Rssa (version 0.13-1)

fossa: Nested Filter-adjusted O-SSA decomposition

Description

Perform nested decomposition by Filter-adjusted O-SSA (FOSSA).

Usage

fossa(x, nested.groups, FILTER = diff, gamma = 1, normalize = FALSE, ...)

Arguments

x

SSA object holding the full one-dimensional SSA (Basic 1D-SSA, Toeplitz SSA) decomposition

nested.groups

vector of numbers of eigentriples from full decomposition for nested decomposition. The argument is coerced to a vector, if necessary

FILTER

function or numeric vector of impulse response coefficients for filter adjustment

gamma

weight of filter adjustment. See `Details' and `References'

normalize

logical, whether to normalize left decomposition vectors before filtering

...

additional arguments passed to decompose routines

Value

Object of class `ossa'. The field `ossa.set' contains the vector of indices of elementary components used in Filter-adjusted O-SSA (that is, used in nested.groups).

Details

FOSSA serves for decomposition of series components that are mixed due to equal contributions of their elementary components, e.g. of sinusoids with equal amplitudes or of complex-form trend and periodics. FOSSA 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. In particular, Filter-adjusted O-SSA performs a nested decomposition specified by a number of eigentriples via Oblique SSA with a specific inner product in the row space: $$% \langle x, y \rangle = (x, y) + \gamma^2(\Phi(x), \Phi(y)), $$ where $(., .)$ denotes conventional inner product and `$Phi$' is function (supposed to be linear operator, but right now no additional checks are performed) or linear filtration which is specified by FILTER argument.

The default value of $Phi$ corresponds to sequential differences, that is, to derivation. Such version of Filter-adjusted O-SSA is called `DerivSSA'. See `References' for more details.

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

See Also

Rssa for an overview of the package, as well as, iossa.

Examples

# Separation of two mixed sine-waves with equal amplitudes
N <- 150
L <- 70
omega1 <- 1/15
omega2 <- 1/10

v <- sin(2*pi*omega1 * (1:N)) + sin(2*pi*omega2 * (1:N))
s <- ssa(v, L)
fs <- fossa(s, nested.groups = 1:4, gamma = 100)

# Rssa does most of the plots via lattice
ws <- plot(wcor(s, groups = 1:4))
wfs <- plot(wcor(fs, groups = 1:4))
plot(ws, split = c(1, 1, 2, 1), more = TRUE)
plot(wfs, split = c(2, 1, 2, 1), more = FALSE)

opar <- par(mfrow = c(2, 1))
plot(reconstruct(s, groups = list(1:2, 3:4)))
plot(reconstruct(fs, groups = list(1:2, 3:4)))
par(opar)

# Real-life example: Australian Wine Sales
data(AustralianWine)
s <- ssa(AustralianWine[1:120, "Fortified"], L = 60)
fs <- fossa(s, nested.groups = list(6:7, 8:9, 10:11), gamma = 10)

plot(reconstruct(fs, groups = list(6:7, 8:9, 10:11)))
plot(wcor(s, groups = 6:11))
plot(wcor(fs, groups = 6:11))

# Real life example: improving of strong separability
data(USUnemployment)
unempl.male <- USUnemployment[, "MALE"]
s <- ssa(unempl.male)
fs <- fossa(s, nested.groups = 1:13, gamma = 1000)

# Comparison of reconstructions
rec <- reconstruct(s, groups = list(c(1:4, 7:11), c(5:6, 12:13)))
frec <- reconstruct(fs, groups <- list(5:13, 1:4))
# Trends
matplot(data.frame(frec$F1, rec$F1, unempl.male), type= 'l',
        col=c("red","blue","black"), lty=c(1,1,2))
# Seasonalities
matplot(data.frame(frec$F2, rec$F2), type = 'l', col=c("red","blue"), lty=c(1,1))

# W-cor matrices before and after FOSSA
ws <- plot(wcor(s, groups = 1:30), grid = 14)
wfs <- plot(wcor(fs, groups = 1:30), grid = 14)
plot(ws, split = c(1, 1, 2, 1), more = TRUE)
plot(wfs, split = c(2, 1, 2, 1), more = FALSE)

# Eigenvectors before and after FOSSA
plot(s, type = "vectors", idx = 1:13)
plot(fs, type = "vectors", idx = 1:13)

# 2D plots of periodic eigenvectors before and after FOSSA
plot(s, type = "paired", idx = c(5, 12))
plot(fs, type = "paired", idx = c(1, 3))

# Compare FOSSA with and without normalize
N <- 150
L <- 70
omega1 <- 1/15
omega2 <- 1/10

v <- 3*sin(2*pi*omega1 * (1:N)) + 2*sin(2*pi*omega2 * (1:N))
s <- ssa(v, L)
fs <- fossa(s, nested.groups = 1:4, gamma = 100)
fs.norm <- fossa(s, nested.groups = 1:4, gamma = 100, normalize = TRUE)
opar <- par(mfrow = c(2, 1))
plot(reconstruct(fs, groups = list(1:2, 3:4)))
plot(reconstruct(fs.norm, groups = list(1:2, 3:4)))
par(opar)