Multi-Partitions Subspace Clustering
Round 1
Reviewer 1 Report
The method proposed in the paper seem interesting and there is a merit in it.
However, it was not clear to see the advantage of the proposed method over the mixture model (I believe there must be a good application).
As the author noted, for high dimensional "x", the optimal choice of "H", "K" etc are difficult to obtain which in practice very important to have the optimal one.
Author Response
Dear Reviewer,
Thank you for taking of your time for carefully reading our article. We hope we addressed in a satisfactory manner most of the issues you raised. Please find bellow our point-by-point responses.
Best Regards,
Vincent Vandewalle
R1: The method proposed in the paper seem interesting and there is a merit in it.
VV : Thank you for taking of your time for carefully reading our article. We are very happy that you find a merit in our proposed method.
R1: However, it was not clear to see the advantage of the proposed method over the mixture model (I believe there must be a good application).
VV: The proposed method is a particular mixture model but with a Cartesian factorisation of the class labels. So the adavantage over classical mixture models is in terms of interpretation as shown on the crabs dataset. This aspect is now emphasied in the text (Section 3.2, page 8, lines 196-211).
R1: As the author noted, for high dimensional "x", the optimal choice of "H", "K" etc are difficult to obtain which in practice very important to have the optimal one.
VV: Thank you for you comment, it is effectively a crucial choice which may have a great impact on the analysis, however as shown on the application in some range of K values the resulting projection map could be the same. Moreover the BIC criterion can be used to find relevant values of the parameters, as shown for example in Table 2 on simulated data. This question is now further discussed in conclusion et perspectives (page 15, lines 324-330).
Reviewer 2 Report
Dear Authors, please find my сomments and suggestions below.
Please, motivate the numbering of equations. You refer to some equations, but you do not refer to other numbered equations. It is no references to equations (2), (5) and (6).
In line 152 you wrote: " the computation would be much more intensive" What do you mean? Can you prove it? Please clarify this statement.
The phrase "Discussion about the model" in line 161 must be a headline. Please fix it.
In lines 196 - 198 you wrote: "In practice the choice of p = 1 seams a good one from an interpretation point of view. Let also notice that choosing K = 2 could also be a good way for finding dimensions partitioning the data." Please explain it more and try to avoid value judgments in scientific texts. Can you prove these sentences? Please add some references.
Have you compared your method with other existing clustering methods besides PCA? Could you describe the advantages and disadvantages of using your method in relation to them? Can you describe the preferred areas of application for your method? I think that this information would be very interesting to readers.
Please check spelling, punctuation, style and grammar of the manuscript. For example, the phrase "point of view" occurs 8 times in the text and the word "however" occurs 16 times. Also, I noticed:
line 11
The behaviour of the model is illustrated on simulated ...
line 33
However, the proposed high dimensional mixture does not properly operates a dimension reduction ...
line 35
... model based data visualiszation. ...
line 38
... This approach allows to apply the same projection on all the data, but does not guarantee ...
line 54
... the data, however, the huge number of possibilities ...
line 69
... Moreover, it is not invariant according to rotation ...
line 70
... data, where or the proposed methodology is ...
line 75
... In Section 5, an conclusion...
line 80
... where xij 80 is the the value of variable j of data i ...
line 95
... This could be done by PCA, however, from a classification perspective ...
line 148
At this step the homoscedasticity could be seen as particularly constraining, however, it is ...
line 168
... an other ... -> another
line 170
... statistical units, but assumes that the sources ...
line 188
... the following steps...
line 231
... BIC criterion are presented in Table 1, where we show that ...
line 236
... the species;, orange or blue, and the sex;, male of or female...
line 255
... Moreover, we have shown the possibility to perform model choice ...
Also, there is no numbering in some lines. Paragraphs between lines 90, 91 and 91, 92 and so on do not have numbers. Please, fix it.
Author Response
Dear Reviewer,
Thank you for taking of your time for carefully reading our article. We appreciate your detailled reivew, which helped us to improve our submitted article. We hope we addressed in a satisfactory manner most of the issues you raised.
Best Regards,
Vincent Vandewalle
R2: Dear Authors, please find my сomments and suggestions below.
R2: Please, motivate the numbering of equations. You refer to some equations, but you do not refer to other numbered equations. It is no references to equations (2), (5) and (6).
VV: Thank you for the comment, the numbers of equations that are not cited have now been removed.
R2: In line 152 you wrote: " the computation would be much more intensive" What do you mean? Can you prove it? Please clarify this statement.
VV: Thank you for the comment, we wanted to say that the computation would be much more intensive since in the framework of heteroscedastic discriminant analysis an iterative algorithm would be required whereas in the FDA framework a closed form formula is available. The statement has now been clarified in the text (Section 2.4, page 7, lines 176-179)
R2: The phrase "Discussion about the model" in line 161 must be a headline. Please
fix it.
VV: Thank you for the remark it has now been modified.
R2: In lines 196 - 198 you wrote: "In practice the choice of p = 1 seams a good one from an interpretation point of view. Let also notice that choosing K = 2 could also be a good way for finding dimensions partitioning the data." Please explain it more and try to avoid value judgments in scientific texts. Can you prove these sentences? Please add some references.
