Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript evaluates Straw-K, where a local chord-to-path criterion applied to the WCSS curve defining a new ratio for cluster-number. The proposed method was empirically compared in depth with various other methods across diverse datasets. However, the manuscript does not clearly specify whether "Straw-K" refers to the proposed method or an existing one; in fact, the term "proposed" is not used at all. Consequently, the manuscript gives the impression that it merely involves an experimental comparison and evaluation of the performance of existing methods. The authors need to clarify the objective and the work accomplished in this manuscript.

Acronyms referring to various methods appear, but the complete lack of explanations for them makes the manuscript very difficult to read and understand. For example, what do terms like "Straw-K" or "ShortStraw" signify? If they refer to specific methods, please provide a brief explanation along with references.

The title suggests that “geometric optimization” is performed; please clearly explain how this optimization was achieved in the manuscript.

The detailed comments are as follows.

What does the acronym “WCSS” stand for? Also, for “Straw-K” and “ShortStraw.”
Further detail descripts on “K-means” is required. What is “K-means seed.” Does this refer to setting the K initial values to begin the clustering iterations in the K-means algorithm? The terminology used is unfamiliar in this field, and the use of such unfamiliar terms makes the manuscript difficult to understand.
On Line 81, are there any related methods for determining the number of clusters using the chord and polyline path?
On Line 108, does “cluster counts” imply the number of clusters or cluster number?
On Line 109, dose “K-means is fitted” imply the clustering iteration converges in the K-means algorithm? Please clarify this sentence. What is “w_k”?
On Line 110, the expression “{(k,w_k): k in K}” is quite awkward. It seems that the expression implies a set of “(k,w_k),” which can be a vector or a range. However, the authors simply say that the set is “WCSS curve.” Please clarify the sentence using standard mathematical notations.
On Line 117, what is the range of M. What is “half-window s”? During the simulations, how did the authors set the s value?
In (3), what is the range of index i.
On Line 119, please define the chord and polyline path.
In Figure 2, the symbol p_i is different from that of Line 117, where p_i is normalized. Match them so that they become the same variable.
On Line 126, what is the meaning of “valid ratios”?
In Algorithm 1, please give further kind explanations on “Algorithm 1: Core Straw-K selection”. Is this mean “Cluster-Number Selection Method: Straw-K”? Note that there are no further algorithms.
On Lines 130-131, please provide further explanations on “it is not a probability and requires empirical validation.” There are many other expressions that are similarly scientifically inaccurate. All sentences must be carefully reviewed to ensure that general readers will have no difficulty reading and understanding them.

Author Response

Dear Editor and Reviewers,

We sincerely thank the Editor and all three reviewers for their careful reading, constructive criticism, and detailed methodological suggestions. The comments were extremely helpful in identifying places where the manuscript required clearer terminology, a fairer boundary construction, more explicit statistical reporting, stronger reproducibility documentation, and a more cautious interpretation of the evidence. Where a reviewer identified an issue that changed the method or analysis rather than only the wording, we revised the code and reran the affected experiments before updating the manuscript.

In response to the reviewers' questions and requested revisions, we reproduce each comment below and provide (i) our response, (ii) the analyses or reasoning used to address it, and (iii) the exact changes made in the revised manuscript and supplementary materials.

Main revisions in this version include:

Boundary correction for Straw-K: K=1 and K=11 are now used only as non-selectable support points, allowing every selectable K=2,...,10 to be scored with the same symmetric window.
A numerical-sensitivity analysis for rounding, perturbations, and K-means initialization.
A separate small-sample sensitivity analysis and a detailed missingness / variable-type preprocessing audit.
A full baseline implementation audit, including correction of the L-method to the Salvador-Chan non-overlapping two-line criterion.
Explicit mean-rank rules and Holm correction across all 21 pairwise Wilcoxon comparisons.
A fixed-seed, non-outcome-selected Figure 5 generated directly from the experiment script.
An independent uncached single-run runtime experiment for Gap, with computational-system specifications.
A complete rewrite of the notation, Algorithm 1, ambiguity interpretation, related-work positioning, and discussion of limitations.

Updated headline results after the methodological corrections: Straw-K exact match = 44.8%, MAE = 1.542, mean rank = 3.167; Friedman chi-square = 13.510, p = 0.0356; no pairwise Wilcoxon comparison remains significant after Holm correction (minimum adjusted p = 0.123).

The responses below are organized exactly by reviewer. Reviewer comments are reproduced first, followed by our response and a concise statement of what changed in the manuscript.

Reviewer 1

We appreciate the reviewer’s emphasis on conceptual clarity and general readability. These comments led us to sharpen the novelty statement, define the terminology and notation more carefully, and rewrite the core algorithm description.

General Comment 1

Reviewer comment: This manuscript evaluates Straw-K, where a local chord-to-path criterion applied to the WCSS curve defining a new ratio for cluster-number. The proposed method was empirically compared in depth with various other methods across diverse datasets. However, the manuscript does not clearly specify whether "Straw-K" refers to the proposed method or an existing one; in fact, the term "proposed" is not used at all. Consequently, the manuscript gives the impression that it merely involves an experimental comparison and evaluation of the performance of existing methods. The authors need to clarify the objective and the work accomplished in this manuscript.

Response: We agree that the previous version did not identify the status of Straw-K explicitly enough. We have revised the Abstract and Introduction to state directly that this manuscript proposes and evaluates Straw-K. We also clarified the exact scope of the contribution. After re-checking the ShortStraw literature, we narrowed the novelty statement: the chord-to-path relation itself is not claimed as new, because ShortStraw already uses related chord/path geometry. The contribution of Straw-K is the adaptation of that local geometry to the normalized WCSS support curve, its discrete use for cluster-number selection, and the accompanying empirical evaluation.

Changes in the revised manuscript: Abstract; Section 1.2 (Study Aim and Main Contributions); Section 1.3 (Related Work). The revised text now uses “we propose and evaluate Straw-K” and explicitly states what is, and is not, claimed as novel.

General Comment 2

Reviewer comment: Acronyms referring to various methods appear, but the complete lack of explanations for them makes the manuscript very difficult to read and understand. For example, what do terms like "Straw-K" or "ShortStraw" signify? If they refer to specific methods, please provide a brief explanation along with references.

