# Evolving Hybrid Cascade Neural Network Genetic Algorithm Space–Time Forecasting

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## Abstract

**:**

## 1. Introduction

_{10}and PM

_{2.5}are some classes of these particulate pollutants [1,2,3,4]. Let us consider a hair: the mean diameter of a single human hair is approximately 70 micrometers. This is roughly 28 times the diameter of PM

_{2.5}. The diameter of particulate matter in PM

_{10}is 10 micrometers or below. Similarly, PM

_{2.5}is normally particles of diameter 2.5 micrometers or below. Both PM

_{10}and PM

_{2.5}are inhalable. We can thus imagine how tiny 2.5 and 10 micrometers are.

_{2}) and nitrogen oxides, originating in PM (NOx) [5,6,7]. All this can be found as a product of building materials, farms, explosions, power stations, industry, and vehicles. PM is seriously damaging, as described above, as it may be opaque and small enough to be inhaled into the lungs or even into the circulation. Therefore, PM contamination affects the cardiovascular system and can cause fatal illnesses such as cardiovascular diseases, erratic heartbeat, and worsening asthma [8,9,10].

## 2. Methods

#### 2.1. Cascade Neural Network

#### 2.2. Genetic Algorithm

Algorithm 1. Scheme of the GA | |

1: | INITIALIZE population and EVALUATE |

2: | while termination condition is not satisfied do |

3: | SELECT parents |

4: | CROSSOVER pairs of parents |

5: | MUTATE the resulting offspring |

6: | EVALUATE new candidates |

7: | REPLACE individuals for the next generation |

8: | end while |

#### 2.3. Cascade Neural Network Genetic Algorithm

Algorithm 2. Function Cascade Neural Network | |

1: | input${n}_{h},m,o$ |

2: | setk = 0 |

3: | calculate Cascade Weighted |

4: | k = 0;for i = 1:nhfor j = 1:mk = k + 1; Wi1(i,j) = W(k); endend |

5: | calculate weighted input and output |

6: | fori = 1:ofor j = 1:mk = k + 1; Wi2(i,j) = W(k); endend |

7: | calculate weighted Bias Input |

8: | fori = 1:nhk = k + 1; Wbi(i,1) = W(k); end |

9: | calculate weighted output |

10: | fori = 1:ofor j = 1:nhk = k + 1; Wo(i,j) = W(k); endend |

11: | calculate weighted Bias Output |

12: | fori = 1:ok = k + 1; Wbo(i,1) = W(k); end |

## 3. Simulation and Results

#### 3.1. Construction of VAR-Cascade

#### 3.2. Study Area

_{2.5}, atmospheric PM

_{10}, and sulfur dioxide (SO

_{2}) levels. Furthermore, the locations of these areas were as established by the Taiwan Environmental Protection Administration Executive Yuan. Table 1 shows statistical summaries of the amounts of air pollution at the four studied locations. The findings typically demonstrate that Taichung has higher concentrations of PM

_{10}, PM

_{2.5}, and NO

_{X}, but in Kaohsiung, SO

_{2}is the greatest pollutant. Figure 1 shows an overview of the genetic algorithm’s training and evaluation phases. Because each type of air pollutant has a different distribution, we trained the same models for each dataset using the same model architecture.

#### 3.3. Air Pollution Forecasting Using VAR-Cascade-GA

_{1}), Taipei (Y

_{2}), Hsinchu (Y

_{3}), and Kaohsiung (Y

_{4}) in Taiwan.

_{2.5}is represented in Figure 3, PM

_{10}is represented in Figure 4, NO

_{X}is represented in Figure 5, and SO

_{2}is represented in Figure 6. In this context, the cascade neural network genetic algorithm model can be used to study nonlinear and nonstationary data on air pollution. The metrics used to evaluate the test set’s result were the root-mean-squared error (RMSE), mean absolute error (MAE), and symmetry mean absolute percentage (sMAPE) between the actual air pollution values and the predicted values. These are metrics that are commonly used in regression problems like our air pollution prediction. If all the metric values are smaller, then the model’s performance is better [25].

