Comparisons Between Frequency Distributions Based on Gini’s Approach: Principal Component Analysis Addressed to Time Series

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This article takes a fresh approach to principal component analysis (PCA) based upon Gini’s proportionality framework and applying that to time series as frequency distributions. While the underlying vision is rich in intellectual beauty, its execution could benefit from being made all the more accessible, clear, and empirically sound. First, the impulse for this reinterpretation of PCA must be better articulated. The present introduction is overly dependent upon methodological zeal explicated in the abstract, with insufficient concern given to implications for practice, and even less for comparison with known PCA methodology. Second, writing is compact and sometimes redundant, and there is a challenge even econometricians and statisticians will experience following the writer's thrust. Less redundancy and a clearer exposition will much enhance the readability of the material without diluting the technical quality of writing. Third, the theoretical foundation of the article would far benefit from more extensive empirical justification. A solitary synthetic data-based simulation is not enough to establish the method’s usefulness and applicability. Comparisons with conventional PCA using actual time series datasets–GDP, stock prices, weather–would create a better demonstration of the utility of the method. A more specific sectionation is additionally necessitated. Existing ones are overly general and notionally overlapping and create confusion. All of them must be assigned clearly defined purpose–definitions, methodology, application, interpretation–and use consistent standardized terminologies. Diagrams and flowcharts may be used to explain complex processes like moving from marginal to joint frequency distributions and the calculation of the α-metric tensor. Mathematical notation is of especial concern. The article introduces a battery of symbols and expressions that are not defined in the proper way. Usage of standard notations with which statistical and econometric communities are accustomed lowers the reader’s learning curve. Adding intuitive explanation and possibly even illustrative examples for formulae will make them even better understood. Moreover, its usefulness is undermined through the omission of algorithmic information or code. Those researchers wishing to implement or test the method are left with no guideline for proceeding. Even a flowchart or a sequence of pseudocode for each step of computing would be significantly better. Of the new ideas in this book—the use of the Fréchet family of distributions—is less developed. The idea is worth a separate illustration in order to demonstrate its capabilities and implications, including assumptions in the model and modularity for combining joint distribution construction. Additionally, while the mathematical elegance of the theoretical foundation in vector space, tensors, and metrics is appealing, the mathematical sophistication will potentially obscure the statistical interpretation. Wherever possible, simplification in this framework or grounding this in simpler constructs like covariance matrices or Euclidean distance will serve to keep things clear. Finally, the article has to conclude with a brief summary of the important results and concrete recommendations for future research. This could involve integration of the method into time series forecasting, extension to unsupervised approaches, or generalization to applications where PCA is traditionally used. As a response to the following considerations, the article could progress from being mathematically ambitious but idea-wise impenetrable to being a more direct and compelling contribution to time series analysis and dimension reduction techniques.

Author Response

Thank you for your comments. Please see the attached author reply.

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Please see attachment.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

The quality of English must be improved.

Author Response

Thank you for your comments 

 

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This is a good paper. However, it contains many issues:

1-The title does not reflect the paper content.

2-The introduction lacks sufficient context, background, and a clear problem statement.

3-The conclusion is brief, and it needs to mention the importance of the findings and the paper's limitations.

Author Response

Thank you for your comments. Please see the attached author reply.

 

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

This paper proposes a novel approach to principal component analysis (PCA) by interpreting time series data as frequency distributions. The author employs Gini's approach to develop a methodological framework that facilitates comparisons between frequency distributions. The paper consists of eight sections that delve into various aspects of this reinterpretation, including theoretical foundations, numerical simulations, and applications within linear systems.

 Strengths:

• Innovative Approach: The author’s attempt to conceptualize time series as frequency distributions is a refreshing perspective that enriches the discussion around PCA. It opens avenues for further research in statistical modeling and data analysis.
• Theoretical Framework:  The integration of Gini's approach into PCA is well-articulated and provides a solid theoretical foundation, which can enhance the understanding of the relationships between marginal and joint frequency distributions.
• Numerical Simulations: The inclusion of numerical simulations to demonstrate the application of the proposed techniques adds empirical weight to the theoretical assertions, illustrating the practical implications of the methodology.
• Comprehensive Coverage: The paper covers a broad range of topics including the definitions of marginal and joint frequency distributions, eigenvalues, and eigenspaces, which are critical to understanding the underlying mathematical structures.

 Weaknesses:

• Complexity and Clarity: The paper is dense and may be difficult to follow for readers not already familiar with advanced statistical concepts. The author could benefit from clearer explanations and more intuitive examples to enhance accessibility.
• Lack of Empirical Validation: While the numerical simulations are a positive aspect, they are limited in scope. The paper would benefit from more extensive empirical validation using real-world data sets to demonstrate the efficacy and applicability of the proposed methods.
• Over-reliance on Mathematical Formalism: The heavy reliance on mathematical notation and formal proofs may alienate practitioners who seek actionable insights without delving deeply into theoretical constructs. A more balanced approach between theory and practical application would be advantageous.
• Literature Review: The paper lacks a thorough review of related literature that situates this work within the broader context of existing research in PCA and time series analysis. A more extensive discussion of how this work builds upon or diverges from previous studies would provide valuable context.
• In conclusion, the paper presents a novel approach to principal component analysis applied to time series, which is valuable and innovative. However, revisions are necessary to improve clarity, enhance empirical validation, and provide a more comprehensive literature review. In my opinion, resolving these issues would strengthen the content of the article and increase the interest of readers more oriented to the empirical aspects of the proposed model.

Comments for author File: Comments.pdf

Author Response

Thank you for your comments. Please see the attached reply.

 

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Please see attached file.

Comments for author File: Comments.pdf

Author Response

Thank you for your comments. Please see the attached file.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

No major issues were noted in the new version of the article.

Author Response

Thank you