Precision in Insurance Forecasting: Enhancing Potential with Ensemble and Combination Models based on the Adaptive Neuro-Fuzzy Inference System in the Egyptian Insurance Industry

Figures & data

Figure 1. The retention ratio of the Egyptian insurance market from 1989 to 2021.

Table 1. Descriptive statistics of the premium retention ratio of Egyptian insurance market.

Descriptive Statistics
N*	33
Minimum	37.7%
Maximum	59.8%
Mean	47
Std. Deviation	6.98

*The number of years from 1989 to 2021.

Figure 2. Block diagram of the study.

Figure 3. ANFIS structure for the sugeno fuzzy model.

Figure 4. The structure of proposed combination models.

Figure 5. The structure of proposed ensemble models.

Table 2. The optimal parameters of statistical time series models.

Model	Parameters	Optimal values
ARIMA	AR (p) I (d) MA (q) The innovation distribution function	p = 2 d = 1 q = 1 Gaussian Distribution
ES	ALPHA (α)	α= 0.9999

Figure 6. The plot of (ACF & PACF) of the market time series.

Figure 7. The plot of the ARIMA (2,1,1) model fit.

Table 3. The goodness of fit measure of ARIMA models.

	The measure
Model	AIC	BIC	Parameters significance
ARIMA (1,0,1)	−121.07	−115.89	AR MA	Significant Not Significant
ARIMA (1,1,1)	−121.02	−115.99	AR MA	Not Significant Not Significant
ARIMA (1,1,2)	−126.20	−119.91	AR MA	Significant Not significant
ARIMA (2,1,1)	−130.89	−124.80	AR MA	Significant Significant
ARIMA (2,1,2)	−129.30	−123.06	AR MA	Significant Not significant

Figure 8. The plots of residual normality of ARIMA (2,1,1) model.

Table 4. The parameters of the ARIMA (2,1,1) model.

Parameters	Value	Standard Error	T Statistic	P-Value
Constant	0.00284	0.0011	2.5808	0.0098
AR {1}	1.07050	0.1731	6.1841	0.0000
AR {2}	−0.5679	0.1476	−3.8471	0.0001
MA {1}	–1	0.1811	–5.5199	0.0000
Variance	0.00038	0.0001	2.6323	0.0084

Table 5. The accuracy measure of the best ARIMA and ES models.

	MAE		RMSE		MAPE
Model	Train	Test	Train	Test	Train	Test	R^2
ARIMA (2,1,1)	0.0144	0.0246	0.0194	0.0307	3.222	4.219	0.885
ES (α= 0.9999)	0.0203	0.0202	0.0264	0.0232	4.383	3.584	0.795

Figure 9. The plot of the ES model fit.

Table 6. The parameters of the best ANFIS model.

	M.F type		M.F number
Model	Input	Output		ANFIS Info
ANFIS (5)	Gaussian	Linear	8	Number of Nodes	36
				Linear Parameter	8
				Non-linear Parameter	32
				Total Parameter	40
				Fuzzy Rules	8

Figure 10. Trained membership functions and the structure of ANFIS model.

Table 7. The accuracy measure of ANFIS model.

	M.F. type			Train FIS		MAE		RMSE		MAPE
Model	Input	Output	No. MF	Optimal. Method	Epoch	Train	Test	Train	Test	Train	Test	R^2
1	Trimf	linear	5	hybrid	500	0.0054	0.0159	0.0073	0.0201	1.249	2.815	0.978
2		constant	6		1000	0.0084	0.0120	0.0135	0.0164	1.822	2.185	0.979
3	Trapmf	linear	3	hybrid	1500	0.0126	0.0270	0.0193	0.0318	2.811	4.636	0.910
4		constant	4		2500	0.0101	0.0238	0.0169	0.0267	2.202	4.114	0.933
5	Gaussmf	linear	8	hybrid	1000	0.0022	0.0119	0.0029	0.0131	0.492	2.118	0.993
6		constant	8		3500	0.0038	0.0144	0.0056	0.0177	0.866	2.498	0.985
7	Gauss2mf	linear	3	hybrid	1500	0.0105	0.0274	0.0170	0.0325	2.218	4.713	0.924
8		constant	8		2000	0.0024	0.0116	0.0038	0.0163	0.538	2.035	0.988

Figure 11. The variation of error and the structure of ARIMA-ANFIS model for pattern (1).

