Empirical mode decomposition (EMD), an adaptive data analysis methodology, has the attractive feature of robustness in the presence of nonlinear and non-stationary time series. Recently, in this journal, Pegram and co-authors (
Pegram
et al
. 2008
Proc. R. Soc. A
464
, 1483–1501), proposed a modification to the EMD algorithm whereby rational splines replaced cubic splines in the extrema envelope-fitting procedure. The modification was designed to reduce variance inflation, a feature frequently observed in cubic spline-based EMD components primarily due to spline overshooting, by introducing a spline tension parameter. Preliminary results there demonstrated the proof of concept that increasing the spline tension parameter reduces the variance of the resultant EMD components. Here, we assess the performance of rational spline-based EMD for a range of tension parameters and two end condition treatments, using a global dataset of 8135 annual precipitation time series. We found that traditional cubic spline-based EMD can produce decompositions that experience variance inflation and have orthogonality concerns. A tension parameter value of between 0 and 2 is found to be a good starting point for obtaining decomposition components that tend towards orthogonality, as measured by an orthogonality index (OI) metric. Increasing the tension parameter generally results in: (i) a decrease in the range of the OI, which is offset by slight increases in (ii) the median value of OI, (iii) the number of intrinsic mode function components, (iv) the average number of sifts per component, and (v) the degree of amplitude smoothing in the components. The two end conditions tested had little influence on the results, with the reflective case being slightly better than the natural spline case as indicated by the OI. The ability to vary the tension parameter to find an orthogonal set of components, without changing any sifting parameters, is a powerful feature of rational spline-based EMD, which we suggest is a significant improvement over cubic spline-based EMD.
A Model for Non-Stationary Time Series and Its Applications in Filtering and Anomaly Detection
