Abstract
The Friedman Test has been proposed in 1937 to analyze tables of ranks, like those arising from a wine contest. If we have N judges and k wines, the standard problem is to analyze a table of N rows and k columns holding the opinion of the judges. The Friedman’s Test is used to accept/reject the null hypothesis that all the wines are equivalent. Friedman offered an asymptotically valid approximation as well as exact tables for low k and N. The accuracy of the asymptotic approximation for moderate k and N was low, and extended tables were required. The published ones were mostly computed using Monte Carlo techniques. The effort required to compute the extended tables for the case without ties was significant (over 100 years of CPU time) and an alternative using many-core processors is described here for the general case with ties. The solution can be used also for other similar tests which yet lack for large enough tables.
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- 1.
Estimated in the order of \(10^{8}\).
- 2.
Set of 10390 pairs (k; N) whose asymptotic estimates differ more than a percentage w.r.t. the Monte Carlo ones.
References
Anzt, H., Dongarra, J.J., Flegar, G., Higham, N.J., Quintana-Ortí, E.S.: Adaptive precision in block-Jacobi preconditioning for iterative sparse linear system solvers. Concurrency Comput.: Practice Experience 31(6), e4460 (2019)
Carballo, P., Perera, P., Rama, S., Pedemonte, M.: A biased random-key genetic algorithm for regression test case prioritization. In: IEEE Latin American Conference on Computational Intelligence, LA-CCI 2018, 7–9 November 2018, Gudalajara, Mexico, pp. 1–6. IEEE (2018)
Corder, G.W., Foreman, D.I.: Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach. Wiley, Hoboken (2009)
Derrac, J., García, S., Molina, D., Herrera, F.: A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol. Comput. 1(1), 3–18 (2011)
Dufrechu, E., Favre, F., Pedemonte, M., Curto, P., Ezzatti, P.: Accelerating radiative heat transfer calculations on modern hardware. In: 2012 XXXVIII Conferencia Latinoamericana En Informatica (CLEI), 1–5 October 2012, Medellin, Colombia, pp. 1–9. IEEE (2012)
Abadi, M., et al.: TensorFlow: Large-scale machine learning on heterogeneous systems (2015). Software available from tensorflow.org
Fawcett, R.F., Salter, K.C.: A Monte Carlo study of the f test and three tests based on ranks of treatment effects in randomized block designs. Commun. Stat. - Simul. Comput. 13, 213–225 (1984)
Friedman, M.: The use of ranks to avoid the assumption of normality implicit in the analysis of variance. J. Am. Stat. Assoc. 32(200), 675–701 (1937)
Friedman, M.: A comparison of alternative tests of significance for the problem of \(m\) rankings. Ann. Math. Stat. 11(1), 86–92 (1940)
Iman, R., Davenport, J.: Approximations of the critical region of the Friedman statistic. Commun. Stat.-Theory Methods 9, 571–595 (1980)
Kirk, D.B., Hwu, W.-M.W.: Programming Massively Parallel Processors: A Hands-on Approach, 3rd edn. Morgan Kaufmann, Burlington (2016)
L’Ecuyer, P., Simard, R.: Testu01: A C library for empirical testing of random number generators. ACM Trans. Math. Softw. 33(4), 22:1–22:40 (2007)
López-Vázquez, C., Hochsztain, E.: Extended and updated tables for the Friedman rank test. Commun. Stat. - Theory Methods 48(2), 268–281 (2019)
Manssen, M., Weigel, M., Hartmann, A.K.: Random number generators for massively parallel simulations on GPU. Eur. Phys. J. Spec. Top. 210(1), 53–71 (2012)
McCool, M., Reinders, J., Robison, A.: Structured Parallel Programming: Patterns for Efficient Computation, 1st edn. Morgan Kaufmann Publishers Inc., San Francisco (2012)
Nvidia Corporation. CUDA Toolkit 10.0 CURAND Library Programming Guide. Nvidia Corporation, September 2018
L’Ecuyer, P., Simard, R.: TestU01 Website (2007). http://simul.iro.umontreal.ca/testu01/tu01.html. Accessed June 2019
Pedemonte, M., Luna, F., Alba, E.: A systolic genetic search for reducing the execution cost of regression testing. Appl. Soft Comput. 49, 1145–1161 (2016)
Quade, D.: Using weighted rankings in the analysis of complete blocks with additive block effects. J. Am. Stat. Assoc. 74(367), 680–683 (1979)
Salmon, J.K., Moraes, M.A., Dror, R.O., Shaw, D.E.: Parallel random numbers: as easy as 1, 2, 3. In: Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis, p. 16. ACM (2011)
Salmon, J.K., Moraes, M.A., Dror, R.O., Shaw, D.E.: Random123 Website (2011). http://www.thesalmons.org/john/random123/. Accessed June 2019
Sheskin, D.J.: Handbook of Parametric and Nonparametric Statistical Procedures, Fifth edn. Chapman and Hall, London (2011)
Stepanov, A., Lee, M.: The Standard Template Library. Hewlett Packard Laboratories (1995)
Acknowledgments
The researchers acknowledges partial support from Programa de Desarrollo de las Ciencias Básicas (PEDECIBA), Sistema Nacional de Investigadores (SNI) and Agencia Nacional de Investigación e Innovación (ANII).
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Irigaray, D., Dufrechou, E., Pedemonte, M., Ezzatti, P., López-Vázquez, C. (2020). Accelerating the Calculation of Friedman Test Tables on Many-Core Processors. In: Crespo-Mariño, J., Meneses-Rojas, E. (eds) High Performance Computing. CARLA 2019. Communications in Computer and Information Science, vol 1087. Springer, Cham. https://doi.org/10.1007/978-3-030-41005-6_9
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