5P model tree, the minimum values of RMSE, ME and PBIAS are related to 10-months in advance predictions (RMSE = 0.828, ME = -0.007, and PBIAS = -2.12). For MLP, the best performance of prediction was obtained one step ahead compared to other models (RMSE = 0.802, ME = -0.002, and PBIAS = -0.47) (Table I). PBIAS indicated that predictions are mostly overestimated (about 85, 54 and 70% for ANFIS, M5P model tree and MLP, respectively).

Table I Performance of models in predicting SPI (from one to 12 months in advance). 

In this paper, we used a ^{Taylor (2001)} diagram (Fig. 2) to evaluate the accuracy of ANFIS, MLP, and the M5P model tree. This diagram provides a visual framework for comparing different model results to a reference model or, mainly, to observations. The Taylor diagram is drawn by standard deviation (STD), centered root mean square error (RMSE) and correlation (COR) between different models and observations. Statistics of STD, RMSE and COR were computed for ANFIS, the M5P model tree and MLP from one to 12 months in advance (Fig. 2). The position of each model in the plot shows how closely the simulated SPI pattern matches with observations. From Fig. 2 it can be seen that predictions of ANFIS and MLP are in agreement with observations, unlike the M5P model tree. Although predictions in ANFIS and MLP are quite similar, some step predictions in MLP are closer to observations (e.g., one-month-in-advance predictions). Standard deviation of prediction data indicated that none of the models was able to predict fluctuations in observation data. Figure 3 compares the observed and predicted SPI for the testing set in a one-month lag time.

Fig. 2 Comparisons of observed and predicted SPIs by MLP, ANFIS and the M5P model with a one-month delay. 

Fig. 3 Comparisons of observed and predicted SPIs for watershed in a one-month lag-time. 

4. Discussion

In this study we used climate indices to predict SPI. Factor analysis was used to determine the most important of large-scale climate signals. Sea surface temperature (SST) on the Pacific Ocean and ENSO (including the BEST, NINO3.4, and NINO4 signals), the southern oscillation index (SOI) and SST on the Atlantic Ocean (including the AMO, AMM, NTA, and TNA signals) were selected as the most important signals. In previous studies, ^{Nazemosadat and Cordey (2000)}, ^{Mariotti et al. (2002)} and ^{Pongracz and Bartholy (2006)} showed the direct effect of ENSO on precipitation. ^{Karabörk et al. (2005)} indicated an inverse relationship between SOI and precipitation.

The ANFIS model was found to have the best performance for the eight-month in advance predictions, whereas the M5P model performed better for 10-month in advance predictions and the MLP network for one-month in advance. Error parameters (Table I) indicated that the MLP network performance was a little better than the other two models, while ^{Dastorani et al. (2010)} showed that the potential of ANNs with the ANFIS model is almost the same in predicting dry land precipitation. The Levenberg-Marquardt training algorithm (used in the MLP network) is more powerful and faster than the standard back-propagation algorithm (used in ANFIS) (Abyaneh et al., 2011). This may be the reason for the better efficiency of the MLP model compared to ANFIS. Also, we used the Taylor diagram (Fig. 2) to investigate the accuracy of the models. It is clear that MLP has a better aptitude in comparison with the M5P model tree and ANFIS.

5. Conclusion

Modeling is important in hydrology. This study investigated the prediction of SPI by using several models based on large-scale climate indices. Results showed that the performance of MLP was better than the M5P model tree and ANFIS (Table I). The best performance of the MLP model for SPI prediction was achieved with eight inputs, two hidden and one output neuron (MLP [8, 2, 1]) for a one-month-in-advance prediction. Also, the Taylor diagram (a very useful tool that compares the performance of different models) indicated that MLP is more efficient than the M5P and ANFIS models. There are many parameters for the determination of models' performances, but hydrologists need a tool that can compare different models. The Taylor diagram would be helpful for this purpose. We hope modelers further use this tool in natural sciences and hydrology modeling.