Time-Varying Filter Empirical Mode Decomposition

Decomposes a signal by successive extraction of components, each obtained by sifting with a local mean that is a B-spline approximation on knots placed by a time-varying cutoff frequency (TVF-EMD).

Usage

decomposeTVFEMD(signal, bandwidthThreshold, splineOrder,
                intermittencyThreshold, maxSift, maxModes = Inf)

Arguments

signal: Numeric vector of finite real values without dim, in the units of the quantity. Positions are sample indices and the cutoff is in radians per sample.
bandwidthThreshold: Dimensionless threshold $\xi > 0$ of the relative instantaneous bandwidth. No default; the original study uses 0.1.
splineOrder: Whole number $n \ge 1$, the order of the B-spline (degree $n - 1$). No default; the original study uses 28.
intermittencyThreshold: Dimensionless threshold $\rho > 0$ of the intermittency test of the cutoff. No default; the original study uses 0.25.
maxSift: Whole number $\ge 1$ of subtractions of one component. No default; pytvfemd stops after five.
maxModes: Whole number $\ge 1$ or Inf, the limit of components; default Inf.

Details

Before extraction $k$ the decomposition ends with "terminalResidue" when $r_{k-1}$ has fewer than four extrema, maxima plus minima, and with "modeLimit" when $k - 1$ equals maxModes. A component is obtained by sifting, $$h_{s+1} = h_s - m_s,$$ where $m_s$ is the periodic least-squares B-spline approximation of order $n$ of $h_s$ on knots at the extrema of $\cos\varphi$, $\varphi$ the cumulative realigned cutoff. Each iteration extends $h_s$ by symmetric reflection of $\mathrm{round}(N/2)$ samples per side and restores the record afterwards. The cutoff follows from the amplitude $A$ and the frequency of the analytic signal, interpolated at the minima and at the maxima of $A$; values above $0.9\pi$ are lowered to $0.9\pi$, undefined or non-positive values are interpolated, and the zones marked by intermittency events, where the relative variation of the cutoff between consecutive maxima of the signal exceeds $\rho$, are replaced by interpolation.

The stopping rule aggregates the relative instantaneous bandwidth $\theta$ over the samples from $\mathrm{round}(0.05N)$ to $\mathrm{round}(0.95N)$: $h_s$ is accepted when the mean of $\theta$ is below $\xi (1 + (s + 1)/4)$, from $s = 1$ in the first component and from $s = 0$ in the later ones, or when the samples with $\theta > \xi$ are fewer than $0.2N$. This aggregation of pytvfemd 0.3.14 replaces the local criterion of the monograph. converged is FALSE when maxSift subtractions end without acceptance or when the cutoff or the knots cannot be built. Acceptance refers to the aggregated rule above; it does not establish the pointwise narrow-band condition. Neither that rule nor a returned component guarantees the two classical IMF conditions.

Value

A list with

modes: Double matrix, samples by components.
residue: Numeric vector, the final residue.
iterations: Integer vector, the subtractions of each component.
converged: Logical vector by component.
stopReason: "terminalResidue" or "modeLimit".

References

The stopping rule and the reflection of the sifting, the end of the decomposition, the interpolation, the bounds and the realignment of the cutoff, the relative bandwidth, the knots and the periodic B-spline approximation are adapted from pytvfemd 0.3.14 (Stefano Bianchi and Alessandro Longo), GPL-3; see the NOTICE file of the installed package.

Examples

Time <- seq(0, 9, length.out = 3000)
Signal <- sin(pi * Time) + sin(2 * pi * Time) + sin(6 * pi * Time) +
    0.5 * Time
OUT <- decomposeTVFEMD(Signal, bandwidthThreshold = 0.1, splineOrder = 28,
                       intermittencyThreshold = 0.25, maxSift = 5)
dim(OUT$modes)
#> [1] 3000    4
OUT$iterations
#> [1] 1 2 1 1
OUT$stopReason
#> [1] "terminalResidue"