Response: We agree. WCSS is now expanded at first use as “within-cluster sum of squares.” Straw-K and ShortStraw are clarified as method names rather than acronyms. ShortStraw is briefly described as the published Wolin et al. polyline corner-detection method, with the manuscript reference retained. Straw-K is described as the coined name of the proposed cluster-number selector.

Changes in the revised manuscript: Abstract; Sections 1.1-1.3; Section 2.1. The first uses of WCSS, Straw-K, and ShortStraw are now accompanied by definitions or explanations.

General Comment 3

Reviewer comment: The title suggests that “geometric optimization” is performed; please clearly explain how this optimization was achieved in the manuscript.

Response: We have clarified this wording. In the title, “geometric optimization” refers to the discrete minimization of Straw-K’s normalized local chord-to-path score over the candidate values of K. It does not refer to changing the optimization objective or iterative update equations inside K-means itself. The optimization problem is now written explicitly as i* = argmin R_i, with K* equal to the corresponding candidate cluster number.

Changes in the revised manuscript: Section 1.2 and Section 2.2 (Straw-K Criterion), including the explicit discrete optimization statement.

Detailed Comments

Comment 1

Reviewer comment: What does the acronym “WCSS” stand for? Also, for “Straw-K” and “ShortStraw.”

Response: WCSS now stands for “within-cluster sum of squares” at first use. Straw-K is not an acronym; it is the coined name of the proposed method. ShortStraw is likewise treated as the published method name, not expanded as an acronym, and is briefly explained with its reference.

Changes in the revised manuscript: Abstract; Sections 1.1-1.3; Section 2.1.

Comment 2

Reviewer comment: Further detail descripts on “K-means” is required. What is “K-means seed.” Does this refer to setting the K initial values to begin the clustering iterations in the K-means algorithm? The terminology used is unfamiliar in this field, and the use of such unfamiliar terms makes the manuscript difficult to understand.

Response: We expanded the K-means description and replaced the ambiguous phrase “K-means seed” with “random-state seed.” The revised Methods explains that K-means alternates assignment and centroid-update steps until the implementation’s convergence criterion is satisfied or the 300-iteration limit is reached. The eight seeds are random states 0-7 controlling pseudo-random centroid initialization; they are not initial values of K. For each candidate K, five initializations are evaluated (n_init=5), and the solution with the smallest WCSS is retained.

Changes in the revised manuscript: Section 2.1 (Workflow and Notation) and Section 2.6 (Repeated Experimental Protocol).

Comment 3

Reviewer comment: On Line 81, are there any related methods for determining the number of clusters using the chord and polyline path?

Response: We expanded the related-work discussion. Kneedle uses a normalized difference curve and a sensitivity rule; the L-method uses a weighted two-line fit; angle-based elbow criteria use local angular changes; and ShortStraw uses chord/path geometry in sketch corner processing. We now state cautiously that, among the cluster-number methods reviewed for this study, we did not identify a prior method that applies the same local chord-to-polyline path-length ratio to a normalized WCSS curve and selects K by minimizing it. This is intentionally phrased as a literature-review finding rather than an absolute historical-first claim.

Changes in the revised manuscript: Section 1.3 (Related Work).

Comment 4

Reviewer comment: On Line 108, does “cluster counts” imply the number of clusters or cluster number?

Response: Yes; the intended meaning was the number of clusters. We removed the ambiguous phrase “cluster counts” where it could be misread as cluster sizes or frequencies and now use “number of clusters” or “candidate numbers of clusters.”

Changes in the revised manuscript: Terminology revised throughout the manuscript, particularly Sections 2.1-2.2.

Comment 5

Reviewer comment: On Line 109, dose “K-means is fitted” imply the clustering iteration converges in the K-means algorithm? Please clarify this sentence. What is “w_k”?

Response: We now define both expressions explicitly. “Fitting K-means” means running the iterative assignment/update procedure until convergence or the maximum iteration limit. We define w(k) as the within-cluster sum of squared Euclidean distances for the fitted solution with k clusters; on the ordered support grid we write w_i = w(q_i).

Changes in the revised manuscript: Section 2.1 now contains the explicit WCSS equation and the definition of fitting/convergence.

Comment 6

Reviewer comment: On Line 110, the expression “{(k,w_k): k in K}” is quite awkward. It seems that the expression implies a set of “(k,w_k),” which can be a vector or a range. However, the authors simply say that the set is “WCSS curve.” Please clarify the sentence using standard mathematical notations.

Response: We agree that the earlier set notation obscured the ordered geometry. It has been replaced by an ordered support grid q_1,...,q_M and the ordered WCSS polyline (q_1,w_1),...,(q_M,w_M). This makes clear that Straw-K evaluates geometry along a sequence/polyline rather than over an unordered set.

Changes in the revised manuscript: Section 2.1 (Workflow and Notation).

Comment 7

Reviewer comment: On Line 117, what is the range of M. What is “half-window s”? During the simulations, how did the authors set the s value?

Response: The revised manuscript defines M as the number of Straw-K support points and s as the positive integer half-window. Following the boundary correction described in response to Reviewer 2, the support grid is q=1,...,11, so M=11, while the selectable outputs remain K=2,...,10. The main experiments used s=1 for every dataset; it was fixed in advance and was not tuned by dataset. With s=1, the immediate neighbor on each side forms the smallest symmetric local window.

Changes in the revised manuscript: Sections 2.1-2.2. The value s=1, the support size M=11, and the selectable range are stated explicitly.

Comment 8

Reviewer comment: In (3), what is the range of index i.

Response: We now define the scoreable index set formally as I_s = {i : q_i is selectable, i-s >= 1, and i+s <= M}. Under the revised support grid and s=1, I_1 corresponds exactly to selectable K=2,...,10.

Changes in the revised manuscript: Section 2.2 (Straw-K Criterion) and Algorithm 1.

Comment 9

Reviewer comment: On Line 119, please define the chord and polyline path.

Response: The chord and path now have separate mathematical and plain-language definitions. The chord c_i is the direct Euclidean distance between the two endpoints of the local window. The polyline path a_i is the sum of the consecutive Euclidean segment lengths along the normalized WCSS polyline through that same window.

