Trigonometric function transformation and its application in software reliability modeling

Dai-Nghia Vy; Van-Thuan Nguyen; Quyet-Thang Huynh; Trung-Nghia Phung; Hung-Cuong Nguyen

doi:10.1371/journal.pone.0339286

Peer Review History

Original SubmissionOctober 2, 2025
27 Oct 2025 Decision Letter - Mallikarjuna Reddy Kesama, Editor PONE-D-25-53209Trigonometric function transformation and its application in software reliability modelingPLOS ONE Dear Dr. Nguyen, Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process. Please submit your revised manuscript by Dec 11 2025 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file. 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There is no requirement to cite these works unless the editor has indicated otherwise. [Note: HTML markup is below. Please do not edit.] Reviewers' comments: Reviewer's Responses to Questions Comments to the Author 1. Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #1: Yes Reviewer #2: No Reviewer #3: Partly ******** 2. Has the statistical analysis been performed appropriately and rigorously? Reviewer #1: N/A Reviewer #2: Yes Reviewer #3: N/A ****** 3. Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified. Reviewer #1: Yes Reviewer #2: Yes Reviewer #3: Yes ****** 4. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #1: No Reviewer #2: Yes Reviewer #3: No ****** 5. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters.) Reviewer #1: Dear authors, In this study, the transformation of four types of S-shaped functions using trigonometric functions is demonstrated. The mean value function has also been calculated by considering these functions. The idea of using a transformed function in software reliability is novel, but it needs improvements: 1. The literature review is too brief. 2. Writing improvements are needed: the paper contains some lengthy sentences, and paragraphs are not justified. Condensing and improving logical flow is recommended. 3. It would be better to write the limitations of the transformation, i.e., write the cases when such a transformation is not possible. 4. The representation of Figure 9 may be improved if proper parameters are considered, specially for the red line of b1(t) and orange line of b3(t). Reviewer #2: Please see the attached file for the details. The authors propose using the trigonometric sine function transformation to model the S-shaped curve of the cumulative number of detection faults during software testing. But the authors did not verify them through experiments. Reviewer #3: This article seems to be an extended study of the following manuscript. "Hung-Cuong, N., Quyet-Thang, H., Trung-Nghia, P., Dai-Nghia, V. (2025). Transformating trigonometric function to apply in software reliability modeling. In: Nghia, P.T., Thai, V.D., Thuy, N.T., Huynh, VN., Van Huan, N. (eds) Advances in Information and Communication Technology. ICTA 2024. Lecture Notes in Networks and Systems, vol 1205. Springer, Cham. https://doi.org/10.1007/978-3-031-80943-9_13". The novel contribution of this paper is limited. This paper contains many grammatical mistakes. English writing need to be improved, severely. ****** 6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. Reviewer #1: No Reviewer #2: No Reviewer #3: No ******** [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] To ensure your figures meet our technical requirements, please review our figure guidelines: https://journals.plos.org/plosone/s/figures You may also use PLOS’s free figure tool, NAAS, to help you prepare publication quality figures: https://journals.plos.org/plosone/s/figures#loc-tools-for-figure-preparation. NAAS will assess whether your figures meet our technical requirements by comparing each figure against our figure specifications. Attachments Attachment Submitted filename: Review report.docx Attachment Submitted filename: comments-1.pdf https://doi.org/10.1371/journal.pone.0339286.r001
Revision 1
15 Nov 2025 Author Response Firstly, the authors would like to thank the reviewers for their valuable comments. We answer your question/comment as follows: A. Academic editor Comment 1: 1. Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at https://journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and https://journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf Our answer: We would like to thank you for the comment of the academic editor. We prepare our manuscript using the PLOS ONE style template. Comment 2: 2. Please note that PLOS One has specific guidelines on code sharing for submissions in which author-generated code underpins the findings in the manuscript. In these cases, we expect all author-generated code to be made available without restrictions upon publication of the work. Please review our guidelines at https://journals.plos.org/plosone/s/materials-and-software-sharing#loc-sharing-code and ensure that your code is shared in a way that follows best practice and facilitates reproducibility