Natural Test for Random Numbers Generator Based on Exponential Distribution

Round 1

Reviewer 1 Report

First of all, I would like to begin by stating that my engineering background is not a very good match for this highly theoretical paper so you should treat my comments critically.

Regarding the abstract (and several places in the paper), I would rephrase "[...] numbers sorted by size" to "numbers sorted by value", it's slightly less confusing.

Also, the first phrase in the abstract states "When random numbers are sorted by size, their successive differences are a random variable." While this is obviously true, it is also somehow incomplete as I believe the claim in the paper is that the successive differences random variable is exponentially distributed for uniformly distributed original values. You should state this clearly in the abstract, also because it clears up the next phrase that talks about exponential distributions, otherwise the entire abstract makes little sense. A suggestion for rephrasing: "We will prove that when uniformly distributed random numbers are sorted by value, their successive differences are a exponentially distributed random variable."

The term "mathematical expectancy" should be changed to "mathematical expectation".

The phrase "This proven characteristic with Theorem proof, can be used for testing of generated random numbers" makes little sense. It should be either rephrased or removed altogether.

A very important part of the abstract and / or introduction that is missing, is the addressed issue. Why is this method of verifying the exponential distribution of the sorted values differences better that existing RNGs statistical tests (like NIST, TestU01, Diehard(er), etc), and why is it better than testing the original sequence for uniformity.

The first paragraph of the introduction has absolutely no relevance to the rest of the paper, it should be completely removed. You can add a phrase when talking about RNG application domains in order to mention medical applications (there you can cite all papers [1 - 6] ;))

The abstract and introduction also need a careful read to fix missing symbols (like PI) and some English spelling and grammar issues.

Unfortunately I couldn't follow the proof, so I will assume it is correct. 

In the last section, where you present the results and conclusions, I would add a comparison between your method and established statistical tests and verify that your approach gives similar results and try to explain the differences (if there are any).

Author Response

Dear reviewer, thank you very much for the useful suggestions. We accepted all of the suggestions and we are sure that his will improve the quality and contribute to a better understanding of the paper.

First of all, I would like to begin by stating that my engineering background is not a very good match for this highly theoretical paper so you should treat my comments critically.

Comment 1: Regarding the abstract (and several places in the paper), I would rephrase "[...] numbers sorted by size" to "numbers sorted by value", it's slightly less confusing. Also, the first phrase in the abstract states "When random numbers are sorted by size, their successive differences are a random variable." While this is obviously true, it is also somehow incomplete as I believe the claim in the paper is that the successive differences random variable is exponentially distributed for uniformly distributed original values. You should state this clearly in the abstract, also because it clears up the next phrase that talks about exponential distributions, otherwise the entire abstract makes little sense. A suggestion for rephrasing: "We will prove that when uniformly distributed random numbers are sorted by value, their successive differences are a exponentially distributed random variable."

Response to comment 1: Your remarks for abstract have been fully accepted.

Comment 2: The term "mathematical expectancy" should be changed to "mathematical expectation".

Response to comment 2: We have corrected this in the whole paper.

Comment 3: The phrase "This proven characteristic with Theorem proof, can be used for testing of generated random numbers" makes little sense. It should be either rephrased or removed altogether.

Response to comment 3: We have removed this phrase.

Comment 4: A very important part of the abstract and / or introduction that is missing, is the addressed issue. Why is this method of verifying the exponential distribution of the sorted values differences better that existing RNGs statistical tests (like NIST, TestU01, Diehard(er), etc), and why is it better than testing the original sequence for uniformity.

Response to comment 4: As has been mentioned, your remarks for abstract have been fully accepted. The manuscript was created on an engineering platform for traffic flow analysis. In the search for a theoretical explanation of the old phenomenon (see the literature Kinzer 1933, Adams 1936), we developed a theorem, and noticed the possibility of application in randomization testing. On your suggestions "A very important part of introduction ...." we have supplemented the text of the introduction with IV new paragraphs (lines from 60 to 78). By introducing the CLT into context, a new way for explaining the entire manuscript is opened. The potential significance of the exponential distribution is fully explained.

