Statistical analysis and forecasting monthly temperature of Sanandaj synoptic station with the application of SARIMA model

Introduction
Analysis and modeling of temperature time series are important challenges in predicting the behavior of the climate. Temperature is one of the basic elements of climate formation and the most basic factor in determining the role and distribution of other climatic elements. The purpose of this research is to statistically model the monthly temperature of the Sanandaj synoptic station and forecast the temperature in order to know in advance the change in weather conditions for environmental planning.
Materials and Methods
Sanandaj synoptic station is located in Kurdistan province. The average annual temperature is 12.8 °C and the average annual rainfall is 492 mm. The seasonal autocorrelated integrated moving average model is a time series forecasting method developed in the 1970s by Bucks and Jenkins. The two general forms of ARIMA models are: non-seasonal ARIMA (p,d,q) and multiplicative seasonal ARIMA (P,D,Q) × (p,d,q). The general form of SARIMA is (p,d,q)(P,D,Q)S ~ Z_t, where P, D, q, and d are the degrees of autocorrelation (AR), moving average (MA), and degree of differentiation, and P, Q are also degrees seasonal and S are the number of differentiated seasonal.
Results and Discussion
The average temperature is 14.39, the median is 14.45, and the mean is 7.8 with a low difference, which indicates that the data is almost normal. The data showed that they are homogenous on average with statistics less than the critical limit and p_value of 0.84 and 0.87 respectively. But Van Newman's test with a statistic of 0.30, which is close to zero, showed that the data are not homogeneous in variance. For modeling, based on the minimum of differentiated variance, non-seasonal and one-season zero difference degrees were detected. The differential degree D=1 indicates that the time series oscillates around a non-horizontal line. Then, according to the significant branches of autocorrelation and partial autocorrelation diagrams and fitting different models, two significant final patterns of Sarima were extracted. According to the Akaike criterion, the M1 pattern, that is  SARIMA (0, 0, 2) (0, 1, 1) ₁₂, was determined as the final pattern. According to the obtained coefficients, the model can be written as follows: Z_t=Z_t-12- 0.407a_t-1 - 0.1753at-2-0.95a-12+ a_t, where it is defined as independent normal random variables with zero mean and one variance. According to this model, the monthly temperature of Sanandaj is a function of the average temperature of one and two months before and the corresponding month of the previous year, as well as a function of random phenomena.
Conclusion
According to this model, the monthly temperature of Sanandaj is a function of the average temperature of one and two months before and the corresponding month of the previous year, as well as a function of random phenomena. By fitting the regression equation, a value of 0.007 was observed; but this constant value was not significant and the absence of a constant (θ0) in the fitted model indicates the lack of certainty of the trend in the average monthly temperature of Sanandaj. The trend of temperature increase in Sanandaj is 0.0002 degrees per month. it is suggested to take into account more stations with statistics and more time periods in the study area and the effect of biennial fluctuations, El Nino and Enso, on the western part of the country, which the authors will consider in future research.