# 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: Optional[ndarray] = 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: Optional[ndarray] = 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]

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.

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.