Forecasting the Tuberculosis Incidence Using a Novel Ensemble Empirical Mode Decomposition-Based Data-Driven Hybrid Model in Tibet, China

Figures & data

Table 1 The Identified Nine Plausible SARIMA Methods and Their Information Criteria

Models	AIC	CAIC	BIC	Log-Likelihood
SARIMA(2,0,0)(2,1,0)12	409.22	409.74	423.16	−199.61
SARIMA(1,0,2)(2,1,0)12	410.01	410.75	426.73	−199.00
SARIMA(1,0,1)(2,1,0)12	411.83	412.36	425.77	−200.92
SARIMA(0,0,2)(2,1,0)12	440.42	440.94	454.35	−215.21
SARIMA(2,0,1)(2,1,0)12	410.70	411.44	427.42	−199.35
SARIMA(2,0,1)(0,1,1)12	415.85	416.37	429.78	−202.92
SARIMA(2,0,1)(0,1,2)12	416.17	416.91	432.90	−202.09
SARIMA(2,0,0)(1,1,1)12	415.06	415.59	429.00	−202.53
SARIMA(2,0,0)(1,1,0)12	432.80	433.14	443.95	−212.40

Abbreviations: SARIMA, seasonal autoregressive integrated moving average method; AIC, Akaike’s Information Criterion; BIC, Bayesian Information Criterion.

Table 2 Statistical Test of the Estimated Parameters for the Optimal SARIMA (2,0,0)(2,1,0)₁₂ Method

Variables	Estimates	Standard Error	t	p
AR1	0.519	0.089	5.831	<0.001
AR2	0.329	0.092	3.576	<0.001
SAR1	−0.736	0.086	−8.558	<0.001
SAR2	−0.526	0.089	−5.910	<0.001

Abbreviations: SARIMA, seasonal autoregressive integrated moving average method; AR1, autoregressive method at lag 1; AR2, autoregressive method at lag 2; SAR1, seasonal autoregressive method at lag 1; SAR2, seasonal autoregressive method at lag 2.

Figure 1 Estimated autocorrelogram and partial autocorrelogram of the residual sequence to forecast the TB epidemic patterns using (A) SARIMA model, (B) basic NARNN model, (C) traditional SARIMA-NARNN hybrid model, and (D) novel EEMD-SARIMA-NARNN hybrid model. As seen, the estimated autocorrelations and partial autocorrelations of the errors from the different models almost fell into the 95% confidence intervals (CIs) with few exceptions, such as the autocorrelations at lags 15 and 19 and partial autocorrelations at lags 15 in (A), and autocorrelations and partial autocorrelations at lag 11 in (D), these are also reasonable because higher-order autocorrelations and partial autocorrelations easily exceed the 95 CIs by chance.

Table 3 The Estimated Parameters of the Optimal NARNN Models for the Original Series and Decomposed Series Using the EEMD Method

Target Components	Hidden Units	Delays	MSE			R
			Training	Validation	Testing	Training	Validation	Testing	Overall
Original series	16	6	0.648	4.531	3.760	0.973	0.842	0.848	0.944
Residuals of SARIMA model	19	6	0.537	8.110	2.287	0.806	0.714	0.607	0.715
IMF1	17	5	0.124	0.885	1.806	0.967	0.799	0.827	0.911
IMF2	16	5	0.052	0.101	0.226	0.992	0.979	0.971	0.988
IMF3	15	5	0.001	0.001	0.002	1.000	1.000	1.000	1.000
IMF4	16	6	0.000	0.000	0.000	1.000	1.000	1.000	1.000
IMF5	14	4	0.000	0.000	0.000	1.000	1.000	1.000	1.000
IMF6	14	5	0.000	0.000	0.000	1.000	1.000	1.000	1.000

Abbreviations: SARIMA, seasonal autoregressive integrated moving average method; NARNN, nonlinear autoregressive artificial neural network.

Table 4 Statistical Test of the Estimated Parameters for the Optimal SARIMA (1,2,0)(1,0,1)₁₂ Method

Variables	Estimates	Standard Error	t	p
AR1	1.000	0.002	578.647	<0.001
SAR1	0.984	0.029	34.493	<0.001
SMA1	0.692	0.152	4.564	<0.001

Abbreviations: SARIMA, seasonal autoregressive integrated moving average method; AR1, autoregressive method at lag 1; SAR1, seasonal autoregressive method at lag 1; SMA1, seasonal moving average method at lag 1.

Figure 2 Intrinsic mode functions (IMFs) and residue components of the original TB incidence series using the EEMD.

Figure 3 Estimated autocorrelogram of the residual sequence to forecast the six intrinsic mode functions (IMFs) decomposed by the EEMD method using the SARIMA-NARNN hybrid model. (A) IMF1, (B) IMF2, (C) IMF3, (D) IMF4, (E) IMF5, and (F) IMF6. For the estimated autocorrelations of the residual series from the different IMFs, all of them are inside of the 95 CIs except for the one at lag zero, meaning that there is no correlated serial in the residuals of the best-fitting SARIMA-NARNN hybrid model. Thereby, there appears to be adequate and reasonable for forecasting upcoming epidemiological trends using these best-fitting models.

Table 5 Comparisons of the Fitted and Forecasted Accuracy Levels Among the Optimal Four Methods

Models	Simulated Part					Predicted Part
	MAD	MAPE	RMSE	MER	RMSPE	MAD	MAPE	RMSE	MER	RMSPE
SARIMA model	0.977	11.222	1.216	0.100	2.702	1.444	11.486	1.658	0.114	3.805
NARNN model	0.770	12.301	1.170	0.078	0.258	1.183	9.335	1.309	0.093	0.101
SARIMA-NARNN hybrid model	0.648	8.634	0.856	0.066	0.155	1.443	11.281	1.803	0.114	0.143
EEMD-SARIMA-NARNN hybrid model	0.438	6.814	0.669	0.045	0.141	0.965	7.958	1.133	0.076	0.096
Reduced percentage
D vs A	55.169	39.280	44.984	55.000	94.782	33.172	30.716	31.665	33.333	97.477
D vs B	43.117	44.606	42.821	42.308	45.349	18.428	14.751	13.445	18.280	4.950
D vs C	32.407	21.079	21.846	31.818	9.032	33.125	29.457	37.160	33.333	32.867

Notes: A refers to the SARIMA method, B takes for the NARNN method, C signifies the traditional SARIMA-NARNN mixture method, D is the novel EEMD-SARIMA-NARNN mixture method.

Abbreviations: SARIMA, seasonal autoregressive integrated moving average method; EEMD, ensemble empirical mode decomposition; NARNN, nonlinear autoregressive artificial neural network; MAD, mean absolute deviation; MAPE, mean absolute percentage error; RMSE, root-mean-square error; MER, mean error rate; RMSPE, root-mean-square percentage error.

Figure 4 Time series plot showing the comparative results between original observations and fitted and predicted values using the selected best-fitting (A) SARIMA model, (B) NARNN model, (C) traditional SARIMA-NARNN hybrid model, and (D) novel EEMD-SARIMA-NARNN hybrid model (The curve at the left of the vertical black dotted line represents the fitted values, whereas the curve at the right denotes the forecasted values). As shown, the produced curve from the EEMD-SARIMA-NARNN hybrid model is closer to the actual both in the fitted and predicted aspects compared with the curves from the other models.