Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I would like to congratulate the authors for their original article. The article deals with filling in the gap caused by the dependence on heavy-tailed distributions, which can affect the performance of necessary parameter adjustments. To address this challenge, they propose an innovative distributed consensus multi-distribution state estimation technique integrating Gaussian and student-t filters. The method incorporates both Gaussian and student-t distributions, enabling to formulate a multi-distribution filter for individual sensors, allocating probabilities to Gaussian and student-t noise models.

In the introduction, the authors carry out a varied, coherent, and consistent bibliographic review. The methodology, results and numerical simulation are well-grounded; However, I would suggest adapting the contents to follow the author guidelines according to the article structure advised.

 Line 37: I would say “estimation error of the CM method”.

Lines 43, 45, 49, 55…: I would suggest to add a space before the references. For example, in line 43 write “observer [17]” instead of “observer[17]”. Please, review the whole manuscript following this criterion.

Line 73: Please, review if the correct form is “parameter”, or “parameters”.

Line 90: Please, review if the correct form is “validate”, or “validates”.

Line 94: dis-tributed  à distributed

Line 111: Please, remove the space after n; (N , A).  à (N, A). 

Line 100: It is said the first time in the article “η is dof”. I assume that dof means “degrees of freedom”, but I would suggest indicating the meaning of dof the first time you use it, for example “degrees of freedom (dof).

Line 102: Please, review if there is a space before the comma “When η = 1”, and if so, remove it.

Line 117: Please, remove the space after “time k”

Line 163: You used for the first time the acronym PDF. Please, explain what it means

Line 165: Please, review if there is a space before the comma “µr ,”, and if so, remove it  

Line 186: Please, remove the space after approximate “p(xk)”

Line 206: Please, remove the space and pi(·) . à pi(·).

Line 251: Review if there is a space before the point, and if so, please remove it  “St(x; xˆ, P, η) .”

Line 255: Please review if “for singe” can be substituted for “for a single”

Line 258: Please, remove the space after “node i “, before the comma, and the same for “H .” before the point.

Line 295: Please, review if in this part of the sentence is correct “with is the state and“ it would be better to use “are”.

Line 300: dis-cussed à discussed

Line 324: Please, review if the spelling of “selected from a in each simulation”.

Caption of Figures. Please, specify the meaning of the acronyms used in the graps, so the reader do not have to look for them in the whole text. (Figures 3, 4, 5, 6, and 7). Review the same aspect in the caption of the tables.

 

Lines 310 to 313. You indicate in the 3.4 Discussion section that “In a broad sense, the algorithm proposed in this paper can be regarded as a special case of the multi-model method. Therefore, the concepts of probability  transfer and input interaction of multi-model method can also be applied to the algorithm proposed in this paper. How to integrate these technologies is one of the problems to be solved in the future”. It is a little bit confusing, as after the discussion section there is a section called “4. Numerical simulation”, so you still keep on describing part of your work after the discussion. According to the instructions for authors of the JSAN journal, the Manuscript should have the following sections “Introduction, Materials and Methods, Results, Discussion, Conclusions (optional).”. Therefore, I would suggest to add an expanded version of section 3.4, but before the conclusions, in which you could add some comments on the numerical simulation.

Author Response

Dear Editor and Reviewers:

We appreciate you very much for your positive and constructive comments and suggestions on our manuscript entitled “Distributed Consensus Multi-distribution Filter for Heavy-tailed Noise” (No: jsan-2978882).

We have studied reviewers’ comments and questions carefully and tried our best to revise over manuscript. The following are the responses and revisions.

Response on the comments of Reviewer #1

Q1: The Abstract section must be provided with some general information where practically the studies of an area of research were used previously. There is no practical destination presented.

Response:

 

In abstract, we add some descriptions about the studies of the research area. The details are presented as follows.

Distributed state estimation is one of the critical technologies in the field of target tracking, where the process noise and measurement noise may have heavy-tailed distribution. (see lines 1-2 in page 1).

 

 

 

Q2: Still, with the Abstract section, some rough results supporting the final conclusion must be raised. In the current form, the Authors do not mention the detailed results which quantify the main advantage.