Changes in the revised manuscript: Section 2.2, Equations defining c_i and a_i, and Figure 2 caption.

Comment 10

Reviewer comment: In Figure 2, the symbol p_i is different from that of Line 117, where p_i is normalized. Match them so that they become the same variable.

Response: We redrew Figure 2 so that its axes and point labels use the same normalized coordinates x_i and y_i that define p_i=(x_i,y_i) in the equations. The former mismatch between raw WCSS coordinates in the figure and normalized p_i in the text has been removed.

Changes in the revised manuscript: Figure 2 and Section 2.2.

Comment 11

Reviewer comment: On Line 126, what is the meaning of “valid ratios”?

Response: The undefined phrase “valid ratios” has been removed. The manuscript now uses “scoreable ratios” and defines them formally as R_i for indices i in I_s for which the local path is finite. In the main experiments, every selectable K=2,...,10 is scoreable under the revised support construction.

Changes in the revised manuscript: Section 2.2; Section 2.3; Algorithm 1.

Comment 12

Reviewer comment: In Algorithm 1, please give further kind explanations on “Algorithm 1: Core Straw-K selection”. Is this mean “Cluster-Number Selection Method: Straw-K”? Note that there are no further algorithms.

Response: We agree that the former title was too terse. Algorithm 1 is now titled “Straw-K Cluster-Number Selection Procedure,” and a sentence before the algorithm explains that it summarizes the complete selection step after the support WCSS values have been computed. Its inputs, output, admissible indices, chord, path, ratio, tie rule, and final selection are now stated explicitly.

Changes in the revised manuscript: Section 2.2 and Algorithm 1.

Comment 13

Reviewer comment: On Lines 130-131, please provide further explanations on “it is not a probability and requires empirical validation.” There are many other expressions that are similarly scientifically inaccurate. All sentences must be carefully reviewed to ensure that general readers will have no difficulty reading and understanding them.

Response: We rewrote this passage substantially. The ambiguity quantity is now described as a deterministic relative separation index based on the smallest and second-smallest Straw-K ratios. We explicitly state that a value such as A=0.99 does not mean 99% uncertainty, a 99% error probability, or a 99% confidence level, because the quantity is not derived from a sampling distribution, posterior probability, or calibration model. We also replaced the vague phrase “requires empirical validation” with the exploratory analysis actually performed: mean ambiguity was compared with seed instability across the 12 datasets using Spearman correlation. After the boundary correction, the association was weak and non-significant (rho=0.325, p=0.303), so the index is retained only as a descriptive flag for close alternatives. We also reviewed the manuscript globally for similarly over-compressed or potentially misleading statistical/scientific phrasing.

Changes in the revised manuscript: Section 2.3 (Geometric Ambiguity Margin), Section 3.6 (Geometric Ambiguity Margin and Numerical Robustness), Discussion, and Conclusions.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript presents the development and evaluation of a new geometric method, Straw-K, for automated cluster-number selection in K-means based on the local geometry of the WCSS curve. The study is of interest from the perspective of advancing clustering methods and automating the selection of the number of clusters. The proposed approach is compared with several established criteria using a collection of empirical datasets. The manuscript is carefully prepared, well structured, and generally makes a positive impression. Its strengths include a clear presentation of the main idea of the method and a comparative computational study. At the same time, several aspects of the methodology and experimental design require additional clarification and more detailed justification, particularly regarding the choice of algorithm parameters, benchmark construction, stability analysis, and interpretation of the obtained results. After the corresponding clarifications and methodological details have been added, the manuscript may be considered suitable for publication. Specific questions and comments are provided below.

1. Line 117: Please specify the value of the half-window s used in the main computational experiments. Since s directly determines the local interval over which R_iis calculated, this parameter is one of the key parameters of the algorithm and should be explicitly stated in the manuscript.

2. Line 121: The values of the local chord-to-path ratio in the figures presented later in the manuscript are very close to one. To what extent can small changes in WCSS resulting from a different K-means initialization, numerical error, or rounding alter the position of the minimum of R_i?

3. Algorithm 1: From the definition of the admissible indices i=s+1, . . . , M−s, it follows that the boundary values of the considered K range cannot be selected by Straw-K. Table 1 contains datasets with Reference K=2, while Digits has Reference K=10. How appropriate is it to compare exact-match rates under this construction of the algorithm? In addition, Algorithm 1 does not specify how ties or numerically almost indistinguishable values of R_i are handled. How is K^∗ determined when two or more candidates have identical or nearly identical values of R_i?

4. Table 1: Two of the datasets contain only 32 observations, while values up to K=10 are considered. With such an N/K ratio, some clusters may contain only a few observations. The manuscript should further discuss whether it is appropriate to include such datasets in the comparative benchmark or, alternatively, provide a separate analysis for small-sample datasets.

5. Line 142: Missing values were replaced by the feature median; however, it is not reported which datasets contained missing values or what proportion of observations or features was affected. This information is necessary to assess the potential influence of imputation on the geometry of the data and the clustering results. In addition, the manuscript states that all features were represented numerically, but it does not clarify whether any of the original variables were categorical or ordinal. If categorical codes were directly treated as numerical features, the use of Euclidean distance and K-means may be methodologically questionable. The preprocessing of such variables should therefore be described in greater detail.

6. Line 148: The implementation details for Kneedle are described in sufficient detail, whereas corresponding information is not provided for the L-method, Silhouette, Calinski-Harabasz, and Davies-Bouldin. Please specify the exact implementations, software/library versions, criteria used to select the optimal K, and rules for handling ties for all baseline methods.

7. Line 168: It is not fully clear how the mean rank by dataset was calculated, particularly when several methods produced identical errors. Please describe the ranking procedure and the treatment of ties in greater detail.

8. Line 170: Was a correction for multiple comparisons applied to the paired Wilcoxon tests reported in the Supplementary Material?

9. Figure 5: For Glass Identification, Table 3 reports a modal Straw-K selection of K=9, whereas Figure 5 shows an individual run in which Straw-K selects K=6, matching the Reference K. What objective criterion was used to select this particular representative run?

