Correction: Yun et al. A Novel Extended Higher-Order Moment Multi-Factor Framework for Forecasting the Carbon Price: Testing on the Multilayer Long Short-Term Memory Network. Sustainability 2020, 12, 1869

Yun, Po; Zhang, Chen; Wu, Yaqi; Yang, Xianzi; Wagan, Zulfiqar Ali

doi:10.3390/su16156318

Open AccessCorrection

Correction: Yun et al. A Novel Extended Higher-Order Moment Multi-Factor Framework for Forecasting the Carbon Price: Testing on the Multilayer Long Short-Term Memory Network. Sustainability 2020, 12, 1869

by

Po Yun

^1,*

,

Chen Zhang

^1,2,*,

Yaqi Wu

¹

,

Xianzi Yang

¹ and

Zulfiqar Ali Wagan

³

¹

School of Management, Hefei University of Technology, Hefei 230009, China

²

Key Laboratory of Process Optimization and Intelligent Decision-Making, Ministry of Education, Hefei 230009, China

³

Department of Education and Literacy, Government of Sindh, Hyderabad 70060, Tando Jam, Pakistan

^*

Authors to whom correspondence should be addressed.

Sustainability 2024, 16(15), 6318; https://doi.org/10.3390/su16156318

Submission received: 18 March 2024 / Accepted: 22 May 2024 / Published: 24 July 2024

(This article belongs to the Special Issue Forecasting Financial Markets and Financial Crisis)

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(1): The authors wish to delete the expression of “where ⊗ means the Kronecker product and ⊕ means the XOR logic operation” throughout the entire paper.
(2): The authors wish to replace the original Table 2 with a new Table 2 and modify some of the content presented in the original Table 2.

Correspondingly, in the research of Yun et al. (2020) [1], the original sentence in 4.2 “The results shown in Table 2 report that the best performance regarding RMSE and MAPE occurs with two hidden layers, because the values of RMSE and MAE are 0.13435 and 1.91017, respectively, which is the lowest value of the entire alternate hidden layer. Further analysis shows that there exists the smallest RMSE and MAE when the number of hidden layer nodes is 64, with the values of 0.1075 and 1.55172, respectively” should be replaced by the following sentence: “The results shown in Table 2 report that the best performance regarding RMSE and MAE occurs with two hidden layers, because the average errors of RMSE and MAE are 2.4050 and 1.7502, respectively, which is the lowest value of the entire alternate hidden layer. Further analysis shows that there exists the smallest RMSE and MAE when the number of hidden layer nodes is 64, with the values of 2.0421 and 1.4617, respectively”.

The new corrected Table 2 should be as follows:

Table 2. Effects of pricing factors on the performance of the Multi-LSTM: hidden layers and hidden nodes.

Hidden- Layer	Nodes	Multi-LSTM (Under no Shock)		Multi-LSTM (Under the Shock of Higher-Order Moment)		Hidden- Layer	Nodes	Multi-LSTM (Under no Shock)		Multi-LSTM (Under the Shock of Higher-Order Moment)
Hidden- Layer	Nodes	RMSE	MAE	RMSE	MAE	Hidden- Layer	Nodes	RMSE	MAE	RMSE	MAE
1	4	2.6027	1.8642	2.3008	1.6883	4	4	2.6054	1.8679	2.7110	2.0115
1	8	2.7738	1.9811	2.3511	1.7284	4	8	2.8014	2.0080	2.5701	1.9251
1	16	3.0549	2.2533	2.9232	2.1577	4	16	3.1356	2.2792	2.8978	1.9707
1	32	3.1812	2.3373	3.3008	2.3939	4	32	2.9678	2.2051	2.2419	1.7047
1	64	3.1672	2.3977	3.4379	2.2215	4	64	2.9555	2.1767	2.1881	1.6140
1	128	3.3809	2.5575	3.4574	2.3864	4	128	2.7090	2.0079	2.0904	1.5173
	Avg	3.0268	2.2319	2.9619	2.0960		Avg	2.8625	2.0908	2.4499	1.7906
2	4	2.5663	1.8323	2.1735	1.2996	5	4	2.5740	1.8678	3.0184	2.2529
2	8	2.8162	2.0227	2.2539	1.8677	5	8	2.8282	2.0490	2.5820	1.8255
2	16	2.2780	2.1883	3.0330	2.3142	5	16	2.9542	2.0689	2.6657	1.9784
2	32	3.2003	2.1760	2.5224	1.8080	5	32	2.9948	2.2283	2.5892	1.8557
2	64	2.9110	2.0783	2.0421	1.4617	5	64	2.9298	2.1921	2.2412	1.6505
2	128	2.2726	1.7919	2.6011	1.9130	5	128	2.7662	2.0339	2.0145	1.4680
	Avg	2.6741	2.0149	2.4050	1.7502		Avg	2.8412	2.0734	2.5185	1.8385
3	4	2.5984	1.9029	2.1707	1.5650	6	4	2.8080	2.1191	2.9981	2.2420
3	8	2.7567	1.9742	2.5407	1.8894	6	8	2.8982	2.1702	2.9087	2.1699
3	16	3.2501	2.3370	3.7999	2.6162	6	16	3.0927	2.2961	2.9134	2.1647
3	32	3.0480	2.2785	2.3051	1.6909	6	32	3.1477	2.2186	2.7400	1.9495
3	64	2.8728	2.0891	2.2269	1.6058	6	64	2.8509	2.1234	2.1736	1.6301
3	128	2.7009	2.0227	2.2900	1.6745	6	128	2.7064	1.9932	2.1564	1.5960
	Avg	2.8712	2.1007	2.5555	1.8403		Avg	2.9173	2.1534	2.6483	1.9587

Note: Bold numbers are the minimum root-mean-square error (RMSE) and mean absolute error (MAE).

The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.

Reference

Yun, P.; Zhang, C.; Wu, Y.; Yang, X.; Wagan, Z.A. A Novel Extended Higher-Order Moment Multi-Factor Framework for Forecasting the Carbon Price: Testing on the Multilayer Long Short-Term Memory Network. Sustainability 2020, 12, 1869. [Google Scholar] [CrossRef]

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© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

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Yun, P.; Zhang, C.; Wu, Y.; Yang, X.; Wagan, Z.A. Correction: Yun et al. A Novel Extended Higher-Order Moment Multi-Factor Framework for Forecasting the Carbon Price: Testing on the Multilayer Long Short-Term Memory Network. Sustainability 2020, 12, 1869. Sustainability 2024, 16, 6318. https://doi.org/10.3390/su16156318

AMA Style

Yun P, Zhang C, Wu Y, Yang X, Wagan ZA. Correction: Yun et al. A Novel Extended Higher-Order Moment Multi-Factor Framework for Forecasting the Carbon Price: Testing on the Multilayer Long Short-Term Memory Network. Sustainability 2020, 12, 1869. Sustainability. 2024; 16(15):6318. https://doi.org/10.3390/su16156318

Chicago/Turabian Style

Yun, Po, Chen Zhang, Yaqi Wu, Xianzi Yang, and Zulfiqar Ali Wagan. 2024. "Correction: Yun et al. A Novel Extended Higher-Order Moment Multi-Factor Framework for Forecasting the Carbon Price: Testing on the Multilayer Long Short-Term Memory Network. Sustainability 2020, 12, 1869" Sustainability 16, no. 15: 6318. https://doi.org/10.3390/su16156318

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Correction: Yun et al. A Novel Extended Higher-Order Moment Multi-Factor Framework for Forecasting the Carbon Price: Testing on the Multilayer Long Short-Term Memory Network. Sustainability 2020, 12, 1869

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