Table 8. The accuracy measure of first pattern of combined model (ARIMA-ANFIS).

	M.F. type			Train FIS		MAE		RMSE		MAPE
Model	Input	Output	No. MF	Optimal. Method	Epoch	Train	Test	Train	Test	Train	Test
1	Trimf	Linear	5.5	hybrid	50	3.2E–5	0.0631	6.2E–5	0.0743	0.008	11.12
2		Constant	4.2		50	0.0170	0.0109	0.0224	0.0142	3.732	1.893
3	Trapmf	Linear	2.2	hybrid	50	0.0152	0.0494	0.0203	0.0564	3.420	8.609
4		Constant	3.2		50	0.0153	0.0167	0.0193	0.0195	3.380	2.879
5	Gaussmf	Linear	4.6	hybrid	50	3.5E–5	0.0355	7.4E–5	0.0447	0.008	6.321
6		Constant	9.2		50	0.0058	0.0110	8.7E–3	0.0152	1.213	1.917
7	Gauss2mf	Linear	6.4	hybrid	50	1.6E–5	0.0153	2.8E–5	0.0211	0.004	2.727
8		Constant	10.2		50	0.0089	0.0108	0.0119	0.0136	2.009	1.884

Figure 12. The variation of error and the structure of ES-ANFIS model for pattern (1).

Table 9. The accuracy measure of first pattern of combined model (ES-ANFIS).

	M.F. type			Train FIS		MAE		RMSE		MAPE
Model	Input	Output	No. MF	Optimal. Method	Epoch	Train	Test	Train	Test	Train	Test
1	Trimf	Linear	8.6	hybrid	50	7.9E–8	0.0563	1.6E–7	0.0680	1.9E–5	9.957
2		Constant	4.2		50	0.0181	0.0127	0.0229	0.0173	3.979	2.212
3	Trapmf	Linear	5.3	hybrid	50	2.5E–4	0.0481	4.6E–4	0.0562	0.059	8.423
4		Constant	3.3		50	0.0152	0.0122	0.0200	0.0176	3.405	2.090
5	Gaussmf	Linear	4.6	hybrid	50	2.6E–5	0.0526	5.3E–5	0.0659	0.006	9.279
6		Constant	9.2		50	0.0037	0.0124	5.5E–3	0.0171	0.819	2.165
7	Gauss2mf	Linear	3.3	hybrid	50	0.0043	0.0145	5.7E–3	0.0193	0.984	2.551
8		constant	3.3		50	0.0155	0.0096	0.0199	0.0148	3.470	1.635

Figure 13. The variation of error and the structure of ARIMA-ANFIS model for pattern (2).

Table 10. The accuracy measure of second pattern of combined model (ARIMA-ANFIS).

	M.F. type			Train FIS		MAE		RMSE		MAPE
Model	Input	Output	No. MF	Optimal. Method	Epoch	Train	Test	Train	Test	Train	Test
1	Trimf	Linear	5.3	hybrid	50	7.9E–4	0.0254	1.4E–3	0.0269	0.189	4.443
2		constant	2.2		50	0.0144	0.0170	0.0188	0.0234	3.195	2.908
3	Trapmf	Linear	4.7	hybrid	50	7.5E–4	0.0324	1.5E–3	0.0437	0.178	5.602
4		constant	2.3		50	0.0120	0.0201	0.0180	0.0264	2.684	3.440
5	Gaussmf	Linear	6.4	hybrid	50	2.1E–5	0.0105	3.5E–5	0.0154	0.005	1.831
6		constant	10.2		50	0.0027	0.0119	0.0040	0.0164	0.604	2.092
7	Gauss2mf	Linear	2.9	hybrid	50	0.0003	0.0224	6.5E–4	0.0263	0.071	3.896
8		constant	10.4		50	0.0011	0.1198	1.9E–3	0.1718	0.253	20.73

Figure 14. The variation of error and the structure of ES-ANFIS model for pattern (2).