10. Table 4: The runtime for the Gap statistic is calculated by distributing the cached reference-generation cost across eight seeds. This approach may substantially underestimate the computational cost of Gap in a typical scenario in which a user performs a single analysis rather than a series of eight experiments sharing the same cache. It would therefore be useful to additionally report the runtime for an independent single-run scenario and to provide the specifications of the computational system used for the timing experiments.

Comments for author File: Comments.pdf

Author Response

Dear Editor and Reviewers,

Main revisions in this version include:

Boundary correction for Straw-K: K=1 and K=11 are now used only as non-selectable support points, allowing every selectable K=2,...,10 to be scored with the same symmetric window.
A numerical-sensitivity analysis for rounding, perturbations, and K-means initialization.
A separate small-sample sensitivity analysis and a detailed missingness / variable-type preprocessing audit.
A full baseline implementation audit, including correction of the L-method to the Salvador-Chan non-overlapping two-line criterion.
Explicit mean-rank rules and Holm correction across all 21 pairwise Wilcoxon comparisons.
A fixed-seed, non-outcome-selected Figure 5 generated directly from the experiment script.
An independent uncached single-run runtime experiment for Gap, with computational-system specifications.
A complete rewrite of the notation, Algorithm 1, ambiguity interpretation, related-work positioning, and discussion of limitations.

The responses below are organized exactly by reviewer. Reviewer comments are reproduced first, followed by our response and a concise statement of what changed in the manuscript.

Reviewer 2

Reviewer’s overall assessment: This manuscript presents the development and evaluation of a new geometric method, Straw-K, for automated cluster-number selection in K-means based on the local geometry of the WCSS curve. The study is of interest from the perspective of advancing clustering methods and automating the selection of the number of clusters. The proposed approach is compared with several established criteria using a collection of empirical datasets. The manuscript is carefully prepared, well structured, and generally makes a positive impression. Its strengths include a clear presentation of the main idea of the method and a comparative computational study. At the same time, several aspects of the methodology and experimental design require additional clarification and more detailed justification, particularly regarding the choice of algorithm parameters, benchmark construction, stability analysis, and interpretation of the obtained results. After the corresponding clarifications and methodological details have been added, the manuscript may be considered suitable for publication. Specific questions and comments are provided below.

We thank the reviewer for the positive assessment and for identifying several methodological issues that required more than textual clarification. In particular, the boundary-selection concern led to a structural correction of Straw-K and a complete rerun of the benchmark, and the baseline audit led to a correction of the L-method implementation. The revised manuscript reports the resulting changes transparently.

Comment 1

Reviewer comment: Line 117: Please specify the value of the half-window s used in the main computational experiments. Since s directly determines the local interval over which Ri is calculated, this parameter is one of the key parameters of the algorithm and should be explicitly stated in the manuscript.

Response: The main experiments used half-window s=1 for every dataset. The value was fixed in advance and not tuned to individual datasets. It uses the immediate neighboring support point on each side of a candidate and is the smallest symmetric local window available on the short support grid.

Changes in the revised manuscript: Section 2.2 now states s=1 explicitly and defines its role; Algorithm 1 uses the formally defined scoreable index set.

Comment 2

Reviewer comment: Line 121: The values of the local chord-to-path ratio in the figures presented later in the manuscript are very close to one. To what extent can small changes in WCSS resulting from a different K-means initialization, numerical error, or rounding alter the position of the minimum of Ri?

Response: We added a numerical sensitivity analysis that separates three issues: rounding, small perturbations of a fixed WCSS profile, and genuine changes caused by K-means initialization. Rounding the WCSS values to 12, 10, 8, or 6 significant digits changed 0 of the 96 Straw-K selections. Across 500 perturbation replicates for each of the 96 dataset-seed cases, independent relative perturbations up to 10^-6 preserved every selected K. At the larger stress-test level of 10^-4, the mean preservation rate was 99.5%, and only 3 of the 96 cases showed any change. By contrast, several difficult datasets did vary across random-state seeds because different K-means initializations produced different fitted WCSS profiles. Thus, ratios being close to one do not by themselves imply floating-point or display-rounding instability.

Changes in the revised manuscript: Section 3.6 (Geometric Ambiguity Margin and Numerical Robustness); supplementary sensitivity outputs and script.

Comment 3

Reviewer comment: Algorithm 1: From the definition of the admissible indices i=s+1, . . . , M−s, it follows that the boundary values of the considered K range cannot be selected by Straw-K. Table 1 contains datasets with Reference K=2, while Digits has Reference K=10. How appropriate is it to compare exact-match rates under this construction of the algorithm? In addition, Algorithm 1 does not specify how ties or numerically almost indistinguishable values of Ri are handled. How is K∗ determined when two or more candidates have identical or nearly identical values of Ri?

Response: We agree that the original construction created a structural fairness problem at the endpoints. We changed the method so that the selectable range remains K=2,...,10, but Straw-K additionally computes WCSS at K=1 and K=11 solely as non-selectable geometric support points. With s=1, K=2 is scored using support {1,2,3} and K=10 using {9,10,11}; K=1 and K=11 can never be returned. We reran the entire benchmark after this correction. Straw-K’s exact-match rate changed from the previous 32.3% to 44.8%, and its MAE is now 1.542, so all headline results, tables, figures, discussion, and supplementary outputs were updated rather than retaining the old values.

For ties, numerical equality is now defined using np.isclose with rtol=1e-12 and atol=1e-12; if multiple candidates are tied at that tolerance, the smallest K is selected deterministically. Broader near-ties are not collapsed; they remain distinct and are reflected in the descriptive ambiguity index.

Changes in the revised manuscript: Sections 2.1-2.2; Algorithm 1; Abstract; all affected Results tables/figures; Discussion; Conclusions; revised experiment script and supplementary outputs.

Comment 4

Reviewer comment: Table 1: Two of the datasets contain only 32 observations, while values up to K=10 are considered. With such an N/K ratio, some clusters may contain only a few observations. The manuscript should further discuss whether it is appropriate to include such datasets in the comparative benchmark or, alternatively, provide a separate analysis for small-sample datasets.

