Evidential reasoning rule for environmental governance cost prediction with considering causal relationship and data reliability

Environmental governance cost prediction can avoid blind investment and waste of resources and achieve effective cost planning for sustainable development of resources and environment. For the sake of solving the problem that most previous studies failed to consider the causal relationship and data reliability of environmental governance inputs and outputs, a new environmental governance cost prediction method is proposed under the framework of the evidential reasoning (ER) rule with three improvements comparing to existing methods: (1) the causal relationship of environmental governance inputs and outputs is embedded into evidence representation for better extracting knowledge from data; (2) the efficiency about the minimum inputs to achieve the maximum outputs is used to evaluate the data reliability of environmental governance inputs and outputs; and (3) a new analytical ER rule is investigated to optimize the process of evidence combination. Hence, the new method includes the calculation of belief distributions, evidence reliabilities, and evidence weights, as well as the combination of evidences to predict environmental governance costs. In the case study, the data of 30 provinces in Mainland China from 2005 to 2020 are collected to verify the effectiveness of the new method. Results show a high level of accuracy of the new method over other existing methods.

Enquiries about data availability should be directed to the authors.

This research was supported by the National Natural Science Foundation of China (No. 72001043), the Natural Science Foundation of Fujian Province, China (Nos. 2022J01178 and 2020J05122), and the Humanities and Social Science Foundation of the Ministry of Education, China (No. 20YJC630188). Prof. Lu gratefully acknowledges the funding support from Hong Kong SustainTech Foundation and PolyU Project of Strategic Importance P0039723.

Authors and Affiliations

1. School of Cultural Tourism and Public Administration, Fujian Normal University, Fuzhou, 350117, China

   Fei-Fei Ye

2. Decision Sciences Institute, Fuzhou University, Fuzhou, 350108, China

   Long-Hao Yang & Ying-Ming Wang

3. School of Computing, Ulster University at Belfast Campus, Belfast, Northern Ireland, UK

   James Uhomoibhi & Jun Liu

4. School of Accounting and Finance, The Hong Kong Polytechnic University, Hung Hom, HKSAR, China

   Haitian Lu

Corresponding author

Correspondence to Long-Hao Yang.

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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Appendix A

Derivation of the analytical aggregation formulas of the ER rule

In order to provide the analytical aggregation formulas of the ER rule, the formulas derivation inspired by the analytical ER algorithm (Wang et al. 2006) is provided when only single propositions \(\theta\) and global ignorance \({{\varvec{\Theta}}}\) are assumed to be focal elements. First of all, the aggregation formulas of two evidences are discussed according to Eqs. (7) and (8) shown in Sect. 3 as follows:

1. (1)

   For the power set \(P({{\varvec{\Theta}}})\), because of \(\tilde{m}_{{P({{\varvec{\Theta}}}),i}} = 1 - \tilde{w}_{i}\), Eq. (8) can be simplified as follows:

   $$\hat{m}_{{P({{\varvec{\Theta}}}),e(2)}} = (1 - \tilde{w}_{2} )(1 - \tilde{w}_{1} ) = \tilde{m}_{{P({{\varvec{\Theta}}}),2}} \tilde{m}_{{P({{\varvec{\Theta}}}),1}} = \prod\limits_{i = 1}^{2} {\tilde{m}_{{P({{\varvec{\Theta}}}),i}} }$$

   (A1)
2. (2)

   For the global ignorance \({{\varvec{\Theta}}}\), Eq. (7) can be simplified as follows:

   $$ \begin{aligned} \hat{m}_{{{{\varvec{\Theta}}},e(2)}} & = [(1 - \tilde{w}_{2} )\tilde{m}_{{{{\varvec{\Theta}}},1}} + (1 - \tilde{w}_{1} )\tilde{m}_{{{{\varvec{\Theta}}},2}} ] + \sum\limits_{{B \cap C = {{\varvec{\Theta}}}}} {\tilde{m}_{B,1} \tilde{m}_{C,2} } \hfill \\& = (1 - \tilde{w}_{2} )\tilde{m}_{{{{\varvec{\Theta}}},1}} + (1 - \tilde{w}_{1} )\tilde{m}_{{{{\varvec{\Theta}}},2}} + \tilde{m}_{{{{\varvec{\Theta}}},1}} \tilde{m}_{{{{\varvec{\Theta}}},2}} \hfill \\& = (1 - \tilde{w}_{2} )\tilde{m}_{{{{\varvec{\Theta}}},1}} + (1 - \tilde{w}_{1} )\tilde{m}_{{{{\varvec{\Theta}}},2}} + \tilde{m}_{{{{\varvec{\Theta}}},1}} \tilde{m}_{{{{\varvec{\Theta}}},2}} \hfill \\ \end{aligned} $$

