Elsevier

Journal of Hydrology

Volume 510, 14 March 2014, Pages 293-298
Journal of Hydrology

Comparison of the MK test and EMD method for trend identification in hydrological time series

https://doi.org/10.1016/j.jhydrol.2013.12.039Get rights and content

Highlights

  • Pre-whitening cannot really improve trend identification when using the MK test.

  • Series’ trend magnitudes greatly influence trend identification of series.

  • EMD method can be an effective alternative for trend identification of series.

  • EMD method can identify the specific shape of the analyzed series’ trend.

Summary

Trend identification is an important issue in hydrological time series analysis, but it is also a difficult task due to the diverse performances of methods. This paper mainly investigated the performances between the Mann–Kendall (MK) test and the empirical mode decomposition (EMD) method for trend identification of series. Analyses of both synthetic and observed series indicate the better performance of EMD compared with the other. The results show that pre-whitening cannot really improve trend identification when using the MK test, but cause wrong results sometimes. It can be due to the good correlation of trend, so pre-whitening would weaken trend’s magnitude. If the trend of the analyzed series has small magnitude, it cannot be accurately identified by the MK test, because the trend would be submerged too severely by other components of series to accurately identify trend. When the analyzed series has short length, its trend cannot be accurately identified by the MK test. However, the EMD method can eliminate the influences of trends’ magnitude and series’ length, so it has more effective power for trend identification. As a result, it is suggested that series’ trend can be directly identified by the MK test but need not do pre-whitening; moreover, the influences of trends’ magnitude should be carefully considered for trend identification. Comparatively, the EMD method can adaptively determine the specific shape of the nonlinear and non-stationary trend of series by considering statistical significance, so it can be an effective alternative for trend identification of hydrological time series.

Introduction

Trend identification is a required task in hydrological series analysis, because it is the basis not only for understanding the long-term variations of hydrological processes, but also for revealing periodicities and other characteristics of hydrological processes (Kallache et al., 2005, Gao et al., 2007, Hamed, 2008). However, trend identification is a difficult problem in practice, and diverse performances of those methods for trend identification are presented (Kahya and Kalayci, 2004, Macdonald et al., 2005, Adam and Lettenmaier, 2008). Two problems should be answered in trend identification of series: evaluation of the statistical significance of trend, and determination of the specific shape of trend.

Presently, the rank-based nonparametric Mann–Kendall (MK) test is used commonly for assessing the statistical significance of series’ trend. It is simple and can handle missing values and values below certain detection limit (Mann, 1945, Kendall, 1975, Kundzewicz et al., 2005). However, the MK test is not robust against serial correlation (Yue et al., 2002, Shao and Li, 2011), and also depends on the sample size, serial correlation, as well as magnitude of the trend to be identified (Adamowski et al., 2009, Sang et al., 2012b). For overcoming the defects of the MK test, various approaches were suggested to handle the affects of serial correlation, such as pre-whitening and variance correction (Hamed, 2009, Khaliq et al., 2009, Rivard and Vigneault, 2009). Besides serial correlation, seasonal components and other periodic fluctuations of series would also affect the MK test statistics (Shao and Li, 2011). Therefore, annual series are used commonly to eliminate the affects of seasonal components. Since observed annual hydrological series usually do not have enough sample size, the influences of sample size and seasonal components cannot be eliminated simultaneously. Moreover, the MK test only can assess the statistical significance of trend, but cannot determine its specific shape, no matter the trend is linear or nonlinear.

To accurately identify different components in hydrological series, many new methods have been employed in hydrological time series analysis. Among those methods, the empirical mode decomposition (EMD) method performs better comparatively, since it can reveal the non-stationary and nonlinear characteristics of a series under multi-temporal scales (Wu and Huang, 2004, Sang et al., 2012a). Contrary to almost all previous decomposition methods (such as Fourier transform, wavelet analysis), EMD is empirical, intuitive, direct and adaptive, without requiring any predetermined basis functions, but being based on the principle of local scale separation (Huang et al., 1998). The EMD method is developed to decompose a series into a set of components called intrinsic mode functions (IMFs), which become the basis representing data. Because of the adaptive nature of the basis, there is no need for harmonics (Kim and Oh, 2009). Therefore, EMD is ideally suitable for analyzing data from non-stationary to nonlinear processes (Huang and Wu, 2006).

The main objective of this paper is to compare the performances between the MK test and the EMD method for trend identification, and further to improve the understanding about trend identification of series. To begin with, mathematical properties of the two methods were described in Section 2. In Section 3, both synthetic and observed series were analyzed by the two methods for comparison. Several conclusions and suggestions for trend identification were given in the final section.

Section snippets

The MK test

The MK test is essentially limited to test the null hypothesis that the data are independent and identically distributed (Mann, 1945, Kendall, 1975). It searches for a trend in a series without specifying whether the trend is linear or nonlinear. Given a series x(t) with the length of n, the null hypothesis of no trend assumes that the series x(t) are independently distributed. The MK test is based on the test statistic S:S=i=1n-1j=i+1nsgn(x(j)-x(i))withsgn(x)=1ifx>0sgn(0)=0sgn(x)=-1ifx<0

A

Data sets

Both synthetic and observed series were used for the study (Table 1). Three types of synthetic series were considered here (Fig. 2). The type-I series were generated using the first-order autocorrelation (AR(1)) model with different correlations, the type-II series were generated using the exponential function with different bases (i.e., different trend magnitudes), and the type-III series were generated using exponential function with different lengths. The first two types of series were used

Conclusion

Trend identification is an important task in hydrological series analysis. In this paper, the performances between the MK test and the EMD method for trend identification were compared. Analyses of both synthetic and observed series indicate the better performances of the EMD method compared with the MK test. The results indicate that pre-whitening cannot really improve trend identification in some situations when using the MK test, and even would cause wrong results. The reason can be due to

Acknowledgements

The authors gratefully acknowledged the most appropriate comments and suggestions given by the Editors and the anonymous reviewers. The authors also thank Ms. Feifei Liu for her assistances in the preparation of the manuscript. This project was financially supported by the National Natural Science Foundation of China (Nos. 41201036 and 41330529).

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