Long-term runoff study using SARIMA and ARIMA models in the United States

METEOROLOGICAL APPLICATIONS Meteorol. Appl. 22: 592 598 (2015) Published online 9 February 2015 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/met.1491 Long-term runoff study using SARIMA and ARIMA models in the United States Mohammad Valipour* Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran ABSTRACT: In this study, the ability of the seasonal autoregressive integrated moving average (SARIMA) and autoregressive integrated moving average (ARIMA) models was investigated for long-term runoff forecasting in the United States. In the first stage, the amount of runoff is forecasted for 2011 in each US state using the data from 1901 to 2010 (mean of all stations in each state). The results show that the accuracy of the SARIMA model is better than that of the ARIMA model. The relative error of the SARIMA model for all states is <5%. In the second stage, the runoff is forecasted for 2001 to 2011 by using the average annual runoff data from 1901 to 2000. The SARIMA model with periodic term equal to 20, R 2 = 0.91, and mean bias error (MBE) = 1.29 mm is the best model in this stage. According to the obtained results, a trend is observed between annual runoff data in the United States every 20 years or almost a quarter century. KEY WORDS ARIMA; hydrological processes; periodic term; runoff forecasting; seasonal trend Received 8 July 2014; Revised 9 November 2014; Accepted 18 November 2014 1. Runoff forecasting history Schar et al. (2004) forecasted a seasonal runoff using precipitation from meteorological data assimilation systems. The high correlations suggest that a reliable seasonal runoff forecasting system can be constructed from the statistical relationship between the model-assimilated precipitation and subsequent runoff. Artificial neural network is one of the other approaches for runoff forecasting. In addition, seasonal runoff was simulated using Hydrologic Engineering Center (HEC) models (Anderson et al., 2002). Patry and Marino (1984c) presented a two-stage urban runoff forecast model. This approach was found to be particularly useful for lead times of up to 30 min. Wang et al. (2011) developed a modified rational equation for arid-region runoff estimation successfully. Habib et al. (2007) investigated the effect of local systematic and random errors of the commonly used tipping-bucket (TB) rain gauges on the accuracy of runoff predictions. The computed runoff differences caused by the TB random errors were dependent on the magnitude of the runoff discharge, and on the temporal resolution of the rainfall input. In recent decades, radar information helps researchers in accurate forecasting of runoff data (Rahimi et al., 2014). Patry and Marino (1984a, 1984b) investigated the sensitivity and application of different equation models to real-time urban runoff forecasting. 2. SARIMA and ARIMA applications in hydrological forecasting Among the many methods and models available for investigating long-term forecasts, such as stochastic models, fuzzy theory, artificial neural network (Banihabib et al., 2012; Valipour et al., 2012a), and rough theory, autoregressive integrated moving average (ARIMA) and seasonal autoregressive integrated moving average (SARIMA) models have been applied based on stochastic theory; considering the serial correlation among observations, and by providing for systematic searching in each stage (identification, estimation and diagnostic check) an appropriate model was always applied to ensure that the hydrological forecast exhibited a strong random component (Zhang et al., 2011). Valipour et al. (2013) compared autoregressive moving average (ARMA), ARIMA and the autoregressive artificial neural network models in forecasting the monthly inflow of the Dez reservoir. The inflow to the reservoir shows that the ARIMA model s results match the forecast compared with the ARMA model. Valipour (2012b), using time series models, determined the required observation data for rainfall forecasting according to the climate conditions. By comparing the R 2 of the models, it was determined that time series