Comparison of the MK test and EMD method for trend identification in 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:with
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).
References (26)
- et al.
Deriving the respiratory sinus arrhythmia from the heartbeat time series using empirical mode decomposition
Chaos Soliton. Fract.
(2004) Trend detection in hydrological data: the Mann–Kendall trend test under the scaling hypothesis
J. Hydrol.
(2008)Enhancing the effectiveness of prewhitening in trend analysis of hydrological data
J. Hydrol.
(2009)- et al.
Trend analysis of streamflow in Turkey
J. Hydrol.
(2004) - et al.
Identification of hydrological trends in the presence of serial and cross correlations: a review of selected methods and their application to annual flow regimes of Canadian rivers
J. Hydrol.
(2009) - et al.
Recent climate change in the Arctic and its impact on contaminant pathways and interpretation of temporal trend data
Sci. Total Environ.
(2005) - et al.
Period identification in hydrological time series using empirical mode decomposition and maximum entropy spectral analysis
J. Hydrol.
(2012) - et al.
A new trend analysis for seasonal time series with consideration of data dependence
J. Hydrol.
(2011) - et al.
Application of new precipitation and reconstructed streamflow products to streamflow trend attribution in northern Eurasia
J. Clim.
(2008) - et al.
Development of a new method of wavelet aided trend detection and estimation
Hydrol. Process.
(2009)
Empirical mode decomposition as a filter bank
IEEE Signal Process. Lett.
Trend of estimated actual evapotranspiration over China during 1960–2002
J. Geophys. Res. – Atmos.
A review on Hilbert-Huang transform: method and its applications to geophysical studies
Rev. Geophys.
Cited by (147)
Quantitative and qualitative assessment of groundwater resources for drinking water supply in the peri-urban area of Dhaka, Bangladesh
2024, Groundwater for Sustainable DevelopmentStatistical significance of PM<inf>2.5</inf> and O<inf>3</inf> trends in China under long-term memory effects
2023, Science of the Total EnvironmentTrends of inorganic sulfur and nitrogen species at an urban site in western Canada (2004–2018)
2023, Environmental PollutionIce monitoring of aluminum conductor steel-reinforced cables using guided waves
2023, Cold Regions Science and Technology