Iterative O-SSA nested decomposition
Perform Iterative O-SSA (IOSSA) algorithm.
## S3 method for class 'ssa' iossa(x, nested.groups, ..., tol = 1e-5, kappa = 2, maxiter = 100, norm = function(x) sqrt(mean(x^2)), trace = FALSE, kappa.balance = 0.5)
x: SSA object holding SSA decomposition
nested.groups: list or named list of numbers of eigentriples from full decomposition, describes initial grouping for IOSSA iterations
tol: tolerance for IOSSA iterations
kappa: kappa' parameter for sigma-correction (see Details' and `References') procedure. If 'NULL', sigma-correction will not be performed
maxiter: upper bound for the number of iterations
norm: function, calculates a norm of a vector; this norm is applied to the difference between the reconstructed series at sequential iterations and is used for convergence detection
trace: logical, indicates whether the convergence process should be traced
kappa.balance: sharing proportion of sigma-correction multiplier between column and row inner products
...: additional arguments passed to decompose
routines
Object of ossa' class. In addition to usual ssa' class fields, it also contains the following fields:
iossa.result' class, a list which contains algorithm parameters, condition numbers, separability measures, the number of iterations and convergence status (see iossa.result`)nested.groups. Otherwise, iossa.groups.all coincides with iossa.groupsnested.groups)See Golyandina N. and Shlemov A. (2015) and Section 2.4 in Golyanina et al (2018) for full details in the 1D case and p.250-252 from the same book for an example in the 2D case.
Briefly, Iterative Oblique SSA (IOSSA) is an iterative (EM-like) method for improving separability in SSA. In particular, it serves for separation of mixed components, which are not orthogonal, e.g., of sinusoids with close frequencies or for trend separation for short series. IOSSA performs a new decomposition of a part of the ssa-object, which is given by a set of eigentriples. Note that eigentriples that do not belong to the chosen set are not changed.
Oblique SSA can make many series orthogonal by the choice of inner product. Iterative O-SSA find the separating inner products by iterations that are hopefully converges to a stationary point. See References for more details.
Sigma-correction procedure does the renormalization of new inner products. This prevents the mixing of the components during the next iteration. Such approach makes the whole procedure more stable and can solve the problem of lack of strong separability.
Details of the used algorithms can be found in Golyandina et al (2018), Algorithms 2.7 and 2.8.
Golyandina N., Korobeynikov A., Zhigljavsky A. (2018): Singular Spectrum Analysis with R. Use R!. Springer, Berlin, Heidelberg.
Golyandina N. and Shlemov A. (2015): Variations of Singular Spectrum Analysis for separability improvement: non-orthogonal decompositions of time series, Statistics and Its Interface. Vol.8, No 3, P.277-294. https://arxiv.org/abs/1308.4022
Rssa for an overview of the package, as well as, ssa-object, fossa, owcor, iossa.result.