#### 3.4. Does the Activation Function Provide High Accuracy and Speed Up the Time Lapse?

_{10}was logsig, that for PM

_{2.5}was SoftMax, that for NO

_{x}was radbas, and that for SO

_{2}was tribas. The SoftMax activation function provided a shorter time lapse than other activation functions.

_{x}with the radial basis activation function, Equation (22) for PM

_{2.5}with the SoftMax activation function, Equation (20) for PM

_{10}with the logsig activation function, and Equation (23) for SO

_{2}with the tribas activation function. We provide the results of forecasting in Figure 7 for the next 30 steps. The results show Taichung constantly leading with the highest pollutant score compared to other cities in Taiwan.

_{x}using the radial basis activation function:

_{2.5}using the SoftMax activation function:

_{10}using the logsig activation function:

_{10}using the tribas activation function:

## 4. Conclusions

_{2}and overall suspended particulate matter was very effective when ever-increasing cars threatened city atmospheres with NOx and particulates. A space–time air pollution analysis over the last 10 years using the monitoring data clearly showed that with urban planning and countermeasure policies, air quality has improved. The analysis should be used to make future policy decisions. Air pollution temporal features were examined herein for Taiwan. The pattern from pollutants to particulates differs in air quality for each location. In a nutshell, the PM, SO

_{2}, and NO

_{x}levels have drastically increased. Future research should examine using VAR-SARIMA, VAR-ARCH, and other traditional time series as input.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

_{2.5}: fine particulate matter with a diameter of 2.5 μm, PM

_{10}: fine particulate matter 10 micrometers or less in diameter, SO

_{2}: sulfur dioxide, NO

_{x}: nitrogen dioxide, BP: backpropagation.

