eXtended Empirical Mode Decomposition

Signals are decomposed into oscillatory modes by empirical mode decomposition and its ensemble, complementary ensemble, complete ensemble, masked, bivariate and multivariate variants, by variational mode decomposition and its multivariate and successive variants, and by local mean decomposition, iterative filtering, time varying filter empirical mode decomposition and the empirical wavelet transform; the synchrosqueezed wavelet transform is also provided. Empirical mode decomposition is due to Huang et al. (1998) , its ensemble variant to Wu and Huang (2009) and variational mode decomposition to Dragomiretskiy and Zosso (2014) . The modes can be described by their instantaneous amplitude and frequency and their Hilbert spectrum, tested as intrinsic mode functions or against white noise, selected by energy and compared through orthogonality, mode splitting, reconstruction quality, ensemble dispersion and similarity measures. Acceleration, velocity and displacement records stored as a long table of time, value, quantity and component are decomposed for every record and quantity in one call, with a warning and a regularization of the time grid when it is not uniform.

xemd

xemd provides R functions for signal decomposition and mode analysis. The library and the xemd command-line interface are installed together from the repository root with the product installer.

In empirical mode decomposition (EMD), the signal's extrema define the local scales used to extract components. Each extracted component is subtracted from the remaining signal, leaving a final residue. Hilbert analysis describes each component through its analytic signal: its modulus gives instantaneous amplitude, and the derivative of its continuous phase divided by $2\pi$ gives instantaneous frequency where that phase is defined and differentiable. The introduction develops these concepts alongside an EMD example and a variational mode decomposition (VMD) example.

A decomposition gives a representation of the observed signal. Its modes do not automatically identify separate physical sources or isolate noise. The methods guide explains how the methods define their components; the interpretation guide distinguishes numerical convergence, reconstruction, mode diagnostics and evidence for retaining a component.

Guides

EMD example

This signal combines sinusoidal oscillations with a trend. The sifting controls specify the envelope-centering test, the boundary treatment and the iteration limit; they are the configuration of this example.

library(xemd)

Time <- seq(0, 9, length.out = 3000)
Signal <- sin(pi * Time) + sin(2 * pi * Time) + sin(6 * pi * Time) +
    0.5 * Time
Sifting <- list(stopRule = "doubleThreshold", thresholds = c(0.05, 0.5),
                fraction = 0.05, maxSift = 2000, boundary = "reflection",
                maxModes = Inf)
OUT <- decomposeEMD(Signal, sifting = Sifting)

OUT$modes contains the extracted components and OUT$residue contains what remains. Their sum reconstructs the input up to rounding. This additive identity does not establish whether an extracted mode recovers one of the original sinusoids. OUT$converged reports acceptance by the chosen sifting rule, while OUT$stopReason reports why extraction ended. The worked introduction shows those outputs, a reconstruction check and the IMF diagnostics.