models are more appropriate for rainfall forecasting in semi-arid climates. In addition, considering the importance of an accurate estimation of evapotranspiration (Psilovikos and Elhag 2013, Valipour 2014a; Valipour 2015; Valipour and Eslamian, 2014), ARIMA can be a powerful model for forecasting evapotranspiration in hydrometeorology and irrigation water requirement (Mahdizadeh Khasraghi et al., 2014; Valipour, 2014b, 2014c; Valipour et al., 2014). Valipour (2013a, 2013b) forecasted snow water equivalent using the ARIMA model with a relation error of <10%. Valipour (2012a, 2012c) determined the critical areas in Iran for agricultural water management according to the annual rainfall by using ARIMA model. Using the collected data, a rainfall forecast for the next 1 year was made by the ARIMA model. 3. Necessity and novelties of this study * Correspondence: M. Valipour, Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran. E-mail: vali-pour@hotmail.com Some of the limitations (according to the previous studies) are as follows: (1) most of the studies focused on short or medium term (daily, weekly or monthly) runoff forecasting; (2) the results 2015 Royal Meteorological Society

Runoff forecasting using SARIMA 593 obtained were applicable for specified climate conditions but not for various climate conditions; (3) the ARMA and ARIMA models were considered more than the SARIMA model among the time series models used for hydrological forecasting (however, the accuracy of the SARIMA model can be increased compared to other time series models by using an appropriate periodic term); (4) although the use of other hydrological parameters such as rainfall improves accuracy of forecasting, these data are not collected in all regions, and (5) in the cases where runoff is forecasted for basins that are spread across more than one state, decision making is difficult for the governments. In this study, the ARIMA and SARIMA (with different periodic terms) models were applied to long-term runoff forecasting (useable for macroeconomic management decisions) as well as for dealing with all of the mentioned limitations across the United States (with different climate conditions). 4. Materials and methods In this study, the SARIMA and ARIMA models were used for runoff forecasting as follows: Seasonal autoregressive integrated moving average = SARIMA (p, d, q)(p, D, Q) s (1) Autoregressive integrated moving average = ARIMA (p, d, q) (2) where p is the order of the non-seasonal autoregressive model, q is the order of non-seasonal moving average model, P is the order of seasonal autoregressive model, Q is the order of seasonal moving average model, d is the number of non-seasonal differences, D is the number of seasonal differences and s is the periodic term. All the ARIMA and SARIMA models were checked using Akaike information criterion (AIC). The necessary stages for determining the mentioned parameters were created using the MINITAB software. It provides a simple, effective way to input statistical data, manipulate that data, identify trends and patterns and then extrapolate the answers to the problem at hand. That is a rather simplistic way of describing this vital and extremely effective tool (http://www.6sigma.us/minitab-training.php). The periodic term mentioned earlier is obtained by using the revision of the Valipour s equation (Valipour et al., 2012b) as follows: s = {A B} {1} (3) where A is the set of divisors of the calibration data number and B is the set of multiples of 1, but not 12 because in this study, forecasting has been done for the annual term (long term) using yearly runoff data; thus the assumption that each 12 month period has a trend, was not usable. In Equation (1), the relationship between each data and itself is not significant, therefore, minus one is necessary. In this study, the first runoff is forecasted for 2011 in each US state using the annual runoff data from 1901 to 2010 (calibration data include mean of all stations in each state) then the runoff for 2001 to 2011 was forecasted for the entire United States using average annual runoff data from 1901 to 2000 (calibration