decomposeIF: Iterative Filtering

In xemd: eXtended Empirical Mode Decomposition

Iterative Filtering

Description

Decomposes a signal by successive extraction of components, each obtained by iterating the subtraction of a moving average whose filter is a supplied prototype resampled to a support set by the extrema of the residue.

Usage

decomposeIF(signal, filter, boundary, tolerance, maxIterations, stretch,
            maxModes = Inf)

Arguments

signal	Numeric vector of finite values without dim, in the units of the quantity. Positions are sample indices.
filter	Prototype filter, a numeric vector of finite weights of odd length, at least 3. No default. The package tests use the 13 weights that fifpy obtains from its prototype for a half width of 6.
boundary	"zero" extends the record with zeros; "periodic" extends it periodically and evaluates the iteration spectrally (fast iterative filtering). No default.
tolerance	Bound of the relative change between consecutive iterates, greater than zero and less than one. No default; the package tests use \sqrt{0.001}.
maxIterations	Whole number \ge 1 of iterations of one extraction. No default; the package tests use 200.
stretch	Number greater than zero that scales the support of the filter. No default; the package tests use 1.6.
maxModes	Whole number \ge 1 or Inf, the limit of extractions; default Inf.

Details

Before extraction k the decomposition ends with "terminalResidue" when r_{k-1} has at most one extremum and with "modeLimit" when k - 1 equals maxModes. Otherwise the half width is l_k = 2\,\mathrm{round}(\mathrm{stretch}\, N / E) for the E extrema of r_{k-1}; from the second extraction a value that does not exceed the previous half width becomes \lceil 1.1\, l_{k-1} \rceil. The prototype is resampled to 2 l_k + 1 weights and the iteration

f^{(q)} = f^{(q-1)} - \mathcal{L}_w f^{(q-1)}, \qquad f^{(0)} = r_{k-1},

with \mathcal{L}_w the moving average under boundary, stops at the first q with \|f^{(q)} - f^{(q-1)}\|_2 / \|f^{(q-1)}\|_2 \le tolerance (converged TRUE) or after maxIterations (converged FALSE). Then c_k = f^{(Q)} and r_k = r_{k-1} - c_k. A shrinking prototype is integrated over rescaled bins; an enlarged one is interpolated linearly and normalized to a unit absolute sum. The components are not shown to satisfy the counting conditions of intrinsic mode functions.

Value

A list with

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

References

Cicone, A. and Zhou, H. (2018). Numerical Analysis for Iterative Filtering with New Efficient Implementations Based on FFT. arXiv:1802.01359.

The support rule, the resampling of the prototype and both iterations are adapted from fifpy, revision 26e51e7d45e551950dea8be37ece447fc4340296, copyright (c) 2022 Emanuele Papini, MIT License; see the NOTICE file of the installed package.

See Also

decomposeLMD, decomposeEMD

Examples

Time <- seq_len(1000) / 1000
Signal <- cos(2 * pi * 2 * Time) + 0.25 * cos(2 * pi * 24 * Time) +
    0.0625 * cos(2 * pi * 288 * Time)
Filter <- c(6.6306270449357867e-06, 0.00053827687799438354,
            0.0082858777882808092, 0.044245208647455882, 0.11927281750086954,
            0.20513332487239222, 0.24503568511284984, 0.20513332487239222,
            0.11927281750086952, 0.044245208647455903, 0.0082858777882808109,
            0.00053827687799438213, 6.6306270449358087e-06)
OUT <- decomposeIF(Signal, filter = Filter, boundary = "periodic",
                   tolerance = sqrt(0.001), maxIterations = 200,
                   stretch = 1.6, maxModes = 3)
OUT$iterations
OUT$stopReason
Zero <- decomposeIF(Signal, filter = Filter, boundary = "zero",
                    tolerance = sqrt(0.001), maxIterations = 200,
                    stretch = 1.6, maxModes = 3)
Zero$iterations

xemd documentation built on Nov. 9, 2026, 1:09 a.m.