Issues with the Application of Empirical Mode Decomposition Analysis

Hydroclimatic variability occurs over time scales ranging from seconds through to millennia. Fluctuations at certain time scales, like monthly, seasonal, annual, interannual and interdecadal, are particularly important for the sustainable management of land and water resources systems. Quantifying the proportion of variation in a hydroclimatic time series due to fluctuations at different time scales is usually done using spectral techniques, like Fourier analysis, which assume a time series to be both linear and stationary. Empirical mode decomposition (EMD), a relatively new form of time series analysis for quantifying the proportion of variation at different time scales, is introduced and key aspects of its application are discussed in this paper. EMD was originally developed as a form of adaptive time series decomposition, used prior to spectral analysis using the Hilbert transform, for non-linear and non-stationary time series data. In this paper the spectral analysis component (Hilbert transform) is not discussed, only the EMD procedure is addressed. EMD has several advantages over other spectral techniques, in that it is relatively easy to understand and use, the fluctuations within a time series are automatically and adaptively selected from the time series and it is robust in the presence of non-linear and non-stationary data. EMD allows the data to speak for themselves. Robustness in the presence of non-linear and non-stationary data is particularly important for hydroclimatology applications where time series are generally non-stationary (or not long enough to categorically prove stationarity) and they exhibit non-linear characteristics like amplitude and frequency modulation with time. In this paper the process of applying EMD to a time series is demonstrated using 10 years of monthly precipitation data from Melbourne Regional Office. The Melbourne monthly precipitation time series is decomposed into three intrinsic mode functions (IMFs) and a residual. The conditions defining an IMF are presented. The sifting process used to obtain each IMF and the residual is described. The general features of IMFs are described along with the ability to combine IMFs and the residual to form low frequency or high frequency filters. Two key decisions in the EMD application process, the rule for deciding when to stop sifting for an IMF and the choice of cubic spline end condition rule are reviewed and discussed in detail. A new cubic spline end condition rule, based on the assumption that the slope of the cubic spline at the end point is equal to zero, is proposed and compared to two other end condition rules from the literature. The comparison is based on three applications of the EMD algorithm, each application with a different end condition rule, to 8135 annual precipitation time series from around the world. The annual precipitation time series have periods of record ranging from 30 up to 299 years and represent a wide range of climatic zones. The proposed end condition rule is found to be the most efficient of the three rules tested, due to the EMD algorithm producing less IMFs when using the proposed rule. The end condition rule proposed in this paper is recommended for future EMD applications.