Integration Learning of Neural Network Training with Swarm Intelligence and Meta-heuristic Algorithms for Spot Gold Price Forecast

Figures & data

Figure 1. The pseudo-code for the proposed IGACO algorithm.

Figure 2. Illustrative schema of the unit matrix.

Figure 3. The design of decoding convention for the matrix form.

Figure 4. Illustrative schema of two-point crossover between $C_t$ and $C_{t+1}$ through each pair of rows individually alternating their values with P_c.

Table 1. The parameters setting for several benchmark problems in the experiment

Description	Continuous test function
	Griewank	Sphere	Rosenbrock	Mackey-Glass time series	B2
Search domain	[−100, 100]	[−100, 100]	[−30, 30]	[0.4, 1.6]	[−100, 100]
The widths on RBFNuNet	[43000, 44200]	[0.1, 0.9]	[200, 700]	[0.1, 0.3]	[23000, 26000]

Table 2. Parameter values setting for the IGACO algorithm

Parameter	Description	Value
E	The maximum number of iterations	1000
$g_t$	The number of the RBFNuNet hidden neuron centers	[1, 50]
$d_i^t$	The width of RBFNuNet hidden neuron	[1100, 39000]
δ	The learning rate of the RBFNuNet	0.34
T	Population size	50
K	The number of ant	100
$\eta$	The pheromone exam	0.48
$\mathrm{\partial }$	The pheromone impressibility	0.72
$\phi$	The pheromone diminish rate	0.37
$H$	A constant	0.68
Pc	Crossover rate (2-point crossover)	[0.45, 0.65]
Pm	Mutation rate (2-point mutation)	[0.1, 0.18]

Table 3. Result comparison among relevant algorithms employed in this experiment

Benchmark problem Competitive algorithm	Griewank		Sphere		Rosenbrock		Mackey-Glass time series		B2
	Training set	Validation set	Training set	Validation set	Training set	Validation set	Training set	Validation set	Training set	Validation set
RBFNuNet (Chen, Wu, and Luk 1999)	27.64 ± 2.82	28.34 ± 3.68	2746.24 ± 334.61	2432.34 ± 320.35	12058.19 ± 1360.41	12969.45 ± 2623.37	703 ± 41.6	712 ± 53.6	5803.27 ± 413.66	5935.23 ± 690.28
GA (Holland 1992)	19.76 ± 16.31	54.43 ± 17.55	1272.21 ± 109.30	1507.23 ± 373.41	1674.13 ± 376.29	2591.39 ± 252.34	2.75E-3 ± 1.52E-4	2.53E-3 ± 1.95E-4	1131.06 ± 394.38	3874.13 ± 539.20
GA-based (Sarimveis et al. 2004)	17.23 ± 16.31	52.65 ± 16.23	1137.54 ± 104.71	1463.28 ± 314.86	1552.56 ± 335.47	2362.67 ± 236.33	2.67E-3 ± 1.45E-4	2.51E-3 ± 1.36E-4	1088.39 ± 324.72	3529.44 ± 505.26
ACO (Dorigo, Maniezzo, and Colorni 1996)	7.57 ± 13.49	51.34 ± 10.25	958.14 ± 11.39	1204.71 ± 221.25	1233.21 ± 146.43	1984.16 ± 360.30	2.70E-3 ± 1.59E-4	2.46E-3 ± 1.92E-4	1014.39 ± 134.27	2782.31 ± 451.63
ACO-based (Tabakhi et al., 2014)	7.26 ± 11.09	48.51 ± 10.24	913.86 ± 10.73	1239.09 ± 207.35	1181.23 ± 117.81	1897.11 ± 315.05	2.88E-3 ± 1.56E-4	2.41E-3 ± 1.79E-4	2147.82 ± 119.14	2502.03 ± 430.63
GA-ACO (Luan et al. 2019)	6.69 ± 3.53	15.28 ± 4.37	413.51 ± 40.67	361.02 ± 25.39	912.31 ± 20.27	626.54 ± 28.23	2.07E-3 ± 1.81E-4	1.95E-3 ± 1.46E-4	298.16 ± 72.81	356.77 ± 16.95
IGACO	5.15 ± 2.61	13.30 ± 2.43	362.04 ± 28.29	297.24 ± 19.72	705.14 ± 12.13	587.17 ± 21.36	1.69E-3 ± 1.41E-4	1.52E-3 ± 1.28E-4	251.48 ± 50.24	317.39 ± 15.52

Table 4. Contrast of the best learning expression among relevant algorithms in the experiment

