| Title: | Ensemble Empirical Mode Decomposition (EEMD) and Its Complete Variant (CEEMDAN) |
|---|---|
| Description: | An R interface for libeemd (Luukko, Helske, Räsänen, 2016) <doi:10.1007/s00180-015-0603-9>, a C library of highly efficient parallelizable functions for performing the ensemble empirical mode decomposition (EEMD), its complete variant (CEEMDAN), the regular empirical mode decomposition (EMD), and bivariate EMD (BEMD). Due to the possible portability issues CRAN version no longer supports OpenMP, you can install OpenMP-supported version from GitHub: <https://github.com/helske/Rlibeemd/>. |
| Authors: | Jouni Helske [aut, cre] (R interface, <https://orcid.org/0000-0001-7130-793X>), Perttu Luukko [aut] (Original libeemd C library, <https://orcid.org/0000-0003-3786-9685>) |
| Maintainer: | Jouni Helske <[email protected]> |
| License: | GPL-3 |
| Version: | 1.4.3 |
| Built: | 2024-11-25 06:36:26 UTC |
| Source: | CRAN |
Function bemd implements the Bivariate EMD (Scheme 2 in the cited article).
bemd(input, directions = 64L, num_imfs = 0L, num_siftings = 50L)bemd(input, directions = 64L, num_imfs = 0L, num_siftings = 50L)
input |
Complex vector of length N. The input signal to decompose. |
directions |
Vector of directional angles (in radians) to use for the decomposition, or an integer defining the number of equally spaced angles to use. |
num_imfs |
Number of Intrinsic Mode Functions (IMFs) to compute. If num_imfs is set to zero, a value of num_imfs = emd_num_imfs(N) will be used, which corresponds to a maximal number of IMFs. Note that the final residual is also counted as an IMF in this respect, so you most likely want at least num_imfs=2. |
num_siftings |
Use a maximum number of siftings as a stopping criterion. If
|
Time series object of class "mts" where series corresponds to
IMFs of the input signal, with the last series being the final residual.
@references
G. Rilling, P. Flandrin, P. Goncalves and J. M. Lilly, "Bivariate Empirical Mode Decomposition", IEEE Signal Processing Letters, Vol. 14 (2007) 936–939
N <- 512 t <- 2 * pi * (0:(N-1))/N input <- cos(0.3 * t) * exp(2i * t) + 0.3 * abs(sin(2.3 * t)) * exp(17i * t) # Use evenly spaced angles as directions num_directions <- 64 directions <- 2 * pi * 1:num_directions / num_directions imfs <- bemd(input, directions, num_imfs = 4, num_siftings = 10) # plot the data plot(Re(input), Im(input), xlim = c(-1, 2)) # plot signal and the imfs for(i in 1:4) points(Re(imfs[,i]), Im(imfs[,i]), col = 1 + i) legend("bottomright", col = 1:5, legend = c("signal", paste0("IMF ",1:4)), pch = 1) data("float") plot(float, type = "l") signal <- float[, 1] + float[, 2] * 1i imfs <- bemd(signal, num_siftings = 10, num_imfs = 4) # plot the data and the imfs oldpar <- par() par(mfrow = c(5, 1), mar = c(0.5, 4.5, 0.5, 0.5), oma = c(4, 0, 2, 0)) ts.plot(float, col = 1:2, lty = 1:2, ylab = "signal", gpars = list(xaxt = "n")) for(i in 1:4) { ts.plot(Re(imfs[, i]), Im(imfs[, i]), col = 1:2, lty = 1:2, ylab = if(i < 4) paste("IMF", i) else "residual", gpars = list(xaxt = "n")) } axis(1) title(xlab = "Time (days)", main = "Bivariate EMD decomposition", outer = TRUE) par(oldpar)N <- 512 t <- 2 * pi * (0:(N-1))/N input <- cos(0.3 * t) * exp(2i * t) + 0.3 * abs(sin(2.3 * t)) * exp(17i * t) # Use evenly spaced angles as directions num_directions <- 64 directions <- 2 * pi * 1:num_directions / num_directions imfs <- bemd(input, directions, num_imfs = 4, num_siftings = 10) # plot the data plot(Re(input), Im(input), xlim = c(-1, 2)) # plot signal and the imfs for(i in 1:4) points(Re(imfs[,i]), Im(imfs[,i]), col = 1 + i) legend("bottomright", col = 1:5, legend = c("signal", paste0("IMF ",1:4)), pch = 1) data("float") plot(float, type = "l") signal <- float[, 1] + float[, 2] * 1i imfs <- bemd(signal, num_siftings = 10, num_imfs = 4) # plot the data and the imfs oldpar <- par() par(mfrow = c(5, 1), mar = c(0.5, 4.5, 0.5, 0.5), oma = c(4, 0, 2, 0)) ts.plot(float, col = 1:2, lty = 1:2, ylab = "signal", gpars = list(xaxt = "n")) for(i in 1:4) { ts.plot(Re(imfs[, i]), Im(imfs[, i]), col = 1:2, lty = 1:2, ylab = if(i < 4) paste("IMF", i) else "residual", gpars = list(xaxt = "n")) } axis(1) title(xlab = "Time (days)", main = "Bivariate EMD decomposition", outer = TRUE) par(oldpar)
Decompose input data to Intrinsic Mode Functions (IMFs) with the Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) algorithm [1], a variant of EEMD.
