# Statistical Graph Empirical Mode Decomposition This function performs statistical graph empirical mode decomposition. ```r 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. ## Returns - **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. 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") ## See Also `gextrema`, `gsmoothing`, `ginterpolating`. ## Examples ```r #### 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) ```

sgemd function