An interpretable prediction of FCM driven by small samples for energy analysis based on air quality prediction

Figures & data

Table 1. Comparisons of FCM.

Authors	Methods	Application
Yang et al. (2019)	Niching-based multi-modal multi-agent genetic algorithm of FCM	Apply to GRN reconstruction
Shen et al. (2020)	Multitasking multiobjective memetic algorithm of FCM	Apply to GRN reconstruction
Wang et al. (2021)	Many-task evolutionary algorithm of FCM	Solving multi task optimization problem
Bağdatli and Dokuz(2021)	FCM developed through the SigAFE algorithm	Autonomous lane change decision
Kazerooni et al. (2021)	FCM built based on Fuzzy AHP and Fuzzy TOPSIS	Improving construction labor productivity
Vidal et al. (2015)	FCM constructed based on questionnaire	Eco-innovative design tasks
Anezakis et al. (2016)	FCM built through correlation analysis	Air pollution prediction
Peng et al. (2020)	Multi-dimensional time-series gradient decent learning algorithm of PS-FCM	Air pollution prediction
Bakhtavar et al. (2020)	Multi-layer artificial neural network; hybrid nonlinear Hebbian and differential evolution learning algorithm of FCM	Prediction of vertical and horizontal distribution of dust emission caused by explosion
Pereira et al. (2020)	FCM obtained through expert evaluation; System dynamics	The impact of energy change on the sustainability of small and medium-sized enterprises
Tu et al. (2021)	Genetic algorithm of FCM	Exploring the interactive relationship between urban residents’ expenditure and energy consumption of production sectors
Salazar and Sanz-Calcedo (2019)	FCM obtained through expert evaluation	The impact of energy consumption, energy cost and emission on building efficiency

Table 2. Comparisons of methods on air quality prediction and resources analysis.

Methods	Air quality prediction and resources analysis	Data driven	Prediction performance
Numerical simulation	Air quality prediction	No	General
Emission inventory	Industrial resources analysis	No	No
Source apportionment of air pollutant	Energy resources analysis	No	No
Machine learning methods based on linear regression	One pollutant concentration prediction and energy resources analysis at a time without considering time delay between the both	Data driven	Low
Machine learning methods based on nonlinear mechanisms	Air quality prediction	Needing a large amount data	High
FCM	Various energy resources analysis based on multiple pollutants in air quality predicted	Data driven (small sample data)	High

Figure 1. The procedure of the research.

Figure 2. FCM structure of various types of energy resources affecting air quality.

Table

Algorithm 1. The FCM optimization algorithm

Input: initial FCM construction and the data processed. Output: FCM model optimized. 1: to initialize randomly the weight matrix $w$ of FCM. 2: repeat 3: for each weight ${\mathrm{w}}_{\mathrm{i}\mathrm{j}}$ 4: to compute the $f$ and $f\prime$ according to Eq.(1)(1) ${\hat{A}}_j^t=f\left(A_j^{t-1}+\sum _{i=1}^n{A}_i^{t-1}{w}_{ij}\right)$(1) and Eq.(7)(7) $where,f{ }^{\mathrm{\prime }}=\frac{c{e}^{-c\cdot ne{t}_{ij}^{t-1}}}{1+e^{-c\cdot ne{t}_{ij}^{t-1}}},andne{t}_{ij}^{t-1}=A_j^{t-1}+\sum _{i=1}^n{A}_i^{t-1}{w}_{ij}$(7) 5: to update ${\mathrm{w}}_{\mathrm{i}\mathrm{j}}$ according to Eq.(7) and Eq.(8)(8) $w_{ij}=w_{ij}+\alpha \mathrm{\nabla }{w}_{ij}$(8) 6: end for 7: until convergence shown as Eq.(5(5) ${\mathrm{A}}_{\mathrm{j}}^{\mathrm{t}}=\frac{\left({\mathrm{X}}_{\mathrm{j}}^{\mathrm{t}}-\mu \right)}{\sigma }$(5) ).

