Bounded-Error Parameter Estimation Using Integro-Differential Equations for Hindmarsh–Rose Model
Round 1
Reviewer 1 Report
Due to the fact that this submission heavily relies on previous work of the authors, I have carefully read the paper "Parameter estimation procedure based on input-output integro-differential polynomials. Application to the Hindmarsh-Rose model." (published in the Proceedings of the 2020 European Control Conference - ECC) and the new submission "Bounded-error parameter estimation using integro-differential equations for Hindmarsh-Rose model" (submitted newly to the journal Algorithms).
Both papers deal with identifying the parameters of dynamic systems by firstly deriving an input-output relation that eliminates non-measured internal system states and that may depend on high-order derivatives of the measured system outputs. Estimating the derivatives of noisy measurements is well known to be a challenging task. Due to the influence of noise, these derivatives become easily imprecise and may significantly influence the result of a parameter identification.
Interestingly, the authors suggest to perform temporal integrations of the input-output relations to attenuate the effect of noisy measurements and the make the parameter identification more reliable.
Especially for the application scenario at hand (the Hindmarsh-Rose model), slight variations of the estimated system parameter may lead to totally different dynamics if the true value is closely located to a Hopf bifurcation point. Therefore, it is essential to quantify the reliability of the estimation accuracy by using sophisticated means.
Although the discussion above is similar in both articles (I recommend not to remove this because this fundamental information is essential to understand the intention of the work), the algorithms used for parameter identification are totally different.
In the paper published at ECC 2020, classical floating-point methods were used to create stochastically disturbed measurements after evaluating the model M times with a subsequent parameter identification. the results are then used to determine the number of runs leading to estimation results that are within a probabilistically motivated bound around the "true" system behavior.
In contrast to this, the new submission does not use a probabilistic interpretation of measurement noise. Instead, the system outputs are assumed to be disturbed by bounded uncertainty with unknown probability distributions in their interior. To handle this task, the concept of interval arithmetic is used. It is employed to analyze the accuracy of the parameter identification algorithm (relying on the same integration relations as in the ECC paper). The results of the proposed new approach are interval bounds for all parameters to be identified in which the true parameters are located with 100% certainty if the measurement noise is assumed to be bounded. This kind of result is impossible to be obtained with the old algorithm published in ECC'20. Moreover, these results allow for detecting the structural changes of the system dynamics with certainty, by checking whether the parameter values corresponding to the bifurcation point are included or not in the estimated parameter ranges.
Due to this significantly different methodology, I strongly recommend to accept this paper for publication. It deals with a methods that is mathematically correct and not yet considered in this form.
Two changes, however, should be made before publishing the article:
a) I recommend that the authors add either structure diagrams or flow charts of the old stochastic procedure (ECC'20) and of the new version to clearly visualize the differences that I summarized above. It may be tedious for a reader (who is not an expert on interval methods) to directly see the fundamental differences of both papers.
b) Mention a further alternative to the parameter identification proposed by the authors, namely, the use of interval methods with a subsequent subdivision of parameter domains in order to reliably identify not plausible parameter subintervals. This method, however, is much more computationally demanding and may significantly be affected by the wrapping effect of interval analysis if specific properties such as cooperativity are not satisfied, cf.
http://dx.doi.org/10.3182/20120711-3-BE-2027.00374
http://dx.doi.org/10.1016/j.ifacol.2018.03.058
During the recommended revision, please also check again the paper from a language point of view. Moreover, please make sure that interval notations (which are not necessarily standard for all readers) are clearly introduced at the first place of appearance.
Author Response
Dear Editor in Chief, Dear Reviewers,
This cover note concerns the article with Manuscript ID: algorithms-1695737, entitled "Bounded-error parameter estimation using integro-differential equations for Hindmarsh-Rose model", by Carine Jauberthie and Nathalie Verdière. The decision was that this paper can not be accepted in its first form and may be resubmitted with revision.
After reading the comments of the Associate Editor and the three reviewers, we have revised our manuscript (the revised parts are in red color) and we resubmit the revision.
We thank you very much for your suggestions and we thank the Associate Editor and rewievers for their worthwhile comments.
You have suggested in the cover paper uploading a document that details the changes (our responses are in blue color) and explains how the comments of the Associate Editor and the referees have been addressed.
We are happy to provide our answers in attachement.
With best regards,
Carine Jauberthie
Nathalie Verdière
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The paper investigates a numerical parameter estimation method based on input-output integro-differential polynomials within a bounded-error framework. The measurement noise and parameters are connected sets (in the proposed work, intervals). This method is based on the Rosenfeld-Groebner elimination algorithm presenting differential equations containing derivatives, sometimes of a high order. The differential relations are pretreated and some integrations are performed in order to improve the numerical results. The resulting relations contain essentially only integrals that depend on the outputs. The initial relations are less sensitive to noise on the measurements than the initial relations. This method is then applied to the Hindmarsh-Rose model, a slow-fast model capable of reproducing the main behaviors of a neuron. The impact of the size of the measurement noise domain on the estimated intervals is also investigated.
The manuscripts is well written and includes some interesting results in parameter estimation problem. The author should consider the following comments:
- The role of the parameter u(t) is not clear in the discussion. Please clarify this control variable.
- The comparison and advantages of the underlying method (ID-IO polynomials) with other methods such as Least Squares Approach should be mentioned.
- The calculation seams correct. But the authors may need to address some related issues such as the nonlinearity and uniqueness of the estimates.
- Clarify the required restrictions on the observations/data.
- I recommend having some discussions about the sensitivity analysis of estimates to the noisy data; see “Sensitivity analysis for dynamic systems with time-lags, Journal of Computational and Applied Mathematics 151 (2), 2003, 445-462”
- Insert some Remarks to highlight the main findings.
Author Response
Dear Editor in Chief, Dear Reviewer 2,
This cover note concerns the article with Manuscript ID: algorithms-1695737, entitled "Bounded-error parameter estimation using integro-differential equations for Hindmarsh-Rose model", by Carine Jauberthie and Nathalie Verdière. The decision was that this paper can not be accepted in its first form and may be resubmitted with revision.
After reading the comments of the Associate Editor and the three reviewers, we have revised our manuscript (the revised parts are in red color) and we resubmit the revision.
We thank you very much for your suggestions and we thank the Associate Editor and rewievers for their worthwhile comments.
You have suggested in the cover paper uploading a document that details the changes (our responses are in blue color) and explains how the comments of the Associate Editor and the referees have been addressed.
We are happy to provide our answers in attachement.
With best regards,
Carine Jauberthie
Nathalie Verdière
Author Response File: Author Response.pdf
Reviewer 3 Report
There is no theoretical support for the proposed method, and the referee did not find enough new insights to support the acceptance of this work for publication in this journal.
Author Response
Dear Editor in Chief, Dear Reviewer 3,
This cover note concerns the article with Manuscript ID: algorithms-1695737, entitled "Bounded-error parameter estimation using integro-differential equations for Hindmarsh-Rose model", by Carine Jauberthie and Nathalie Verdière. The decision was that this paper can not be accepted in its first form and may be resubmitted with revision.
After reading the comments of the Associate Editor and the three reviewers, we have revised our manuscript (the revised parts are in red color) and we resubmit the revision.
We thank you very much for your suggestions and we thank the Associate Editor and rewievers for their worthwhile comments.
You have suggested in the cover paper uploading a document that details the changes (our responses are in blue color) and explains how the comments of the Associate Editor and the referees have been addressed.
We are happy to provide our answers in attachement.
With best regards,
Carine Jauberthie
Nathalie Verdière
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
The revision looks good.
