Multistep-Ahead Air Passengers Traffic Prediction with Hybrid ARIMA-SVMs Models

Table 2

Prediction accuracy measures for hold-out sample.

Data preprocess	Strategy	Prediction horizon (ℎ)						Average 1−ℎ			Average rank
		1	2	4	6	8	12	18	24	1–24
Non-deseasonalization-detrending		MAPE
	I-ARIMA	22.235	24.580	24.127	25.328	22.485	26.369	28.059	31.254	25.555	5.444
	I-SVM	7.079	10.549	13.013	13.580	11.768	10.045	18.510	10.506	11.881	3.889
	D-ARIMA	26.025	25.895	23.745	24.765	25.014	25.956	27.654	32.547	26.450	5.556
	D-SVM	7.096	7.521	𝟕.𝟕𝟓𝟒^∗	8.353	𝟕.𝟏𝟒𝟎^∗	𝟕.𝟏𝟗𝟏^∗	9.553	9.830	𝟖.𝟎𝟓𝟓^∗	2.111
	I-ARIMA-SVM	𝟔.𝟐𝟗𝟏^∗	7.895	8.025	8.281	8.781	8.943	9.295	9.728	8.405	2.444
	D-ARIMA-SVM	6.341	𝟕.𝟐𝟏𝟑^∗	7.984	𝟖.𝟎𝟔𝟏^∗	8.632	8.259	𝟖.𝟖𝟗𝟒^∗	𝟗.𝟔𝟐𝟏^∗	8.126	𝟏.𝟓𝟓𝟔^∗
		SMAPE
	I-ARIMA	22.204	24.319	23.547	24.412	21.541	25.849	26.462	30.146	24.810	5.444
	I-SVM	7.075	10.287	12.384	12.509	11.284	9.623	16.158	10.009	11.166	3.889
	D-ARIMA	25.884	24.998	23.017	24.251	24.932	25.687	26.199	31.968	25.867	5.556
	D-SVM	7.089	7.452	𝟕.𝟓𝟖𝟎^∗	8.084	𝟔.𝟖𝟗𝟓^∗	𝟔.𝟗𝟗𝟕^∗	8.853	𝟗.𝟏𝟔𝟔^∗	𝟕.𝟕𝟔𝟓^∗	1.889
	I-ARIMA-SVM	𝟔.𝟒𝟓𝟕^∗	7.644	7.995	𝟕.𝟓𝟒𝟑^∗	7.687	8.787	8.783	9.964	8.108	2.444
	D-ARIMA-SVM	6.549	𝟔.𝟗𝟓𝟔^∗	7.885	7.983	7.564	7.784	𝟖.𝟓𝟗𝟓^∗	9.589	7.863	𝟏.𝟕𝟕𝟖^∗
		MASE
	I-ARIMA	10.024	11.581	10.992	12.067	11.125	12.694	14.553	16.005	12.380	5.333
	I-SVM	2.718	4.259	4.079	4.193	3.197	𝟐.𝟐𝟔𝟕^∗	3.847	𝟑.𝟐𝟖𝟐^∗	3.480	3.111
	D-ARIMA	12.099	11.867	11.261	11.794	11.803	12.043	13.216	17.258	12.668	5.667
	D-SVM	2.731	2.728	2.607	𝟐.𝟔𝟎𝟓^∗	𝟏.𝟗𝟎𝟔^∗	2.759	2.902	4.964	2.900	2.222
	I-ARIMA-SVM	𝟏.𝟗𝟖𝟐^∗	𝟏.𝟗𝟔𝟖^∗	3.058	2.953	2.051	3.352	3.872	5.942	3.147	2.778
	D-ARIMA-SVM	2.058	2.181	𝟐.𝟑𝟗𝟑^∗	3.186	2.011	2.421	𝟐.𝟔𝟒𝟑^∗	5.027	𝟐.𝟕𝟒𝟎^∗	𝟏.𝟖𝟖𝟗^∗
deseasonalization-detrending		MAPE