Table

Algorithm 2. The FCM prediction algorithm

Input: FCM model optimized and original data of various energy consumptions and concentration of air pollutants. Output: the prediction concentration of air pollutants and the prediction error. 1: to preprocess the data according to Eq.(5)(5) ${\mathrm{A}}_{\mathrm{j}}^{\mathrm{t}}=\frac{\left({\mathrm{X}}_{\mathrm{j}}^{\mathrm{t}}-\mu \right)}{\sigma }$(5) 2: to infer the air quality ${\hat{A}}_j^{t+1}$ at $t+1$ time according to the FCM model and data processed at $t$ time according to Eq.(1)(1) ${\hat{A}}_j^t=f\left(A_j^{t-1}+\sum _{i=1}^n{A}_i^{t-1}{w}_{ij}\right)$(1) . 3: to transform the state ${\hat{A}}_j^{t+1}$ predicted to original data ${\hat{X}}_j^{t+1}$ according to Eq.(9)(9) ${\hat{X}}_j^t={\hat{A}}_j^t\sigma +\mu$(9) . 4: to evaluate the error according to the value predicted and actual value according to Eq.(10)(10) $RMS{E}_j=\sqrt{\frac{1}{T}\sum _{t=1}^T{\left(X_j^t-{\hat{X}}_j^t\right)}^2}$(10) . 5: to add the ${\hat{X}}_j^{t+1}$ to the dataset for the prediction at next time. 6: to call Algorithm 1 for weights optimization of FCM model 7: t = t + 1 and go to step 2.

Table 3. Data statistics.

Variable	Unit	City	Mean	Std. Dev.	Min	Max
Coal	ten thousand ton	Beijing	2328.505	751.905	490.460	3068.970
		Tianjin	4130.890	816.458	2635.000	5298.000
		Hebei	24,900.972	5805.270	13,739.000	31,663.270
Coke	ten thousand ton	Beijing	233.121	189.845	0.180	455.730
		Tianjin	626.035	299.013	128.770	955.480
		Hebei	6211.254	2223.654	1808.020	8402.280
Crude oil	ten thousand ton	Beijing	905.125	157.355	700.500	1162.930
		Tianjin	1189.065	422.235	675.580	1759.150
		Hebei	1287.877	311.662	697.590	1761.930
Gasoline	ten thousand ton	Beijing	308.456	133.262	92.700	489.850
		Tianjin	181.476	63.866	94.760	274.490
		Hebei	288.797	114.602	147.410	494.860
Kerosene	ten thousand ton	Beijing	326.672	174.913	110.380	644.000
		Tianjin	34.765	27.435	11.420	101.500
		Hebei	10.131	8.440	2.710	29.460
Diesel oil	ten thousand ton	Beijing	168.387	54.566	70.690	241.120
		Tianjin	290.526	67.665	183.420	378.170
		Hebei	579.744	235.920	169.520	843.590
Fuel oil	ten thousand ton	Beijing	50.371	33.476	2.810	114.400
		Tianjin	97.745	28.573	40.680	149.960
		Hebei	49.649	12.089	24.530	65.820
Natural gas	ten thousand ton	Beijing	67.394	50.716	7.580	164.560
		Tianjin	28.606	24.613	6.480	83.310
		Hebei	34.111	28.376	2.810	96.700
Power	ten thousand ton	Beijing	695.460	231.395	344.130	1066.890
		Tianjin	565.418	204.437	247.940	807.930
		Hebei	2390.429	869.949	965.830	3441.740
SO2	Mg /m^3	Beijing	0.0414	0.022	0.008	0.080
		Tianjin	0.053	0.018	0.016	0.077
		Hebei	0.061	0.026	0.027	0.128
NO2	Mg /m^3	Beijing	0.061	0.010	0.046	0.077
		Tianjin	0.047	0.005	0.038	0.054
		Hebei	0.037	0.009	0.028	0.051
PM2.5	Mg /m^3	Beijing	0.132	0.028	0.084	0.180
		Tianjin	0.115	0.023	0.088	0.167
		Hebei	0.111	0.033	0.076	0.190

Table 4. Air quality prediction errors (RMSE) of Beijing-Tianjin-Hebei regions based on FCM with different transformation function.

