Moment Approach for Quantitative Evaluation of Randomness Based on RMT Formula

Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 16)

Abstract

We develop in this article a quantitative formulation of the randomness-test based on the random matrix theory (RMT-test), in order to compare a subtle difference of randomness between given random sequences. Namely, we compare the moments of the actual eigenvalue distribution to the corresponding theoretical expression that we derive from the formula theoretically derived by the random matrix theory. We employ the moment analysis in order to compare the eigen-value distribution of the cross correlation matrix between pairs of sequences. Using this method, we compare the randomness of five kinds of random data generated by two pseudo-random generators (LCG and MT) and three physical generators. Although the randomness of the individual sequence can be quantified in a precise manner using this method, we found that the measured values of randomness fluctuate significantly. Taking the average over 100 independent samples each, we conclude that the randomness of the random data generated by the five generators are indistinguishable by the proposed method, while the same method can detect the randomness of the derivatives of the sequences, or the initial part of LCG, which are distinctly lower.

Keywords

Randomness measure RMT-test Moment analysis Eigenvalues of cross correlation matrix Marcenko-Pastur distribution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Mehta, M.: Random Matrices, 3rd edn. Academic Press (2004)Google Scholar
  2. 2.
    Edelman, A., Rao, N.R.: Random Matrix Theory. Acta Numerica, 1–65 (2005)Google Scholar
  3. 3.
    Wigner, E.P.: Ann. Math. 67, 325–327 (1958)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Plerou, V., Gopikrishnan, P., Rosenow, B., Amaral, L.A.N., Stanley, H.E.: Physical Review Letters 83, 1471–1474 (1999)CrossRefGoogle Scholar
  5. 5.
    Plerou, V., Gopikrishnan, P., Rosenow, B., Amaral, L., Stanley, H.E.: Random Matrix Approach to Cross Correlation in Financial Data. Physical Review E 65(066126) (2002)Google Scholar
  6. 6.
    Laloux, L., Cizeaux, P., Bouchaud, J., Potters, M.: Noise Dressing of Financial Correlation Matrices. Physical Review Letters 83, 1467–1470 (1998)CrossRefGoogle Scholar
  7. 7.
    Tanaka-Yamawaki, M.: Cross Correlation of Intra-day Stock Prices in Comparison to Random Matrix Theory. Intelligent Information Management 3, 65–70 (2011)CrossRefGoogle Scholar
  8. 8.
    Tanaka-Yamawaki, M., Kido, T., Itoi, R.: Trend-Extraction of Stock Prices in the American Market by Means of RMT-PCA. In: Watada, J., Phillips-Wren, G., Jain, L.C., Howlett, R.J. (eds.) Intelligent Decision Technologies. SIST, vol. 10, pp. 637–646. Springer, Heidelberg (2011), doi:10.1007/973-642-22194-1CrossRefGoogle Scholar
  9. 9.
    Yang, X., Itoi, R., Tanaka-Yamawaki, M.: Testing Randomness by Means of RMT Formula. In: Watada, J., Phillips-Wren, G., Jain, L.C., Howlett, R.J., et al. (eds.) Intelligent Decision Technologies. SIST, vol. 10, pp. 589–596. Springer, Heidelberg (2011), doi:10.1007/973-642-22194-1CrossRefGoogle Scholar
  10. 10.
    Knuth, D.E.: The Art of Computer Programming. Seminumerical Algorithms, vol. 2. Addison-Wesley (1980)Google Scholar
  11. 11.
    Matsumoto, M., Nishimura, T.: Mersenne Twister: A 623-Dimensionally Equidistributed Uniform Pseudorandom Number Generator. ACM Trans. Modeling and Computer Simulation 8, 3–30 (1998)MATHCrossRefGoogle Scholar
  12. 12.
    Marcenko, V., Pastur, L.: Distribution of Eigenvalues for Some Sets of Random Matrices. Mathematics of the USSR-Sbornik 1, 457–483 (1994)CrossRefGoogle Scholar
  13. 13.
    Sengupta, A., Mitra, P.: Distribution of Singular Values for Some Random Matrices. Physical Review E 60, 3389–3392 (1999)CrossRefGoogle Scholar
  14. 14.
    Random Number Library: http://random.ism.ac.jp/random
  15. 15.
    Yang, X., Itoi, R., Tanaka-Yamawaki, M.: Testing Randomness by Means of Random Matrix Theory. Accepted for Progress of Theoretical Physics (2012) (supplement)Google Scholar
  16. 16.

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Information and Knowledge Engineering, Graduate School of EngineeringTottori UniversityTottoriJapan

Personalised recommendations