## References

- Querol, X.; Alastuey, A.; Ruiz, C.R.; Artiñano, B.; Hansson, H.C.; Harrison, R.M.; Buringh, E.; Ten Brink, H.M.; Lutz, M.; Bruckmann, P.; et al. Speciation and origin of PM10 and PM2.5 in selected European cities. Atmos. Environ.
**2004**, 38, 6547–6555. [Google Scholar] [CrossRef] - Fan, J.; Wu, L.; Ma, X.; Zhou, H.; Zhang, F. Hybrid support vector machines with heuristic algorithms for prediction of daily diffuse solar radiation in air-polluted regions. Renew. Energy
**2020**, 145, 2034–2045. [Google Scholar] [CrossRef] - Masseran, N.; Safari, M.A.M. Intensity–duration–frequency approach for risk assessment of air pollution events. J. Environ. Manag.
**2020**, 264, 110429. [Google Scholar] [CrossRef] [PubMed] - Masseran, N.; Safari, M.A.M. Modeling the transition behaviors of PM 10 pollution index. Environ. Monit. Assess.
**2020**, 192, 441. [Google Scholar] [CrossRef] [PubMed] - De Vito, S.; Piga, M.; Martinotto, L.; Di Francia, G. CO, NO
_{2}and NOx urban pollution monitoring with on-field calibrated electronic nose by automatic bayesian regularization. Sens. Actuators B Chem.**2009**, 143, 182–191. [Google Scholar] [CrossRef] - Winarso, K.; Yasin, H. Modeling of air pollutants SO
_{2}elements using geographically weighted regression (GWR), geographically temporal weighted regression (GTWR) and mixed geographically temporalweighted regression (MGTWR). ARPN J. Eng. Appl. Sci.**2016**, 11, 8080–8084. [Google Scholar] - Zhang, J.J.; Wei, Y.; Fang, Z. Ozone pollution: A major health hazard worldwide. Front. Immunol.
**2019**, 10, 2518. [Google Scholar] [CrossRef][Green Version] - Bernstein, J.A.; Alexis, N.; Barnes, C.; Bernstein, I.L.; Bernstein, J.A.; Nel, A.; Peden, D.; Diaz-Sanchez, D.; Tarlo, S.M.; Williams, P.B. Health effects of air pollution. J. Allergy Clin. Immunol.
**2004**, 114, 1116–1123. [Google Scholar] [CrossRef] - Xing, Y.F.; Xu, Y.H.; Shi, M.H.; Lian, Y.X. The impact of PM2.5 on the human respiratory system. J. Thorac. Dis.
**2016**, 8, E69–E74. [Google Scholar] - Rossati, A. Global warming and its health impact. Int. J. Occup. Environ. Med.
**2017**, 8, 7–20. [Google Scholar] [CrossRef][Green Version] - Suhartono, S.; Subanar, S. Development of model building procedures in wavelet neural networks for forecasting non-stationary time series. Eur. J. Sci. Res.
**2009**, 34, 416–427. [Google Scholar] - Suhermi, N.; Suhartono; Prastyo, D.D.; Ali, B. Roll motion prediction using a hybrid deep learning and ARIMA model. Procedia Comput. Sci.
**2018**, 144, 251–258. [Google Scholar] [CrossRef] - McCulloch, W.S.; Pitts, W. A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophys.
**1943**, 5, 115–133. [Google Scholar] [CrossRef] - Chen, R.C.; Dewi, C.; Huang, S.W.; Caraka, R.E. Selecting critical features for data classification based on machine learning methods. J. Big Data
**2020**, 7, 52. [Google Scholar] [CrossRef] - Caraka, R.E.; Lee, Y.; Chen, R.C.; Toharudin, T. Using Hierarchical Likelihood towards Support Vector Machine: Theory and Its Application. IEEE Access
**2020**, 8, 194795–194807. [Google Scholar] [CrossRef] - Mueller, J.-A.; Lemke, F. Self-Organising Data Mining: An Intelligent Approach to Extract Knowledge from Data. 1999. Available online: https://www.knowledgeminer.eu/pdf/sodm.pdf (accessed on 6 May 2021).
- De Gooijer, J.G.; Hyndman, R.J. 25 years of time series forecasting. Int. J. Forecast.
