Abstract
Generating random numbers is prerequisite to any Monte Carlo method implemented in a computer program. Therefore, identifying a good random number generator is important to guarantee the quality of the output of the Monte Carlo method. However, sequences of numbers generated by means of algorithms are not truly random, but having certain control on its randomness essentially makes them pseudo-random. What then matters the most is that the simulation of a physical variable with a probability distribution, needs to have the same distribution generated by the algorithm itself. In this perspective, considering the example of gauge block calibration given in "Evaluation of measurement data—Supplement 1 to the “Guide to the expression of uncertainty in measurement”—Propagation of distributions using a Monte Carlo method", we explore the properties and output of three commonly used random number generators, namely the linear congruential (LC) generator, Wichmann-Hill (WH) generator and the Mersenne-Twister (MT) generator. Extensive testing shows that the performance of the MT algorithm transcends that of LC and WH generators, particularly in its time of execution. Further, these generators were used to estimate the uncertainty in the measurement of the length, with input variables having different probability distributions. While, in the conventional GUM approach the output distribution appears to be Gaussian-like, we from our Monte-Carlo calculations find it to be a students' t-distribution. Applying the Welch-Satterthwaite equation to the result of the Monte Carlo simulation, we find the effective degrees of freedom to be 16. On the other hand, using a trial–error fitting method to determine the nature of the output PDF, we find that the resulting distribution is a t-distribution with 46 degrees of freedom. Extending these results to calculate the expanded uncertainty, we find that the Monte-Carlo results are consistent with the recently proposed mean/median-based unbiased estimators which takes into account the artifact of transformation distortion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
The Joint Committee for Guides in Metrology (JCGM) ‘Evaluation of Measurement Data — Supplement 1 to the “Guide to the Expression of Uncertainty in Measurement” — Propagation of Distributions Using a Monte Carlo Method’ (2008).
M. Wei, Y. Zeng, C. Wen, X. Liu, C. Li and S. Xu ‘Comparison of MCM and GUM Method for Evaluating Measurement Uncertainty of Wind Speed by Pitot Tube’ MAPAN-J. Metrol. Soc India (2019).
X. Tian and K. Benkrid ‘Mersenne Twister Random Number Generation on FPGA, CPU and GPU’. NASA/ESA Conference on Adaptive Hardware and Systems (2009).
B. Wichmann and D. Hill ‘Algorithm AS183: An Efficient and Portable Pseudo-Random Number Generator’ Journal of the Royal Statistical Society (1982).
M. Matsumoto and T. Nishimura 'Mersenne Twister: a 623-Dimensionally Equidistributed Uniform Pseudo-Random Number Generator’. ACM Transactions on Modelling and Computer Simulation (1997).
S. Sinha, SKH. Islam and MS. Obaidat. ‘A Comparative Study and Analysis of some Pseudorandom Number Generator Algorithms’ Security and Privacy (2018).
A. Roy and T. Gamage. ‘Recent Advances in Pseudorandom Number Generation’ (2007).
K. Marton, A. Suciu, C. Sacarea and O. Cret. ‘Generation and Testing of Random Numbers for Cryptographic Applications’. Proceedings of the Romanian Academy, volume 13 (2012).
V. Ristovka and V. Bakeva. ‘Comparison of the Results Obtained by Pseudo Random Number Generator based on Irrational Numbers’ International Scientific Journal "Mathematical Modeling".
IIT Kanpur (online) ‘Autocorrelation’. Available at: http://home.iitk.ac.in/~shalab/econometrics/Chapter9-Econometrics-Autocorrelation.pdf
P. Burgoine ‘Testing of Random Number Generators’ University of Leeds. http://www1.maths.leeds.ac.uk/~voss/projects/2012-RNG/Burgoine.pdf.
C.M. Grinstead and J.L. Snell ‘Introduction to Probability’. Chapter 7 (1998).
K.N. Koide ‘Analysis of Random Generators in Monte Carlo Simulation: Mersenne Twister and Sobol’.
A. Gaeini, A. Mirghadri, G. Jandaghi and B. Keshavarzi ‘Comparing Some Pseudo-Random Number Generators and Cryptography Algorithms Using a General Evaluation Pattern’. Modern Education and Computer Science Press (2016).
F. Sepehri, M. Hajivaliei and H. Rajabi ‘Selection of Random Number Generators in GATE Monte Carlo Toolkit’. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment (2020).
S. Wang, X. Ding, D. Zhu, H. Yu and H. Wang ‘Measurement Uncertainty Evaluation in Whiplash Test Model Via Neural Network and Support Vector Machine-Based Monte Carlo Method’. Measurement 119. (2018).
A.Y. Mohammed and A.G. Jedrzejczyk ‘Estimating the Uncertainty of Discharge Coefficient Predicted for Oblique Side Weir Using Monte Carlo Method’ Flow Measurement and Instrumentation 73 (2020).
H. Huang ‘Why the Scaled and Shifted t-distribution Should not be used in the Monte Carlo Method for Estimating Measurement Uncertainty?’ Measurement 136 (2019).
A. Doğanaksoy, F. Sulak, M. Uğuz, O. Şeker, and Z. Akcengiz ‘New Statistical Randomness Test Based on Length of Runs’ Hindawi Article 626408 (2015).
J.H. Zar ‘Biostatistical Analysis-5th Edition’ Prentice-Hall, Inc. Division of Simon and Schuster One Lake Street Upper Saddle River, NJ United States (2007).
The Joint Committee for Guides in Metrology (JCGM) ‘Evaluation of Measurement Data — Guide to the Expression of Uncertainty in Measurement’ (2008).
R. Willink ‘A Generalization of the Welch–Satterthwaite Formula for Use with Correlated Uncertainty Components’ Metrologia 44(5):340 (2007).
R. Kacker, B. Toman and D. Huang ‘Comparison of ISO-GUM, Draft GUM Supplement 1 and Bayesian Statistics Using Simple Llnear Calibration’ Metrologia 43 S167 (2006).
R. Kacker ‘Bayesian Alternative to the ISO-GUM’s Use of the Welch–Satterthwaite Formula’ Metrologia 43 1 (2005).
H. Huang ‘Uncertainty-Based Measurement Quality Control’ Accred Qual Assur 19(2):65–73 (2014).
H. Huang ‘On the Welch-Satterthwaite Formula for Uncertainty Estimation: A Paradox and Its Resolution’ Cal Lab Magazine, October – December (2016).
M. Ballico ‘Limitations of the Welch-Satterthwaite Approximation for Measurement Uncertainty Calculations’ Metrologia (2000).
H. Huang ‘A Paradox in Measurement Uncertainty Analysis’ Measurement Science Conference, Pasadena (2010).
J.D. Jenkins ‘The Student’s t-distribution Uncovered’ Measurement Science Conference (2007).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Malik, K., Pulikkotil, J. & Sharma, A. Comparison of Pseudorandom Number Generators and Their Application for Uncertainty Estimation Using Monte Carlo Simulation. MAPAN 36, 481–496 (2021). https://doi.org/10.1007/s12647-021-00443-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12647-021-00443-3