In recent years, many studies have been done on forecasting fuzzy time series. First order fuzzy time series forecasting methods with first-order lagged variables and high order fuzzy time series forecasting methods with consecutive lagged variables constitute the considerable part of these studies. However, these methods are not effective in forecasting fuzzy time series which contain seasonal structures. In this respect, it would be more appropriate to use methods that consider the seasonal relations in seasonal fuzzy time series forecasting. Although seasonal fuzzy time series forecasting methods exist in literature, these methods use equal interval lengths in partition of the universe of discourse. This situation incapacitates the performance of the method in forecasting time series including seasonality and trend. In this study, a new fuzzy time series forecasting method in which intervals constituting partition of the universe of discourse increase in time at a rate that obtained based on optimization was proposed. The proposed method was applied to two real time series and obtained results were compared with other methods and the superior performance of the proposed method was proved.

Tourism industry plays a major role in creating job opportunities and income generation in all countries and it has grown remarkably in recent decades. Iran has a unique situation in tourism industry due to its amazing ancient monuments and natural attractions. Therefore، developing the tourism industry can be a suitable way to improve Iranian economy and can reduce its dependence on oil income. The purpose of this paper is modeling and forecasting seasonal flows of tourist arrivals into Iran. Moreover the forecasting accuracy of methods is compared. There are different methods that can be used to forecast the economic variables. Today forecasting is regarded as an important instrument for economic policymakers. In order to deal with seasonality، Autoregressive Integrated Moving Average Approach (ARIMA) processes have been generalized. When modeling time series with systematic seasonal movements، Box and Jenkins recommend the use of Seasonal Autoregressive (SAR) and Seasonal Moving Average (SMA) terms. Therefore، we utilized the Seasonal Autoregressive Integrated Moving Average Approach (SARIMA) and seasonal integration model based on seasonal unit root test. We used Hylleberg et al. HEGY test is used for unit root testing. HEGY developed separate regression based T and F tests for unit roots at various frequencies in the quarterly data. The seasonal data about the number of tourist arrivals to Iran was obtained from the Iranian Cultural Heritage and Tourism Organization. Time scope covers 44 seasons، from 2001 to 2010. Discussion and Quarterly series of tourist arrivals shows the periodic behavior. HEGY test results indicate the presence of non-stationary tourist arrivals series. Therefore SARIMA and seasonal integration models are fitted into the data. In order to achieve stationary series، these time series need to be seasonally differentiated. In the next step، SARIMA and seasonal integration models were estimated. ARIMA (1،1،0) (1،1،1) 4 model identified as the best model among the SARIMA candidates. Finally، two indicators including RMSE (root mean squared error)، MAPE (mean absolute percentage error) were employed in order to measure the performance of models. This study has discussed about two kinds of seasonal models including seasonal autoregressive integrated moving average approach (SARIMA) and seasonal integration model. Results of HEGY’s seasonal unit root test demonstrated that seasonality unit root exists in the tourist arrivals to Iran. Furthermore، the comparison of forecasting accuracy revealed that seasonal integration model has high accuracy more than seasonal ARIMA model. Thus، the seasonal integration model was selected as best model to forecast of tourism arrivals to Iran. This result is important to decision makers to evaluate tourism arrivals.

