Spline manipulations for empirical mode decomposition (EMD) on bounded intervals and beyond

Abstract

Empirical mode decomposition (EMD), introduced by N.E. Huang et al. in 1998, is perhaps the most popular data-driven computational scheme for the decomposition of a non-stationary signal or time series $f\left(t\right)$, with time-domain $R:=(-\infty ,\infty )$, into finitely many oscillatory components $\left\{f_1\right(t),\cdots ,f_K(t\left)\right\}$, called intrinsic mode functions (IMFs), and some “almost monotone” remainder $r\left(t\right)$, called the trend of $f\left(t\right)$. The core of EMD is the iterative “sifting process” applied to each function $m_{k-1}\left(t\right)$ to compute $f_k\left(t\right)$, for $k=1,\cdots ,K$, where $m_0\left(t\right):=f\left(t\right)$ and $m_k\left(t\right):=m_{k-1}\left(t\right)-f_k\left(t\right)$, with trend $r\left(t\right):=m_K\left(t\right)$. For the computation of each IMF, the sifting process depends on cubic spline interpolation of the local maxima and local minima for computing the upper and lower envelopes, respectively, and on subtracting the mean of the two envelopes from the result of the previous iterative step. Since it is not feasible to search for all local extrema in the entire time-domain $(-\infty ,\infty )$, implementation of the sifting process is commonly performed on some desired truncated bounded interval $[a,b]$. The main objective of this paper is to introduce and develop four “cubic spline manipulation engines”, called “quasi-interpolation (QI)”, “enhanced quasi-interpolation (EQI)”, “local interpolation (LI)”, and “improved global interpolation (IGI)” cubic spline manipulation engines, in order to significantly improve the performance of EMD on the truncated time-domains with minimal boundary artifacts, computational efficiency, accuracy, and consistency. Introduction and construction of the “fundamental quasi-interpolation” (FQI) splines as basis functions of the QI manipulation engine eliminates the need of matrix inversion for computing (global) cubic spline interpolation, since the local maximum values and local minimum values are used as coefficients of their FQI spline series representations, respectively. For the EQI spline manipulation engine, the FQI functions are formulated in terms of the same cubic B-spline basis for both the upper and lower envelopes; and for the LI spline manipulation engine, the “cubic spline blending” operation is applied to further modify the FQI splines to enable true cubic spline interpolation by “correcting the approximate interpolation error” of the EQI engine. As a consequence, the EQI and LI manipulation engines have the common property that in computing the means of the upper and lower envelopes, the only computation is averaging the B-spline coefficients, instead of computing the upper and lower envelopes separately. Furthermore, fast cubic spline pre-processing of the given $f\left(t\right)$ is also introduced to assure numerical stability in the computation of the Hilbert transform of the first IMF $f_1\left(t\right)$ on the truncated time-domain. The theory, along with methods and explicit formulas, developed in this paper are intended for other applications beyond EMD.