Response:

In abstract, we add the detailed results which quantify the main advantage. The details are presented as follows.

Simulation results demonstrate that estimation accuracy of the proposed algorithm has been improved by at least 20% compared to traditional algorithm. (see lines 17-19 in page 1).

Q3: In the Introduction section, each of the cited items must be mentioned separately to present the advantages and disadvantages of the previous methods. Each of the references listed must add some value to the paper, if not, should not be cited.

Response:

We add some analysis about the references. The details are presented as follows.

For Gossip-based method, each sensor selects one or several connected neighbors randomly to send information. Although having low communication cost, gossip-based methods show slow convergence. The diffusion method uses a one-step convex combination of the received data, so it cannot converge to a consensus result. (see lines 34-38 in page 2).

Meanwhile, we delete some references.

 

 

Q4: From the literature review in lines 21-75, there is no critical review which motivates the proposal presented in further lines. Usually, the motivations and study requirements are based on the lack of the current stage of knowledge. In the current form, this lack is negligible or does not exist.

Response:

We add the motivations and study requirements in paper. The details are presented as follows.

 

We can find that many researches focus on the robustness. However, in practical scenarios, the heavy-tailed noise is a low probability event, and the model noise still can be presented as Gaussian distribution in most cases. Although the mentioned robust filter can provide a robustness performance in the presence of outliers, it will lose estimation accuracy when the noise is normal. Therefore, how to balance the robustness and estimation accuracy as well as obtain consensus results are significant problems.

Motivated by these, the paper proposes a distributed consensus state estimation method based on the Gaussian distribution and the student-t distribution.

(see lines 81-89 in page 3).

Q5: Subsection 2.1. Student-t distribution, it is not clear if the equations (1) and (2) are newly proposed by Authors or received from previous studies. If are not newly calculated, must be referenced to the primary sources.

Response:

We add the reference about the equations (1) and (2) in paper. (see lines 122 in page 3).

 

 

Q6: Considering the 2.2. Problem statement section, it is not clear what is proposed by the Authors and what is from previous researchers' publications. Authors must emphasize their work against already widely known studies.

 

Response:

 

 We merge the section 2.1 and section 2.2 of original manuscript. Specifically, we first provide the basic state estimation model like process equation and measurement equation. Then, we introduce the problem of heavy-tailed noise and traditional molding methods, like student’s t distribution. Finally, we analyze the disadvantages with existing methods and introduce the key problem that this paper focus on.

 

The details are presented as section 2. (see lines 107-141 in pages 3-4).

 

Q7: Referring to the words ‘H0 indicates that the process and measurement noises obey the Gaussian distribution as follows’, lines 139-140, it is not justified and difficult to follow how the Authors confirmed this statement. In its current form, it is more like an assumption and conclusion. Similar to the H1 description. It must be supported by results.

Response:

H0 and H1 are really two hypotheses.

For the traditional method, it is always supposed that the noises obey Gaussian distribution. In this paper, we added another hypothesis that the noises obey Student’s t distribution. Similar to traditional multiple-model algorithm, we compute the state estimation results based on the two different distributions. Then, the two results are fused based on different hypotheses weights.

 

Q8: There is no critical discussion in section 3.4. Discussion. Authors must provide some information on the limitation and disadvantages of the proposal.

Response:

We adjust the section 3.4 of original manuscript. The discussion in original manuscript has changed to Outlook, and we add some information on the limitation and disadvantages of the proposal in Outlook. (see lines 371-380 in page 17).

 

Q9: To the previous comment, if the discussion is not critical, an additional section The Outlook must be added.

Response:

We adjust the section 3.4 of original manuscript. First, part of section 3.4 is presented as comment of the algorithm in the end of section 4. (see lines 371-380 in page 16-17).

And the other part is presented at section 5. The section 5 of original manuscript has changed to “Conclusion and Outlook”. We add some information on the limitation and disadvantages of the proposal in Outlook. (see lines 401-410 in page 17).

 

Q10: There are many shortcuts and abbreviations that an additional section of Nomenclature must be provided.