and reuse. Our answer: We would like to thank you for the comment of the academic editor. Our manuscript does not have any source code. It provides a theoretical analysis of the application of the transformation of trigonometric functions in Software reliability modelling. Comment 3: 3. Please update your submission to use the PLOS LaTeX template. The template and more information on our requirements for LaTeX submissions can be found at http://journals.plos.org/plosone/s/latex. Our answer: We would like to thank you for the comment of the academic editor. We prepare our manuscript using the PLOS ONE LaTeX template. Comment 4: 4. Thank you for stating the following financial disclosure: “This research is supported by Hung Vuong University under grant number HV33.2024” Please state what role the funders took in the study. If the funders had no role, please state: "The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript." If this statement is not correct you must amend it as needed. Please include this amended Role of Funder statement in your cover letter; we will change the online submission form on your behalf. Our answer: We would like to thank you for the comment of the academic editor. We have feedback on your comment as follows: - We move the Role of Funder to the Cover letter as you require. - Authors Vy Dai-Nghia, Nguyen Van-Thuan, and Nguyen Hung-Cuong are lecturers of Hung Vuong University. This work was done in the lab at our University with the financial support of our University. So we state: "This work was funded by Hung Vuong University (grant HV33.2024)" in the Cover letter. Comment 5: 5. Please note that funding information should not appear in the Acknowledgments section or other areas of your manuscript. We will only publish funding information present in the Funding Statement section of the online submission form. Please remove any funding-related text from the manuscript. Our answer: We would like to thank you for the comment of the academic editor. We will remove the funding information in the Acknowledgments section as you require. Comment 6: 6. When completing the data availability statement of the submission form, you indicated that you will make your data available on acceptance. We strongly recommend all authors decide on a data sharing plan before acceptance, as the process can be lengthy and hold up publication timelines. Please note that, though access restrictions are acceptable now, your entire data will need to be made freely accessible if your manuscript is accepted for publication. This policy applies to all data except where public deposition would breach compliance with the protocol approved by your research ethics board. If you are unable to adhere to our open data policy, please kindly revise your statement to explain your reasoning and we will seek the editor's input on an exemption. Please be assured that, once you have provided your new statement, the assessment of your exemption will not hold up the peer review process. Our answer: We would like to thank you for the comment of the academic editor. Our manuscript does not have any source code. It provides a theoretical analysis of the application of the transformation of trigonometric functions in Software reliability modelling. Comment 7: 7. PLOS requires an ORCID iD for the corresponding author in Editorial Manager on papers submitted after December 6th, 2016. Please ensure that you have an ORCID iD and that it is validated in Editorial Manager. To do this, go to ‘Update my Information’ (in the upper left-hand corner of the main menu), and click on the Fetch/Validate link next to the ORCID field. This will take you to the ORCID site and allow you to create a new iD or authenticate a pre-existing iD in Editorial Manager. Our answer: We would like to thank you for the comment of the academic editor. I updated my ORCID in My Information in the Editorial Manager system. My ORCID is 0009-0002-5974-8518. Comment 8: 8. Please include a separate caption for each figure in your manuscript. Our answer: We would like to thank you for the comment of the academic editor. We provide a separate caption for each figure in our manuscript. Comment 9: 9. Please upload a new copy of Figures 1 to 9 as the detail is not clear. Please follow the link for more information: https://journals.plos.org/plosone/s/figures Our answer: We would like to thank you for the comment of the academic editor. We update the new version of figures in .tiff and .eps format to improve the quality of figures. Comment 10: 10. We note you have included a table to which you do not refer in the text of your manuscript. Please ensure that you refer to Table 4 in your text; if accepted, production will need this reference to link the reader to the Table. Our answer: We would like to thank you for the comment of the academic editor. It is our mistake. We must refer to Tables 3 and 4 instead of Tables 2 and 3. We update the reference of Table 4 as