semd: Statistical Empirical Mode Decomposition
In EMD: Empirical Mode Decomposition and Hilbert Spectral Analysis

Description Usage Arguments Details Value References See Also Examples

This function performs empirical mode decomposition using spline smoothing not interpolation for sifting process. The smoothing parameter is automatically detemined by cross-validation.

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semd(xt, tt=NULL, cv.kfold, cv.tol=0.1^1, cv.maxiter=20, 
    emd.tol=sd(xt)*0.1^2, max.sift=20, stoprule="type1", boundary="periodic", 
    smlevels=1, max.imf=10)

`xt`	observation or signal observed at time `tt`
`tt`	observation index or time index
`cv.kfold`	the number of fold of cross-validation
`cv.tol`	tolerance for cross-validation
`cv.maxiter`	maximum iteration for cross-validation
`emd.tol`	tolerance for stopping rule of sifting. If `stoprule=type5`, the number of iteration for S stoppage criterion.
`max.sift`	the maximum number of sifting
`stoprule`	stopping rule of sifting. The `type1` stopping rule indicates that absolute values of envelope mean must be less than the user-specified tolerance level in the sense that the local average of upper and lower envelope is zero. The stopping rules `type2`, `type3`, `type4` and `type5` are the stopping rules given by equation (5.5) of Huang et al. (1998), equation (11a), equation (11b) and S stoppage of Huang and Wu (2008), respectively.
`boundary`	specifies boundary condition from “none", “wave", “symmetric", “periodic" or “evenodd". See Zeng and He (2004) for `evenodd` boundary condition.
`smlevels`	specifies which level of the IMF is obtained by smoothing spline.
`max.imf`	the maximum number of IMF's

This function performs empirical mode decomposition using spline smoothing not interpolation for sifting process. The smoothing parameter is automatically detemined by cross-validation. Optimization is done by golden section search. See Kim et al. (2012) for details.

`imf`	IMF's
`residue`	residue signal after extracting IMF's from observations `xt`
`nimf`	the number of IMF's
`optlambda`	smoothing parameter minimizing prediction errors of cross-validation
`lambdaconv`	a sequence of smoothing parameters for searching optimal smoothing papameter
`perr`	prediction errors of cross-validation according to `lambdaconv`

Huang, N. E., Shen, Z., Long, S. R., Wu, M. L. Shih, H. H., Zheng, Q., Yen, N. C., Tung, C. C. and Liu, H. H. (1998) The empirical mode decomposition and Hilbert spectrum for nonlinear and nonstationary time series analysis. Proceedings of the Royal Society London A, 454, 903–995.

Huang, N. E. and Wu, Z. (2008) A review on Hilbert-Huang Transform: Method and its applications to geophysical studies. Reviews of Geophysics, 46, RG2006.

Kim, D., Kim, K.-O. and Oh, H.-S. (2012) Extending the Scope of Empirical Mode Decomposition using Smoothing. EURASIP Journal on Advances in Signal Processing, 2012:168, doi: 10.1186/1687-6180-2012-168.

Zeng, K and He, M.-X. (2004) A simple boundary process technique for empirical mode decomposition. Proceedings of 2004 IEEE International Geoscience and Remote Sensing Symposium, 6, 4258–4261.

extractimf, emd.

ndata <- 2048
tt <- seq(0, 9, length=ndata)                 
xt <- sin(pi * tt) + sin(2* pi * tt) + sin(6 * pi * tt)  + 0.5 * tt 
set.seed(1)
xt <- xt + rnorm(ndata, 0, sd(xt)/5)

## Not run: 
### Empirical Mode Decomposition by Interpolation
emdbyint <- emd(xt, tt, max.imf = 5, boundary = "wave")
### Empirical Mode Decomposition by Smoothing
emdbysm <- semd(xt, tt, cv.kfold=4, boundary="wave", smlevels=1, max.imf=5)

par(mfcol=c(6,2), mar=c(2,2,2,1), oma=c(0,0,2,0))                              
rangext <- range(xt); rangeimf <- rangext - mean(rangext)
plot(tt, xt, xlab="", ylab="", main="signal", ylim=rangext, type="l")
mtext("Decomposition by EMD", side = 3, line = 2, cex=0.85, font=2)
plot(tt, emdbyint$imf[,1], xlab="", ylab="", main="imf 1", ylim=rangeimf,  type="l")
abline(h=0, lty=2)
plot(tt, emdbyint$imf[,2], xlab="", ylab="", main="imf 2", ylim=rangeimf,  type="l")
abline(h=0, lty=2)
plot(tt, emdbyint$imf[,3], xlab="", ylab="", main="imf 3", ylim=rangeimf,  type="l")
abline(h=0, lty=2)
plot(tt, emdbyint$imf[,4], xlab="", ylab="", main="imf 4", ylim=rangeimf,  type="l")
abline(h=0, lty=2)
plot(tt, emdbyint$imf[,5]+emdbyint$residue, xlab="", ylab="", main="remaining signal",
    ylim=rangext, type="l")

plot(tt, xt, xlab="", ylab="", main="signal", ylim=rangext, type="l")
mtext("Decomposition by SEMD", side = 3, line = 2, cex=0.85, font=2)
plot(tt, emdbysm$imf[,1], xlab="", ylab="", main="noise", ylim=rangeimf,  type="l")
abline(h=0, lty=2)
plot(tt, emdbysm$imf[,2], xlab="", ylab="", main="imf 1", ylim=rangeimf,  type="l")
abline(h=0, lty=2)
plot(tt, emdbysm$imf[,3], xlab="", ylab="", main="imf 2", ylim=rangeimf,  type="l")
abline(h=0, lty=2)
plot(tt, emdbysm$imf[,4], xlab="", ylab="", main="imf 3", ylim=rangeimf,  type="l")
abline(h=0, lty=2)
plot(tt, emdbysm$residue, xlab="", ylab="", main="residue", ylim=rangext, type="l")
## End(Not run)

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