Comment 5: The first paragraph of the introduction has absolutely no relevance to the rest of the paper, it should be completely removed. You can add a phrase when talking about RNG application domains in order to mention medical applications (there you can cite all papers [1 - 6] ;))

Response to comment 5: Paragraphs I and II have changed positions. The first impression is that the introduction text has a better continuum. From the importance of random numbers, it moves to humans (brains) and computers. Please accept the retention of II paragraphs of the introduction, without modification. Yet, human is the basis for generating random numbers.

Comment 6: The abstract and introduction also need a careful read to fix missing symbols (like PI) and some English spelling and grammar issues.

Response to comment 6: We have corrected all mistakes such nature in the whole paper. Also, all typographical errors are corrected. All figures were restored to their original high resolution format.

Comment 7: In the last section, where you present the results and conclusions, I would add a comparison between your method and established statistical tests and verify that your approach gives similar results and try to explain the differences (if there are any).

Response to comment 7: In the end we found the results needed to compare the new test with the previous tests, first of all the Diehard battery of test. We just listed the results. We did not comment on them, because in the conclusion the limitations for new test was is given - 10^6 decimals and the comparative results are referred to 10^9 decimals. After all, we did not set standards for the new method. If researchers accept the new test, they will set standards depending on many factors in the application, such as the size of a set of random decimals, required verification speeds, etc.

Thank you for the great guidance that, we believe, improved the manuscript especially in the introduction.

Reviewer 2 Report

Authors of manuscript discussed successive differences between random numbers and proposed test of quality of given generator of random numbers. The discussed problems seems to be quite interested and are valuable for publish to a wide audience who are interested in area of application of random number generators.

 

However, some elements of paper should be improved, the list of typographic errors is as follows:

 

+ line 24, the pi symbol is missing,

+ line 38, 43, the use of apostrophes and quotation should be corrected,

+ line 46 word “Phusical”,

+ Figure 1, quality of picture is not good (picture is bitmap?), other figures should be also check,

+ line 107, the dot after equation (3), in others equations punctuation marks should be also corrected,

+ line 128 arrow and dots in enumerations.

Author Response

Dear reviewer, thank you very much for the useful suggestions. We accepted your suggestions and we are sure that his will improve the quality of the paper.

Authors of manuscript discussed successive differences between random numbers and proposed test of quality of given generator of random numbers. The discussed problems seems to be quite interested and are valuable for publish to a wide audience who are interested in area of application of random number generators.

However, some elements of paper should be improved, the list of typographic errors is as follows:

Comments:

 

line 24, the pi symbol is missing,

line 38, 43, the use of apostrophes and quotation should be corrected,

line 46 word “Phusical”,

Figure 1, quality of picture is not good (picture is bitmap?), other figures should be also check,

line 107, the dot after equation (3), in others equations punctuation marks should be also corrected,

line 128 arrow and dots in enumerations.

Response to comments: All typographical errors are corrected. The final text of the manuscript is undergoing to proofreading from MDPI. The final text will be further improved.

Please note that random variable T and  T with dot are not the same. Random variable T with dot is convolution of two random intervals f(A) and f(b). Because of this, we have adopted an offensive expression with big dot. We tried with *,  but * is not noticeable as dot. We agree that it is cumbersome, but please accept the dot mark.

All figures were restored to their original high resolution format, while Figures 1 and 2 are replaced with new.

Reviewer 3 Report

This is an interesting paper on a topic of technical interest in computational statistics.
However, I am missing a reference to standard works on testing for randomness in computational statistics.

Also, the paper should be revised stylistically (please number the theorems and provide clear statement to
theorems with all definitions beeing explicit) and from the language perspective. The overall readability is
fine, but some phrases come across as not being idiomatic.

Follows a short list of suggestions for improvement.

1.) In the abstract in "second on the consecutive decimals of the"
you are missing the constant \pi before the equality sign
2.) Rather than "Random numbers are applicable in various areas"
say something like >>Random number generators are applicable in various areas..
Are random numbers useful or something that generates them, analyzes their randomness , .... is actually useful?
3.) "Phusical" should probably be Physical
4.) "equal expectancy" >> equal expectation
5.) Define T_i:=t_{i+1}-t_i formally and state it explicitly in the statement of Theorem on pg. 3.
Figure 1 is not a definition!
6.) Also, theorems should be numbered for easier reference.

