MIXED SEASONAL ONE-DIMENSIONAL BILINEAR MODEL FOR SEASONAL TIME SERIES DATA

Abstract

The seasonal bilinear time series model is an effective tool in statistical analysis of seasonal time series data. The existing Seasonal Autoregressive Integrated Moving Average (SARIMA) bilinear model focused only on series at the peak of seasons, leaving out the pre-peak and post-peak of the seasons. This usually results in loss of vital information about the historical behaviour of the series, such as the best planting time for a farmer to ensure optimal crop growth. There is need to obtain a nonlinear seasonal time series model to capture the behaviour of a series pre- and post-peak of seasons as a way of providing historical pattern information. The aim of this study was to develop a seasonal time series model capable of explaining the behaviour of a series pre- and post-peak of seasons. Autoregressive Moving Average (ARMA) of order (p,q), integrated series, seasonal ARMA of order (P,Q), length of seasons (s) and bilinear terms were used to develop Mixed SARIMA One-Dimensional Bilinear (MSARIMAODBL) model with Mixed Seasonal Autoregressive Integrated One-Dimensional Bilinear (MSARIODBL), Pure Seasonal Autoregressive Integrated One-Dimensional Bilinear (PSARIODBL) and Pure SARIMA One-Dimensional Bilinear (PSARIMAODBL) as its subset models. The MSARIMAODBL and MSARIODBL were used to study the series at pre-peak, peak and post-peak of seasons, while PSARIODBL and PSARIMAODBL for the peak of seasons. Nonlinear least squares method of minimising errors and Newton-Raphson iterative procedure were employed in estimating their parameters. Stationarity conditions were stated in order to prevent non-convergency of the models. Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) were used to determine the optimum model order in each case. The Residual Variance (RV) was used to determine the best model among the estimated models. Lagos monthly rainfall series from the Nigerian Meteorological Agency between 1984-2016 were used to validate the models at s= 1, 2, 3, 4, 6 and 12. Monte Carlo simulation procedure was employed at sample sizes, n=250, 500 and 1000, each replicated 50 times. Forecast values arising from the best model were compared with the original series using the Mean Absolute Error (MAE). The developed model was

where is a stochastic process and is a white-noise process. The , , and were the nonseasonal AR, seasonal AR and moving average components, while are the bilinear components. Its subsets were obtained when Q, (p,q,Q) and (p,q) were zero. Its estimated parameters were =0.0835, =0.1062, =-0.4812, =-0.4703, =-0.6159, =0.6159, =0.6813, while its stationarity condition was

The respective optimal models, AIC, BIC and RV for the MSARIMAODBL, MSARIODBL, PSARIODBL, and PSARIMAODBL were [(1,0,1)(2,2,2,1,1)4, (6.56), (5.70), (0.000000094)]; [(2,0,0)(3,2,0,3,1)1, (6.80), (6.38), (0.000019)]; [(0,0,0)(3,2,0,3,1)1, (8.74), (7.91), (0.000044)]; and [(0,0,0)(1,2,2,4,1)1, (7.50), (6.89), (0.00021)]. The MSARIMAODBL emerged the best with the lowest RV, and revealed the performance of the series pre-peak, peak and post-peak. Monte Carlo technique yielded estimates which compared favourably with those of the MSARIMAODBL, with a better performance as the sample size increased. Sample forecast values for 2015 and 2016 derived from the best model compared favourably with the original data with MAE of 4.404. The developed mixed seasonal autoregressive integrated moving average one-dimensional bilinear model explained the behaviour of a nonlinear seasonal time series during, pre-peak and after the peak of seasons.