ceemdan( input, num_imfs = 0, ensemble_size = 250L, noise_strength = 0.2, S_number = 4L, num_siftings = 50L, rng_seed = 0L, threads = 0L )ceemdan( input, num_imfs = 0, ensemble_size = 250L, noise_strength = 0.2, S_number = 4L, num_siftings = 50L, rng_seed = 0L, threads = 0L )
input |
Vector of length N. The input signal to decompose. |
num_imfs |
Number of Intrinsic Mode Functions (IMFs) to compute. If num_imfs is set to zero, a value of num_imfs = emd_num_imfs(N) will be used, which corresponds to a maximal number of IMFs. Note that the final residual is also counted as an IMF in this respect, so you most likely want at least num_imfs=2. |
ensemble_size |
Number of copies of the input signal to use as the ensemble. |
noise_strength |
Standard deviation of the Gaussian random numbers used as additional noise. This value is relative to the standard deviation of the input signal. |
S_number |
Integer. Use the S-number stopping criterion for the EMD procedure with the given
values of $S$. That is, iterate until the number of extrema and zero crossings in the signal
differ at most by one, and stay the same for S consecutive iterations. Typical values are in
the range 3–8. If |
num_siftings |
Use a maximum number of siftings as a stopping criterion. If
|
rng_seed |
A seed for the GSL's Mersenne twister random number generator. A value of zero
(default) denotes an implementation-defined default value. For |
threads |
Non-negative integer defining the maximum number of parallel threads (via OpenMP's
|
The size of the ensemble and the relative magnitude of the added noise are
given by parameters ensemble_size and noise_strength, respectively. The
stopping criterion for the decomposition is given by either a S-number [2] or
an absolute number of siftings. In the case that both are positive numbers,
the sifting ends when either of the conditions is fulfilled.
Time series object of class "mts" where series corresponds to
IMFs of the input signal, with the last series being the final residual.
M. Torres et al, "A Complete Ensemble Empirical Mode Decomposition with Adaptive Noise" IEEE Int. Conf. on Acoust., Speech and Signal Proc. ICASSP-11, (2011) 4144–4147
N. E. Huang, Z. Shen and S. R. Long, "A new view of nonlinear water waves: The Hilbert spectrum", Annual Review of Fluid Mechanics, Vol. 31 (1999) 417–457
imfs <- ceemdan(UKgas, threads = 1) # trend extraction ts.plot(UKgas, imfs[, ncol(imfs)], col = 1:2, main = "Quarterly UK gas consumption", ylab = "Million therms") # CEEMDAN for logarithmic demand, note that increasing ensemble size # will produce smoother results imfs <- ceemdan(log(UKgas), ensemble_size = 50, threads = 1) plot(ts.union("log(obs)" = log(UKgas), Seasonal = imfs[, 1], Irregular = rowSums(imfs[, 2:5]), Trend = imfs[, 6]), main = "Quarterly UK gas consumption")imfs <- ceemdan(UKgas, threads = 1) # trend extraction ts.plot(UKgas, imfs[, ncol(imfs)], col = 1:2, main = "Quarterly UK gas consumption", ylab = "Million therms") # CEEMDAN for logarithmic demand, note that increasing ensemble size # will produce smoother results imfs <- ceemdan(log(UKgas), ensemble_size = 50, threads = 1) plot(ts.union("log(obs)" = log(UKgas), Seasonal = imfs[, 1], Irregular = rowSums(imfs[, 2:5]), Trend = imfs[, 6]), main = "Quarterly UK gas consumption")
Electrocardiogram Data
Example ECG data from MIT-BIH Normal Sinus Rhythm Database, ECG1 of record 16265, first 2049 observations (0 to 16 seconds with sampling interval of 0.0078125 seconds)
A time series object.
