Sensitivity Analysis of Mathematical Models

Round 1

Reviewer 1 Report

Please see the attached review report.

Comments for author File: Comments.pdf

Author Response

Dear colleague,

Thank you for the provided review. Your remarks are very valuable and could help me to make the article better.

Regarding to the questions,

Q1. In introduction section line 37-40, what are the other factors on which the partial derivative depends.

Here I mean that when there are connections between factors of the model (e.g., in case when correlations exist, or adding multiplication of factors could increase the quality of model) the partial derivative transitively depends not only on chosen factor but on other factors related to the chosen one. May be the formulation was not clear. I have re-write this sentence.

Q2. Why the partial derives can not work for non-linear models.

It is also wrong formulation. Of course, local technique can be applied to the non-linear model. The main idea to receive the accurate result is that factors are independent on each other, which can be seen in non-linear models. It means that there are limitations on correct application of local SA techniques. This point was also re-formulated.

Q3. The introduction is very short. The author needs to add two to three paragraphs.

Q7. The author may also add some recants work about the current study related to sensitivity analysis of some real-world mathematical models and its applications in introduction part. They may also cite some other related work to increase the number of references as the current references are very few.

There have been added references to 13 applied studies related to applications of different SA approaches to particular problems. There are highlighted medicine and environmental modeling as the most appliable areas of SA, also mentioned applications in Economics, Robotics and Psychology.

Q4. The introduction should also make a compelling case for why the study is useful along with a clear statement of its novelty or originality by providing relevant information and providing answers to basic questions such as:

1. What is already known in the literature?
2. What was done and how it was done?

In the introduction part it was added the novelty points on the proposed technique. Comparing to the existing approaches our technique does not use approximation of the original model, thus it does not cause the lack of the accuracy of  obtained results; the approach permits the interconnection between factors, the only limitation is the existing of the first partial derivatives. Our technique also takes into account all available information both of factors and parameters of the initial model, which makes  obtained results of the analysis sustainable.

Q5. There is no numbering of equations from page 2-4, which make it hard to understand about which tern and equation the author is taking, for example in the last two equation on page 4, where the author says that the integral of the expression from 0 to 1 is zero, the expression is unique? I guess the author is

taking about the last term in the second last equation on page 4? What about the second term in the same equation?

Thank you for the comment. The numbering in this part of the paper was added, it makes the understanding better.

Q6. Page 4 first equation what is the reason for considering only the term above the main diagonal in the second term on the right-hand side of the equation.

The obtained matrix is symmetric, it is enough to take under investigation the only half of the matrix.

Q8. In the error table on page 12, when t increases the absolute error also increases, while in example 1 page 10, it is opposite, why?

Could you please clarify the question? Table on page 12 contains sensitivity measures, not errors.

Q9. Is the first mathematical expression on page 7 is correct? One of the expression may need equal sign (h_ij=).

The expression is correct. The Mann-Whitney U-statistic defines the exact number of pairs of values x1_i and x_2j, for which x_1i < x_2j.

Q10. Even though the article is well written, but the author may still look for some punctuation and editing issues.

Thank you! The grammar is checked by English-speaking person.

Reviewer 2 Report

In this article, the author proposes an alternative approach for sensitivity analysis based on applying Analysis of Finite Fluctuations, which uses the Lagrange mean value theorem to
estimate the contribution of changes in the variables of a function to the output change. Author investigate the presented approach on the example of a class of fully connected neural network models. As a result of Sensitivity Analysis, a set of sensitivity measures for each input is obtained. For their averaging it is proposed to use a point and
interval estimation algorithm using Tukey's weighted average. A comparison of the described method with the computation of Sobol's indices is given; the consistency of the proposed method is shown. The computational robustness of the procedure for finding sensitivity measures of inputs is investigated. Numerical experiments are carried
out on the neuraldat data set of the NeuralNetTools library of the R data processing language.

I have the following comments

1. Authors should expand the background and motivation for this article explicitly. 

2. I observed a lot of grammatical errors. It should be fixed in the revised version

3. Beta_j should be defined in the formula for SRC_j

4. It will be nice for authors to add code for their analysis for reproducibility and also to encourage open science

5. The article is difficult to follow. The author should work on the flow.

6. The article lacks in-depth discussion, limitations and conclusion. This should be extended

Author Response

Dear colleague,

Thank you for the provided review. Your remarks are very valuable and could help me to make the article better.

Regarding to the questions,

Q1. Authors should expand the background and motivation for this article explicitly.

I have added the quick review on applied studies involving Sensitivity Analysis, namely in the field of Medicine and environmental field, basically in Economics, Robotics and Psychology (lines 52-72). I also put into introduction part issues that contains the novelty of the study. They are namely presented in lines 74-82.

Q2. I observed a lot of grammatical errors. It should be fixed in the revised version

Thank you for the comment. The article was checked by English speaking person.

Q3. Beta_j should be defined in the formula for SRC_j

The explanation of beta parameters is added into the article (line 113).

Q4. It will be nice for authors to add code for their analysis for reproducibility and also to encourage open science

Now we are preparing R package to be posted. It will be announced as soon as possible. Next studies will have links to the repository.

Q5. The article is difficult to follow. The author should work on the flow.

I have tried to reformulate some passages to make the understanding of the paper better.

Q6. The article lacks in-depth discussion, limitations and conclusion. This should be extended

In the conclusion section the limitation on applying the proposed approach is added as well as advantages of the technique comparing with the existing approaches.

Round 2

Reviewer 2 Report

Satisfied with the new version of the article