# Separate three non-separable sine series with different amplitudes N <- 150 L <- 70 omega1 <- 0.05 omega2 <- 0.06 omega3 <- 0.07 F <- 4*sin(2*pi*omega1 * (1:N)) + 2*sin(2*pi*omega2 * (1:N)) + sin(2*pi*omega3 * (1:N)) s <- ssa(F, L) ios <- iossa(s, nested.groups = list(1:2, 3:4, 5:6), kappa = NULL, maxiter = 100, tol = 1e-3) plot(reconstruct(ios, groups = ios$iossa.groups)) summary(ios) # Separate two non-separable sines with equal amplitudes N <- 200 L <- 100 omega1 <- 0.07 omega2 <- 0.06 F <- sin(2*pi*omega1 * (1:N)) + sin(2*pi*omega2 * (1:N)) s <- ssa(F, L) # Apply FOSSA and then IOSSA fs <- fossa(s, nested.groups = 1:4) ios <- iossa(fs, nested.groups = list(1:2, 3:4), maxiter = 100) summary(ios) opar <- par(mfrow = c(3, 1)) plot(reconstruct(s, groups = list(1:2, 3:4))) plot(reconstruct(fs, groups = list(1:2, 3:4))) plot(reconstruct(ios, groups = ios$iossa.groups)) par(opar) wo <- plot(wcor(ios, groups = 1:4)) gwo <- plot(owcor(ios, groups = 1:4)) plot(wo, split = c(1, 1, 2, 1), more = TRUE) plot(gwo, split = c(2, 1, 2, 1), more = FALSE) data(USUnemployment) unempl.male <- USUnemployment[, "MALE"] s <- ssa(unempl.male) ios <- iossa(s, nested.groups = list(c(1:4, 7:11), c(5:6, 12:13))) summary(ios) # Comparison of reconstructions rec <- reconstruct(s, groups = list(c(1:4, 7:11), c(5:6, 12:13))) iorec <- reconstruct(ios, groups <- ios$iossa.groups) # Trends matplot(data.frame(iorec$F1, rec$F1, unempl.male), type='l', col=c("red","blue","black"), lty=c(1,1,2)) # Seasonalities matplot(data.frame(iorec$F2, rec$F2), type='l', col=c("red","blue"),lty=c(1,1)) # W-cor matrix before IOSSA and w-cor matrix after it ws <- plot(wcor(s, groups = 1:30), grid = 14) wios <- plot(wcor(ios, groups = 1:30), grid = 14) plot(ws, split = c(1, 1, 2, 1), more = TRUE) plot(wios, split = c(2, 1, 2, 1), more = FALSE) # Eigenvectors before and after Iterative O-SSA plot(s, type = "vectors", idx = 1:13) plot(ios, type = "vectors", idx = 1:13) # 2D plots of periodic eigenvectors before and after Iterative O-SSA plot(s, type = "paired", idx = c(5, 12)) plot(ios, type = "paired", idx = c(10, 12), plot.contrib = FALSE) data(AustralianWine) Fortified <- AustralianWine[, "Fortified"] s <- ssa(window(Fortified, start = 1982 + 5/12, end = 1986 + 5/12), L = 18) ios <- iossa(s, nested.groups = list(trend = 1, 2:7), kappa = NULL, maxIter = 1) fs <- fossa(s, nested.groups = 1:7, gamma = 1000) rec.ssa <- reconstruct(s, groups = list(trend = 1, 2:7)) rec.iossa <- reconstruct(ios, groups = ios$iossa.groups); rec.fossa <- reconstruct(fs, groups = list(trend = 7, 1:6)) Fort <- cbind(`Basic SSA trend` = rec.ssa$trend, `Iterative O-SSA trend` = rec.iossa$trend, `DerivSSA trend` = rec.fossa$trend, `Full series` = Fortified) library(lattice) xyplot(Fort, superpose = TRUE, col = c("red", "blue", "green4", "black")) # Shaped 2D I. O-SSA separates finite rank fields exactly mx1 <- outer(1:50, 1:50, function(i, j) exp(i/25 - j/20)) mx2 <- outer(1:50, 1:50, function(i, j) sin(2*pi * i/17) * cos(2*pi * j/7)) mask <- matrix(TRUE, 50, 50) mask[23:25, 23:27] <- FALSE mask[1:2, 1] <- FALSE mask[50:49, 1] <- FALSE mask[1:2, 50] <- FALSE mx1[!mask] <- mx2[!mask] <- NA s <- ssa(mx1 + mx2, kind = "2d-ssa", L = c(10, 10)) plot(reconstruct(s, groups = list(1, 2:5))) ios <- iossa(s, nested.groups = list(1, 2:5), kappa = NULL) plot(reconstruct(ios, groups = ios$iossa.groups)) # I. O-SSA for MSSA N.A <- 150 N.B <- 120 L <- 40 omega1 <- 0.05 omega2 <- 0.055 tt.A <- 1:N.A tt.B <- 1:N.B F1 <- list(A = 2 * sin(2*pi * omega1 * tt.A), B = cos(2*pi * omega1 * tt.B)) F2 <- list(A = 1 * sin(2*pi * omega2 * tt.A), B = cos(2*pi * omega2 * tt.B)) F <- list(A = F1$A + F2$A, B = F1$B + F2$B) s <- ssa(F, kind = "mssa") plot(reconstruct(s, groups = list(1:2, 3:4)), plot.method = "xyplot") ios <- iossa(s, nested.groups = list(1:2, 3:4), kappa = NULL) plot(reconstruct(ios, groups = ios$iossa.groups), plot.method = "xyplot")