sgemd: 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)

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.

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.

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. tools:::Rd_expr_doi("https://doi.org/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. tools:::Rd_expr_doi("https://doi.org/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. tools:::Rd_expr_doi("https://doi.org/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. tools:::Rd_expr_doi("https://doi.org/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. tools:::Rd_expr_doi("https://doi.org/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. tools:::Rd_expr_doi("https://doi.org/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. tools:::Rd_expr_doi("https://doi.org/10.1109/TSP.2017.2733489")

Examples

Run this code

#### 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

# \donttest{
# 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) 
# }

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