A Hierarchical Approach for Joint Parameter and State Estimation of a Bilinear System with Autoregressive Noise

Round 1

Reviewer 1 Report

This work is interesting and the topic extremely important from the application point of view, in my opinion. It is also well-written and well-structured in the main parts.  However, I have some suggestions to increase the quality and the impact of the paper. See below.

- Regarding the assumption 1: are you assuming Gaussian noise? or not? for the Kalman filtering this assumption is needed. Please, clarify it.

- Regarding the assumption 2: can you extend your work doing  also model selection? Discuss it as future line (see also second reference below).

- Please improve (with more explanations) the caption of Figures 1 and 2.

- It is important to improve the state-of-the-art to increase the impact and appealing of the paper.   

I strongly suggest to discuss about in the introduction related particle Monte Carlo methods (for tracking and positioning) where also also static parameters are estimated (as in your paper) and/or a model selection is performed, for instance consider:

 

- N. Chopin, P. E. Jacob, O. Papaspiliopoulos. SMC2: an efficient algorithm for sequential analysis of state-space models. arXiv:1101.1528, 2013.

- L. Martino, J. Read, V. Elvira, F. Louzada, Cooperative Parallel Particle Filters for on-Line Model Selection and Applications to Urban Mobility, Digital Signal Processing Vol. 60, pp. 172-185, 2017.

-  I. Urteaga, M. F. Bugallo, and P. M. Djuric. Sequential Monte Carlo methods under model uncertainty, IEEE Statistical Signal Processing Workshop (SSP), pages 15, 2016.

- L. Martino, V. Elvira, G. Camps-Valls, Distributed Particle Metropolis-Hastings schemes, IEEE Statistical Signal Processing Workshop, (SSP), 2018.

- C. M. Carvalho, M. S. Johannes, H. F. Lopes, and N. G. Polson. Particle Learning and Smoothing. Statist. Sci., Volume 25, Number 1 (2010), 88-106.

This discussion can substantially increase the number of interested readers.

- Please, upload the final version of your manuscript in Arxiv and/or ResearchGate when/if published, to increase the diffusion and the possible citations of this work.

Author Response

Response to review

 

This work is interesting and the topic extremely important from the application point of view, in my opinion. It is also well-written and well-structured in the main parts. However, I have some suggestions to increase the quality and the impact of the paper. See below.

The authors' response: Thank you for your positive comments.

 

- Regarding the assumption 1: are you assuming Gaussian noise? or not? for the Kalman filtering this assumption is needed. Please, clarify it.

The authors' response: Yes. $v_t$ is a Gaussian noise with zero mean and variance $\sigma^2$. According to your advice, we have clarified it. Please see Assumption 1 on Page 4.

 

- Regarding the assumption 2: can you extend your work doing also model selection? Discuss it as future line (see also second reference below).

The authors' response: According to your suggestion, we will combine the particle Monte Carlo methods to study the model selection. Please see Conclusions.

 

- Please improve (with more explanations) the caption of Figures 1 and 2.

The authors' response: After taking consideration of your advice, we have improved the caption and added more explanations of Figures 1 and 2. Please see the third paragraph on Page 11.

 

- It is important to improve the state-of-the-art to increase the impact and appealing of the paper.  

I strongly suggest to discuss about in the introduction related particle Monte Carlo methods (for tracking and positioning) where also static parameters are estimated (as in your paper) and/or a model selection is performed, for instance consider:

 

- N. Chopin, P. E. Jacob, O. Papaspiliopoulos. SMC2: an efficient algorithm for sequential analysis of state-space models. arXiv:1101.1528, 2013.

 

- L. Martino, J. Read, V. Elvira, F. Louzada, Cooperative Parallel Particle Filters for on-Line Model Selection and Applications to Urban Mobility, Digital Signal Processing Vol. 60, pp. 172-185, 2017.

 

-  I. Urteaga, M. F. Bugallo, and P. M. Djuric. Sequential Monte Carlo methods under model uncertainty, IEEE Statistical Signal Processing Workshop (SSP), pages 15, 2016.

 

- L. Martino, V. Elvira, G. Camps-Valls, Distributed Particle Metropolis-Hastings schemes, IEEE Statistical Signal Processing Workshop, (SSP), 2018.

 

- C. M. Carvalho, M. S. Johannes, H. F. Lopes, and N. G. Polson. Particle Learning and Smoothing. Statist. Sci., Volume 25, Number 1 (2010), 88-106.

This discussion can substantially increase the number of interested readers.

The authors' response: In the revised paper, we have discussed about the particle Monte Carlo methods and added the related references you mentioned (e.g. Refs. [45],[48],[49],[56],[57]). Please see the fourth paragraph in Introduction.

 

- Please, upload the final version of your manuscript in Arxiv and/or ResearchGate when/if published, to increase the diffusion and the possible citations of this work.

The authors' response: Thank you for you positive advice.

Reviewer 2 Report

To the best of my knowledge, the derivation of the estimators appear to be correct. 

I  have two questions for the authors:

1. Why does the matrix A has to have this specific form? How restrictive is this choice?

2. Does it make sense to consider a bigger system in the numerical example? How does algorithm behaves if n=O(100)?

Author Response

Response to review

To the best of my knowledge, the derivation of the estimators appear to be correct.

The authors' response: Thank you for your positive comments.

I have two questions for the authors:

1. Why does the matrix A has to have this specific form? How restrictive is this choice?

The authors' response: The canonical structure has such a form. Because any observable system is equivalent to a canonical structure. Please see Reference [53].

The bilinear system can be transformed to an observer canonical bilinear state space model on condition that a bilinear system is controllable and observable.

 

2. Does it make sense to consider a bigger system in the numerical example? How does algorithm behaves if n=O(100)?

The authors' response: The proposed algorithms is suitable for a large $n$ system. In simulation, a large-scale system has more parameters. For n=(100), the computational amount is large and the computation time is longer.

The proposed F-TS-RGLS algorithm can greatly reduce the computational burden, especially for large-scale systems. Please see Table 1.

Reviewer 3 Report

Honestly, I do not understand the structure of the article. Some figures are in section reference.

Figures 4, 5, 6, 6 are for me illegible, marked estimates in different color suggest that we are dealing with these only values, unless this is true and the algorithm calculates only the values in the marked points?

From the text it does not follow, I suggest a thorough review of the manuscript.

Author Response

Response to review

 

The authors' response: Thank you for your positive comments.

Honestly, I do not understand the structure of the article. Some figures are in section reference.

The authors’ reply: Thank you for your careful reading. We have improved the structure of this article and moved the figures in section reference.

 

Figures 4, 5, 6, 7 are for me illegible, marked estimates in different color suggest that we are dealing with these only values, unless this is true and the algorithm calculates only the values in the marked points?

The authors’ reply: According to your advice, we have improved the simulation results. The solid line represents the true value and the point line represents the estimated value. Please see Figures 4-7 on Pages 14-15.

 

From the text it does not follow, I suggest a thorough review of the manuscript.

The authors’ reply: After taking consideration of your advice, we have reviewed this manuscript carefully and have improved the writing of the paper and corrected typos.

Round 2

Reviewer 3 Report

I do not understand the yellow underlining of the sentence beginning with "Assumption 1 ..." since it is in the same form as it was? In the earlier sentence, we have numerical values, while in the underlined the values are expressed as words, in my opinion it should be unified. But if the Editor does not question this, I will also not pay any attention to it either.