Response: We retained the two N=32 datasets as explicit small-sample stress cases, but added the separate analysis requested by the reviewer. At K=10, singleton or two-observation clusters can indeed occur. Excluding Motor Trend Cars and Spector Student Performance leaves 10 datasets and 80 evaluations per method. In that sensitivity analysis, Straw-K’s exact-match rate is 53.8% and MAE is 1.525; the Friedman test is significant (chi-square=15.622, p=0.0159), but none of the 21 pairwise Wilcoxon comparisons remains significant after Holm correction (minimum adjusted p=0.082). We therefore report the sensitivity transparently rather than deleting the small datasets after seeing the results.

Changes in the revised manuscript: Section 3.3 (Sensitivity to Small Samples and Preprocessing), Section 4.5 (Limitations), and supplementary small-sample outputs.

Comment 5

Reviewer comment: Line 142: Missing values were replaced by the feature median; however, it is not reported which datasets contained missing values or what proportion of observations or features was affected. This information is necessary to assess the potential influence of imputation on the geometry of the data and the clustering results. In addition, the manuscript states that all features were represented numerically, but it does not clarify whether any of the original variables were categorical or ordinal. If categorical codes were directly treated as numerical features, the use of Euclidean distance and K-means may be methodologically questionable. The preprocessing of such variables should therefore be described in greater detail.

Response: We performed a dataset-level preprocessing audit. Missing selected feature values occurred only in Palmer Penguins and Mammalian Sleep. Penguins contained 8 missing feature cells out of 1,376 (0.58%), affecting 2 of 344 observations. Mammalian Sleep contained 90 missing feature cells out of 456 (19.74%) after 7 observations with missing target were removed; 47 of the remaining 76 observations had at least one missing selected feature. Complete-case sensitivity left the Penguins result unchanged, but Mammalian Sleep was materially sensitive: the median-imputed analysis had Straw-K modal K=3, whereas complete cases gave modal K=4 and 87.5% exact agreement.

We also audited coded variables. ANES includes ordinal-coded predictors; Motor Trend Cars includes binary/discrete variables; Spector includes a binary indicator; and Fuel Economy MPG includes discrete numeric variables. The main benchmark keeps their documented numerical representation after standardization, but the manuscript now states the Euclidean-distance assumption explicitly and reports restricted-feature sensitivity analyses. For example, removing the binary indicator from Spector changed Straw-K’s modal selection from K=4 to K=2, with 62.5% exact agreement in the restricted analysis. These analyses are reported as sensitivities, not as replacements chosen because they improve agreement with the external labels.

Changes in the revised manuscript: Section 2.4 (Benchmark Datasets), Section 3.3, Section 4.5, and supplementary preprocessing audit files.

Comment 6

Reviewer comment: Line 148: The implementation details for Kneedle are described in sufficient detail, whereas corresponding information is not provided for the L-method, Silhouette, Calinski-Harabasz, and Davies-Bouldin. Please specify the exact implementations, software/library versions, criteria used to select the optimal K, and rules for handling ties for all baseline methods.

Response: We expanded the implementation description for all baselines and audited the code. Silhouette selects the largest scikit-learn silhouette score; Calinski-Harabasz selects the largest score; Davies-Bouldin selects the smallest score. The environment uses scikit-learn 1.8.0 and NumPy 2.3.5, and numerical ties are resolved with np.isclose(rtol=1e-12, atol=1e-12) in favor of the smallest K. Kneedle uses kneed 0.8.5 with the stated KneeLocator parameters; if no elbow is returned, a deterministic maximum-curvature fallback is used (6 of 96 Kneedle evaluations, all on Digits). Gap uses B=50 and the one-standard-error rule.

During this audit, we found that our previous L-method code did not exactly match the Salvador-Chan two-segment criterion because the breakpoint was effectively shared by both fitted segments. We corrected it to non-overlapping left and right segments with the published weighted RMSE criterion and reran the benchmark. This materially changed the L-method results and the omnibus statistics; all manuscript values were therefore regenerated from the corrected implementation.

Changes in the revised manuscript: Section 2.5 (Baselines and Reproducible Implementations); corrected experiment script; all affected results, statistical outputs, and figures.

Comment 7

Reviewer comment: Line 168: It is not fully clear how the mean rank by dataset was calculated, particularly when several methods produced identical errors. Please describe the ranking procedure and the treatment of ties in greater detail.

Response: Mean rank is now defined explicitly. For each dataset-method pair, absolute error is first averaged across the eight seeds. The seven methods are then ranked within that dataset by this dataset-level mean absolute error, with lower error receiving the better rank. Exact ties receive the average of the occupied ranks. The reported mean rank is the arithmetic mean of these 12 dataset-level ranks.

Changes in the revised manuscript: Section 2.6 (Repeated Experimental Protocol) and the caption/notes for the aggregate performance table.

Comment 8

Reviewer comment: Line 170: Was a correction for multiple comparisons applied to the paired Wilcoxon tests reported in the Supplementary Material?

Response: The earlier supplementary output contained unadjusted pairwise Wilcoxon p-values. In the revision, all 21 two-sided paired Wilcoxon signed-rank comparisons are adjusted with the Holm step-down procedure to control the family-wise error rate. After the corrected L-method implementation, the Friedman omnibus test is significant (chi-square=13.510, p=0.0356), but none of the 21 pairwise comparisons remains significant after Holm correction; the minimum adjusted p-value is 0.123. We therefore do not claim pairwise statistical superiority of Straw-K or any other method.

Changes in the revised manuscript: Section 2.6; Section 3.1 (Aggregate Empirical Performance); Discussion; supplementary Wilcoxon-Holm output.

Comment 9

Reviewer comment: Figure 5: For Glass Identification, Table 3 reports a modal Straw-K selection of K=9, whereas Figure 5 shows an individual run in which Straw-K selects K=6, matching the Reference K. What objective criterion was used to select this particular representative run?

Response: The displayed Glass run was not selected because it matched Reference K=6. The experiment code used the same pre-specified random-state seed, seed 0, for all four Figure 5 panels. Under the corrected benchmark, Glass seed 0 selects K=6, whereas the modal selection across eight seeds is K=9. We made this selection rule explicit and replaced the manually reconstructed profile figure with the figure generated directly by the experiment script. The caption now states that every panel uses seed 0 and that no panel was chosen based on agreement with the external label count.