   (A2)

Because of \(\tilde{m}_{{P({{\varvec{\Theta}}}),i}} = 1 - \tilde{w}_{i}\), Eq. (A2) can be further simplified as follows:

1. (3)

   For the single proposition θ (\(\theta \in {{\varvec{\Theta}}}\)), Eq. (8) can be simplified as follows:

   $$ \begin{aligned} \hat{m}_{\theta ,e(2)} &= [(1 - \tilde{w}_{2} )\tilde{m}_{\theta ,1} + (1 - \tilde{w}_{1} )\tilde{m}_{\theta ,2} ] + \sum\limits_{B \cap C = \theta } {\tilde{m}_{B,1} \tilde{m}_{C,2} } \hfill \\& = (1 - \tilde{w}_{2} )\tilde{m}_{\theta ,1} + (1 - \tilde{w}_{1} )\tilde{m}_{\theta ,2} + \tilde{m}_{\theta ,1} \tilde{m}_{\theta ,2} + \tilde{m}_{{{{\varvec{\Theta}}},1}} \tilde{m}_{\theta ,2} + \tilde{m}_{\theta ,1} \tilde{m}_{{{{\varvec{\Theta}}},2}} \hfill \\& = (1 - \tilde{w}_{2} + \tilde{m}_{{{{\varvec{\Theta}}},2}} )\tilde{m}_{\theta ,1} + (1 - \tilde{w}_{1} + \tilde{m}_{{{{\varvec{\Theta}}},1}} )\tilde{m}_{\theta ,2} + \tilde{m}_{\theta ,1} \tilde{m}_{\theta ,2} \hfill \\ \end{aligned} $$

   (A4)

When assume \(\tilde{m}^{\prime}_{{P({{\varvec{\Theta}}}),i}} = 1 - \tilde{w}_{i} + \tilde{m}_{{{{\varvec{\Theta}}},i}}\), Eq. (A4) can be further simplified as follows:

Because of \(\tilde{m}^{\prime}_{{P({{\varvec{\Theta}}}),i}} = 1 - \tilde{w}_{i} + \tilde{m}_{{{{\varvec{\Theta}}},i}} = \tilde{m}_{{P({{\varvec{\Theta}}}),i}} + \tilde{m}_{{{{\varvec{\Theta}}},i}}\), Eq. (A5) can be written as below:

Based on the above formulas deviations, the following three equations are assumed to be true for aggregate the first i-1 evidences:

Owing to two facts: (1) the normalization is the bridge to associate \(\hat{m}_{\theta ,e(i - 1)}\), \(\hat{m}_{{{{\varvec{\Theta}}},e(i - 1)}}\), and \(\hat{m}_{{P({{\varvec{\Theta}}}),e(i - 1)}}\) to \(\tilde{m}_{\theta ,e(i - 1)}\), \(\tilde{m}_{{{{\varvec{\Theta}}},e(i - 1)}}\), and \(\tilde{m}_{{P({{\varvec{\Theta}}}),e(i - 1)}}\)(Yang and Xu 2013); (2) the normalization in the evidence aggregation rule of the D–S theory of evidence can be applied at the end of the evidence aggregation process without changing the aggregation result (Yen, 1990), \(\hat{m}_{\theta ,e(i - 1)}\), \(\hat{m}_{{{{\varvec{\Theta}}},e(i - 1)}}\), and \(\hat{m}_{{P({{\varvec{\Theta}}}),e(i - 1)}}\) shown in Eqs. (A7)–(A9) are used as \(\tilde{m}_{\theta ,e(i - 1)}\), \(\tilde{m}_{{{{\varvec{\Theta}}},e(i - 1)}}\), and \(\tilde{m}_{{P({{\varvec{\Theta}}}),e(i - 1)}}\) to aggregate the ith evidence according to Eqs. (10)–(11).