MIT-BIH Normal Sinus Rhythm Database, PhysioBank ATM, https://archive.physionet.org/cgi-bin/atm/ATM
data("ECG") plot(ECG)data("ECG") plot(ECG)
Decompose input data to Intrinsic Mode Functions (IMFs) with the Ensemble Empirical Mode Decomposition algorithm [1].
eemd( input, num_imfs = 0, ensemble_size = 250L, noise_strength = 0.2, S_number = 4L, num_siftings = 50L, rng_seed = 0L, threads = 0L )eemd( input, num_imfs = 0, ensemble_size = 250L, noise_strength = 0.2, S_number = 4L, num_siftings = 50L, rng_seed = 0L, threads = 0L )
input |
Vector of length N. The input signal to decompose. |
num_imfs |
Number of Intrinsic Mode Functions (IMFs) to compute. If num_imfs is set to zero, a value of num_imfs = emd_num_imfs(N) will be used, which corresponds to a maximal number of IMFs. Note that the final residual is also counted as an IMF in this respect, so you most likely want at least num_imfs=2. |
ensemble_size |
Number of copies of the input signal to use as the ensemble. |
noise_strength |
Standard deviation of the Gaussian random numbers used as additional noise. This value is relative to the standard deviation of the input signal. |
S_number |
Integer. Use the S-number stopping criterion for the EMD procedure with the given
values of $S$. That is, iterate until the number of extrema and zero crossings in the signal
differ at most by one, and stay the same for S consecutive iterations. Typical values are in
the range 3–8. If |
num_siftings |
Use a maximum number of siftings as a stopping criterion. If
|
rng_seed |
A seed for the GSL's Mersenne twister random number generator. A value of zero (default) denotes an implementation-defined default value. |
threads |
Non-negative integer defining the maximum number of parallel threads (via OpenMP's
|
The size of the ensemble and the relative magnitude of the added noise are given by parameters
ensemble_size and noise_strength, respectively. The stopping criterion for the
decomposition is given by either a S-number [2] or an absolute number of siftings. In the case
that both are positive numbers, the sifting ends when either of the conditions is fulfilled.
Time series object of class "mts" where series corresponds to IMFs of the input
signal, with the last series being the final residual.
Z. Wu and N. Huang, "Ensemble Empirical Mode Decomposition: A Noise-Assisted Data Analysis Method", Advances in Adaptive Data Analysis, Vol. 1 (2009) 1–41
N. E. Huang, Z. Shen and S. R. Long, "A new view of nonlinear water waves: The Hilbert spectrum", Annual Review of Fluid Mechanics, Vol. 31 (1999) 417–457
x <- seq(0, 2*pi, length.out = 500) signal <- sin(4*x) intermittent <- 0.1 * sin(80 * x) y <- signal * (1 + ifelse(signal > 0.7, intermittent, 0)) plot(x = x,y = y,type = "l") # Decompose with EEMD imfs <- eemd(y, num_siftings = 10, ensemble_size = 50, threads = 1) plot(imfs) # High frequencies ts.plot(rowSums(imfs[, 1:3])) # Low frequencies ts.plot(rowSums(imfs[, 4:ncol(imfs)]))x <- seq(0, 2*pi, length.out = 500) signal <- sin(4*x) intermittent <- 0.1 * sin(80 * x) y <- signal * (1 + ifelse(signal > 0.7, intermittent, 0)) plot(x = x,y = y,type = "l") # Decompose with EEMD imfs <- eemd(y, num_siftings = 10, ensemble_size = 50, threads = 1) plot(imfs) # High frequencies ts.plot(rowSums(imfs[, 1:3])) # Low frequencies ts.plot(rowSums(imfs[, 4:ncol(imfs)]))
Decompose input data to Intrinsic Mode Functions (IMFs) with the Empirical Mode Decomposition algorithm.
emd(input, num_imfs = 0, S_number = 4L, num_siftings = 50L)emd(input, num_imfs = 0, S_number = 4L, num_siftings = 50L)
input |
Vector of length N. The input signal to decompose. |
num_imfs |
Number of Intrinsic Mode Functions (IMFs) to compute. If num_imfs is set to zero, a value of num_imfs = emd_num_imfs(N) will be used, which corresponds to a maximal number of IMFs. Note that the final residual is also counted as an IMF in this respect, so you most likely want at least num_imfs=2. |
S_number |
Integer. Use the S-number stopping criterion [1] for the EMD procedure with the given values of S.
That is, iterate until the number of extrema and zero crossings in the
signal differ at most by one, and stay the same for S consecutive
iterations. Typical values are in the range 3–8. If |
num_siftings |
Use a maximum number of siftings as a stopping criterion. If
|
This is a wrapper around eemd with ensemble_size = 1 and noise_strength = 0.
Time series object of class "mts" where series corresponds to
IMFs of the input signal, with the last series being the final residual.
