semd {EMD}

Statistical Empirical Mode Decomposition

Description

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

Usage

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)

Arguments

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

Details

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.

Value

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

References

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.

See Also

extractimf, emd.

Examples

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)