iossa: Iterative O-SSA nested decomposition

Description

Perform Iterative O-SSA (IOSSA) algorithm.

Usage

iossa(x, nested.groups, ..., tol = 1e-5, kappa = 2, maxiter = 100, norm = function(x) sqrt(mean(x^2)), trace = FALSE, kappa.balance = 0.5)

Arguments

SSA object holding the full one-dimensional SSA (classic 1D-SSA, Toeplitz 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

Value

Object of `ossa' class. In addition to usual `ssa' class fields, it also contains the following fields:

iossa.result: object of `iossa.result' class, a list which contains algorithm parameters, condition numbers, separability measures, the number of iterations and convergence status (see iossa.result)
iossa.groups: list of groups within the nested decomposition; numbers of components correspond to their numbers in the full decomposition
iossa.groups.all: list, describes cumulative grouping after after sequential Iterative O-SSA decompositions in the case of non-intersecting nested.groups. Otherwise, iossa.groups.all coincides with iossa.groups
ossa.set: vector of the indices of elementary components used in Iterative O-SSA (that is, used in nested.groups)

Details

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 (see References).

References

Golyandina N. and Shlemov A. (2013): Variations of Singular Spectrum Analysis for separability improvement: non-orthogonal decompositions of time series, http://arxiv.org/abs/1308.4022

Examples

Run this code

# 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")

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