Forecasting of 125 years annual precipitation time series in Mashhad

One of custom methods for forecasting climatic variables is time series approach. We used this method for long term precipitation in Mashhad synoptic station. The annual precipitation series has three components: trend, cyclic variations and random fluctuations. The two components of trend and cyclic are not showed in statistical period of less than 100 years in arid and semi-arid regions .Therefore, linear, nonlinear, heuristic, meta-heuristic methods of time series patterns cannot explain this phenomenon well. In this study, the annual and long-term precipitation time series of Mashhad station with a statistical duration of 125 years was considered. The trend in data, cyclic variation are considered. Box and Jenkins’s method (1976) for time series is fitted on data and optimal selection of parameters and best performance are studied. Also, test for outlier in data and diagnostic analysis for residual are done.

In this research, the modeling of annual and long-term precipitation time series of Mashhad synoptic station with a statistical duration of 125 years (1894-2018) was investigated. At first, the trend of data in mean is tested using Mann-Kendall and Sen Approaches. Then, the variance of data was de-trended by box-cox transformation. Cyclic changes were considered by fitting polynomials from six to 12 degree. Optimal selection of the number of pattern parameters was based on Autocorrelation function (ACF), Partial Autocorrelation function (PACF), Extended Autocorrelation function (EACF) and Akaike (AIC) and Bayesian Criterion (BIC) criterions. Because ACF and PACF are good for distinguishing ARI (p, d) and IMA (d, q) patterns and if we have ARIMA(p, d, q) pattern we should use EACF. The patterns performance was evaluated by criterions such as the Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), Mean Absolute Percentage Error (MAPE), Mean Percentage Error (MPE), Normalized Mean Squared Error (NMSE), Signed Mean Squared Error (SMSE) and U.Tow type of outlier test, Additive Outlier and Innovational Outlier are done on all observation. Also, Diagnostic analysis is done on residuals to consider their behavior over time, normality and independent.

First, the significant trend was evaluated by using the non-parametric Man-Kendall and Sen Tests at a significant level of 95%. The results showed that the annual precipitation in Mashhad does not have a significant trend in mean. But, Data has trend in variance which stabilized by box-cox transformation. Cyclic changes by fitting polynomials from six to 12 degrees displayed that none of them were significant. But they can approximately show the wet and drought cycle in 1984-2018 years. Optimal selection of the number of pattern parameters was based on Autocorrelation function (ACF), Partial Autocorrelation function (PACF), Extended Autocorrelation function (EACF). Also, Akaike (AIC) and Bayesian Criterion (BIC) criterions are used. But result showed that IMA (1, 1), ARIMA (1,2,3) and IMA(1,3) have significant parameters. The patterns performance was evaluated by criterions such as the Mean Absolute Error (MAE), Mean Percentage Error (MPE), Normalized Mean Sauared Error(NMSE), Signed Mean Squared Error(SMSE), Root Mean Squared Error (RMSE), Mean Absolute Percentage Error (MAPE) and U. The results presented that the IMA (1, 1) model has the optimal number of parameter in the model, significant parameter and the best performance. The observations do not have AO and IO outlier. The results of diagnostic analysis also demonstrated that residuals are stable over time, follow normal distribution and are independent. Therefore, the long-term annual precipitation sequence of Mashhad follows the white noise pattern and the best prediction of the amount of precipitation is the average of data.

The annual and long-term precipitation time series of Mashhad station with a statistical duration of 125 years was modeled by time series pattern. We found that data doesn’t have trend in mean but has trend in variance. There are not significant cyclic changes but using the long term data we can see wet and drought cycles better. IMA (1, 1), ARIMA (1,2,3) and IMA(1,3) have significant parameters. IMA (1, 1) model has the optimal number of parameter in the model, significant parameter and the best performance. The observations do not have any outlier and residuals are stable over time, follow normal distribution and are independent. The long-term annual precipitation time series of Mashhad synoptic station follows the white noise pattern.