highlighted on page 10. Comment 11: 11. If the reviewer comments include a recommendation to cite specific previously published works, please review and evaluate these publications to determine whether they are relevant and should be cited. There is no requirement to cite these works unless the editor has indicated otherwise. Our answer: We would like to thank you for the comment of the academic editor. Reviewer 3 mentioned that this work is extended from "Hung-Cuong, N., Quyet-Thang, H., Trung-Nghia, P., Dai-Nghia, V. (2025). Transformating trigonometric function to apply in software reliability modeling. In: Nghia, P.T., Thai, V.D., Thuy, N.T., Huynh, VN., Van Huan, N. (eds) Advances in Information and Communication Technology. ICTA 2024. Lecture Notes in Networks and Systems, vol 1205. Springer, Cham. https://doi.org/10.1007/978-3-031-80943-9_13". So we cite this paper in subsection 3.2 on page 4 to inherit this previous work. B. Reviewer #1 Comment 1: Dear authors, In this study, the transformation of four types of S-shaped functions using trigonometric functions is demonstrated. The mean value function has also been calculated by considering these functions. The idea of using a transformed function in software reliability is novel, but it needs improvements: 1. The literature review is too brief. Our answer: We would like to thank you for the comment of reviewer. We add Table 1 on page 4 to list some studied NHPP SRM that use the S-shaped function. Those models provide an overview of previous works. Comment 2: 2. Writing improvements are needed: the paper contains some lengthy sentences, and paragraphs are not justified. Condensing and improving logical flow is recommended. Our answer: We would like to thank you for the comment of reviewer. We have checked the grammatical mistakes in English and edited our manuscript carefully. Comment 3: 3. It would be better to write the limitations of the transformation, i.e., write the cases when such a transformation is not possible. Our answer: We would like to thank you for the comment of reviewer. In our work, the most famous trigonometric function, sin(t), is studied. The domain of this function is the full set of real numbers, i.e., t ∈ R. So this function can be transformed in all cases. Comment 4: 4. The representation of Figure 9 may be improved if proper parameters are considered, specially for the red line of b1(t) and orange line of b3(t). Our answer: We would like to thank you for the comment of reviewer. We have feedback on your comment as follows: As mentioned on page 10: "We can see that this form is not fully S-shaped because it only contains the two last phases of the S-shaped curve as described in sub-section 2.2". Furthermore, b1(t) must pass through the origin of the Cartesian coordinate system, i.e., t = 0, then b1(t) = 0. So b1(t) is illustrated by b = 2.0. - If β < 0, b3(t) has the similar shape with b1(t) which is "not fully S-shaped". So b3(t) is illustrated by (b, β, k) = (2.0, −4.0, 1.5). So we propose to retain this figure to illustrate the S-shape pattern of the 4 types of S-shaped functions. C. Reviewer #2 Comment 1: The authors propose using the trigonometric sine function transformation to model the S-shaped curve of the cumulative number of detection faults during software testing. But the authors did not verify them through experiments. Our answer: We would like to thank you for the comment of reviewer. Our manuscript provides a theoretical analysis of the application of the transformation of trigonometric functions in Software reliability modelling without any experimental evaluation. The experimental results will be studied and discussed in future works. Comment 2: The problems of this paper are as follows, 1) Although the authors introduced the property of trigonometric sine function transformation, such as bsin(t) and b sin(βt + φ) + k, they indeed ignored the limitation of t and t ≥ 0 during software testing. This restriction will seriously weaken the accuracy of the fault detection rate based on the sine function. I suggest the author compare the fitting and predictive performance of the model proposed in Table 1 with other models. In addition, previous studies have used the sine function to establish software reliability models. I suggest the author to include those models for comparison when conducting experiments. Our answer: We would like to thank you for the comment of reviewer. We have feedback on your comment as follows: - In this work, we evaluate the mathematical properties of 4 types of S-shaped functions. All of the previous types of S-shaped function, i.e., b1(t), b2(t), and b3(t), are not considered in the case of t < 0. - Our manuscript provides a theoretical analysis of the application of the transformation of trigonometric functions in Software reliability modelling without any experimental evaluation. The experimental results will be studied and discussed in future works. - We do not know any SRM