	I-ARIMA	20.278	22.569	22.127	23.315	20.483	24.434	26.146	29.348	23.587	5.444
	I-SVM	5.150	8.566	11.041	11.595	9.793	8.138	16.625	8.628	9.942	3.889
	D-ARIMA	24.068	23.884	21.745	22.752	23.012	24.021	25.741	30.641	24.483	5.556
	D-SVM	5.276	5.647	𝟓.𝟖𝟗𝟏^∗	6.477	𝟓.𝟐𝟕𝟓^∗	𝟓.𝟑𝟗𝟑^∗	7.777	8.061	𝟔.𝟐𝟐𝟓^∗	2.111
	I-ARIMA-SVM	𝟒.𝟒𝟕𝟏^∗	6.021	6.162	6.405	6.916	7.145	7.519	7.959	6.575	2.444
	D-ARIMA-SVM	4.521	𝟓.𝟑𝟑𝟗^∗	6.121	𝟔.𝟏𝟖𝟓^∗	6.767	6.461	𝟕.𝟏𝟏𝟖^∗	𝟕.𝟖𝟓𝟐^∗	6.295	𝟏.𝟓𝟓𝟔^∗
		SMAPE
	I-ARIMA	20.136	22.197	21.436	22.289	19.428	23.803	24.438	28.129	22.732	5.444
	I-SVM	5.242	8.400	10.509	10.621	9.406	7.813	14.370	8.228	9.324	4.000
	D-ARIMA	23.836	22.896	20.926	22.147	22.839	23.661	24.195	29.971	23.809	5.556
	D-SVM	5.233	5.542	𝟓.𝟔𝟖𝟏^∗	6.173	𝟒.𝟗𝟗𝟒^∗	𝟓.𝟏𝟔𝟑^∗	7.042	𝟕.𝟑𝟔𝟐^∗	𝟓.𝟖𝟗𝟗^∗	𝟏.𝟕𝟕𝟖^∗
	I-ARIMA-SVM	𝟒.𝟔𝟎𝟏^∗	5.734	6.096	𝟓.𝟔𝟑𝟐^∗	5.786	6.953	𝟔.𝟗𝟕𝟐^∗	8.160	6.242	2.444
	D-ARIMA-SVM	4.693	𝟓.𝟎𝟒𝟔^∗	5.986	6.072	5.663	5.950	6.784	7.785	5.997	𝟏.𝟕𝟕𝟖^∗
		MASE
	I-ARIMA	8.533	10.037	9.459	10.521	9.589	11.226	13.107	14.566	10.880	5.333
	I-SVM	1.747	3.235	3.066	3.167	2.181	𝟏.𝟑𝟏𝟗^∗	2.921	𝟐.𝟑𝟔𝟑^∗	2.500	3.333
	D-ARIMA	11.056	10.370	9.775	10.295	10.314	10.622	11.817	15.866	11.264	5.667
	D-SVM	1.657	1.600	1.490	𝟏.𝟒𝟕𝟓^∗	𝟎.𝟕𝟖𝟕^∗	1.707	1.872	3.941	1.816	2.111
	I-ARIMA-SVM	𝟎.𝟗𝟎𝟖^∗	𝟎.𝟖𝟒𝟎^∗	1.941	1.823	0.932	2.300	2.842	4.919	2.063	2.667
	D-ARIMA-SVM	0.984	1.053	𝟏.𝟐𝟕𝟔^∗	2.056	0.892	1.369	𝟏.𝟔𝟏𝟑^∗	4.004	𝟏.𝟔𝟓𝟔^∗	𝟏.𝟖𝟖𝟗^∗

Note: I-ARIMA means ARIMA model which employs iterated strategy; D-ARIMA means ARIMA model which employs direct strategy, the same as I-SVM, D-SVM, I-ARIMA-SVM, and D-ARIMA-SVM. For each column of table, the entry with the smallest value is set in boldface and marked with an asterisk, and the entry with second smallest value is set in boldface type.