**2006**, 22, 443–473. [Google Scholar] [CrossRef][Green Version] - Kaimian, H.; Li, Q.; Wu, C.; Qi, Y.; Mo, Y.; Chen, G.; Zhang, X.; Sachdeva, S. Evaluation of different machine learning approaches to forecasting PM2.5 mass concentrations. Aerosol Air Qual. Res.
**2019**, 19, 1400–1410. [Google Scholar] [CrossRef][Green Version] - Guo, Y.; Liu, Y.; Oerlemans, A.; Lao, S.; Wu, S.; Lew, M.S. Deep learning for visual understanding: A review. Neurocomputing
**2016**, 187, 27–48. [Google Scholar] [CrossRef] - Szandała, T. Review and comparison of commonly used activation functions for deep neural networks. arXiv
**2020**. Available online: https://arxiv.org/abs/2010.09458 (accessed on 6 May 2021). - Sony, S.; Dunphy, K.; Sadhu, A.; Capretz, M. A systematic review of convolutional neural network-based structural condition assessment techniques. Eng. Struct.
**2021**, 226, 111347. [Google Scholar] [CrossRef] - Caraka, R.E.; Chen, R.C.; Yasin, H.; Pardamean, B.; Toharudin, T.; Wu, S.H. Prediction of Status Particulate Matter 2.5 using State Markov Chain Stochastic Process and HYBRID VAR-NN-PSO. IEEE Access
**2019**, 7, 161654–161665. [Google Scholar] [CrossRef] - Kuster, C.; Rezgui, Y.; Mourshed, M. Electrical load forecasting models: A critical systematic review. Sustain. Cities Soc.
**2017**, 35, 257–270. [Google Scholar] [CrossRef] - Cios, K.J.; Pedrycz, W.; Swiniarski, R.W.; Kurgan, L.A. Data Mining: A Knowledge Discovery Approach; Springer: Boston, MA, USA, 2007; ISBN 9780387333335. [Google Scholar]
- Makridakis, S.; Spiliotis, E.; Assimakopoulos, V. The M4 Competition: 100,000 time series and 61 forecasting methods. Int. J. Forecast.
**2020**, 36, 54–74. [Google Scholar] [CrossRef] - Makridakis, S.G.; Wheelwright, S.C.; Hyndman, R.J. Forecasting: Methods and Applications. J. Forecast.
**1998**, 1–656. [Google Scholar] [CrossRef] - Wong, K.W.; Wong, P.M.; Gedeon, T.D.; Fung, C.C. Rainfall prediction model using soft computing technique. Soft Comput.
**2003**, 7, 434–438. [Google Scholar] [CrossRef] - Hochreiter, S.; Schmidhuber, J. Long Short-Term Memory. Neural Comput.
**1997**, 9, 1735–1780. [Google Scholar] [CrossRef] [PubMed] - Mislan, M.; Haviluddin, H.; Hardwinarto, S.; Sumaryono, S.; Aipassa, M. Rainfall Monthly Prediction Based on Artificial Neural Network: A Case Study in Tenggarong Station, East Kalimantan—Indonesia. Procedia Comput. Sci.
**2015**, 59, 142–151. [Google Scholar] [CrossRef][Green Version] - Darwin, C. The Correspondence of Charles Darwin: 1821–1860; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
- Pfeiffer, J.R. Evolutionary theory. In George Bernard Shaw in Context; Cambridge University Press: Cambridge, UK, 2015; ISBN 9781107239081. [Google Scholar]
- Wuketits, F.M. Charles darwin and modern moral philosophy. Ludus Vitalis
**2009**, 17, 395–404. [Google Scholar] - García-Martínez, C.; Rodriguez, F.J.; Lozano, M. Genetic algorithms. In Handbook of Heuristics; Springer: Cham, Switzerland, 2018; ISBN 9783319071244. Available online: https://www.springer.com/gp/book/9783319071237 (accessed on 6 May 2021).
- Sivanandam, S.; Deepa, S. Introduction to Genetic Algorithms; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Gupta, J.N.D.; Sexton, R.S. Comparing backpropagation with a genetic algorithm for neural network training. Omega
**1999**, 27, 679–684. [Google Scholar] [CrossRef] - Caraka, R.E.; Chen, R.C.; Yasin, H.; Lee, Y.; Pardamean, B. Hybrid Vector Autoregression Feedforward Neural Network with Genetic Algorithm Model for Forecasting Space-Time Pollution Data. Indones. J. Sci. Technol.
**2021**, 6, 243–266. [Google Scholar] - Kubat, M.; Kubat, M. The Genetic Algorithm. In An Introduction to Machine Learning; Springer International Publishing: Cham, Switzerland, 2017. [Google Scholar]
- Moscato, P.; Cotta, C. A Modern Introduction to Memetic Algorithms. In Handbook of Metaheuristics; Springer: Boston, MA, USA, 2010; pp. 141–183. Available online: https://link.springer.com/chapter/10.1007/978-1-4419-1665-5_6 (accessed on 6 May 2021).
- Makridakis, S.; Wheelwright, S.C. Forecasting Methods for Management. Oper. Res. Q.
