Iterative Filtering

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.

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$iterations
#> [1] 3 3 1
OUT$stopReason
#> [1] "modeLimit"
Zero <- decomposeIF(Signal, filter = Filter, boundary = "zero",
                    tolerance = sqrt(0.001), maxIterations = 200,
                    stretch = 1.6, maxModes = 3)
Zero$iterations
#> [1]  6  5 22