Changes in the revised manuscript: Figure 5 and its caption; code constant PROFILE_DISPLAY_SEED=0; supplementary panel-selection audit.

Comment 10

Reviewer comment: Table 4: The runtime for the Gap statistic is calculated by distributing the cached reference-generation cost across eight seeds. This approach may substantially underestimate the computational cost of Gap in a typical scenario in which a user performs a single analysis rather than a series of eight experiments sharing the same cache. It would therefore be useful to additionally report the runtime for an independent single-run scenario and to provide the specifications of the computational system used for the timing experiments.

Response: We agree. The original table reflected an amortized repeated-experiment accounting and was not a good proxy for a one-off user analysis. We added a second timing experiment in which observed seed 0 is fixed in advance and all B=50 Gap reference datasets are regenerated from scratch for each dataset. In this independent scenario, mean Gap selection/reference work is approximately 0.903 s and mean end-to-end runtime is 0.963 s across the 12 datasets; Straw-K’s corresponding mean end-to-end runtime is approximately 0.071 s. For comparison, the original eight-seed amortized Gap end-to-end mean was approximately 0.181 s.

The revised manuscript also reports the timing environment: AMD EPYC 9V74 80-Core Processor host, 5 logical CPUs available to the process, 5.81 GiB RAM, Linux 6.18.44 x86_64, Python 3.13.5, scikit-learn 1.8.0, NumPy 2.3.5, and SciPy 1.17.0. We explicitly note that runtime values are hardware- and environment-dependent and do not present a general speedup factor.

Changes in the revised manuscript: Section 3.5 (Runtime under Amortized and Independent Gap Accounting), revised runtime table/figure, and supplementary independent-runtime script and system metadata.

Reviewer 3 Report

Comments and Suggestions for Authors

1) What is the main question addressed by the research?
The main question is whether Straw-K, a local chord-to-trajectory ratio applied to the WCSS curve, can provide a useful and reproducible recommendation regarding the number of clusters after applying the k-means method. In the course of the study, we tested whether this simple geometric criterion can compete with seven other recognized selection methods on twelve empirical data sets and eight initial points, and whether its research indicator of ambiguity can indicate the instability of the initial point.

2) Do you consider the topic original or relevant to the field? Does it address a specific gap in the field?
The topic is moderately original and relevant. The ratio of the chord to the length of the curve, proposed by Straw-K, is a new criterion, although it is based on the same sliding window idea as in the ShortStraw method. The problem being addressed is of a practical nature: an automated, reproducible selection of the inflection point that is cheaper than the “break curve” or “silhouette” method, provided that the method of weighted squares of curves is available. However, this method makes a negligible contribution to science, as there are already many curve‑based selectors, and the claimed advantage is not statistically significant.

3) What does it add to the subject area compared with other published material?
This adds a transparent and inexpensive assessment of local curvatures, which ranks WCSS curvatures without threshold values, normalization tricks, or reference distributions, and also allows for research ambiguity. The benchmark also compares seven selectors from eight source data sets with cached references to gaps and full compliance with K‑means, and honestly reports Friedman’s insignificant results. This contrasts with works that claim a clear winner. The Digits/PCA diagnostics and discussion of failure patterns further clarify in which cases WCSS elbows should not be trusted.

4) What specific improvements should the authors consider regarding the methodology?
A number of improvements are needed. The sample of 12 datasets is too small for stable ranking; add UCI/OpenML datasets and simulations with various indicators of overlap, imbalance, and dimensionality. Present the results of the paired Wilcoxon test in the main text, not in the supplementary materials. Use a sample with a larger tolerance for uncertainty, calibrate it, and justify the choice of window. Also, provide confidence intervals for each data set and evaluate the Straw‑K method in comparison with WCSS/the k‑means method, since the current conclusions are limited.

5) Are the conclusions consistent with the evidence and arguments presented and do they address the main question posed?
Yes, overall. The conclusions do not correctly emphasize the superiority of the method; it is noted that the Friedman test did not reveal statistically significant differences and that Straw-K only allows for a quick and visual ranking of curves. The failures in the “Numbers,” “Glass,” and “Breast Cancer” tests are openly discussed. However, the emphasis in the annotation on the best exact match and the mean absolute error may somewhat overstate the practical value of the method, given that the differences are not statistically significant. The uncertainty is still correctly accounted for.

6) Are the references appropriate?
In most cases, this is acceptable and relevant. The main recommendations cover K-means, elbow, knee, L-shaped methods, Silhouette, CH, DB, Gap and ShortStraw, supporting the method and basic indicators. However, there are several problems: in reference 4, there is an error in the spelling of Rousseeuw as “Ruseu”; in reference 9, the list of authors is incomplete (“Xiong, Y.; Laviola, J.; Jr.”); and the Krzhanovsky‑Lai rule is cited but not applied. Additional recent surveys on cluster number selection would make it possible to expand the coverage.

7) Any additional comments on the tables and figures.
The tables and graphs are generally clear and informative, but there are some shortcomings. In Table 1, the columns are combined and the formatting is disrupted, which makes it difficult to read. Table 3 lacks data on the frequency of modal values. In Figure 5, the Y‑axes on different panels differ, which makes comparison difficult. In Figure 6, the axis of the logarithmic scale is unreadable. Standard deviations or confidence intervals should be added to Table 2, and all figures should use a larger font and a consistent system of labels.

Author Response

Dear Editor and Reviewers,

Main revisions in this version include:

Boundary correction for Straw-K: K=1 and K=11 are now used only as non-selectable support points, allowing every selectable K=2,...,10 to be scored with the same symmetric window.
A numerical-sensitivity analysis for rounding, perturbations, and K-means initialization.
A separate small-sample sensitivity analysis and a detailed missingness / variable-type preprocessing audit.
A full baseline implementation audit, including correction of the L-method to the Salvador-Chan non-overlapping two-line criterion.
Explicit mean-rank rules and Holm correction across all 21 pairwise Wilcoxon comparisons.
A fixed-seed, non-outcome-selected Figure 5 generated directly from the experiment script.
An independent uncached single-run runtime experiment for Gap, with computational-system specifications.
A complete rewrite of the notation, Algorithm 1, ambiguity interpretation, related-work positioning, and discussion of limitations.

