Probabilistic Rank Correlation - A New Rank and Comparison Based Correlation Coefficient with a Simple, Pragmatic Transitivity Condition

Probabilistic Rank Correlation - A New Rank and Comparison Based Correlation Coefficient with a Simple, Pragmatic Transitivity Condition

Ruiting Lian Changle Zhou Ben Goertzel 

Fujian Provincial Key Laboratory of Brain-like Intelligent Systems, Department of Cognitive Science, Xiamen University, Xiamen, China

Hanson Robotics Open Cog Foundation709C, 7/F Bio-Informatics Centre 2 Science Park West Avenue Shatin, Hong Kong, China

Page: 
476-496
|
DOI: 
https://doi.org/10.18280/ama_a.540403
Received: 
10 August 2017
| |
Accepted: 
30 October 2017
| | Citation

OPEN ACCESS

Abstract: 

A novel measure of correlation between data sets is proposed based on applying the notion of “probabilistic support” to compare the pairwise comparisons of measurements. Probabilistic Rank Correlation (PRC) is a crisp instantiation of this idea, in the spirit of traditional rank correlations. It is shown that, under broad conditions, Probabilistic Rank Correlations has a strong, elegant transitivity property. The practical application of the PRC is also illustrated.

Keywords: 

Correlation, Transitivity, Probabilistic support, Probabilistic Rank Correlation, Correlation measure

1. Introductions
2. Transitivity of Correlation
3. Probabilistic Rank Correlation
4. Algebraic Properties of Symmetric and Asymmetric Probabilistic Support
5. Comparison with Kendall Correlation
6. A Simple Sufficient Condition for Transitivity of the Probabilistic Rank Correlation
7. A Less Stringent Transitivity Condition
8. Handling Equally Ranked Values
9. Practical Examples
10. Conclusion
  References

1. Abdi H. The Kendall rank correlation coefficient. 2007. Encyclopedia of Measurement and Statistics. Sage, Thousand Oaks, CA, pp. 508-510.

2. J. Benesty, J. Chen, Y. Huang, I. Cohen. Pearson correlation coefficient. 2009. Noise Reduction in Speech Processing. Springer Berlin Heidelberg, pp. 1-4.

3. Castro Sotos, Ana Elisa, Andreas Vesaliusstraat, Vanhoof Stijn, Onghena Patrick Van Den Noortgate. The non-transitivity of Pearson’s correlation coefficient: an educational perspective. 2007. Proc. 56th Session of the Int. Statistical Institute 2007, pp. 22-29.

4. G.W. Corder, D.I. Foreman. Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach. 2009. Wiley.

5. C. Croux, C. Dehon. Influence functions of the Spearman and Kendall correlation measures. 2010. Statistical Methods & Applications, vol. 19, no. 4, pp. 497-515.

6. Eric Langford, Neil Schwertman, Margret Owens. Is the Property of Being Positively Correlated Transitive? 2001. The American Statistican, vol. 55, no. 4, pp. 322-325.

7. A. Gonzalez-Serna, R. A. McGovern, P. R. Harrigan, F. Vidal, A. F. Y. Poon, S. Ferrando-Martinez, E. Ruiz-Mateos, Correlation of the virological response to short-term maraviroc monotherapy with standard and deep-sequencing-based genotypic tropism prediction methods. 2012. Antimicrobial Agents and Chemotherapy, vol. 56, no. 3, pp. 1202-1207.

8. M. G. Kendall. A new measure of rank correlation. 1938. Biometrika, vol. 30, no. 1/2, pp. 81-93.

9. E. Langford, N. Schwartzman, M. Owens, Is the property of being positively correlated transitive? 2001. The American Statistician, vol. 55, pp. 322-325.

10. C. T. Le. A New Rank Test Against Ordered Alternatives in K-Sample Problems. 2007. Biometrical Journal, vol. 30, no. 1, pp. 87-92.

11. Wei Li, Reform on Mathematical Modelling Teaching Contents in the Era of Big Data. 2016. AMSE Journals-2016-Series: Advances A; vol. 59, no. 1, pp. 129-144.

12. D. Liu, W. Y. Dong, B. X. Wang, Methods for Supplier Library Construction and Parts Similarity Measurement in Web-Based Parts Library Platform. 2017. AMSE JOURNALS-AMSE IIETA publication-2017-Series: Advances A; vol. 54, no. 1, pp. 87-105.

13. A. Mahmood, S. Khan, Exploiting transitivity of correlation for fast template matching. 2010. IEEE Transactions on Image Processing, vol. 19, no. 8, pp. 2190-2200.

14. P. E. McKight, J. Najab. Kruskal‐Wallis Test. 2010. Corsini Encyclopedia of Psychology.

15. Thomas L. Moore, Paradoxes in film ratings. 2006. Journal of Statistics Education, vol. 14, no. 1): n1.

16. K. Popper, D. W. Miller, Why probabilistic support is not inductive. 1987. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 321, no. 1562, pp. 569-591.

17. M. H. Quenouille. Approximate tests of correlation in time-series. 1949. Journal of the Royal Statistical Society. Series B (Methodological). vol. 11, no. 1, pp. 68-84.

18. Tomoji Shogenji. A Condition for Transitivity in Probabilistic Support. 2003. British Journal for the Philosophy of Science, vol. 54, pp. 613-616.

19. R. S. Tsay. Analysis of financial time series, 2005. vol. 543, John Wiley & Sons. 

20. Emine Yilmaz, Aslam Javed, Robertson Stephen. A new Rank Correlation coefficient for information retrieval. Proceedings of the 31st annual international ACM SIGIR conference on Research and development in information retrieval.