Statistical Papers

, Volume 60, Issue 6, pp 1827–1847 | Cite as

On circular correlation for data on the torus

  • Xiaoping ZhanEmail author
  • Tiefeng Ma
  • Shuangzhe Liu
  • Kunio Shimizu
Regular Article


The circular correlation topic for data on the torus is studied. Firstly, the order for two points on the circumference is considered and an order function is defined. Then, an alternative moment coefficient to measure T-linear association between two circular variables based on the order function is proposed. After the concordant on the torus is explained, an alternative rank correlation coefficient on the torus is also proposed. A number of properties for the two coefficients are investigated and their comparisons with the existing alternatives are made. Two examples of real data analysis are presented to illustrate our results.


Circular correlation Circular data Kendall’s \(\tau \) Circular order T-linear 



We would like to thank very much the Editor and reviewers for their constructive and useful comments which led to an improved presentation of the manuscript. The work of the first two authors is supported by the National Natural Science Foundation of China (No. 11471264, 11401148, 11571282).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Xiaoping Zhan
    • 1
    Email author
  • Tiefeng Ma
    • 1
  • Shuangzhe Liu
    • 2
  • Kunio Shimizu
    • 3
  1. 1.School of Statistics, Center of Statistical ResearchSouthwestern University of Finance and EconomicsChengduChina
  2. 2.Program of Mathematics and StatisticsUniversity of CanberraCanberraAustralia
  3. 3.School of Statistical ThinkingThe Institute of Statistical MathematicsTokyoJapan

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