Function	Beijing			Tianjin			Hebei
	SO2	NO2	PM2.5	SO2	NO2	PM2.5	SO2	NO2	PM2.5
tanh	0.0083	0.0052	0.0119	0.0157	0.0063	0.0109	0.0136	0.0033	0.0143
sigmoid	0.0266	0.0151	0.0484	0.0249	0.0048	0.0156	0.0343	0.0028	0.0198
relu	0.0264	0.0106	0.0336	0.0259	0.0080	0.0239	0.0242	0.0070	0.0447

Figure 3. The predicted value and actual value of air pollutants in Beijing-Tianjin-Hebei.

Table 5. Air quality prediction results of Beijing-Tianjin-Hebei regions based on different methods.

Method	Beijing			Tianjin			Hebei
	SO2	NO2	PM2.5	SO2	NO2	PM2.5	SO2	NO2	PM2.5
FCM (tanh)	0.0083	0.0052	0.0119	0.0157	0.0063	0.0109	0.0136	0.0033	0.0143
SVR	0.0130	0.0022	0.0181	0.0174	0.0089	0.0300	0.0296	0.0119	0.0440
LR	0.0102	0.0109	0.0143	0.0137	0.0180	0.1898	0.1281	0.0189	0.1040
CNN	0.0440	0.0134	0.0525	0.0460	0.0229	0.0750	0.0181	0.0388	0.1215
LSTM	0.0376	0.0115	0.0486	0.0473	0.0200	0.0654	0.0327	0.0341	0.1001
PCA-based forecasting	0.1540	0.2105	0.4650	0.0232	0.0150	0.1718	0.1985	0.0470	0.3840

Figure 4. Energy consumption trend of Beijing-Tianjin-Hebei.

Table 6. Affecting of energy resources on air quality in Beijing.

Relationship	SO2	NO2	PM2.5
Coal	−0.1015	−0.9801	0.4271
Coke	2.0288	1.8573	0.3234
Crude oil	−0.3672	0.3095	−1.6490
Gasoline	−2.3639	−0.4891	0.1403
Kerosene	0.7299	−0.6949	−0.4980
Diesel oil	−1.3990	−0.1804	−0.4278
Fuel oil	1.3149	1.2563	−0.1509
Natural gas	0.4038	1.9107	−2.8908
Power	−0.3442	0.0272	−0.7345

Table 7. Affecting of energy resources on air quality in Tianjin.

Relationship	SO2	NO2	PM2.5
Coal	0.2138	1.4858	2.0441
Coke	−1.7264	−0.1586	0.4451
Crude oil	0.1268	−0.8739	0.3536
Gasoline	0.7843	−0.1038	−0.2237
Kerosene	−1.0672	−1.7426	−0.8915
Diesel oil	−0.2051	−0.8263	0.1695
Fuel oil	1.2286	−0.6212	−0.3761
Natural gas	1.6558	1.3784	−1.5573
Power	0.0020	0.2226	0.5149

Table 8. Affecting of energy resources on air quality in Hebei.

relationship	SO2	NO2	PM2.5
Coal	−0.8857	1.6932	0.5375
Coke	0.5888	−0.3793	1.6377
Crude oil	0.6495	−0.0795	0.5021
Gasoline	−0.8114	0.5947	−0.9524
Kerosene	0.0002	1.3081	0.6552
Diesel oil	−0.2474	0.4595	−0.1052
Fuel oil	0.3098	0.3526	−0.3648
Natural gas	−0.7473	−0.3773	−0.0086
Power	1.2774	−0.2485	−0.8444

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Data Availability Statement

Please refer to the National Bureau of statistics (https://data.stats.gov.cn/) for energy production and consumptions data; Beijing Municipal Ecology and Environment Bureau (sthjj.beijing.gov.cn/) for air pollution.