Response:

We add a section about Nomenclatures in paper. (see lines 22-23 in page 1).

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors, please find below some suggestions concerning the submission Distributed Consensus Multi-distribution Filter for Heavy-tailed Noise, with the manuscript number: jsan-2978882, that must be resolved before any further considerations:

1.      The Abstract section must be provided with some general information where practically the studies of an area of research were used previously. There is no practical destination presented.

2.      Still, with the Abstract section, some rough results supporting the final conclusion must be raised. In the current form, the Authors do not mention the detailed results which quantify the main advantage.

3.      In the Introduction section, each of the cited items must be mentioned separately to present the advantages and disadvantages of the previous methods. Each of the references listed must add some value to the paper, if not, should not be cited.

4.      From the literature review in lines 21-75, there is no critical review which motivates the proposal presented in further lines. Usually, the motivations and study requirements are based on the lack of the current stage of knowledge. In the current form, this lack is negligible or does not exist.

5.      Subsection 2.1. Student-t distribution, it is not clear if the equations (1) and (2) are newly proposed by Authors or received from previous studies. If are not newly calculated, must be referenced to the primary sources.

6.      Considering the 2.2. Problem statement section, it is not clear what is proposed by the Authors and what is from previous researchers' publications. Authors must emphasize their work against already widely known studies.

7.      Referring to the words ‘H0 indicates that the process and measurement noises obey the Gaussian distribution as follows’, lines 139-140, it is not justified and difficult to follow how the Authors confirmed this statement. In its current form, it is more like an assumption and conclusion. Similar to the H1 description. It must be supported by results.

8.      There is no critical discussion in section 3.4. Discussion. Authors must provide some information on the limitation and disadvantages of the proposal.

9.      To the previous comment, if the discussion is not critical, an additional section The Outlook must be added.

10.  There are many shortcuts and abbreviations that an additional section of Nomenclature must be provided.

Generally, the submitted manuscript is interesting by the subject, however includes many weaknesses that make it generally weak and required proper improvements.

Author Response

Dear Editor and Reviewers:

We appreciate you very much for your positive and constructive comments and suggestions on our manuscript entitled “Distributed Consensus Multi-distribution Filter for Heavy-tailed Noise” (No: jsan-2978882).

We have studied reviewers’ comments and questions carefully and tried our best to revise over manuscript. The following are the responses and revisions.

Response on the comments of Reviewer #2

Q1: Line 37: I would say “estimation error of the CM method”.

Lines 43, 45, 49, 55…: I would suggest to add a space before the references. For example, in line 43 write “observer [17]” instead of “observer[17]”. Please, review the whole manuscript following this criterion.

Line 73: Please, review if the correct form is “parameter”, or “parameters”.

Line 90: Please, review if the correct form is “validate”, or “validates”.

Line 94: dis-tributed  à distributed

Line 111: Please, remove the space after n; (N , A).  à (N, A).

Line 100: It is said the first time in the article “η is dof”. I assume that dof means “degrees of freedom”, but I would suggest indicating the meaning of dof the first time you use it, for example “degrees of freedom (dof).

Line 102: Please, review if there is a space before the comma “When η = 1”, and if so, remove it.

Line 117: Please, remove the space after “time k”

Line 163: You used for the first time the acronym PDF. Please, explain what it means

Line 165: Please, review if there is a space before the comma “µr ,”, and if so, remove it 

Line 186: Please, remove the space after approximate “p(xk)”

Line 206: Please, remove the space and pi(·) . à pi(·).

Line 251: Review if there is a space before the point, and if so, please remove it  “St(x; xˆ, P, η) .”

Line 255: Please review if “for singe” can be substituted for “for a single”

Line 258: Please, remove the space after “node i “, before the comma, and the same for “H .” before the point.

Line 295: Please, review if in this part of the sentence is correct “with is the state and“ it would be better to use “are”.

Line 300: dis-cussed à discussed

Line 324: Please, review if the spelling of “selected from a in each simulation”.

Caption of Figures. Please, specify the meaning of the acronyms used in the graps, so the reader do not have to look for them in the whole text. (Figures 3, 4, 5, 6, and 7). Review the same aspect in the caption of the tables.