The responses below are organized exactly by reviewer. Reviewer comments are reproduced first, followed by our response and a concise statement of what changed in the manuscript.

Reviewer 3

We thank Reviewer 3 for the careful assessment. Several observations referred to values or formulations from the earlier manuscript version. Where the preceding reviewer-driven methodological corrections changed those facts, we state the updated results explicitly below and have synchronized the manuscript accordingly.

Question 1

Reviewer question/comment: What is the main question addressed by the research?
The main question is whether Straw-K, a local chord-to-trajectory ratio applied to the WCSS curve, can provide a useful and reproducible recommendation regarding the number of clusters after applying the k-means method. In the course of the study, we tested whether this simple geometric criterion can compete with seven other recognized selection methods on twelve empirical data sets and eight initial points, and whether its research indicator of ambiguity can indicate the instability of the initial point.

Response: We agree with the reviewer’s characterization of the central question, with two terminology corrections. Straw-K uses a local chord-to-path ratio on the normalized WCSS curve, and the study asks whether this geometric criterion can provide a useful and reproducible cluster-number recommendation after the candidate K-means fits have been obtained. The final benchmark contains 12 empirical datasets and eight K-means random-state seeds, not eight “initial points.” Seven selectors are compared in total: Straw-K plus six baselines. The ambiguity index is evaluated only as an exploratory descriptive signal of seed instability and is not treated as calibrated uncertainty.

Changes in the revised manuscript: Abstract; Section 1.2 (Study Aim and Main Contributions); Sections 2.3 and 2.6. The revised wording consistently uses “random-state seed” and clarifies the seven-selector design.

Question 2

Reviewer question/comment: Do you consider the topic original or relevant to the field? Does it address a specific gap in the field?
The topic is moderately original and relevant. The ratio of the chord to the length of the curve, proposed by Straw-K, is a new criterion, although it is based on the same sliding window idea as in the ShortStraw method. The problem being addressed is of a practical nature: an automated, reproducible selection of the inflection point that is cheaper than the “break curve” or “silhouette” method, provided that the method of weighted squares of curves is available. However, this method makes a negligible contribution to science, as there are already many curve-based selectors, and the claimed advantage is not statistically significant.

Response: We appreciate the reviewer’s balanced assessment and have further narrowed the novelty claim. After re-checking the ShortStraw literature, we do not claim that the chord-to-path relation itself is new; ShortStraw already uses related chord/path geometry. The contribution of Straw-K is the adaptation of that local geometric relation to the normalized WCSS support curve, the discrete selection rule over K, and the reproducible evaluation protocol. We also avoid claiming a unique computational advantage over all curve-based alternatives: Kneedle and the L-method are likewise inexpensive once their required curve is available. Statistically, the corrected analysis now yields a significant Friedman omnibus test, but no individual pairwise comparison remains significant after Holm correction. Accordingly, the manuscript describes Straw-K as competitive and inspectable rather than superior.

Changes in the revised manuscript: Abstract; Sections 1.2-1.3; Section 4.1; Section 4.3; Conclusions.

Question 3

Reviewer question/comment: What does it add to the subject area compared with other published material?
This adds a transparent and inexpensive assessment of local curvatures, which ranks WCSS curvatures without threshold values, normalization tricks, or reference distributions, and also allows for research ambiguity. The benchmark also compares seven selectors from eight source data sets with cached references to gaps and full compliance with K-means, and honestly reports Friedman’s insignificant results. This contrasts with works that claim a clear winner. The Digits/PCA diagnostics and discussion of failure patterns further clarify in which cases WCSS elbows should not be trusted.

Response: We agree with the reviewer’s emphasis on transparency, but several factual details in this comment correspond to the earlier manuscript version and have now been corrected. The final benchmark uses 12 datasets, not eight source datasets, and compares seven selectors in total. Straw-K does use explicit min-max normalization of the candidate axis and WCSS values, so we no longer characterize it as avoiding normalization. Gap reference datasets are cached only in the repeated eight-seed benchmark; an independent uncached single-run timing is now reported separately. Finally, after the boundary correction and corrected L-method implementation, the Friedman omnibus test is significant (chi-square=13.510, p=0.0356), although none of the 21 pairwise Wilcoxon tests remains significant after Holm adjustment. The Digits/PCA diagnostic and failure-pattern discussion are retained.

Changes in the revised manuscript: Sections 2.2, 2.5-2.7, 3.1, 3.4-3.6, and Discussion. The runtime and statistical results were regenerated rather than merely reworded.

Question 4

Reviewer question/comment: What specific improvements should the authors consider regarding the methodology?
A number of improvements are needed. The sample of 12 datasets is too small for stable ranking; add UCI/OpenML datasets and simulations with various indicators of overlap, imbalance, and dimensionality. Present the results of the paired Wilcoxon test in the main text, not in the supplementary materials. Use a sample with a larger tolerance for uncertainty, calibrate it, and justify the choice of window. Also, provide confidence intervals for each data set and evaluate the Straw-K method in comparison with WCSS/the k-means method, since the current conclusions are limited.

Response: We addressed the recommendations that can be incorporated without changing the scope of the study after review. First, all Holm-adjusted pairwise Wilcoxon results are now shown in the main manuscript as a compact p-value matrix. Second, Table 2 now includes 95% dataset-level cluster-bootstrap confidence intervals for exact-match rate and MAE (20,000 dataset-resampling replicates), which avoids treating the eight seeds within a dataset as independent sampling units. The modal-selection table now reports modal frequencies out of eight seeds, providing dataset-level information about stability.

We also strengthened the rationale for the half-window. The main analysis uses s=1, fixed in advance. Because K=2 is an eligible boundary value, s=1 is the only symmetric half-window that can score K=2 while using only valid positive cluster counts; any s>=2 would require a support value at K<=0. The ambiguity index remains deliberately uncalibrated: a 12-dataset sample is too small to support a credible probability calibration, so the manuscript explicitly treats calibration as future work on a larger benchmark.