data). In the first stage, A = 110 and therefore s = {2, 5, 11, 22, 55, 110} and in the second stage A = 100 and therefore s = {2, 4, 5, 10, 20, 25, 50, 100}. When s = 110 and s = 100, it means that there are no seasonal trends in the first and second stages, respectively, and these values 110 and 100 are equivalent to the ARIMA model findings. Therefore, six different models including ARIMA, SARIMA2, SARIMA5,, and SARIMA55 were used for forecasting the runoff in 52 US states and also eight different models, ARIMA, SARIMA2, SARIMA4, SARIMA5, SARIMA10, SARIMA20, SARIMA25 and SARIMA50 were used for forecasting the average runoff across the United States. To investigate the accuracy of the SARIMA and ARIMA models in the runoff forecasting for each state, the relative error is used as follows: RE = 100 R O R F R O (4) where RE is the relative error (%), R O is the observed runoff for each state in 2011 and R F is the forecasted runoff for each state in 2011. In this study, two different indices are used to assess the ability of the SARIMA and ARIMA models in runoff forecasting for the entire United States as follows: R 2 = 1 (R O R F ) 2 (R O R A ) 2 (5) MBE = (R O R F ) n (6) where R A is the average annual observed runoff of 237.20 mm from 2001 to 2011 and n is the number of forecasted datasets equal to 11 (2001 2011) (MBE, mean bias error). In this study only runoff data are applied for annual runoff forecasting but annual average data of temperature ( C), relative humidity (%) and rainfall were also collected from all the US states and the distribution maps of these hydrological variables were plotted for all the US states using these data. It is useful to find reasons of the best periodic term for each state. Figure 1 shows the map plotted based on the collected annual hydrological data of temperature, relative humidity, rainfall and runoff. 5. Results and discussion As Figure 1 shows, temperature increases from north to south. The increasing trend is slow (e.g. compare Wyoming with Montana) as a result, in New Mexico the temperature is lower than that of other southern states. Apart from the 8 states (Montana, Wyoming, Nevada, Idaho, New Mexico, Utah, Colorado and Arizona), the relative humidity is >64% in the remaining 44 states. Amounts of rainfall for eastern states, especially southeast states, are higher than the other states. In the maps showing rainfall and runoff data the values of these parameters increase as one goes from the centre to the periphery. However, comparing these two maps shows some conflict: in Washington and Hawaii states, the rainfall values are <800 and 600 mm respectively, but the runoffs are >800 and 1000 mm, respectively. These cases indicate that the amount of snowmelt in these states is high. In Texas and in the central and eastern states, the runoff values are much lower than the rainfall. This can be due to natural causes such as high evaporation due to hot and sultry weather or controlling the runoff by government policy and management. 5.1. Runoff forecasting for each state of the United States After running 72 different structures for the ARIMA model and more than 5000 different structures (using MINITAB software) for each of the SARIMA models (based on the periodic term), the best structure for each of the models was obtained.

594 M. Valipour Figure 1. The map designed based on gathered annual hydrological data for temperature, relative humidity, rainfall and runoff. Note that ranges in the figure legend include values greater than or equal to the lower bound and less than the upper bound indicated. See http://en.wikipedia.org/ wiki/list_of_u.s._state_abbreviations for definition US state abbreviations. In the ARIMA models, the minimum and maximum relative errors are related to the District of Columbia and Texas states, respectively. This finding indicates the wide range of changes in the relative error of ARIMA model due to different trends in the runoff data for each state. In this study, accuracy of the ARIMA model increased when using five autoregressive and moving average parameters (p = 5andq = 5) instead of the two parameters, similar to previous