Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise

Computes each residue as the average local mean of the previous residue perturbed by a mode of supplied noise realizations, and each mode as the difference between consecutive residues.

Usage

decomposeICEEMDAN(signal, noise, amplitudeRatio = 0.2, normalizeNoise,
                  sifting)

Arguments

signal: Numeric vector of at least two finite values without dim, in the units of the quantity.
noise: Finite numeric matrix with one row per sample of signal and one column per realization, $M \ge 1$; column $m$ is a realization $w^{(m)}$ of unit variance, used as supplied. The function draws no random numbers.
amplitudeRatio: Dimensionless ratio $\varepsilon_0 \ge 0$. Default 0.2, as in decomposeEEMD.
normalizeNoise: TRUE or FALSE, without default. From the second stage on, TRUE divides each noise mode by its standard deviation before scaling; FALSE keeps its amplitude. normalizeNoise = TRUE corresponds to SNRFlag = 2 of ceemdan.m.
sifting: Named list of sifting controls, as in decomposeEMD; the rule "consecutive" is rejected. The same list decomposes the noise realizations.

Details

Each column of noise is decomposed once with sifting, $\eta_k^{(m)} = E_k(w^{(m)})$. From $r_0 = x$, before stage $k$ the common residue $r_{k-1}$ ends the decomposition, in this order, with "terminalResidue" when it has at most one extremum, with "modeLimit" when $k - 1$ equals sifting$maxModes and with "insufficientExtrema" when its envelopes cannot be built. Stage $k$ perturbs $r_{k-1}$ with $p^{(m)} = \beta_{k-1}^{(m)} \eta_k^{(m)}$, where $\beta_0^{(m)} = \varepsilon_0 \, \mathrm{sd}(x) / \mathrm{sd}(\eta_1^{(m)})$ and, for $k \ge 2$, $\beta_{k-1} = \varepsilon_0 \, \mathrm{sd}(r_{k-1})$, divided by $\mathrm{sd}(\eta_k^{(m)})$ when normalizeNoise is TRUE; standard deviations use the divisor $N - 1$. With $A$ the average first mode of $r_{k-1} + p^{(m)}$ and $\bar p$ the mean perturbation, the new residue is the mean local mean and the mode the difference of residues: $$r_k = r_{k-1} + \bar p - A, \qquad d_k = r_{k-1} - r_k.$$ A realization without $\eta_k$ contributes a null perturbation and stays in the divisor $M$.

Value

A list with

modes: Double matrix, samples by modes.
residue: Numeric vector, the final common residue.
iterations: Integer vector by mode, the sum of the subtractions of the $M$ extractions of the stage.
converged: Logical vector by mode: TRUE when the $M$ extractions accepted, FALSE when one ended without accepting or could not begin, NA when all completed fixed sifting.
stopReason: "terminalResidue", "modeLimit" or "insufficientExtrema".

Conditions

A noise mode of zero standard deviation leaves its scale undefined and stops the function with an error of class xemdZeroEnergy.

References

Colominas, M. A., Schlotthauer, G. and Torres, M. E. (2014). Improved complete ensemble EMD: A suitable tool for biomedical signal processing. Biomedical Signal Processing and Control, 14, 19–29. doi:10.1016/j.bspc.2014.06.009

The selection of the reflected extrema and the spline interpolation of the envelope nodes are adapted from emd.m (G. Rilling, March 2007), distributed with CEEMDAN revision af9a0f83, copyright (c) 2020 Marcelo A. Colominas, MIT License; see the NOTICE file of the installed package.

Examples

Time <- seq(0, 9, length.out = 1000)
Signal <- sin(pi * Time) + sin(2 * pi * Time) + sin(6 * pi * Time) +
    0.5 * Time
set.seed(628)
Noise <- t(matrix(stats::rnorm(10L * length(Signal)), nrow = 10L,
                  byrow = TRUE))
Sifting <- list(stopRule = "doubleThreshold", thresholds = c(0.05, 0.5),
                fraction = 0.05, maxSift = 2000, boundary = "reflection",
                maxModes = Inf)
# Two of the ten realizations keep the example short.
OUT <- decomposeICEEMDAN(Signal, noise = Noise[, 1:2], amplitudeRatio = 0.2,
                         normalizeNoise = TRUE, sifting = Sifting)
dim(OUT$modes)
#> [1] 1000    7
OUT$stopReason
#> [1] "insufficientExtrema"
max(abs(rowSums(OUT$modes) + OUT$residue - Signal))
#> [1] 4.440892e-16