We agree that a broader UCI/OpenML benchmark and controlled simulations varying overlap, imbalance, dimensionality, covariance structure, and elbow sharpness would strengthen external validity. Rather than adding a new large experimental program after review, we now state this explicitly as future work. We also clarify that WCSS itself is an objective curve, not a deterministic selector until a decision rule or human reader is added; therefore, a direct algorithmic comparison against “WCSS” alone is not well-defined. We added a future-work proposal for a controlled reader study comparing Straw-K with manual visual elbow judgments.

Changes in the revised manuscript: Section 2.2; Section 2.6; Table 2; new main-text Wilcoxon table; modal-frequency table; Section 4.6 (Future Work).

Question 5

Reviewer question/comment: Are the conclusions consistent with the evidence and arguments presented and do they address the main question posed?
Yes, overall. The conclusions do not correctly emphasize the superiority of the method; it is noted that the Friedman test did not reveal statistically significant differences and that Straw-K only allows for a quick and visual ranking of curves. The failures in the “Numbers,” “Glass,” and “Breast Cancer” tests are openly discussed. However, the emphasis in the annotation on the best exact match and the mean absolute error may somewhat overstate the practical value of the method, given that the differences are not statistically significant. The uncertainty is still correctly accounted for.

Response: We agree with the reviewer’s concern about overstatement and have kept the conclusion deliberately qualified. There are also two important updates relative to the earlier version referenced in the comment. First, after the methodological corrections the Friedman omnibus test is significant, but no pairwise comparison remains significant after Holm correction; the Abstract now presents both facts together immediately after the descriptive exact-match and MAE values. Second, Breast Cancer Diagnostic is no longer a Straw-K failure: the boundary correction makes K=2 scoreable with the same symmetric window as interior candidates, and Straw-K selects K=2 in all eight seeds. Digits and Glass remain important failure/instability examples. The conclusions therefore support Straw-K as a competitive, inspectable option rather than a generally superior selector.

Changes in the revised manuscript: Abstract; Section 3.1; Section 4.1-4.2; Conclusions.

Question 6

Reviewer question/comment: Are the references appropriate?
In most cases, this is acceptable and relevant. The main recommendations cover K-means, elbow, knee, L-shaped methods, Silhouette, CH, DB, Gap and ShortStraw, supporting the method and basic indicators. However, there are several problems: in reference 4, there is an error in the spelling of Rousseeuw as “Ruseu”; in reference 9, the list of authors is incomplete (“Xiong, Y.; Laviola, J.; Jr.”); and the Krzhanovsky-Lai rule is cited but not applied. Additional recent surveys on cluster number selection would make it possible to expand the coverage.

Response: We rechecked the bibliography carefully. The revised reference list already gives the correct spelling “Rousseeuw, P.J.” and the complete ShortStraw-based reference as “Xiong, Y.; LaViola, J.J., Jr.” with the corrected DOI. We retain Krzanowski-Lai because it is relevant methodological context for sum-of-squares-based cluster-number selection, but the revised Related Work now states explicitly that it is cited as context and was not one of the seven benchmarked selectors. To broaden current coverage, we also added two recent/modern validation references: Hassan et al. (2024), “From A-to-Z Review of Clustering Validation Indices,” Neurocomputing, and Gagolewski et al. (2021), “Are Cluster Validity Measures (In)valid?”, Information Sciences.

Changes in the revised manuscript: Section 1.3 (Related Work) and References. References 4 and 9 were verified/corrected, the role of Krzanowski-Lai was clarified, and two broader recent validation references were added.

Question 7

Reviewer question/comment: Any additional comments on the tables and figures.
The tables and graphs are generally clear and informative, but there are some. shortcomings. In Table 1, the columns are combined and the formatting is disrupted, which makes it difficult to read. Table 3 lacks data on the frequency of modal values. In Figure 5, the Y-axes on different panels differ, which makes comparison difficult. In Figure 6, the axis of the logarithmic scale is unreadable. Standard deviations or confidence intervals should be added to Table 2, and all figures should use a larger font and a consistent system of labels.

Response: We implemented the requested presentation improvements. Table 1 was reformatted to separate Dataset, N, d, Reference K, and Source more clearly. The modal-selection table (renumbered in the revised manuscript because a Wilcoxon table was added before it) now reports each modal K together with its frequency out of eight seeds, including explicit ties. Figure 5 was regenerated with normalized WCSS on a common 0-1 left axis and a common ratio-axis range across all four panels, while retaining the same fixed seed 0 in every panel. Figure 6 uses a linear, not logarithmic, runtime axis and its labels/ticks were enlarged. Table 2 now includes 95% dataset-level cluster-bootstrap confidence intervals for both exact-match rate and MAE. Figure labels were enlarged and standardized where applicable.

Changes in the revised manuscript: Table 1; Table 2; main-text Wilcoxon table; modal-selection table; Figures 3-6, especially the regenerated normalized Figure 5.

Closing Note

We again thank the Editor and all reviewers for the detailed comments. The revised manuscript and supplementary package have been synchronized to the corrected analysis outputs, and stale values from the earlier boundary construction, earlier L-method implementation, and unadjusted statistical reporting have been removed. The additional Reviewer 3 comments also led us to add dataset-level bootstrap confidence intervals, place the Holm-adjusted Wilcoxon results in the main text, report modal frequencies, normalize Figure 5 for cross-panel comparison, reformat Table 1, and expand the recent validation literature. We believe the revised version is substantially clearer, more reproducible, and more appropriately qualified.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The revised manuscript has been amended to properly incorporate the reviewers' comments.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have satisfactorily addressed all critical comments and provided the necessary clarifications. I believe that the manuscript is now suitable for publication and recommend its acceptance in its present form.

Review Reports

Reviewer 1

General Comment 1

General Comment 2

General Comment 3

Detailed Comments

Comment 1

Comment 2

Comment 3

Comment 4

Comment 5

Comment 6

Comment 7

Comment 8

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Comment 10

Comment 11

Comment 12

Comment 13

Reviewer 2

Comment 1

Comment 2

Comment 3

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Comment 5

Comment 6

Comment 7

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Comment 10

Reviewer 3

Question 1

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Question 4

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Question 6

Question 7

Closing Note