**1974**, 25, 648–649. [Google Scholar] [CrossRef] - Makridakis, S. A Survey of Time Series. Int. Stat. Rev. Rev. Int. Stat.
**1976**, 44, 29. [Google Scholar] [CrossRef] - Warsito, B.; Santoso, R.; Suparti; Yasin, H. Cascade Forward Neural Network for Time Series Prediction. J. Phys. Conf. Ser.
**2018**, 1025, 012097. [Google Scholar] [CrossRef] - Schetinin, V. A learning algorithm for evolving cascade neural networks. Neural Process. Lett.
**2003**, 17, 21–31. [Google Scholar] [CrossRef] - Ding, S.; Zhao, H.; Zhang, Y.; Xu, X.; Nie, R. Extreme learning machine: Algorithm, theory and applications. Artif. Intell. Rev.
**2015**, 44, 103–115. [Google Scholar] [CrossRef] - Suhartono; Prastyo, D.D.; Kuswanto, H.; Lee, M.H. Comparison between VAR, GSTAR, FFNN-VAR and FFNN-GSTAR Models for Forecasting Oil Production Methods. Mat. Malays. J. Ind. Appl. Math.
**2018**, 34, 103–111. [Google Scholar] - Prastyo, D.D.; Nabila, F.S.; Lee, M.H.S.; Suhermi, N.; Fam, S.F. VAR and GSTAR-based feature selection in support vector regression for multivariate spatio-temporal forecasting. In Communications in Computer and Information Science; Springer: Singapore, 2018; pp. 46–57. [Google Scholar]
- Zhang, P.G. Time series forecasting using a hybrid ARIMA and neural network model. Neurocomputing
**2003**, 50, 159–175. [Google Scholar] [CrossRef] - Geurts, M.; Box, G.E.P.; Jenkins, G.M. Time Series Analysis: Forecasting and Control. J. Mark. Res.
**2006**. [Google Scholar] [CrossRef] - McLeod, A.I.; Yu, H.; Mahdi, E. Time Series Analysis with R. Handb. Stat.
**2011**, 30, 661–672. [Google Scholar] [CrossRef] - Liao, W.T. Clustering of time series data—A survey. Pattern Recognit.
**2005**, 38, 1857–1874. [Google Scholar] [CrossRef] - Subba Rao, T. Time Series Analysis. J. Time Ser. Anal.
**2010**, 31, 139. [Google Scholar] [CrossRef] - Mudelsee, M. Climate Time Series Analysis: Regression; Springer: Dordrecht, The Netherlands, 2010; Volume 42, ISBN 978-90-481-9481-0. [Google Scholar]
- Zhu, X.; Pan, R.; Li, G.; Liu, Y.; Wang, H. Network vector autoregression. Ann. Stat.
**2017**, 45, 1096–1123. [Google Scholar] [CrossRef][Green Version] - Nourani, V.; Baghanam, A.H.; Adamowski, J.; Gebremichael, M. Using self-organizing maps and wavelet transforms for space-time pre-processing of satellite precipitation and runoff data in neural network based rainfall-runoff modeling. J. Hydrol.
**2013**, 476, 228–243. [Google Scholar] [CrossRef] - Ippoliti, L.; Valentini, P.; Gamerman, D. Space-time modelling of coupled spatiotemporal environmental variables. J. R. Stat. Soc. Ser. C Appl. Stat.
**2012**. [Google Scholar] [CrossRef] - Sharma, S.; Sharma, S. Understanding Activation Functions in Neural Networks. Int. J. Eng. Appl. Sci. Technol.
**2017**, 4, 310–316. [Google Scholar] - Apicella, A.; Donnarumma, F.; Isgrò, F.; Prevete, R. A survey on modern trainable activation functions. Neural Netw.
**2021**, 138, 14–32. [Google Scholar] [CrossRef] - Al-Rikabi, H.M.H.; Al-Ja’afari, M.A.M.; Ali, A.H.; Abdulwahed, S.H. Generic model implementation of deep neural network activation functions using GWO-optimized SCPWL model on FPGA. Microprocess. Microsyst.
**2020**, 77, 103141. [Google Scholar] [CrossRef] - Boob, D.; Dey, S.S.; Lan, G. Complexity of training ReLU neural network. Discret. Optim.
**2020**, 100620. [Google Scholar] [CrossRef] - Liu, B. Understanding the loss landscape of one-hidden-layer ReLU networks. Knowl. Based Syst.
**2021**, 220, 106923. [Google Scholar] [CrossRef] - Bouwmans, T.; Javed, S.; Sultana, M.; Jung, S.K. Deep neural network concepts for background subtraction: A systematic review and comparative evaluation. Neural Netw.
**2019**, 117, 8–66. [Google Scholar] [CrossRef][Green Version]