For efficient organization and effective management of tourism and the pertinent activities, modeling and forecasting the tourist destination areas are vital issues for good performance. It helps make a better policy and plan for supplying tourist requirements. The number of tourists is related to the market supply and demand. Different services are cooperated in supplying tourism productions, such as reception, entertainment, residential, health and information services. On the other hand, regarding demand, there are many factors affecting the tourists’ destination. Forexample, economic-social conditions, language, culture and motivation that form the request process tourists. Undoubtedly, demand prediction is a drastic factor especially for activities related to tourism. In one hand, manager and planners relevant to tourism make attempt to fulfill tourism's demands. On the other hand, many of tourisms products like hotel’s rooms, airplane seats, rent car, museum or cultural plans are not being reserved or stored naturally. A hotel room that is not reserved for a night, an airplane seat that has no passenger and a restaurant table that remains empty, are the benefits that have spoiled and they may not be reserved for the future. Therefore, the tourists demand shall be predicted. Alongside the prediction process and tourism entry demand model, the governments can organize their strategies better and prepare appropriate infrastructure for serving the tourists; the private sectors could make appropriate marketing strategies for obtaining the maximum benefits from tourist entry increase, as well. The forecasting of tourism demand is an essential tool for determining the required supply and the appropriate distribution method of tourism services. When services (like tourism) achieve desirable market, its current amount and the future potential volume shall be estimated precisely. Market underestimation or overestimation makes the supplier lose the main part of his/her interest. Hence, planning and development of tourism require identifying such these kinds of motivations and demands. Accordingly, what is vitally important for the tourism management is the amount of accuracy of prediction model that led to development and diversity of tools and new methods in prediction. In this article, the plan is to forecast the number of tourist arrival for the historical - cultural site of Perspolis in south Iran. The time series involves monthly data that were collected for both domestic and international tourists. In order to testify the performance of forecasting method, the collected data were divided into two sets, training (Farvardin 1376- Esfand 1387) and testing (Farvardin1388- Esfand 1389). We used seasonal ARIMA model to detect the hidden structure of data and finally forecast the arrivals for both data sets. Based on the Box & Jenkins approach, both time series data were analyzed. In this approach, stationarity of time series is a preliminary condition. Therefore, before any attempts, the time series were made stationary by differencing. The result of data analysis of Persepolis- domestic tourism Since the number of visitors in Farvadin (April) of each year has considerable difference from the other menthes, therefore, it is likely that the forecasting model would be seasonal. The great amount of autocorrelation function in the lags 12, 24 & 36 confirms the existence of the seasonal model. Since the seasonal data are not stationary, differencing can help to make a steady time series. The results showed that, seasonal differencing in order 12, and then first differencing make the time series in an acceptable stationary form. Thus, we could determine the seasonal model of ARIMA (p,1,q) (P,1,Q)12 according to the ACF and PACF of the final series. Exponential decay of PACF in some of the first lags (figure 3, right frame) and the fact that autocorrelation amount in lag 1,r1,is significantly different from zero, shows no seasonal moving average model of order 1, MA(1), i.e. p=0, q=1. It is also observed in autocorrelation function (figure 3, left frame) that the amount of r24 is significant and this means a seasonal MA (2) (P=0, Q=2). Therefore, the final model of ARIMA (0,1,1) (01,2)12 may be written as the following: 1) 􁈺1 − 􀜤􁈻􁈺1 − 􀜤12􁈻􀜻􀯧 = 􁈺1 − 􀟠1􀜤􁈻 􁉀1 − 􀟙1􀜤12 − 􀟙􀬶􀜤24􁉁 􀝁􀯧 2) 􀜻􀯧 − 􀜻􀯧􀬿1 − 􀜻􀯧􀬿24 + 􀜻􀯧􀬿25 = 􀝁􀯧 − 􀟠1􀝁􀯧􀬿1 − 􀟙1􀝁􀯧􀬿12 + 􀟙1􀟠 􀝁􀯧􀬿13 − 􀟙2􀝁􀯧􀬿24 + 􀟙2􀟠1􀝁􀯧􀬿25 The result of data analysis on Persepolis- international tourism The plot of this time series implies that it is non-stationary. However, seasonality is not obvious in the last example, but since the amount of r6 and r12 in autocorrelation diagram are located out of the 95% confidence interval, a seasonal differencing with a six-month course is suggested. The results show that the six-month seasonal differentiation series is not stationary, but if this series be re-differencing (first order) we may observe an approximately stationary series. In order to determine the order and the kind of series in non-seasonal part of ARIMA (p,1,q)(P,1Q)6, we could consider the amount of autocorrelation as an evidence of damping sine wave to zero and since the two first amount of partial autocorrelation are significant and different from zero, the unseasonal autoregressive model, p=2, q=0, is suggested. In the seasonal part, (P,1,Q), r6, r12,r18,…., are damping to zero and since the amount of partial autocorrelation in lag 6 is significant, the seasonal AR model with Q=0 & P=1 seems to be more appropriate. ARIMA (2,1,0)(1,1,0)6 is as the following.3) 􁈺1 − 􀟚1􀜤 − 􀟚2􀜤2􁈻􁈺1 − 􀟜􀜤6􁈻􁈺1 − 􀜤􁈻􁈺1 − 􀜤6􁈻􀜻􀯧 = 􀝁􀯧 Evaluation of the suggested model was made by comparing real test data versus the forecasted data. Figures 5 and 9 successfully showed that both real and forecasted values of tourist arrival have the same variation in different months. In this research, we conclude that, the tourist arrival time series can be stationary by two differentiations (seasonal and first order differencing). In other words, the seasonal factor of this series is the inseparable part of them, with this difference that, the seasonal course for domestic and foreign visitors is 12 & 6 months, respectively. The results also show that the seasonal ARIMA model is an appropriate estimation for forecasting the number of tourists.

