iossa: Iterative O-SSA nested decomposition
In asl/rssa: A Collection of Methods for Singular Spectrum Analysis

iossa

R Documentation

Iterative O-SSA nested decomposition

Description

Perform Iterative O-SSA (IOSSA) algorithm.

Usage

## 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)

Arguments

`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

Details

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.

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)

References

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

Examples

# 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")

asl/rssa documentation built on Aug. 29, 2022, 10:16 a.m.