that uses the trigonometric sine function. If there is any model that uses the trigonometric sine function, we will use it to compare with our proposed models in our next work. Comment 3: 2) In Subsection 2.2, “Phase 3. The function goes through inflection point, which is the point that the derivation of the function changes from positive to negative quantities.” I disagree with this statement. The S-shaped function (i.e. mean value function) in the software reliability model is a non-decreasing function in any situation. In other words, the derivative of the S-shaped function (mean value function) in the software reliability model is never negative under any circumstances. Our answer: We would like to thank you for the comment of reviewer. We agree with your statement: "The S-shaped function (i.e. mean value function) in the software reliability model is a non-decreasing function in any situation.". However, subsection 2.2 discusses the value of the derivation of the function. For example, b2(t) has form: b_2 (t)=b 1/(1+βe^(-bt) ) And the first derivation of the function: ∂/∂t b_2 (t)=(b^2 βe^(-bt))/(1+βe^(-bt) )^2 And in subsection 4.2, we present that the inflection point of b2(t) is (ln⁡β/b, b/2). This means that when t<ln⁡β/b, we have ∂/∂t b_2 (t)>0. And vice versa, when t>ln⁡β/b, we have ∂/∂t b_2 (t)<0. Furthermore, we change from "the derivation of the function" to "the first derivation of the function" on page 3 to emphasize our opinion. Comment 4: 3) Some grammar errors are as follows, Under equations (7,9,11), “Subtitute...” should be “Substitute...”, you must check it in the whole paper. In the last paragraph of Subsection 3.1, “sin(t) increase...” should be “sin(t) increases...”. Our answer: We would like to thank you for the comment of reviewer. We have checked the grammatical mistakes in English and edited our manuscript carefully. D. Reviewer #3 Comment 1: This article seems to be an extended study of the following manuscript. "Hung-Cuong, N., Quyet-Thang, H., Trung-Nghia, P., Dai-Nghia, V. (2025). Transformating trigonometric function to apply in software reliability modeling. In: Nghia, P.T., Thai, V.D., Thuy, N.T., Huynh, VN., Van Huan, N. (eds) Advances in Information and Communication Technology. ICTA 2024. Lecture Notes in Networks and Systems, vol 1205. Springer, Cham. https://doi.org/10.1007/978-3-031-80943-9_13". Our answer: We would like to thank you for the comment of reviewer. So we cite this paper in subsection 3.2 on page 4 to inherit this previous work. Comment 2: The novel contribution of this paper is limited. Our answer: We would like to thank you for the comment of reviewer. We have feedback on your comment as follows. In abstract, we address our contribution in this manuscript as follows: "The first contribution is a deep mathematical appreciation of three well-known S-shaped functions. The second is the mathematical transformation of the trigonometric function to meet the real assumption. The last are the advantages and the applicability of this transformation in software reliability modeling.". All of those contributions have not been studied in any previous works. Comment 3: This paper contains many grammatical mistakes. English writing need to be improved, severely. Our answer: We would like to thank you for the comment of reviewer. We have checked the grammatical mistakes in English and edited our manuscript carefully. Attachments Attachment Submitted filename: Response to Reviewers.pdf https://doi.org/10.1371/journal.pone.0339286.r002
4 Dec 2025 Decision Letter - Mallikarjuna Reddy Kesama, Editor Trigonometric function transformation and its application in software reliability modeling PONE-D-25-53209R1 Dear Dr. Van Thuan Nguyen, We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements. Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication. An invoice will be generated when your article is formally accepted. Please note, if your institution has a publishing partnership with PLOS and your article meets the relevant criteria, all or part of your publication costs will be covered. 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Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified. Reviewer #1: Yes ****** 5. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #1: Yes ****** 6. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #1: Since the authors answered all my queries, the article may be accepted for publication after checking the format. ****** 7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. Reviewer #1: No ******** <qb-div data-qb-element="re-enable-flow" style="z-index: 2147483647; max-width: 1px; max-height: 1px; box-sizing: border-box; position: fixed; top: 10px; right: 10px;"><qb-div style="all: initial !important;"></qb-div></qb-div> Attachments Attachment Submitted filename: Review report R1.docx https://doi.org/10.1371/journal.pone.0339286.r003
Formally Accepted
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