Mathematics and Statistics Vol. 9(4), pp. 511 - 520
DOI: 10.13189/ms.2021.090410
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Time Series Forecasting with Trend and Seasonal Patterns using NARX Network Ensembles

Hermansah 1,2, Dedi Rosadi 1,*, Abdurakhman 1, Herni Utami 1
1 Mathematics Study Program, Gadjah Mada University, Yogyakarta, 55281, Daerah Istimewa Yogyakarta, Indonesia
2 Mathematics Education Study Program, Riau Kepulauan University, Batam, 29424, Kepulauan Riau, Indonesia


In this research, we propose a Nonlinear Auto-Regressive network with exogenous inputs (NARX) model with a different approach, namely the determination of the main input variables using a stepwise regression and exogenous input using a deterministic seasonal dummy. There are two approaches in making a deterministic seasonal dummy, namely the binary and the sine-cosine dummy variables. Approximately half the number of input variables plus one is contained in the neurons of the hidden layer. Furthermore, the resilient backpropagation learning algorithm and the tangent hyperbolic activation function were used to train each network. Three ensemble operators are used, namely mean, median, and mode, to solve the overfitting problem and the single NARX model's weakness. Furthermore, we provide an empirical study using actual data, where forecasting accuracy is determined by Mean Absolute Percent Error (MAPE). The empirical study results show that the NARX model with binary dummy exogenous is the most accurate for trend and seasonal with multiplicative properties data patterns. For trend and seasonal with additive properties data patterns, the NARX model with sine-cosine dummy exogenous is more accurate, except the fact that the NARX model uses the mean ensemble operator. Besides, for trend and non-seasonal data patterns, the most accurate NARX model is obtained using the mean ensemble operator. This research also shows that the median and mode ensemble operators, which are rarely used, are more accurate than the mean ensemble operator for data that have trend and seasonal patterns. The median ensemble operator requires the least average computation time, followed by the mode ensemble operator. On the other hand, all of our proposed NARX models' accuracy consistently outperforms the exponential smoothing method and the ARIMA method.

Time Series, Forecasting, NARX Network, Stepwise Regression, Deterministic Seasonal Dummy, Ensemble Mean, Ensemble Median, Ensemble Mode

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Hermansah , Dedi Rosadi , Abdurakhman , Herni Utami , "Time Series Forecasting with Trend and Seasonal Patterns using NARX Network Ensembles," Mathematics and Statistics, Vol. 9, No. 4, pp. 511 - 520, 2021. DOI: 10.13189/ms.2021.090410.

(b). APA Format:
Hermansah , Dedi Rosadi , Abdurakhman , Herni Utami (2021). Time Series Forecasting with Trend and Seasonal Patterns using NARX Network Ensembles. Mathematics and Statistics, 9(4), 511 - 520. DOI: 10.13189/ms.2021.090410.