مقایسۀ مهارت مدل¬های استوکاستیک و شبکه¬ها¬ی عصبی مصنوعی در مدل‎سازی و پیش‎بینی مقادیر و طبقات شاخص بارندگی استاندارد شده

Introduction
Drought is a temporary and recurring meteorological event which results from the lack of precipitation over an unusual extended period of time. Early indication of possible droughts can help set out drought mitigation strategies and measures, in advance. Therefore, the drought forecasting plays an important role in the planning and management of water resource systems.
Stochastic models have been extensively used for forecasting hydrologic variables such as annual and monthly stream flow, precipitation, and etc. in the past. But they are basically linear models assuming that data are stationary, and have a limited ability to capture non-stationarities and nonlinearities in the hydrologic data. However, it is necessary to consider alternative models when nonlinearity and non-stationarity play a significant role in the forecasting. In the recent decades, artificial neural networks have shown great ability in modeling and forecasting nonlinear and non-stationary time series due to their innate nonlinear property and flexibility for modeling.
The aim of this study is to compare the stochastic and artificial neural network models in forecasting the standardized precipitation index (SPI) in some stations of Iran. This is because of the multiplicity of drought occurrences in Iran and the necessity to determine the best forecasting model.
 
Methodology
The monthly total precipitation data (1973-2007) related to four synoptic stations of Iran including Bandar Anzali (with very wet climate), Hamedan Nojeh (with semi arid climate), and Bushehr (with arid climate) and Zahedan (with hyper arid climate) have been used after the homogeneity and adequacy of data have been confirmed by statistical tests.
In the present study standardized precipitation index (SPI) time series (at 3-, 6- and 12-month timescales) have been calculated for the period of 1973-2007. The most suitable distribution function for precipitation at 3- , 6- and 12- month timescales has been determined by Easyfit software on the basis of kolmogorov-Smirnov statistic. This is performed separately for each month. Then, each cumulative probability density function is transformed into a cumulative standardized normal distribution. The SPI values for the period of 1973-2000 are used to calibrate the models and the rest of the data to be tested.
Development of stochastic model consists of three stages of identification, estimation, and diagnostic checking (Box and Jenkins, 1976, 19). During the identification stage the candidate forms of the models are determined using the autoregressive function (ACF) and partial autoregressive function (PACF) and general forms of the models are determined on the basis of Schwarz Bayesian information criterion (Schwartz, 1978, 461–464) and Akaike information criterion (Akaike, 1974, 716–723). In the estimation stage the model parameters were calculated using Minitab14 software. Finally, diagnostic checks of the model are performed using kolmogorov-Smirnov (K-S) and Portmanteau test (Makridakis et al., 2003, 185) to reveal possible model inadequacies and to assist in selecting the best model.
In the present paper two different approaches of neural networks including recursive multi-step neural network approach (RMSNN) and direct multi-step neural network approach (DMSNN) are used for forecasting several time steps ahead. The RMSNN approach based on one output node forecasts a single step ahead, and the network is applied recursively, using the previous predictions as inputs for the subsequent forecasts. DMSNN is based on the multiple outputs, when several nodes are included in the output layer, and each output node represents one time step to be forecasted.
 The models are evaluated with statistical tests, correlation coefficient, and error index for 1- to 12-lead time ahead forecasting over the period of 2001- 2007. Also, the capability of the models in forecasting the SPI classes is investigated using Cohen’s Kappa statistic (Cohen, 1960, 37–46).
 
Results and Discussion
The results of stochastic modeling of SPI time series showed that the null hypothesis related to the normality of residuals is accepted for 3- and 6- month time scales but rejected for 12-month time scales at 1% significant level in all stations. The results of Portmanteau test signify that the chosen stochastic models are adequate on the available data at 1% significant level.
The results of artificial neural networks (RMSNN and DMSNN) modeling of each SPI time series are presented as optimal architectures of the best number of input and hidden neurons.
The significance lead times of drought forecasting are determined based on correlation coefficient and Kappa statistic between the observed and forecasted values of the SPI time series in the stations of interest. Accordingly, the most appropriate models for SPI values and classes have been determined by a comparison of three models for each time series.
 
Conclusion
The results have revealed that generally, for 3-, 6- and 12-month time scales, stochastic models (with average error of 0.678, 0569 and 0.344 and average correlation coefficient of 0.682, 0.777 and 0.919, respectively) are more accurate than artificial neural network models to forecast SPI values. The comparison of models in forecasting SPI classes also showed that the most accurate model for forecasting SPI classes for 3-, 6- and 12-month time scales is DMSNN, RMSNN and stochastic model (with average Kappa of 0.397, 0530 and 0.750) in sequence.