**Figure 7.**Forecasting all pollution datasets using the CFNN with a genetic algorithm and backpropagation.

Pollution | Location | N | Mean | SE Mean | StDev | Variance | Minimum | Q_{1} | Median | Q_{3} | Maximum | Range |
---|---|---|---|---|---|---|---|---|---|---|---|---|

PM_{10} | TAICHUNG | 3632 | 50.642 | 0.419 | 24.949 | 622.476 | 5 | 32 | 45.5 | 65 | 173 | 168 |

TAIPEI | 3632 | 21.244 | 0.208 | 12.412 | 154.052 | 1 | 12 | 19 | 27 | 100 | 99 | |

HSINCHU | 3632 | 22.46 | 0.219 | 13.135 | 172.521 | 1 | 13 | 19 | 29 | 103 | 102 | |

KAOHSIUNG | 3632 | 31.719 | 0.31 | 18.477 | 341.414 | 1 | 17 | 29 | 44 | 123 | 122 | |

SO_{2} | TAICHUNG | 3632 | 2.8706 | 0.017 | 1.0208 | 1.042 | 0 | 2.2 | 2.7 | 3.4 | 9.3 | 9.3 |

TAIPEI | 3632 | 2.9835 | 0.0263 | 1.5742 | 2.4781 | 0.4 | 1.9 | 2.6 | 3.7 | 16.2 | 15.8 | |

HSINCHU | 3632 | 2.6778 | 0.0186 | 1.1125 | 1.2377 | 0.1 | 1.9 | 2.5 | 3.2 | 13.2 | 13.1 | |

KAOHSIUNG | 3632 | 5.3129 | 0.0515 | 3.0833 | 9.5066 | 0 | 3.3 | 4.5 | 6.4 | 33.8 | 33.8 | |

PM_{2.5} | TAICHUNG | 3632 | 26.623 | 0.264 | 15.66 | 245.244 | 1 | 15 | 23 | 35 | 106 | 105 |

TAIPEI | 3632 | 23.635 | 0.203 | 12.161 | 147.892 | 3.97 | 15.107 | 20.84 | 29.023 | 109.83 | 105.86 | |

HSINCHU | 3632 | 18.179 | 0.13 | 7.763 | 60.265 | 0.63 | 13.04 | 16.34 | 21.102 | 76.72 | 76.09 | |

KAOHSIUNG | 3632 | 23.854 | 0.163 | 9.747 | 95.014 | 5.27 | 16.49 | 21.495 | 29.697 | 68.96 | 63.69 | |

NO_{X} | TAICHUNG | 3632 | 22.944 | 0.171 | 10.26 | 105.269 | 4.35 | 15.23 | 20.57 | 28.63 | 81.43 | 77.08 |

TAIPEI | 3632 | 6.196 | 0.106 | 6.315 | 39.874 | 0.09 | 1.85 | 4.15 | 8.28 | 65.14 | 65.05 | |

HSINCHU | 3632 | 3.2786 | 0.0546 | 3.2643 | 10.6559 | 0 | 1.52 | 2.3 | 3.79 | 45.65 | 45.65 |

Pollution | Portion | Training | Testing | Average | Elapsed Time | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