VV: Thank you for your comment, the sentence has now been reformulated in order to avoid any value judgments: "In practice the choice of $p=1$ enforces to find clustering which could be visualized in one dimension, which can help the practitioner. Moreover choosing $K=2$ is the minimal requirement in order to investigate a clustering structure. However, if possible we recommend to explore the largest possible number of models and choosing the best one with the BIC."
R2: Have you compared your method with other existing clustering methods besides PCA? Could you describe the advantages and disadvantages of using your method in relation to them? Can you describe the preferred areas of application for your method? I think that this information would be very interesting to readers.
VV: In the experimental part of the article, we especially focused on finding the clustering subspaces and thus on the comparison with the components of the PCA. Comparing the proposed approach with other existing clustering techniques would also be possible, for instance in terms of rand index, however the obtained partitions would rather be accordance. The main adavantage of our proposed approach is to produced a factorised version of the partition space as weel as the related clustering subspaces which is not a standard ouput of clustering methods. This fact has now be made clearer in the discussion about the model (Section 3.2, page 8, lines 196-211).
R2: Please check spelling, punctuation, style and grammar of the manuscript. For example, the phrase "point of view" occurs 8 times in the text and the word "however" occurs 16 times.
VV: Thank for your comment, the article has now been carrefully inspected, and redundant sentences have been removed, and the following remarks have been taken into account. Moreover if accepted the English of the article will be revised by the journal.
R2: Also, I noticed:
R2: line 11
R2: The behaviour of the model is illustrated on simulated ...
VV: replaced by "Model's behavior"
R2: line 33
R2: However, the proposed high dimensional mixture does not properly operates a dimension reduction ...
VV : However, the proposed high-dimensional mixture does not perform dimension reduction ...
R2: line 35
R2: ... model based data visualiszation. ...
VV: ... model-based data visualization. ...
R2: line 38
R2: ... This approach allows to apply the same projection on all the data, but does not guarantee ...
VV: ... This approach allows the same projection to be applied to all data, but does not guarantee ...
R2: line 54
R2: ... the data, however, the huge number of possibilities ...
VV: ... but the tree structure search makes estimation even more difficult as the number of variables increases ... (cf rephrasing of the introduction)
R2: line 69
R2: ... Moreover, it is not invariant according to rotation ...
VV : ... Moreover, it is not invariant up to a rotation ...
R2: line 70
R2: ... data, where or the proposed methodology is ...
VV: .... data, where our proposed methodology is ...
R2: line 75
R2: ... In Section 5, an conclusion...
VV: ... In Section 5, a conclusion ...
R2: line 80
R2: ... where xij 80 is the the value of variable j of data i ...
VV: ... where xij is the value of variable j of data i ...
R2: line 95
R2: ... This could be done by PCA, however, from a classification perspective ...
VV: ... This could be done by using PCA, but from a classification perspective
R2: line 148
R2: At this step the homoscedasticity could be seen as particularly constraining, however, it is ...
R2: At this step the homoscedasticity could be seen as particularly constraining, although it is ...
R2: line 168
R2: ... an other ... -> another
VV: done
R2: line 170
R2: ... statistical units, but assumes that the sources ...
VV: ... statistical units, it assumes that the sources ...
R2: line 188
R2: ... the following steps...
VV: ... the following steps...
R2: line 231
R2: ... BIC criterion are presented in Table 1, where we show that ...
VV: done
R2: line 236
R2: ... the species;, orange or blue, and the sex;, male of or female...
VV: done
R2: line 255
R2: ... Moreover, we have shown the possibility to perform model choice ...
VV: done
R2: Also, there is no numbering in some lines. Paragraphs between lines 90, 91 and 91, 92 and so on do not have numbers. Please, fix it.
VV: Thank you for the comment, all the lines have now numbers.
Reviewer 3 Report
Dear author,
Very interesting article as well as the model itself which allows to combine visualization and clustering with many clustering view points.The article has a correct structure, the research background has been sufficiently presented. In my opinion, the analysis and construction of the model is correct.
He suggested that I consider numbering all the mathematical formulas used.
I still have suggestions for applications. The model itself is very interesting and I wonder to use it in my own research but I lacked in conclusions strongly articulated advantages of this model compared to existing as well as limitations and potential errors that may occur when using it in practice.
Yours sincerely
Reviewer
Author Response
Dear Reviewer,
Thank you for taking of your time for carefully reading our article. We hope we addressed in a satisfactory manner most of the issues you raised. Please find bellow our point-by-point responses.
Best Regards,
Vincent Vandewalle
R3: Very interesting article as well as the model itself which allows to combine visualization and clustering with many clustering view points.
The article has a correct structure, the research background has been sufficiently presented. In my opinion, the analysis and construction of the model is correct.
VV: Thank you, we are very happy that it enjoys you.
R3: He suggested that I consider numbering all the mathematical formulas used.
I still have suggestions for applications. The model itself is very interesting and I wonder to use it in my own research but I lacked in conclusions strongly articulated advantages of this model compared to existing as well as limitations and potential errors that may occur when using it in practice.
VV: Thank you for your interest, the discussion has now been extented to strongly emphasise the pro and the cons of the proposed model (Section 3.2, page 8, lines 196-211).
Round 2
Reviewer 2 Report
Dear authors, thank you for the changes you made according to my comments and suggestions. I have no more questions for you. In my opinion, now the manuscript may be published in Mathematics Journal.