studies. The accuracy of the ARIMA model is not dependent on the amount of observed runoff in 2011. For example, in Arizona, observed runoff in 2011 was 8.55 mm (the minimum among all states), forecasted runoff was 9.36 mm, and RE = 9.46% (a good forecasting); in Puerto Rico observed runoff was around 1000 mm and RE = 15.78% (a moderate forecasting); but in Ohio observed runoff was 593.05 (a moderate runoff according to the Table 1) and the RE = 34.69% (a poor forecasting). The cases that are valid for the ARIMA model are also correct for the SARIMA models. In the SARIMA55 model, the value of the seasonal parameters (P, D and Q) is never more than 1 because of the periodic term (55) and overflow error. Figure 2 can be used for better perception about performance of forecasting models applied in this study. In 44% of cases (23 states), RE was >30% (poor forecasting) in the ARIMA model and only in 10% of cases (5 states), RE was <5% (excellent forecasting). Therefore, the ARIMA model had the minimum accuracy among all models in this study based on relative error. In the SARIMA2 and SARIMA5 models, the percentage related to poor forecasting was reduced and the percentage related to excellent forecasting was higher than in the ARIMA model; however, the majority of the states have RE >30%. In the model, 19 states had RE <5% and thus the percentage related to the excellent forecasting was more than that of other forecasting models. However, the best model of excellent forecasting was with 36 states for which relative error was <5%. This model forecasted the runoff for all US states as excellent or good except for eight states. By increasing the periodic term from 22 to 55, performance of the SARIMA model is reduced significantly. In the SARIMA55 model, the number of states with excellent forecasting was 52% less than. Table 1 shows the best models for each state based on relative error. According to Table 1 at least one of the SARIMA models is capable of runoff forecasting for all the US states with RE <5%. The minimum and maximum relative errors in relate to Delaware and Mississippi states, respectively. Figure 3 shows the best obtained results for runoff forecasting in each of the US state. Figure 3 confirms the capability of each SARIMA model in Figure 2. According to Figure 3 most US states are compatible with model. The model was placed in second order in this respect. The time series related to the annual runoff of each state was drawn for better understanding of Figure 3. In Figure 3, the states that were situated close to each other (and therefore with similar climate and hydrological processes) showed compatibility with a specific SARIMA model, with the exception of Kansas. According to the annual runoff time series,

Runoff forecasting using SARIMA 595 Table 1. The best runoff forecasting models for each US state in 2011. State Best model Observed Forecasted RE (%) State Best model Observed Forecasted RE (%) Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware District of Columbia Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri (5,1,1)(0,1,1) SARIMA55 (4,0,5)(0,0,1) (5,0,1)(0,1,1) SARIMA5 (5,1,3)(0,1,1) (5,3,1)(0,1,1) SARIMA55 (5,0,1)(0,1,1) (4,1,2)(0,1,3) SARIMA5 (2,0,3)(3,0,1) SARIMA2 (1,1,4)(3,0,1) (4,0,4)(1,1,0) (5,1,3)(0,1,2) (1,1,2)(2,0,1) (5,2,0)(4,0,1) (0,1,1)(2,1,1) (3,1,0)(0,1,2) (3,1,1)(3,0,1) (2,0,0)(1,2,0) (0,0,2)(2,0,1) (2,3,1)(1,1,1) (4,1,2)(2,1,0) SARIMA2 (0,0,4)(3,0,1) (0,3,2)(0,0,1) (1,0,2)(2,0,1) (0,2,2)(2,1,1) (2,0,2)(3,1,2) SARIMA5 (0,1,1)(3,1,1) 368.52 370.39 0.51 Montana (2,4,1)(1,1,1) 599.50 586.79 2.12 Nebraska SARIMA5 (0,0,4)(3,0,1) 8.55 8.65 1.17 Nevada (0,3,1)(1,1,1) 377.02 372.10 1.31 New Hampshire (0,1,1)(2,1,1) 338.60 349.11 3.10 New Jersey (0,2,2)(0,1,1) 92.95 90.83 2.28 New Mexico (0,0,4)(1,1,2) 1034.30 1033.60 0.07 New York (3,1,2)(2,1,2) 453.26 453.12 0.03 North Carolina (0,1,1)(5,1,0) 422.22 424.79 0.61 North Dakota (4,4,0)(1,0,2) 145.60 151.48 4.04 Ohio (2,2,1)(0,1,1) 186.84 180.26 3.52 Oklahoma SARIMA55 (1,0,0)(1,0,0) 1550.06 1545.11 0.32 Oregon (2,2,0)(1,0,0) 359.74 360.17 0.12 Pennsylvania (2,2,2)(2,1,1) 412.08 413.07 0.24 Puerto Rico (4,0,2)(0,0,1) 