The Use of the Evenness of Eigenvalues of Similarity Matrices to Test for Predictivity of Ecosystem Classifications

Round 1

Reviewer 1 Report

Comments Feoli and Ganis similarity, classification and diversity “an eternal golden braid” in quantitative ecology

The authors clearly introduced the background of this study, framing the paper well enough for audiences of both mathematics and ecology. The general aim of this work is to use mathematical tools to calculate similarity, to produce classifications and to measure diversity. In specific, they looked for mathematical tools suitable to quantify specific phenomena and relationships within ecological contexts, ultimately to stress how similarity, classification and diversity are belonging to the same conceptual framework where eigen-analysis of similarity matrices can play a very important role. In general, I feel the aim and the particular questions of this study appear to be very interesting to me.

I have some questions with the manuscript:

1.     Lines 127 – 129: I have a hard time to understand this sentence. More explanations are appreciated here.

2.     I would recommend if possible, the authors can condense the section 2, Similarity, classification, diversity and eigenvalues. Especially because most information here are well understood by the audiences of mathematics. Instead, introducing the novelty of this work is more appealing.

3.     Line 227 – 231: the reason to arrange the vegetation types into two associations is not well explained. Line 227 – 230 is vague.

4.     I am not sure how to read the numbers in table 2, what do they mean? I am sorry if I missed some information somewhere.

5.     As shown in table 3, the authors claim separation between the two associations in the space defined by environmental factors and in the space defined by the combination of chorological and structural characters is significant. I am not sure how to tell the significance here. More explanation is expected here in results or discussion.

6.     The way to present data in table 4, 6,7 looks heavy to me, I would recommend to leave the most related results in the main text, table, but move some to appendix.

Some minor comments:

Line 9 – 13: this sentence is too long.

Line space looks wrong 249 – 254. Lines 287 – 300 look off here.

The way to use the codes in table 4 is quite confusing. Not sure if this is the best way.

Line 418 – 420: The broader application of this model to other system is not that convincing to me here, more specific discussion might be helpful.

Author Response

We thank the Reviewers for their comments. Since for all the three reviewers it was interesting the method of measuring the class separation, i.e. the evenness of eigenvalues of similarity matrices,but they suggested substantial revisions of the section 2, we reduced the paper and the number of tables, we changed also the title of the paper and we rewrote the paper focussing on the index E(l) .

Reviewer 2 Report

Remarks in attachment.

Comments for author File: Comments.pdf

Author Response

We thank the Reviewers for their comments. Since for all the three reviewers it was interesting the method of measuring the class separation, i.e. the evenness of eigenvalues of similarity matrices,but they suggested substantial revisions of the section 2, we reduced the paper and the number of tables, we changed also the title of the paper and we rewrote the paper focussing on the index E(l) .

Reviewer 3 Report

This review article proposes that the entropy of the eigenvalues of a similarity matrix can be used to inform analysis of the diversity of ecological communities. This is certainly an interesting idea and, I think, one worthy of publication.

I have two major concerns about the manuscript as it currently stands.

Firstly, the focus on this solitary idea, the use of the entropy of the eigenvalues of a similarity matrix. The ms is presented as a reivew article, which would usually imply a substantially broader view of the subject than this focus can allow. Indeed the Introduction of the ms implies a much broader treatment than is ultimately presented. I would suggest that the authors need to substantially broaden their scope if they wish this ms to be submitted as a review article.

Secondly, the key ideas of the ms are not particularly clearly explained. This is, in part, due to the quality of the written English, the ms would certainly benefit from an examination by a scientific proof-reader. However, there are more fundamental issues with the exposition in the ms. The core of the ms, Section 2, is not particularly clear and requires substantial revision to ensure that these key ideas are clearly presented to the reader.

Author Response

We thank the Reviewers for their comments. Since for all the three reviewers it was interesting the method of measuring the class separation, i.e. the evenness of eigenvalues of similarity matrices,but they suggested substantial revisions of the section 2, we reduced the paper and the number of tables, we changed also the title of the paper and we rewrote the paper focussing on the index E(l) .

Round 2

Reviewer 2 Report

Comments attached.

Comments for author File: Comments.docx

Author Response

“The authors have substantially rewritten their paper, and the new version has addressed many of my comments. Removing the previous Section 2 has made the paper much more focused. Its scope and aims are now clearly outlined, and the methods are explained much more clearly.

I recommend only some minor changes before acceptance for publication:”

(We answer point by point):

1.       The title is not grammatically correct – a verb is missing.

Maybe we can adopt the following title: The use of the evenness of eigenvalues of similarity matrices to test for predictivity of classifications of ecosystem states.