Determining future water consumption of cities and planning proper capacity for municipal water supply system (transfer network and purification complexes) revealed the importance of applying behavioral models and forecasting water consumptions in cities. Establishment of public and private administrations and big industries in Tehran's and encompassing trade and political centers beside population growth, showed the necessity of reconstructing municipal water infrastructures. Present study applied SARIMA model for modeling and forecasting municipal water consumption of Tehran city. Hence, using monthly time series of Tehran municipal water consumption during April 1998 till March 2010, seasonal unit root test was applied. Results showed all non-seasonal and seasonal unit roots exist in mentioned monthly time series. Therefore by choosing proper filter, fitting SARIMA models was considered. Considering the mean error of 2.32 percentages and high forecasting power of this model, forecasting municipal water consumption of Tehran city for 2011 could provide constructive results for municipal managers.

Reference evapotranspiration is one of the most important factors in irrigation timing and field management. Moreover, reference evapotranspiration forecasting can play a vital role in future developments. Therefore in this study, the seasonal autoregressive integrated moving average (ARIMA) model was used to forecast the reference evapotranspiration time series in the Esfahan, Semnan, Shiraz, Kerman, and Yazd synoptic stations.
In the present study in all stations (characteristics of the synoptic stations are given in Table 1), the meteorological data, including mean, maximum and minimum air temperature, relative humidity, dry-and wet-bulb temperature, dew-point temperature, wind speed, precipitation, air vapor pressure and sunshine hours were collected from the Islamic Republic of Iran Meteorological Organization (IRIMO) for the 41 years from 1965 to 2005. The FAO Penman-Monteith equation was used to calculate the monthly reference evapotranspiration in the five synoptic stations and the evapotranspiration time series were formed. The unit root test was used to identify whether the time series was stationary, then using the Box-Jenkins method, seasonal ARIMA models were applied to the sample data.
Table 1. The geographical location and climate conditions of the synoptic stations
Station Geographical location Altitude (m) Mean air temperature (°C) Mean precipitation (mm) Climate, according to the De Martonne index classification
Longitude (E) Latitude (N) Annual Min. and Max.
Esfahan 51° 40' 32° 37' 1550.4 16.36 9.4-23.3 122 Arid
Semnan 53° 33' 35° 35' 1130.8 18.0 12.4-23.8 140 Arid
Shiraz 52° 36' 29° 32' 1484 18.0 10.2-25.9 324 Semi-arid
Kerman 56° 58' 30° 15' 1753.8 15.6 6.7-24.6 142 Arid
Yazd 54° 17' 31° 54' 1237.2 19.2 11.8-26.0 61 Arid
The monthly meteorological data were used as input for the Ref-ET software and monthly reference evapotranspiration were obtained. The mean values of evapotranspiration in the study period were 4.42, 3.93, 5.05, 5.49, and 5.60 mm day−1 in Esfahan, Semnan, Shiraz, Kerman, and Yazd, respectively. The Augmented Dickey-Fuller (ADF) test was performed to the time series. The results showed that in all stations except Shiraz, time series had unit root and were non-stationary. The non-stationary time series became stationary at 1st difference. Using the EViews 7 software, the seasonal ARIMA models were applied to the evapotranspiration time series and R2 coefficient of determination, Durbin–Watson statistic (DW), Hannan-Quinn (HQ), Schwarz (SC) and Akaike information criteria (AIC) were used to determine, the best models for the stations were selected. The selected models were listed in Table 2. Moreover, information criteria (AIC, SC, and HQ) were used to assess model parsimony. The independence assumption of the model residuals was confirmed by a sensitive diagnostic check. Furthermore, the homoscedasticity and normality assumptions were tested using other diagnostics tests.
Table 2- The selected time series models for the stations
Station Seasonal ARIMA model Information criteria R2 DW
SC HQ AIC
Esfahan ARIMA(1, 1, 1)×(1, 0, 1)12 1.2571 1.2840 1.2396 0.8800 1.9987
Semnan ARIMA(5, 1, 2)×(1, 0, 1)12 1.5665 1.5122 1.4770 0.8543 1.9911
Shiraz ARIMA(2, 0, 3)×(1, 0, 1)12 1.3312 1.2881 1.2601 0.9665 1.9873
Kerman ARIMA(5, 1, 1)×(1, 0, 1)12 1.8097 1.7608 1.8097 0.8557 2.0042
Yazd ARIMA(2, 1, 3)×(1, 1, 1)12 1.7472 1.7032 1.6746 0.5264 1.9943
The seasonal ARIMA models presented in Table 2, were used at the 12 months (2004-2005) forecasting horizon. The results showed that the models produce good out-of-sample forecasts, which in all the stations the lowest correlation coefficient and the highest root mean square error were obtained 0.988 and 0.515 mm day−1, respectively.
In the presented paper, reference evapotranspiration in the five synoptic stations, including Esfahan, Semnan, Shiraz, Kerman, and Yazd, were calculated using the FAO Penman-Monteith method for the 41 years, and the time series were formed. The selected models gave good out-of-sample forecasts of the monthly evapotranspiration for all the stations. The models can be used in the short-term prediction of monthly reference evapotranspiration. Note that, the use of models in long-term forecasting was not recommended. The time series model can be used in lost data. Even though more methods are available for model building, the use of time series models in water resources are advocated in modeling and forecasting. Time series can be used as a tool to find lost data.