1. (1)

   For the power set \(P({{\varvec{\Theta}}})\), a new aggregation formula can be obtained from Eq. (11) below:

   $$\hat{m}_{{P({{\varvec{\Theta}}}),e(i)}} = (1 - \tilde{w}_{i} )\tilde{m}_{{P({{\varvec{\Theta}}}),e(i - 1)}} = \tilde{m}_{{P({{\varvec{\Theta}}}),i}} \prod\limits_{l = 1}^{i - 1} {\tilde{m}_{{P({{\varvec{\Theta}}}),l}} } = \prod\limits_{l = 1}^{i} {\tilde{m}_{{P({{\varvec{\Theta}}}),l}} }$$

   (A10)
2. (2)

   For the global ignorance \({{\varvec{\Theta}}}\), a new aggregation formula can be obtained from Eq. (10) below:

   $$ \begin{aligned} \hat{m}_{{{{\varvec{\Theta}}},e(i)}}& = [(1 - \tilde{w}_{i} )\tilde{m}_{{{{\varvec{\Theta}}},e(i - 1)}} + \tilde{m}_{{P({{\varvec{\Theta}}}),e(i - 1)}} \tilde{m}_{{{{\varvec{\Theta}}},i}} ] + \sum\limits_{{B \cap C = {{\varvec{\Theta}}}}} {\tilde{m}_{B,e(i - 1)} \tilde{m}_{C,i} } \hfill \\&= (1 - \tilde{w}_{i} )\tilde{m}_{{{{\varvec{\Theta}}},e(i - 1)}} + \tilde{m}_{{P({{\varvec{\Theta}}}),e(i - 1)}} \tilde{m}_{{{{\varvec{\Theta}}},i}} + \tilde{m}_{{{{\varvec{\Theta}}},e(i - 1)}} \tilde{m}_{{{{\varvec{\Theta}}},i}} \hfill \\& = (\tilde{m}_{{P({{\varvec{\Theta}}}),i}} + \tilde{m}_{{{{\varvec{\Theta}}},i}} )\tilde{m}_{{{{\varvec{\Theta}}},e(i - 1)}} + \tilde{m}_{{{{\varvec{\Theta}}},i}} \tilde{m}_{{P({{\varvec{\Theta}}}),e(i - 1)}} \hfill \\& = (\tilde{m}_{{P({{\varvec{\Theta}}}),i}} + \tilde{m}_{{{{\varvec{\Theta}}},i}} )\left[\prod\limits_{l = 1}^{i - 1} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} )} - \prod\limits_{l = 1}^{i - 1} {\tilde{m}_{{P({{\varvec{\Theta}}}),l}} } \right] + \tilde{m}_{{{{\varvec{\Theta}}},i}} \prod\limits_{l = 1}^{i - 1} {\tilde{m}_{{P({{\varvec{\Theta}}}),l}} } \hfill \\& = (\tilde{m}_{{P({{\varvec{\Theta}}}),i}} + \tilde{m}_{{{{\varvec{\Theta}}},i}} )\prod\limits_{l = 1}^{i - 1} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} )} - (\tilde{m}_{{P({{\varvec{\Theta}}}),i}} + \tilde{m}_{{{{\varvec{\Theta}}},i}} )\prod\limits_{l = 1}^{i - 1} {\tilde{m}_{{P({{\varvec{\Theta}}}),l}} } + \tilde{m}_{{{{\varvec{\Theta}}},i}} \prod\limits_{l = 1}^{i - 1} {\tilde{m}_{{P({{\varvec{\Theta}}}),l}} } \hfill \\& = \prod\limits_{l = 1}^{i} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} )} - \prod\limits_{l = 1}^{i} {\tilde{m}_{{P({{\varvec{\Theta}}}),l}} } \hfill \\ \end{aligned} $$

   (A11)
3. (3)