Benchmark problem Competitive algorithm	Griewank	Sphere	Rosenbrock	Mackey-Glass time series	B2
	RMSE
RBFNuNet (Chen, Cowan, and Grant 1991)	3.421	7.288	5.392	4.173	7.193
GA (Holland 1992)	5.175E-1	42.831E-2	47.816E-5	35.794E-2	25.269E-2
GA-based (Sarimveis et al. 2004)	5.142E-1	39.377E-2	47.211E-5	33.528E-2	24.843E-2
ACO (Dorigo, Maniezzo, and Colorni 1996)	5.092E-1	61.395E-2	43.843E-5	38.271E-2	22.075E-2
ACO-based (Tabakhi et al., 2014)	5.027E-1	53.471E-2	40.349E-5	31.604E-2	19.845E-2
GA-ACO (Luan et al. 2019)	4.828E-1	3.205E-2	3.914E-5	4.383E-2	5.660E-2
IGACO	4.179E-1	2.078E-2	3.242E-5	2.737E-2	4.735E-2

Table 5. Comparison of the time consumed (in seconds) among relevant algorithms arriving at the preset RMSE threshold

Experiment (Benchmark problem)	Griewank	Sphere	Rosenbrock	Mackey-Glass time series	B2
RMSE threshold Competitive algorithm	5.175E-1	61.395E-2	47.81E-5	38.271E-2	25.26E-2
OLS (Chen, Cowan, and Grant 1991)	–	–	–	–	–
GA (Holland 1992)	15.265	4.231	3.793	5.098	663.513
GA-based (Sarimveis et al. 2004)	11.389	6.379	5.285	4.872	349.228
ACO (Dorigo, Maniezzo, and Colorni 1996)	13.749	7.263	3.346	8.271	526.381
ACO-based (Tabakhi et al., 2014)	10.598	11.450	5.813	6.095	276.313
GA-ACO (Luan et al. 2019)	2.489	2.817	1.205	2.337	2.601
IGACO	1.871	2.279	0.748	1.593	1.926

Note. “–” means that it failed to achieve the preset RMSE threshold

Table 6. The data period of the spot gold price forecast exercise

Spot gold price forecast	The observations: month/day/year (number of sampling)
	Learning set (90%)	Forecasting set (10%)
Data Period	Feb 1^st 2008~ Dec-23 2008 (228)	Dec-29 2008~ Feb 2^nd 2009 (24)

Table 7. Parameters setup for the IGACO algorithm in the spot gold price forecast exercise

Parameter	Description	Value
E	The maximum number of iterations	1000
$g_t$	The number of the RBFNuNet hidden neuron centers	[1, 50]
$d_i^t$	The width of RBFNuNet hidden neuron	[1000, 38000]
δ	The learning rate of the RBFNuNet	0.25
T	Population size	50
K	The number of ant	100
$\eta$	The pheromone exam	0.4
$\mathrm{\partial }$	The pheromone impressibility	0.9
$\phi$	The pheromone diminish rate	0.45
$H$	A constant	0.75
Pc	Crossover rate (two-point crossover)	[0.6, 0.7]
Pm	Mutation rate (two-point mutation)	[0.15, 0.25]

Table 8. The forecasting errors comparison for relevant algorithms used in the spot gold price forecast exercise

Competitive algorithm	Error
	RMSE	MAE	MAPE (%)
RBFNuNet (Chen, Wu, and Luk 1999)	8.217	5.729	2.134
GA (Holland 1992)	66.406E-4	68.053E-4	1.957
GA-based (Sarimveis et al. 2004)	53.037E-4	47.738E-4	1.423
ACO (Dorigo, Maniezzo, and Colorni 1996)	41.481E-4	22.391E-4	1.215
ACO-based (Tabakhi et al., 2014)	29.120E-4	18.925E-4	1.064
GA-ACO (Luan et al. 2019)	7.471E-4	11.301E-4	14.186E-2
IGACO	6.236E-4	8.572E-4	11.874E-2
ARIMA (2, 1, 2) model	16.172	11.173	1.328

Table 9. The statistical results for T-test among relevant algorithms

Competitive algorithm	Paired differences
	Mean	Standard deviation	T	Significant (two-tailed)
RBFNuNet (Chen, Wu, and Luk 1999)	−625.1048	641.0314	−3.163	0.017*
GA (Holland 1992)	−306.3764	622.7529	−1.381	0.143*
GA-based (Sarimveis et al. 2004)	−284.0823	551.5489	−1.026	0.172*
ACO (Dorigo, Maniezzo, and Colorni 1996)	−517.2013	633.3081	−2.736	0.034*
ACO-based (Tabakhi et al., 2014)	−369.8834	548.7602	−1.475	0.027*
GA-ACO (Luan et al. 2019)	−204.0923	526.3407	−0.928	0.019*
IGACO	−125.3682	434.1382	−0.647	0.773
ARIMA (2, 1, 2) model	−226.1536	584.4939	−1.349	0.235

*: 5% significance level

Note: Mean is equal to arithmetic average.

Figure 5. The forecasting results comparison of the proposed IGACO algorithm and Box-Jenkins model for the spot gold price forecast exercise.

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