During the last decades long-memory processes have evolved as an important part of time series analysis and long memory parameter in asset returns has important evidences for many paradigms in modern finance theory. If asset returns display long memory, the series realizations are not independent over time, realizations from the remote past can help forecast future returns. Therefore the presence of long memory in asset returns contradicts the weak form of the market efficiency hypothesis. Also this characteristic has fundamental effect on time series prediction methods. This research examines the presence of long memory in series of return and volatility of Tehran Stock Exchange. Significance evidence of long memory is found in first and second moments of Tehran Stock Exchange return series. Also predictive accuracy of AMRA, GARCH, ARFIMA and FIGARCH models compared in variety of forecast horizons with recursive and rolling estimation schemes. The results of this research show that ARMA model perform better in 1-step ahead forecast, while for greater forecast horizons, including weekly, monthly, seasonal and yearly predictions, FIGARCH model outperform other alternatives.

In this study, we develop a neural network with a time shifting approach to forecast time series patterns. We investigate the impact of different layer-weight configurations to capture the trends in the forms of seasonal, chaotic, etc. We also hypothesize the combined effect of the delayed inputs and the forward connections to introduce a dynamical structure. The effect of overfitting issue is procedurally monitored to gain the resistance property from the early stoppage of training process and to reduce the predictions'' error. Finally, the performance of the proposed network is challenged by six well-known deterministic and non-deterministic time series and compared by the autoregression (AR), artificial neural network (ANN), adaptive k-nearest neighbors (AKN), and adaptive neural network (ADNN) models. The results show that the proposed network outperforms the conventional models, particularly in forecasting the chaotic and seasonal time series.

Because of the increasing importance of supplying water for the country، water resource management is of paramount importance. Predicting precipitation، as one of the most important climatic parameters، is especially important in using water supplies. Time series can be used to predict precipitation. Time series analysis seems to be a suitable tool for such forecasting. The present work studies the comparison of the rainfall forecast precision by Box-Jenkins Time Series at three stations in Khuzestan province. The monthly rainfall data of three Meteorological stations in province for 48 years، i. e. (1961-2008) were used. The accuracy and precision of models are evaluated based on AIC statistics and the analysis of the graphs of Autocorrelation Function and Partial Autocorrelation Function. The obtained suitable seasonal rainfall model was (0،1،2) * (2،1،0) 12 for Ahwaz، (0،1،2) * (0،1،1) 12 for Abadan and (1،1،2) * (0،1،1) 12 for Dezful. The results of this study show that these models accurately predict seasonal rainfall almost well.

The Box-Jenkins model is applied as a parametric method for time series analysis and fitting seasonal and non-seasonal autoregressive moving average models. But this procedure is not useful for short length and non stationary time series data. To overcome these problems, two nonparametric methods i.e. Artificial Neural Network and Singular Spectrum Analysis are introduced. These procedures do not require any statistical assumptions about normality of errors and could be used for short time series data. In this article, after introducing the above methods, their accuracy in forecasting sales of four types of food products, pharmaceutical and health care of a distribution Corporation are compared. Then using simulation studies, the effectiveness of these methods for short-term and long-term predictions are evaluated. The results show the superiority of Singular Spectrum Analysis ​​compared to the other two methods in terms of the root mean square error of forecasting.

Nowadays, Box-Jenkins models are considered one of the applicable tools to forecast different time series, especially water table fluctuations. This study addresses the prediction of unit hydrograph fluctuations of Sahneh plain aquifer using Box-Jenkins models. The study area is Sahneh Plain located in the province of Kermanshah. Using monthly data related to 21 piezometers available in the plain, unit hydrograph time-series of Sahneh plain aquifer were computed and plotted for 19 water years (1991-92 to 2009-10). With the use of autocorrelation and partial autocorrelation functions, static time series of multiple seasonal Box-Jenkins models were fitted to the data. After reviewing the results, six top models were introduced. After verifying, ultimately, SARIMA (1,1,4)(3,1,0)12 model was chosen as a best one to forecast unit hydrograph fluctuations in the Sahneh aquifer. The results show that the selected model is capable of predicting seasonal fluctuations, but does not correctly represent the period or cycle.