@references
N. E. Huang, Z. Shen and S. R. Long, "A new view of nonlinear water waves: The Hilbert spectrum", Annual Review of Fluid Mechanics, Vol. 31 (1999) 417–457
Find the local minima and maxima from input data. This includes the artificial extrema added to the ends of the data as specified in the original EEMD article [1]. In the case of flat regions at the extrema, the center point of the flat region will be considered the extremal point [2].
extrema(input)extrema(input)
input |
Numeric vector or time series object. |
a list with matrices minima and maxima which give time points and values of local minima and
maxima of input where time points are transformed to match the sampling times of input.
Z. Wu and N. Huang, "Ensemble Empirical Mode Decomposition: A Noise-Assisted Data Analysis Method", Advances in Adaptive Data Analysis, Vol. 1 (2009) 1–41.
P. Luukko, J. Helske and E. Räsänen, "Introducing libeemd: A program package for performing the ensemble empirical mode decomposition", Computational Statistics (2015).
ext <- extrema(UKgas) plot(UKgas, ylim = range(ext$maxima[, 2], ext$minima[, 2])) points(ext$maxima, col = 2, pch = 19) points(ext$minima, col = 2, pch = 19) # Artificial extremas obtained by extrapolating last two extrema # Beginning of the series lines(ext$minima[1:3, ], col = 4) # This is discarded as it produces smaller extrema than the last observation: b <- lm(c(ext$maxima[2:3, 2]) ~ ext$maxima[2:3, 1])$coef[2] points(x = ext$maxima[1, 1], y = ext$maxima[2, 2] - b, col = 4,pch = 19) lines(x = ext$maxima[1:3, 1], y = c(ext$maxima[2, 2] - b, ext$maxima[2:3, 2]), col = 4) # End of the series # These produce more extreme values than the last observation which is thus disregarded lines(ext$minima[27:29, ],col = 4) lines(ext$maxima[26:28, ],col = 4)ext <- extrema(UKgas) plot(UKgas, ylim = range(ext$maxima[, 2], ext$minima[, 2])) points(ext$maxima, col = 2, pch = 19) points(ext$minima, col = 2, pch = 19) # Artificial extremas obtained by extrapolating last two extrema # Beginning of the series lines(ext$minima[1:3, ], col = 4) # This is discarded as it produces smaller extrema than the last observation: b <- lm(c(ext$maxima[2:3, 2]) ~ ext$maxima[2:3, 1])$coef[2] points(x = ext$maxima[1, 1], y = ext$maxima[2, 2] - b, col = 4,pch = 19) lines(x = ext$maxima[1:3, 1], y = c(ext$maxima[2, 2] - b, ext$maxima[2:3, 2]), col = 4) # End of the series # These produce more extreme values than the last observation which is thus disregarded lines(ext$minima[27:29, ],col = 4) lines(ext$maxima[26:28, ],col = 4)
Float Data
The data are a position record from an acoustically tracked subsurface oceanographic float, used as an example data in Rilling et al (2007).
A time series object.
http://wfdac.whoi.edu
G. Rilling, P. Flandrin, P. Goncalves and J. M. Lilly, "Bivariate Empirical Mode Decomposition", IEEE Signal Processing Letters, Vol. 14 (2007) 936–939
data("float") plot(float, type = "l")data("float") plot(float, type = "l")
Return the number of IMFs extracted from input data of length N, including the final residual. This is just [log_2(N)] for N>3.
emd_num_imfs(N)emd_num_imfs(N)
N |
An integer defining the length of input data. |
The number of IMFs which would be extracted from input data of length N, including the final residual.
Package Rlibeemd contains functions for the ensemble empirical mode decomposition (EEMD), its complete variant (CEEMDAN) or the regular empirical mode decomposition (EMD).
Package is based on the libeemd C library: https://bitbucket.org/luukko/libeemd
P. Luukko, J. Helske and E. Räsänen, "Introducing libeemd: A program package for performing the ensemble empirical mode decomposition", Computational Statistics (2015).
Z. Wu and N. Huang, "Ensemble Empirical Mode Decomposition: A Noise-Assisted Data Analysis Method", Advances in Adaptive Data Analysis, Vol. 1 (2009) 1–41.
N. E. Huang, Z. Shen and S. R. Long, "A new view of nonlinear water waves: The Hilbert spectrum", Annual Review of Fluid Mechanics, Vol. 31 (1999) 417–457.
Torres et al, A Complete Ensemble Empirical Mode Decomposition with Adaptive Noise IEEE Int. Conf. on Acoust., Speech and Signal Proc. ICASSP-11, (2011) 4144–4147.