495.40 504.48 1.83 Rhode Island SARIMA5 (0,0,3)(2,0,3) 305.45 306.63 0.39 South Carolina (3,0,2)(2,0,1) 36.85 35.03 4.94 South Dakota (0,3,1)(2,0,0) 631.48 620.61 1.72 Tennessee (1,1,3)(3,0,1) 111.18 116.34 4.64 Texas SARIMA55 (5,0,0)(1,0,0) 957.13 925.61 3.29 Utah (2,3,2)(2,1,0) 455.51 455.12 0.09 Vermont (2,2,2)(2,1,1) 850.86 812.16 4.55 Virginia SARIMA2 (0,0,4)(4,0,1) 333.96 333.75 0.06 Washington (0,0,5)(4,1,0) 287.77 300.14 4.30 West Virginia (0,0,4)(3,0,1) 299.20 284.35 4.96 Wisconsin (2,1,0)(3,0,1) 331.70 326.17 1.67 Wyoming (2,4,1)(1,1,1) 214.82 216.52 0.79 56.60 56.65 0.09 68.90 71.50 3.78 863.97 865.15 0.14 880.27 843.26 4.20 11.06 11.28 2.01 952.39 914.77 3.95 271.89 276.04 1.53 110.14 105.85 3.89 593.05 613.69 3.48 66.23 67.36 1.71 599.54 576.13 3.90 826.04 864.23 4.62 998.96 1045.94 4.70 642.83 643.03 0.03 171.35 177.90 3.82 91.66 94.61 3.22 581.12 583.22 0.36 12.30 11.79 4.11 105.21 109.68 4.25 1036.29 1024.72 1.12 334.93 345.44 3.14 1103.22 1134.69 2.85 596.16 595.10 0.18 356.72 350.94 1.62 141.23 140.90 0.24 although runoff was always <250 mm for Kansas, in this small range intensity of changes and peaks were more similar for states such as Minnesota, Montana and Wyoming. In addition, amount of runoff in Kansas was 36.85 mm in 2011, which was less than half of the annual runoff mean (from 1901 to 2011) in this state. Therefore, it was required that s > 11 (compared to neighbouring states). In Maryland, District of Columbia and Virginia there is better forecasting with minimum periodic term. In these three states because of almost moderate hydrological processes (Figure 1) and runoff rate between 200 and 600 mm, all runoff events were forecasted by choosing s = 2. However, also obtained acceptable values for these states (Figure 2). For many states more than one model could be used; however, only in five states relative error values were <10% in all models (Tennessee, Virginia, West Virginia, Delaware and Maryland). In Nebraska, Arkansas, Rhode Island and Delaware, although SARIMA55 obtained a better forecasting, was also able to forecast runoff with RE < 10%. However, in

596 M. Valipour Figure 2. Performance of runoff forecasting models applied in this study based on forecasting accuracy. Missouri, the model could not forecast like the SARIMA5 model. In this state, the amount of runoff from 1901 to 1922 (22 years) was <200 mm; however, a time period of 22 years and R O = 200 mm were not repeated in later years. The SARIMA model assumes that periods with s = 22 will occur in the future (calibration stage). This assumption reduced the accuracy of the model. In Oklahoma, Alaska and Texas, performance of the SARIMA55 model was much better than that of the model. Although runoff was always <300 mm (with the exception of 1920) for Oklahoma, in this small range, intensity of changes and peaks were more similar for states such as Minnesota and Wisconsin. In addition, the runoff in Oklahoma was 60.23 mm in 2011 which was less than half of the annual mean runoff (from 1901 to 2011) in this state. Therefore, it was required that s > 22 for better forecasting. In Alaska, the runoff decreased from >4000 mm to <2000 mm in the 1950s. Due to these extreme changes, the SARIMA55 model was more capable of runoff forecasting in this state. In Texas, amount of runoff in 2011 was 12.30 mm. This value was the minimum amount of runoff in the entire statistical period. However, other models had a RE >30% for this state. From the other peak values forecasted by the SARIMA models it can be seen that runoffs in Connecticut, Minnesota, Montana, New York and Vermont were the maximum in the last 111 years. In addition, the accuracy of the models is not acceptable in New Jersey, North Dakota and Ohio during the last century. 5.2. Average runoff forecasting for all over the United States The time series related to the annual runoff of each state show that due to climatic changes the runoff could not be forecasted for more than 1 year with a good accuracy for each state. However, in some situations that need a comprehensive study, for example, for flood control and use of runoff caused by rainfall, it is required to forecast with a time horizon longer than 1 year. While the average annual runoff of all the states was considered as the runoff of the United States, because of neutralization of drastic climatic changes and reduction in the peak points,

