An Optimal and Stable Algorithm for Clustering Numerical Data

Round 1

Reviewer 1 Report

This paper introduces a new seeding mechanism known as zero-point multidimensional spaces that can provide cluster optimality and stability, resolving relevant issues, as well as a corresponding algorithm called the zero k-approximate modal haplotype (Zk-AMH) algorithm.

This work is really well written. However, I suggest following the suggestions above, in order to further enhance it.

Furthermore, I suggest improving the analysis of literature. In particular, I think it could be interesting to consider this related paper:

https://link.springer.com/chapter/10.1007/978-3-030-00084-4_27

Moreover, the authors should focus more on which gaps the paper aims to cover and which are the limits of this paper.

I encourage the authors to refine their work to make it available for publication in the journal.

Author Response

Please see the attachment

Author Response File: Author Response.pdf

Reviewer 2 Report

The manuscript suggests a new seeding method for the k-AMH algorithm. Although the introduction is well organised and properly grounds the subsequent chapters, the following sections contain a lot of errors in the mathematical descriptions.

Main comments:

• Equations contain a lot of errors. The notation of sets and objects are mixed (see for example Equation 1, lines 288, 289, etc.
• I don’t understand why is a partition matrix (W) used in subsection 2.1. As the title shows, this subsection is about the k-means clustering algorithm, but the original k-means does not apply a partition matrix. It is used only in the c-means algorithm.
• Figure 2: it is not defined how is the first, second, and third k-object is selected. It seems the clusters were known a priori.
• Line 317: “… as shown in example above” is not a well-defined description. The algorithm should be given without using an example.

Some errors as examples:

• line 105: “dataset X into cluster C” -> dataset X into C clusterS
• Lines 111 and 112: The dot at the end of the sentence is missing.
• Equation 1: It is suggested to define the distance of two elements from X and Z (X_i and Z_l), and not the sets (X, Z). The same problem can be seen in Equation 9.
• Equation 2: w is not defined
• There are many equations that are not numbered.
• In Equation 6: r is not defined
• In Equation 7: q is not defined
• Equation 8: the third line is not correct.
• Algorithm 2: line 5 is in my opinion wrong. I think it should be standing here maximization instead of minimization.
• Algorithm 2, line 6: Equation (6) does not say anything about updating. It is only an inequality. Same error in Algorithm 3 in line 6.
• Line 246: Equation is not numbered, furthermore as it defines the distance between two objects, the left side notation of the equation (SO) is not correct.
• Line 288: “object X and zero seeding O” -> “object X_l and zero seeding O_i”
• same error in line 289

Author Response

Please see the attachment

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Congratulations to the authors. In my opinion, now the paper is ready to be published.

Reviewer 2 Report

I accept in present form.