2.       What is the ‘sharpness’ of a classification (title & line 34) - is this a technical definition? Do the authors mean the crispness?

Yes! it is a technical definition, a "sharp" classificationis one  that is tendentially crisp, i.e. when the submatrices representing the classes are part of a disjoint similarity matrix.

3.       All the tables are missing. I assume these are the same ones as in the previous version? If not I would need to see these.

The tables were in the zip file word. I hope this time they will be available more easily. They have been further simplified and reduced to three .

4.       The authors have not addressed my previous comment about Tables 3 & 4: what is the ‘probability of the evenness test’? Does this refer to a p-value from a significance test? If so it’s odd to report a p-value as a percentage. This is never explained and not obvious from the discussion, some more details are needed.

The significance of the evenness test is computed by a permutation technique; the probability indicates the relative frequency of the values, obtained by permutation, that are higher than that observed. 10.000 E(l) values are obtained by permuting the scores in the similarity matrix among and between the k classes in which the similarity matrix has been subdivided.

5.       The authors also did not address my previous comment about their choice of the Shannon (Pielou) index for the eigenvalues of the similarity matrix. Every evenness index has advantages and disadvantages, and I would like to see them justify their choice.

Of course there are other measures of evenness. We use the one based on Shannon’s entropy, because it is the most easy to understand and it has long history. We do not see contraindications when it is used for calculating the evenness of the eigenvalues of similarity matrics. In this case it is not used as a diversity measure. There is a strong correlation between the different measures owing to the fact that the majority of them can be obtained by the Reyni’s ( 1961), Hill’s (1973) and Patil and Tallie’s (1976) generalizations (i.e. by changing a power parameter, you can obtain many of the most important indices of diversity including those of Shannon and Gini-Simpson). The literature on the topic is very rich but there is not need to review it in the present paper. What it is important for the paper is not the discussion on the way to measure the evenness but the evenness of the eigenvalues itself.

6.       Again, an additional proofreading is necessary, since there are still many minor English mistakes (particularly grammar).

English has been reviewed by a mother speaking English teacher. If some error persists this is due to the fact that we are continuously reviewing the text.

7.       Based on my previous comment, the authors have changed the word ‘features’ but only in some places. In Section 3, ‘feature’ is still often used when the authors mean ‘featureistics’ or ‘features’.

Character has been substituted by feature everywhere.

Reviewer 3 Report

The English usage in this article remains very difficult to follow particularly in the introduction. It remains unclear to me whether this approach is new, the historical account provided suggests not, or whether this is simply an attempt to promote the authors' other work. (If the latter, it would seem more appropriate to write a true review article including a comparative analysis of other approaches.)

In revising the article the authors have omitted the tables (referenced in the text) making it impossible to assess the scientific soundness of the results.

Author Response

We are conscious that our paper could be improved as any paper published by any author, however we have rewritten it and the tables of the example reduced. Also the title has been slightly changed. We think that now it has reached a level of acceptability. The aim was focussed in presenting the index E(l)  and its background.

In the paper we explain that the index was already proposed to find if the separation between the classes of a classification could be considered significant or not on the basis of the features used for the classification itself (internal features). In this paper we propose the index E(l) for a new aim, that is to test if a given classification is predictive with respect external variables (i.e. variables not used to obtain the classification). The index E(l) could be considered one alternative of ANOVA (MANOVA) and of other non-parametric techniques that were proposed later (e.g. ANOSIM and PERMANOVA). We do not think useful to explain these techniques because they have already a very ‘large’ literature and because they do not say nothing new with respect the index proposed in 1972 by Feoli and Bressan [4]. We thought that for the sake of our paper it would be enough to compare the index with the Kruskal-Wallis test as we have done.

The aim of the paper is not promoting the research of our group as it is supposed by the reviewer, the aim is to promote the idea that eigenanalysis should be used not only for ordination methods as usual, but also for classification purposes in line with what discussed in the Dale’s paper [13] and with the Occam’s razor rule.

Our English will not be Oxfordian, but at the normal level of non-mother English authors who are used to publish in international journals with impact factor and with referees. We tried to correct all the errors.

We do not think that the paper in the version that is now  should be transformed to a review, but we think appropriate to consider it in the category of “article” or better “short notes”. Notwithstanding we are citing several papers of Feoli, the aim of this paper is not to promote the Feoli’s research, but just to show clearly how the idea was born and why.   The paper was revised by a mother language English teacher. If there are errors this is due to the fact that we revisited the paper many times after the English revision by changing some sentences. Sorry for that!

Round 3

Reviewer 3 Report

Fine.