7.) References on the randomness tests should be updated. One of the pioneers of testing randomness is
for instance George Marsaglia with his diehard suite of tests for randomness

See a reference
G. Marsaglia and W. Tsang, Some Difficult-to-pass Tests of Randomness.
Journal of Statistical Software, vol. 7 (2002), no. 3

and some further web resources at
https://web.archive.org/web/20160125103112/http://stat.fsu.edu/pub/diehard/
http://www0.cs.ucl.ac.uk/staff/d.jones/GoodPracticeRNG.pdf

Pleas update the references. Also a very classic work by D. Knuth should be referenced like

D. Knuth, The Art of Computer Programming, Volume 2 (3rd Ed.): Seminumerical Algorithms.
Addison-Wesley Longman Publishing Co., Inc., 1997. ISBN 0-201-89684- 2.

 

Author Response

Dear reviewer, thank you very much for the useful suggestions. We accepted all of the suggestions and we are sure that his will improve the quality and contribute to a better understanding of the paper.

This is an interesting paper on a topic of technical interest in computational statistics. However, I am missing a reference to standard works on testing for randomness in computational statistics.

Also, the paper should be revised stylistically (please number the theorems and provide clear statement to

theorems with all definitions beeing explicit) and from the language perspective. The overall readability is fine, but some phrases come across as not being idiomatic.

Follows a short list of suggestions for improvement.

Comment 1: In the abstract in "second on the consecutive decimals of the" you are missing the constant \pi before the equality sign.

Response to comment 1: Your remarks for abstract have been fully accepted.

Comment 2: Rather than "Random numbers are applicable in various areas" say something like >>Random number generators are applicable in various areas. Are random numbers useful or something that generates them, analyzes their randomness, .... is actually useful?

Response to comment 2: We would like to inform you that we have changed the places of the first and second paragraphs following the instructions of other reviewers. The text of the manuscript now begins with the acceptance of your regular remark.

Comment 3: "Phusical" should probably be Physical.

Response to comment 3: Corrected. Please see line 45.

Comment 4: "equal expectancy" >> equal expectation

Response to comment 4: We have corrected this in the whole paper.

Comment 5: Define T_i:=t_{i+1}-t_i formally and state it explicitly in the statement of Theorem on pg. 3. Figure 1 is not a definition! 

Response to comment 5: Your remark: Define T…. formally and state it explicitly in the statement of theorem on pg. 3! … It remains amazing that an important relation in the text of the theorem has been missed. Special thanks for this remark. The indexes in the text of the theorem are aligned with the indexes in Figures 1, too.

All figures were restored to their original high resolution format, while Figures 1 and 2 are replaced with new.

Comment 6: Also, theorems should be numbered for easier reference.

Response to comment 6: Theorem is numbered.

Comment 7: References on the randomness tests should be updated. One of the pioneers of testing randomness is for instance George Marsaglia with his diehard suite of tests for randomness

See a reference

Marsaglia and W. Tsang, Some Difficult-to-pass Tests of Randomness. Journal of Statistical Software, vol. 7 (2002), no. 3

and some further web resources at

https://web.archive.org/web/20160125103112/http://stat.fsu.edu/pub/diehard/

http://www0.cs.ucl.ac.uk/staff/d.jones/GoodPracticeRNG.pdf

Pleas update the references. Also a very classic work by D. Knuth should be referenced like

Knuth, The Art of Computer Programming, Volume 2 (3rd Ed.): Seminumerical Algorithms.

Addison-Wesley Longman Publishing Co., Inc., 1997. ISBN 0-201-89684- 2.

Response to comment 7: 6 new references have been introduced into the manuscript. You can find significant additions in the introduction text (new paragraphs, lines from 60 to 78). They came from your directing to important references, most notably the Diehard battery test from Marsaglia.

Thank you for the exceptionally valuable remarks. 

Round 2

Reviewer 3 Report

I am satisfied with the revision.