RMSE | MAE | SMAPE | RMSE | MAE | SMAPE | RMSE | MAE | SMAPE | |||

PM_{2.5} | 50:50 | 9.84 | 6.89 | 3.91 | 7.89 | 5.70 | 3.83 | 8.87 | 6.30 | 3.87 | 76.42 |

PM_{10} | 13.53 | 9.57 | 3.84 | 11.37 | 8.05 | 3.86 | 12.45 | 8.81 | 3.85 | 71.86 | |

NO_{X} | 5.53 | 3.55 | 6.94 | 4.37 | 2.73 | 6.95 | 4.95 | 3.14 | 6.95 | 74.23 | |

SO_{2} | 1.72 | 1.18 | 3.60 | 1.15 | 0.80 | 3.61 | 1.44 | 0.99 | 3.61 | 75.51 | |

PM_{2.5} | 60:40 | 9.86 | 6.88 | 3.89 | 7.41 | 5.42 | 3.57 | 8.63 | 6.15 | 3.73 | 76.56 |

PM_{10} | 13.49 | 9.54 | 3.76 | 10.67 | 7.57 | 3.56 | 12.08 | 8.55 | 3.66 | 75.24 | |

NO_{X} | 6.48 | 3.48 | 6.93 | 4.10 | 2.65 | 6.24 | 5.29 | 3.06 | 6.59 | 73.15 | |

SO_{2} | 1.65 | 1.13 | 3.48 | 1.07 | 0.80 | 3.43 | 1.36 | 0.97 | 3.46 | 76.13 | |

PM_{2.5} | 70:30 | 9.43 | 6.62 | 3.83 | 7.13 | 5.27 | 3.50 | 8.28 | 5.95 | 3.67 | 71.13 |

PM_{10} | 13.30 | 9.29 | 3.74 | 10.09 | 7.13 | 3.74 | 11.70 | 8.21 | 3.74 | 77.87 | |

NO_{X} | 5.31 | 3.36 | 6.98 | 4.01 | 2.54 | 7.12 | 4.66 | 2.95 | 7.05 | 74.41 | |

SO_{2} | 1.60 | 1.09 | 3.36 | 1.00 | 0.77 | 3.41 | 1.30 | 0.93 | 3.38 | 80.90 | |

PM_{2.5} | 80:20 | 9.25 | 6.46 | 3.81 | 6.83 | 4.99 | 3.50 | 8.04 | 5.73 | 3.65 | 74.25 |

PM_{10} | 13.10 | 9.07 | 3.74 | 9.19 | 6.56 | 4.13 | 11.14 | 7.82 | 3.93 | 74.25 | |

NO_{X} | 5.25 | 3.29 | 7.11 | 3.79 | 2.49 | 6.68 | 4.52 | 2.89 | 6.90 | 72.37 | |

SO_{2} | 1.54 | 1.05 | 3.42 | 0.92 | 0.70 | 3.36 | 1.23 | 0.88 | 3.39 | 80.24 | |

PM_{2.5} | 90:10 * | 9.03 | 6.34 | 3.77 | 6.78 | 4.94 | 3.47 | 7.90 | 5.64 | 3.62 | 83.83 |

PM_{10} | 12.77 | 8.85 | 3.77 | 8.02 | 5.93 | 4.09 | 10.40 | 7.39 | 3.93 | 75.00 | |

NO_{X} | 5.11 | 3.20 | 6.90 | 3.70 | 2.43 | 6.53 | 4.40 | 2.81 | 6.72 | 80.47 | |

SO_{2} | 1.48 | 1.10 | 3.37 | 0.80 | 0.62 | 2.77 | 1.14 | 0.86 | 3.07 | 71.71 |

**Noted:**Best simulation with low error (*) and yellow highlight represent the lowest value of each information pollution, accuracy measurement, and elapsed time.