 

Response:

For the above problems, we have carefully corrected and checked the corresponding issues.

 

Q2: Lines 310 to 313. You indicate in the 3.4 Discussion section that “In a broad sense, the algorithm proposed in this paper can be regarded as a special case of the multi-model method. Therefore, the concepts of probability transfer and input interaction of multi-model method can also be applied to the algorithm proposed in this paper. How to integrate these technologies is one of the problems to be solved in the future”. It is a little bit confusing, as after the discussion section there is a section called “4. Numerical simulation”, so you still keep on describing part of your work after the discussion. According to the instructions for authors of the JSAN journal, the Manuscript should have the following sections “Introduction, Materials and Methods, Results, Discussion, Conclusions (optional).”. Therefore, I would suggest to add an expanded version of section 3.4, but before the conclusions, in which you could add some comments on the numerical simulation.

 

Response:

 

We adjust the section 3.4 of original manuscript. First, part of section 3.4 is presented as comment of the algorithm in the end of section 4. (see lines 371-380 in page 16-17).

And the other part is presented at section 5. The section 5 of original manuscript has changed to “Conclusion and Outlook”. We add some information on the limitation and disadvantages of the proposal in Outlook. (see lines 401-410 in page 17).

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

All of the raised issues were respond in a required manner and, respectively, the submission improved according to the comments so the revised manuscript can be considered for publication in a quality journal as the Journal of Sensor and Actuator Networks is.

Author Response

Dear Editor:

We appreciate you very much for your positive and constructive comments and suggestions on our manuscript entitled “Distributed Consensus Multi-distribution Filter for Heavy-tailed Noise” (No: jsan-2978882).

We have studied the comments and questions carefully and tried our best to revise over manuscript. The following are the responses and revisions.

 

Q1: Before giving acceptance to this paper, the authors are really expected to add a good number of figures to improve the readability of this paper, e.g.,

1. Figure about the research background and research problem
2. Figure about the studied system model or network model
3. Figure about the organization of this research paper
4. Figure about the workflow of the proposed method

Response:

1. We add the figure about the research background and research problem. (See Figure 1 in page 2).
2. We add the description about the system model in figure 4. (See Figure 4 in page 12).
3. We add the figure about the organization of this research paper. (See Figure 2 in page 4).
4. We add the figure about the workflow of the proposed method. (See Figure 4 in page 12).

 

Q2: Also, please kindly provide one high-quality tabled based comparison with related researches to highlight the novelty and research motivation of this paper.

Response:

 We add a table to present the differences between the proposed algorithm and related researches. (See Table 1 in page 3)

 

Q3: Actually, this paper should be further improved in following aspects:

1. An individual section on related work should be provided. The above mentioned high-quality tabled based comparison should be included in this section.
2. A section on System Model or Network Model should be provided. A new Table in which all used variables should be summarized and clearly explained.
3. The currently presented information on the simulation is not clear enough. How about the transmission range of sensor nodes? How about the research result if the node density is changed in the network?
4. Conducting simulation on only one network toplogy is surely not enough. The authors should conduct many more rounds experiments on different node densities and network toplogies.

 

Response:

 a: The introduction is divided into two parts. The first part is Background, and the second part is Related work, including a table about the comparison with other researches.

b:  The system model or network model has been presented in Section 2. Besides, we add a table to summarize the variables of system model. (See Table 2 in page 4).

c: In this paper, we focus on the estimation problem with heavy-tailed noise, and distributed sensor network is a classical application. Thus, we did not spend too much space introducing some of the characteristics of the network.

 In simulation, the choice of transmission range is to obtain a network topology, and different transmission ranges will lead to different network topologies. Specifical, we set the transmission range as 825.

 

We add another experiments to test the performance of the proposed algorithm in a network with fewer nodes and lower density. The simulation results are presented in figure 11, Table 5 and Table 6.(See page 18).

 

d: We add another experiments to test the performance of the proposed algorithm in a network with fewer nodes and lower density. The network topology is presented in figure 11. The simulation results are presented in Table 5 and Table 6. (See Table 5 and Table 6 in page 18)