sgemd {GSD}

Statistical Graph Empirical Mode Decomposition

Description

This function performs statistical graph empirical mode decomposition.

Usage

sgemd(graph, nimf, smoothing = FALSE, smlevels = c(1),  
    boundary = FALSE, reflperc = 0.3, reflaver = FALSE, 
    connperc = 0.05, connweight = "boundary", 
    tol = 0.1^3, max.sift = 50, verbose = FALSE)

Arguments

graph

an igraph graph object with vertex attributes of coordinates x, y, a signal z, and edge attribute of weight.

nimf

specifies the maximum number of intrinsic mode functions (IMF).

smoothing

specifies whether intrinsic mode functions are constructed by interpolating or smoothing envelopes. When smoothing =TRUE, denoise envelopes utilizing the graph Fourier transform and empirical Bayes thresholding.

smlevels

specifies which level of the IMF is obtained by smoothing other than interpolation.

boundary

When boundary=TRUE, a given graph is reflected for boundary treatment.

reflperc

expand a graph by adding specified percentage of a graph at the boundary when boundary=TRUE.

reflaver

specifies the method assigning signal to reflected vertices. When reflaver=TRUE, the signal on reflected vertices is produced by averaging signals on neighboring vertices on a given graph. Otherwise, signal on reflected vertices is the same to the signal on a given graph.

connperc

specifies percentage of a graph for connecting a given graph and reflected graph when boundary=TRUE.

connweight

specifies the method assigning the edge weights for connecting a given graph and reflected graph when boundary=TRUE. The edge weights are calculated by Gaussian kernel considering the relative distance between vertices. When connweight="graph", the relative distance is calculated based on the maximum distance of all the neighboring edges of a given graph. When connweight="boundary", the relative distance is calculated based on the maximum distance of the connected vertices between a given graph and reflected graph.

tol

tolerance for stopping rule of sifting.

max.sift

the maximum number of sifting.

verbose

specifies whether sifting steps are displayed.

Details

This function performs statistical graph empirical mode decomposition utilizing extrema detection of a graph signal.

Value

imf

list of IMF's according to the frequencies with imf[[1]] the highest-frequency IMF.

residue

residue signal after extracting IMF's.

nimf

the number of IMF's.

n_extrema

Each row specifies the number of local maxima and local minima of the remaining signal after extracting the i-th IMF. The first row represents the number of local maxima and local minima of a given signal.

References

Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., Yen, N.- C., Tung, C. C., and Liu, H. H. (1998). The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 454(1971), 903–-995. doi:10.1098/rspa.1998.0193

Johnstone, I. and Silverman, B.~W. (2004). Needles and straw in haystacks: empirical Bayes estimates of possibly sparse sequences. The Annals of Statistics, 32, 594–1649. doi:10.1214/009053604000000030

Kim, D., Kim, K. O., and Oh, H.-S. (2012a). Extending the scope of empirical mode decomposition by smoothing. EURASIP Journal on Advances in Signal Processing, 2012, 1–17. doi:10.1186/1687-6180-2012-168

Kim, D., Park, M., and Oh, H.-S. (2012b). Bidimensional statistical empirical mode decomposition. IEEE Signal Processing Letters, 19(4), 191–194. doi:10.1109/LSP.2012.2186566

Ortega, A., Frossard, P., Kovačević, J., Moura, J. M. F., and Vandergheynst, P. (2018). Graph signal processing: overview, challenges, and applications. Proceedings of the IEEE 106, 808–828. doi:10.1109/JPROC.2018.2820126

Shuman, D. I., Narang, S. K., Frossard, P., Ortega, A., and Vandergheynst, P. (2013). The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Processing Magazine, 30(3), 83–98. doi:10.1109/MSP.2012.2235192

Tremblay, N., Borgnat, P., and Flandrin, P. (2014). Graph empirical mode decomposition. 22nd European Signal Processing Conference (EUSIPCO), 2350–2354

Zeng, J., Cheung, G., and Ortega, A. (2017). Bipartite approximation for graph wavelet signal decomposition. IEEE Transactions on Signal Processing, 65(20), 5466–5480. doi:10.1109/TSP.2017.2733489

See Also

gextrema, gsmoothing, ginterpolating.

Examples

#### example : composite of two components having different frequencies

## define vertex coordinate
x <- y <- seq(0, 1, length=30)
xy <- expand.grid(x=x, y=y)

## weighted adjacency matrix by Gaussian kernel 
## for connecting vertices within distance 0.04
A <- adjmatrix(xy, method = "dist", 0.04) 

## signal
# high-frequency component
signal1 <- rep(sin(12.5*pi*x - 1.25*pi), 30)

# low-frequency component
signal2 <- rep(sin(5*pi*x - 0.5*pi), 30)

# composite signal
signal0 <- signal1 + signal2

# noisy signal with SNR(signal-to-noise ratio)=5
signal <- signal0 + rnorm(900, 0, sqrt(var(signal0) / 5)) 

# graph with signal
gsig <- gsignal(vertex = cbind(xy, signal), edge = A, edgetype = "matrix")

# graph empirical mode decomposition (GEMD) without boundary treatment
out1 <- sgemd(gsig, nimf=3, smoothing=FALSE, boundary=FALSE)

# denoised signal by GEMD 
dsignal1 <- out1$imf[[2]] + out1$imf[[3]] + out1$residue


# statistical graph empirical mode decomposition (SGEMD) with boundary treatment
out2 <- sgemd(gsig, nimf=3, smoothing=TRUE, boundary=TRUE)
names(out2)

# denoised signal by SGEMD 
dsignal2 <- out2$imf[[2]] + out2$imf[[3]] + out2$residue

# display of a signal, denoised signal, imf2, imf3 and residue by SGEMD
gplot(gsig, size=3) 
gplot(gsig, dsignal2, size=3) 
gplot(gsig, out2$imf[[2]], size=3) 
gplot(gsig, out2$imf[[3]], size=3) 
gplot(gsig, out2$residue, size=3)