EMD

Empirical Mode Decomposition (EMD) is an iterative procedure which decomposes signal into a set of oscillatory components, called Intrisic Mode Functions (IMFs).

class PyEMD.EMD(spline_kind: str = 'cubic', nbsym: int = 2, **kwargs)[source]

Empirical Mode Decomposition

Method of decomposing signal into Intrinsic Mode Functions (IMFs) based on algorithm presented in Huang et al. [Huang1998].

Algorithm was validated with Rilling et al. [Rilling2003] Matlab’s version from 3.2007.

Threshold which control the goodness of the decomposition:

std_thr — Test for the proto-IMF how variance changes between siftings.
svar_thr – Test for the proto-IMF how energy changes between siftings.
total_power_thr — Test for the whole decomp how much of energy is solved.
range_thr — Test for the whole decomp whether the difference is tiny.

References

[Huang1998]

N. E. Huang et al., “The empirical mode decomposition and the Hilbert spectrum for non-linear and non stationary time series analysis”, Proc. Royal Soc. London A, Vol. 454, pp. 903-995, 1998

[Rilling2003]

G. Rilling, P. Flandrin and P. Goncalves, “On Empirical Mode Decomposition and its algorithms”, IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing NSIP-03, Grado (I), June 2003

Examples

>>> import numpy as np
>>> T = np.linspace(0, 1, 100)
>>> S = np.sin(2*2*np.pi*T)
>>> emd = EMD(extrema_detection='parabol')
>>> IMFs = emd.emd(S)
>>> IMFs.shape
(1, 100)

__call__(S: ndarray, T: ndarray | None = None, max_imf: int = -1) → ndarray[source]: Call self as a function.

__init__(spline_kind: str = 'cubic', nbsym: int = 2, **kwargs)[source]

Initiate EMD instance.

Configuration, such as threshold values, can be passed as kwargs (keyword arguments).

Parameters:

FIXEint (default: 0)
FIXE_Hint (default: 0)
MAX_ITERATIONint (default 1000): Maximum number of iterations per single sifting in EMD.
energy_ratio_thrfloat (default: 0.2): Threshold value on energy ratio per IMF check.
std_thr float(default 0.2): Threshold value on standard deviation per IMF check.
svar_thr float(default 0.001): Threshold value on scaled variance per IMF check.
total_power_thrfloat (default 0.005): Threshold value on total power per EMD decomposition.
range_thrfloat (default 0.001): Threshold for amplitude range (after scaling) per EMD decomposition.
extrema_detectionstr (default ‘simple’): Method used to finding extrema.
DTYPEnp.dtype (default np.float64): Data type used.

Examples

>>> emd = EMD(std_thr=0.01, range_thr=0.05)

__weakref__: list of weak references to the object (if defined)

check_imf(imf_new: ndarray, imf_old: ndarray, eMax: ndarray, eMin: ndarray) → bool[source]: Huang criteria for IMF (similar to Cauchy convergence test). Signal is an IMF if consecutive siftings do not affect signal in a significant manner.

emd(S: ndarray, T: ndarray | None = None, max_imf: int = -1) → ndarray[source]

Performs Empirical Mode Decomposition on signal S. The decomposition is limited to max_imf imfs. Returns IMF functions and residue in a single numpy array format.

Parameters:

Snumpy array,: Input signal.
Tnumpy array, (default: None): Position or time array. If None is passed or self.extrema_detection == “simple”, then numpy range is created.
max_imfint, (default: -1): IMF number to which decomposition should be performed. Negative value means all.

Returns:

IMFs and residuenumpy array: A numpy array which cointains both the IMFs and residual, if any, appended as the last slice.

end_condition(S: ndarray, IMF: ndarray) → bool[source]

Tests for end condition of whole EMD. The procedure will stop if:

Absolute amplitude (max - min) is below range_thr threshold, or
Metric L1 (mean absolute difference) is below total_power_thr threshold.

Parameters:

Snumpy array: Original signal on which EMD was performed.
IMFnumpy 2D array: Set of IMFs where each row is IMF. Their order is not important.

Returns:

endbool: Whether sifting is finished.

extract_max_min_spline(T: ndarray, S: ndarray) → Tuple[ndarray, ndarray, ndarray, ndarray][source]

Extracts top and bottom envelopes based on the signal, which are constructed based on maxima and minima, respectively.

Parameters:

Tnumpy array: Position or time array.
Snumpy array: Input data S(T).

Returns:

max_splinenumpy array: Spline spanned on S maxima.
min_splinenumpy array: Spline spanned on S minima.
max_extremanumpy array: Points indicating local maxima.
min_extremanumpy array: Points indicating local minima.

find_extrema(T: ndarray, S: ndarray) → Tuple[ndarray, ndarray, ndarray, ndarray, ndarray][source]

Returns extrema (minima and maxima) for given signal S. Detection and definition of the extrema depends on extrema_detection variable, set on initiation of EMD.

Parameters:

Tnumpy array: Position or time array.
Snumpy array: Input data S(T).

Returns:

local_max_posnumpy array: Position of local maxima.
local_max_valnumpy array: Values of local maxima.
local_min_posnumpy array: Position of local minima.
local_min_valnumpy array: Values of local minima.

get_imfs_and_residue() → Tuple[ndarray, ndarray][source]

Provides access to separated imfs and residue from recently analysed signal.

Returns:

imfsnp.ndarray: Obtained IMFs
residuenp.ndarray: Residue.

get_imfs_and_trend() → Tuple[ndarray, ndarray][source]

Provides access to separated imfs and trend from recently analysed signal. Note that this may differ from the get_imfs_and_residue as the trend isn’t necessarily the residue. Residue is a point-wise difference between input signal and all obtained components, whereas trend is the slowest component (can be zero).

Returns:

imfsnp.ndarray: Obtained IMFs
trendnp.ndarray: The main trend.

prepare_points(T: ndarray, S: ndarray, max_pos: ndarray, max_val: ndarray, min_pos: ndarray, min_val: ndarray)[source]

Performs extrapolation on edges by adding extra extrema, also known as mirroring signal. The number of added points depends on nbsym variable.

Parameters:

Tnumpy array: Position or time array.
Snumpy array: Input signal.
max_positerable: Sorted time positions of maxima.
max_valiterable: Signal values at max_pos positions.
min_positerable: Sorted time positions of minima.
min_valiterable: Signal values at min_pos positions.

Returns:

max_extremanumpy array (2 rows): Position (1st row) and values (2nd row) of minima.
min_extremanumpy array (2 rows): Position (1st row) and values (2nd row) of maxima.

prepare_points_parabol(T, S, max_pos, max_val, min_pos, min_val) → Tuple[ndarray, ndarray][source]

Performs mirroring on signal which extrema do not necessarily belong on the position array.

See EMD.prepare_points().

prepare_points_simple(T: ndarray, S: ndarray, max_pos: ndarray, max_val: ndarray | None, min_pos: ndarray, min_val: ndarray | None) → Tuple[ndarray, ndarray][source]

Performs mirroring on signal which extrema can be indexed on the position array.

See EMD.prepare_points().

spline_points(T: ndarray, extrema: ndarray) → Tuple[ndarray, ndarray][source]

Constructs spline over given points.

Parameters:

Tnumpy array: Position or time array.
extremanumpy array: Position (1st row) and values (2nd row) of points.

Returns:

Tnumpy array: Position array (same as input).
splinenumpy array: Spline array over given positions T.