Runoff forecasting using SARIMA 597 Figure 3. Performance of ARIMA and SARIMA models for runoff forecasting in each US state. Table 2. The best structures of each model for runoff forecasting (2001 2011) in all US states. Model Structure R 2 MBE ARIMA (5,1,3) 0.86 4.06 SARIMA2 (4,1,4)(1,0,1) 0.86 1.72 SARIMA4 (3,1,4)(1,0,1) 0.86 2.90 SARIMA5 (3,1,3)(2,1,1) 0.87 8.09 SARIMA10 (3,1,3)(2,0,2) 0.87 1.86 SARIMA20 (0,1,4)(1,0,1) 0.91 1.29 SARIMA25 (1,1,4)(1,0,1) 0.90 4.39 SARIMA50 (1,1,3)(1,0,1) 0.88 2.49 Figure 4 shows the performance of the SARIMA20 model for runoff forecasting in the United States. Since in 2004 and 2008, amounts of observed runoff were closest to the average runoff of all the years (235.30 mm), accuracy of the SARIMA20 model was better in these 2 years than other forecasted years. Based on the forecast results obtained for each state and for the United States as a whole it can be claimed that, if 1901 is considered as a benchmark, after each 20 years or almost a quarter century, there is a trend in the observed runoff data in the United States. The author claims can be investigated by the Earth move data and other usable information related to the National Aeronautics and Space Administration (NASA), in a separate study. runoff was forecasted for 11 years ahead (Table 2). According to Table 2, all the ARIMA and SARIMA models had a good forecasting due to the trend in average runoff data; however, the SARIMA20 model with the maximum R 2 and minimum MBE was more capable of long-term runoff forecasting. Therefore, while intensity of changes is mild, the forecasting horizon can be increased to more than 1 year in each region. 6. Recommendation Although the results show that the obtained accuracy is currently acceptable (the RE of the SARIMA models for all the states is <5% and R 2 = 0.91 for the runoff estimation in the United States), a hybrid forecasting that combines nonlinear and machine-learning approaches can improve or enhance Figure 4. Performance of the SARIMA20 model in long-term runoff forecasting for the United States. (a) Observed and forecasted runoff versus time and (b) forecasted runoff versus observed runoff.

598 M. Valipour the accuracy of the ARIMA and SARIMA models. Previous research on water resources management showed that most of the hybrid techniques have been applied only for the ARIMA model, and the SARIMA model has not been considered (Zhang et al., 2011). Therefore, a study of the SARIMA hybrid model (instead of ARIMA hybrid model) has been recommended as continuation of current research to improve the accuracy of the SARIMA model. 7. Conclusions In this study, the ability of the SARIMA and ARIMA models was evaluated for long-term runoff forecasting. Results obtained show that SARIMA models forecast the annual runoff better than the ARIMA models. The SARIMA models were very sensitive to the periodic term. By choosing an appropriate periodic term, SARIMA models were able to forecast the annual runoff for each US state with an RE <5%. However, the performance of SARIMA models is sharply declined if a correct periodic term is not used because of underestimation or overestimation in the calibration stage. As a rule but not always, if there were moderate weather changes and a seasonal trend in the calibration data, SARIMA2 and SARIMA5 were superior models. If there were drastic and unrepeatable changes in the calibration data, SARIMA55 was the superior model and in other cases, and were two reliable models for annual runoff forecasting. Also, in those regions where the peak points in the calibration dataset are close to the average of all data, annual runoff data can be forecast for one decade ahead. An important result of this work is the identification of the annual runoff trend using the periodic term of the Box-Jenkins models (ARMA, ARIMA, SARMA and SARIMA). 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