Table 11. The accuracy measure of second pattern of combined model (ES-ANFIS).

	M.F. type			Train FIS		MAE		RMSE		MAPE
Model	Input	Output	No. MF	Optimal. Method	Epoch	Train	Test	Train	Test	Train	Test
1	Trimf	Linear	6.2	hybrid	50	4.9E–4	0.0238	7.0E–4	0.0283	0.115	4.087
2		Constant	6.6		50	1.8E–3	0.0114	3.3E–3	0.0163	0.455	2.001
3	Trapmf	Linear	2.2	hybrid	50	0.0136	0.0169	0.0178	0.0198	2.962	2.914
4		Constant	4.4		50	0.0134	0.0198	0.0189	0.0228	3.039	3.403
5	Gaussmf	Linear	8.2	hybrid	50	1.4E–4	0.0257	1.9E–4	0.0299	0.032	4.433
6		Constant	10.6		50	3.4E–6	0.0175	6.6E–6	0.0163	8.2E–4	2.055
7	Gauss2mf	Linear	2.2	hybrid	50	0.0118	0.0109	0.0166	0.0137	2.544	1.912
8		Constant	10.10		50	1.9E–6	0.0122	2.8E–6	0.0166	4.5E–4	2.119

Table 12. The accuracy measure of ensemble models.

		MAE		RMSE		MAPE
No	Model	Train	Test	Train	Test	Train	Test	R^2
1	ARIMA - ES - ANFIS – MLR	2.16E–3	0.0095	2.76E–3	0.0101	0.489	1.651	0.9956
2	ARIMA - ANFIS – MLR	2.16E–3	0.0131	2.77E–3	0.0140	0.488	2.300	0.9933
3	ES - ANFIS – MLR	2.15E–3	0.0572	2.87E–3	0.0586	0.487	10.00	0.9583

Figure 15. The comparison of all predictive models (a) single models (b) combined models (c) ensemble models.

Table 13. The comparison between all the models.

		MAE		RMSE		MAPE
Model		Train	Test	Train	Test	Train	Test	R^2	Order
Single models	ARIMA	0.0144	0.0246	0.0194	0.0307	3.222	4.219	0.885	9^th
	ES	0.0203	0.0202	0.0264	0.0232	4.383	3.584	0.795	8^th
	ANFIS	2.2E–3	0.0119	2.9E–3	0.0161	0.492	2.118	0.993	6^th
Combined models	ARIMA-ANFIS (pattern 1)	0.0089	0.0108	0.0119	0.0136	2.009	1.884	0.959	4^th
	ES-ANFIS (pattern 1)	0.0155	0.0096	0.0199	0.0148	3.470	1.665	0.881	2^nd
	ARIMA-ANFIS (pattern 2)	2.1E–5	0.0105	3.5E–5	0.0154	0.005	1.831	0.997	3^rd
	ES-ANFIS (pattern 2)	0.0118	0.0109	0.0166	0.0137	2.544	1.912	0.917	5^th
Ensemble models	ARIMA- ES - ANFIS - MLR	2.1E–3	0.0095	2.7E–3	0.0101	0.487	1.651	0.996	1^st
	ARIMA - ANFIS - MLR	2.16E–3	0.0131	2.77E–3	0.0140	0.488	2.300	0.993	7^th
	ES - ANFIS - MLR	2.15E–3	0.0572	2.87E–3	0.0586	0.487	10.00	0.958	10^th

Data Availability Statement

The data that support the findings of this study are publicly available on their website (https://fra.gov.eg).