Local Sensitivity of Failure Probability through Polynomial Regression and Importance Sampling

Round 1

Reviewer 1 Report

This paper proposes a method to calculate the deterministic part of the partial derivative of the degradation function. Based on the Weak approach, they use the change of variables and polynomial regression based on the Tyler expansion for better results.

Comments:

1.       In this paper, it is assumed that g(s,x) is known while many degradation reliability problems are studying the relationship between system failure and s and x. It is necessary to add the literature in the beginning about some engineering examples that the degradation function is known. It would be interesting to study whether this method works well in the setting where the g(s,x) needs to be estimated from the observations.

2.       Figure 1, why in b, P4 has larger difference than P2 for the same sigma? A log-scale y-axis plot (absolute difference) is needed here since the paper is interested in the difference when the sigma is small.

3.       Some hyperparameter tuning issues: How do you choose the importance sampling density function? And it is necessary to show after the importance sampling, you can actually compare the sampling efficiency, which can be done by doing a Kolmogorov–Smirnov test between the sampled distributions with the true distribution. How do you choose different smooth failure domain indicator approximation functions?

4.       The choice of degree n in this paper is quite close to the information criteria (etc., AIC/BIC). Extensive discussion or additional simulation results using AIC/BIC criteria are needed.

 

5.       Table 1, 2, and 3, explain what the percentage numbers mean in the title. 

Author Response

The Reviewer will find our response in the attached PDF document.

Author Response File: Author Response.pdf

Reviewer 2 Report

In this paper a new method to compute the local sensitivity of a failure probability with respect to design parameters or distribution parameters is presented, based on a heteroscedastic polynomial regression. This approach is inspired by the recent Weak approach framework and is presented as an improvement of the latter. The main innovation of the proposed approach is to express the sensitivity estimate as the constant coefficient of a Taylor series expansion, which can be recovered with a polynomial regression. The proposed approach can be applied to various simulation methods and is presented here
with IS. Moreover, this approach is independent of the dimension of the system, and the distribution of the inputs along the shape of the failure domain. Indeed, after a change of variable in the integral of interest, only the scalar response of the system matters.

One of the main outlooks of the proposed method is to improve the stability of
the resulting sensitivity estimate. Indeed, in several applications, the mean value of the test is very close to the reference value but the CV can be slightly too large for the method to be definitely an upgrade of the Weak approach. Another interesting outlook of the proposed approach is to obtain a better estimation of the theoretical variance of the sensitivity estimate. Indeed, with the Weak approach, the theoretical variance is already available with formula Eq. (25) for IS. The availability of an accurate estimation of the
theoretical variance is of great interest.

I think this paper is mainly of theoretical interest until the authors explain the assumption of normal distributions in IS. Could you give some explanations about it?

Author Response

The Reviewer will find our response in the attached PDF document.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments for author File: Comments.pdf

Author Response

The Reviewer will find our response in the attached PDF document.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Thanks again for the interesting paper. Figures 1(b) and (c) may also have different line types to better distinguish the two absolute differences. I don't have other comments.

Author Response

We agree with the Reviewer and have made the proposed changes to the Figure 1. 

Reviewer 2 Report

The authors corrected the paper concerning my recommendation.

Author Response

We thank the Reviewer for his/her comments on the article.