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A New Two-Dimensional Rank Correlation Coefficient

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Abstract

Spearman’s rank correlation coefficient might be the best known nonparametric measure of association currently in use. It assesses the linear relationships between the ranks of monotonically related variables even if the relationship between the variables is not linear. This study presents a new method for calculating two-dimensional (horizontal and vertical) rank correlation coefficients between matrices composed of variables which are not necessarily in linear association. The matrices contain the ranks of measurements instead of raw values. The averages of all rows are used for calculating the horizontal rank correlation and the averages of all columns are used for calculating the vertical rank correlation instead of considering the averages of the whole matrices. This approach enables a separate determination of the degree of horizontal and vertical relationships between the compared data matrices by using the horizontal and vertical variance and covariance values that constitute the base of the two-dimensional correlation method. The presented method is first applied on 5 simple hypothetical matrices and then on the monthly total precipitation records of 6 stations in southwest Turkey. The results have shown that the presented rank correlation approach successfully assesses the two-dimensional associations between the hypothetical matrices and time series data like precipitation, provides a measure for exactly determining the monotonic relationships not determined by the conventional Pearson’s and Spearman’s correlation approaches, it is much more robust to outliers and normality is not a prerequisite. The software developed for calculating two-dimensional rank correlation coefficients is freely provided together with this paper.

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Acknowledgements

The author would like to thank to The General Directorate of State Hydraulic Works of Turkey for providing the data used in this study. The complete data set is restricted for explicit publication and is commercially available at http://www2.dsi.gov.tr/duyuru/bedel.htm (in Turkish). The author would also like to thank the editors and reviewers for their valuable contributions and comments. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Correspondence to Fatih Dikbaş.

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Dikbaş, F. A New Two-Dimensional Rank Correlation Coefficient. Water Resour Manage 32, 1539–1553 (2018). https://doi.org/10.1007/s11269-017-1886-0

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Keywords

  • Two-dimensional variance
  • Two-dimensional covariance
  • Two-dimensional rank correlation
  • Time series analysis
  • Resistance to outliers