Pollution | Activation Function | Training | Testing | Average | Elapsed Time | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

RMSE | MAE | SMAPE | RMSE | MAE | SMAPE | RMSE | MAE | SMAPE | |||

PM_{2.5} | logsig | 9.02 | 6.30 | 3.77 | 6.77 | 4.94 | 3.54 | 7.90 | 5.62 | 3.66 | 78.03 |

PM_{10} | 12.79 | 8.86 | 3.71 | 8.12 | 6.02 | 4.05 | 10.46 | 7.44 | 3.88 | 75.48 | |

NO_{X} | 5.04 | 3.15 | 6.84 | 3.79 | 2.43 | 6.62 | 4.42 | 2.79 | 6.73 | 72.43 | |

SO_{2} | 1.48 | 1.01 | 3.35 | 0.82 | 0.63 | 3.00 | 1.15 | 0.82 | 3.18 | 76.59 | |

PM_{2.5} | radbas | 9.06 | 6.37 | 3.80 | 6.80 | 5.00 | 3.51 | 7.93 | 5.69 | 3.66 | 80.97 |

PM_{10} | 12.79 | 8.85 | 3.79 | 8.11 | 6.03 | 4.19 | 10.45 | 7.44 | 3.99 | 74.77 | |

NO_{X} | 5.09 | 3.16 | 6.83 | 3.75 | 2.40 | 6.19 | 4.42 | 2.78 | 6.51 | 82.35 | |

SO_{2} | 1.49 | 1.01 | 3.36 | 0.83 | 0.64 | 2.93 | 1.16 | 0.83 | 3.15 | 73.33 | |

PM_{2.5} | SoftMax | 9.07 | 6.35 | 3.78 | 6.70 | 4.91 | 3.46 | 7.89 | 5.63 | 3.62 | 75.47 |

PM_{10} | 12.81 | 8.90 | 3.75 | 8.13 | 6.04 | 4.24 | 10.47 | 7.47 | 3.99 | 74.70 | |

NO_{X} | 5.11 | 3.19 | 7.12 | 3.64 | 2.35 | 6.23 | 4.38 | 2.77 | 6.67 | 72.11 | |

SO_{2} | 1.48 | 1.01 | 3.42 | 0.84 | 0.66 | 3.12 | 1.16 | 0.84 | 3.27 | 77.53 | |

PM_{2.5} | tribas | 9.03 | 6.34 | 3.80 | 6.81 | 4.97 | 3.52 | 7.92 | 5.66 | 3.66 | 93.20 |

PM_{10} | 12.81 | 8.90 | 3.75 | 8.13 | 6.04 | 4.24 | 10.47 | 7.47 | 3.99 | 74.70 | |

NO_{X} | 5.13 | 3.21 | 6.98 | 3.68 | 2.37 | 6.53 | 4.41 | 2.79 | 6.75 | 72.29 | |

SO_{2} | 1.49 | 1.01 | 3.36 | 0.84 | 0.65 | 2.94 | 1.16 | 0.83 | 3.15 | 80.35 |

**Noted:**Yellow highlight represent the lowest value of each information pollution, accuracy measurement, and elapsed time.

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Caraka, R.E.; Yasin, H.; Chen, R.-C.; Goldameir, N.E.; Supatmanto, B.D.; Toharudin, T.; Basyuni, M.; Gio, P.U.; Pardamean, B. Evolving Hybrid Cascade Neural Network Genetic Algorithm Space–Time Forecasting. *Symmetry* **2021**, *13*, 1158.
https://doi.org/10.3390/sym13071158

**AMA Style**

Caraka RE, Yasin H, Chen R-C, Goldameir NE, Supatmanto BD, Toharudin T, Basyuni M, Gio PU, Pardamean B. Evolving Hybrid Cascade Neural Network Genetic Algorithm Space–Time Forecasting. *Symmetry*. 2021; 13(7):1158.
https://doi.org/10.3390/sym13071158

**Chicago/Turabian Style**

Caraka, Rezzy Eko, Hasbi Yasin, Rung-Ching Chen, Noor Ell Goldameir, Budi Darmawan Supatmanto, Toni Toharudin, Mohammad Basyuni, Prana Ugiana Gio, and Bens Pardamean. 2021. "Evolving Hybrid Cascade Neural Network Genetic Algorithm Space–Time Forecasting" *Symmetry* 13, no. 7: 1158.
https://doi.org/10.3390/sym13071158