Ensemble Empirical Mode Decomposition of Australian monthly rainfall and temperature data

Abstract

Empirical Mode Decomposition (EMD), developed by Huang et al. (1998), is a form of adaptive time series decomposition. Traditional forms of spectral analysis, like Fourier, assume that a time series (either linear or nonlinear) can be decomposed into a set of linear components. However, as the degree of non-periodic behaviour and non-stationarity in a time series increases, the set of linear components describing that time series increases substantially when using Fourier techniques. In the physical sciences, time series are often non-periodic, more stochastic and even non-stationary, so Fourier based spectral analysis techniques often produce large sets of physically meaningless harmonics when applied to these problems (Huang et al. 1998). In contrast the EMD method does not assume a time series is linear or stationary prior to analysis. EMD adaptively decomposes a time series into a set of independent intrinsic mode functions (IMFs) and a residual component. When the IMFs and residual are summed together they form the original time series. An inconvenient feature of EMD is mode mixing, where a fluctuation of given frequency may split across two IMFs. The data driven adaptive iterative nature of the EMD algorithm means mode mixing is difficult to avoid without subjectively deciding on the likely nature of any signal to be extracted prior to analysis. Mode mixing between IMFs is problematic when investigating the physical significance of IMFs, as an expected physical signal may be present but split across IMFs. Wu and Huang (2009) proposed Ensemble EMD (EEMD), a noise assisted data analysis method, as a way of overcoming the mode mixing problem. In EEMD, an ensemble of EMD trials is obtained by adding finite amplitude normally distributed white noise to the time series prior to each EMD run. The IMFs and residual from each trial are grouped by IMF order into ensembles and the IMF and residual ensemble averages form the EEMD result. Since the white noise is different for each EMD trial the noise cancels out during averaging as the ensemble size increases. However, the noise serves the useful purpose of changing the ordering of local maxima and minima within the time series, thus provoking a different EMD outcome in each trial. Wu and Huang (2009) believe EEMD provides more physically meaningful IMFs and residual than traditional EMD. The EEMD is carried out using monthly rainfall and temperature data from 44 Australian rainfall and ten climate stations respectively. The results from the analysis are presented in terms of dominant frequencies, trends, changes in variance with time (Hilbert spectrum) and relationships between climate indices and rainfall and temperature. For the monthly rainfall data, almost all the stations exhibited an annual cycle while a number of stations showed cycles with periods about 3 and 20 years. The mean monthly temperature data only showed an annual cycle. However, the residuals from the EEMD analysis of the temperature data showed a steadily increasing trend for all ten sites.