   For the single proposition θ (\(\theta \in {{\varvec{\Theta}}}\)), a new aggregation formula can be obtained from Eq. (10) below:

   $$ \begin{aligned} \hat{m}_{\theta ,e(i)} &= [(1 - \tilde{w}_{i} )\tilde{m}_{\theta ,e(i - 1)} + \tilde{m}_{{P({{\varvec{\Theta}}}),e(i - 1)}} \tilde{m}_{\theta ,i} ] + \sum\limits_{B \cap C = \theta } {\tilde{m}_{B,e(i - 1)} \tilde{m}_{C,i} } \hfill \\& = (1 - \tilde{w}_{i} )\tilde{m}_{\theta ,e(i - 1)} + \tilde{m}_{{P({{\varvec{\Theta}}}),e(i - 1)}} \tilde{m}_{\theta ,i} + \tilde{m}_{\theta ,e(i - 1)} \tilde{m}_{\theta ,i} + \tilde{m}_{{{{\varvec{\Theta}}},e(i - 1)}} \tilde{m}_{\theta ,i} + \tilde{m}_{\theta ,e(i - 1)} \tilde{m}_{{{{\varvec{\Theta}}},i}} \hfill \\& = (\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{\theta ,i} + \tilde{m}_{{{{\varvec{\Theta}}},i}} )\tilde{m}_{\theta ,e(i - 1)} + \tilde{m}_{\theta ,i} \tilde{m}_{{{{\varvec{\Theta}}},e(i - 1)}} + \tilde{m}_{\theta ,i} \tilde{m}_{{P({{\varvec{\Theta}}}),e(i - 1)}} \hfill \\& = (\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{\theta ,i} + \tilde{m}_{{{{\varvec{\Theta}}},i}} )[\prod\limits_{l = 1}^{i - 1} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} + \tilde{m}_{\theta ,l} )} - \prod\limits_{l = 1}^{i - 1} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} )} ] \hfill \\& \quad + \tilde{m}_{\theta ,i} [\prod\limits_{l = 1}^{i - 1} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} )} - \prod\limits_{l = 1}^{i - 1} {\tilde{m}_{{P({{\varvec{\Theta}}}),l}} } ] + \tilde{m}_{\theta ,i} \prod\limits_{l = 1}^{i - 1} {\tilde{m}_{{P({{\varvec{\Theta}}}),l}} } \hfill \\& = (\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{\theta ,i} + \tilde{m}_{{{{\varvec{\Theta}}},i}} )\prod\limits_{l = 1}^{i - 1} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} + \tilde{m}_{\theta ,l} )} - (\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{\theta ,i} + \tilde{m}_{{{{\varvec{\Theta}}},i}} )\prod\limits_{l = 1}^{i - 1} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} )} \hfill \\ & \quad + \tilde{m}_{\theta ,i} \prod\limits_{l = 1}^{i - 1} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} )} - \tilde{m}_{\theta ,i} \prod\limits_{l = 1}^{i - 1} {\tilde{m}_{{P({{\varvec{\Theta}}}),l}} } + \tilde{m}_{\theta ,i} \prod\limits_{l = 1}^{i - 1} {\tilde{m}_{{P({{\varvec{\Theta}}}),l}} } \hfill \\& = \prod\limits_{l = 1}^{i} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} + \tilde{m}_{\theta ,l} )} - (\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},i}} )\prod\limits_{l = 1}^{i - 1} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} )} - \tilde{m}_{\theta ,i} \prod\limits_{l = 1}^{i - 1} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} )} \hfill \\& \quad + \tilde{m}_{\theta ,i} \prod\limits_{l = 1}^{i - 1} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} )} \hfill \\& = \prod\limits_{l = 1}^{i} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} + \tilde{m}_{\theta ,l} )} - \prod\limits_{l = 1}^{i} {(\tilde{m}_{{P({{\varvec{\Theta}}}),l}} + \tilde{m}_{{{{\varvec{\Theta}}},l}} )} \hfill \\ \end{aligned} $$

   (A12)

Therefore, the equations shown in Eqs. (A7)–(A9) are true for the aggregation of the first i (i = 1,…, L) evidences. For i = L, the following non-normalized aggregation formula can be obtained below:

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