Cadzow Iterations

Description

Perform the finite rank approximation of the series via Cadzow iterations

Usage

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## S3 method for class 'ssa'
cadzow(x, rank, correct = TRUE, tol = 1e-6, maxiter = 0,
         norm = function(x) max(abs(x)),
         trace = FALSE, ..., cache = TRUE)

Arguments

x

input SSA object

rank

desired rank of approximation

correct

logical, if 'TRUE' then additional correction as in (Gillard et al, 2013) is performed

tol

tolerance value used for convergence criteria

maxiter

number of iterations to perform, if zero then iterations are performed until the convergence

norm

distance function used for covergence criterion

trace

logical, indicates whether the convergence process should be traced

...

further arguments passed to reconstruct

cache

logical, if 'TRUE' then intermediate results will be cached in the SSA object.

Details

Cadzow iterations aim to solve the problem of the approximation of the input series by a series of finite rank. The idea of the algorithm is quite simple: alternating projections of the trajectory matrix to Hankel and low-rank matrices are performed which hopefully converge to a Hankel low-rank matrix.

Note that the results of one Cadzow iteration with no correction coincides with the result of reconstruction by the leading rank components.

Unfortunately, being simple, the method often yields the solution which is far away from the optimum.

References

Cadzow J. A. (1988) Signal enhancement a composite property mapping algorithm, IEEE Transactions on Acoustics, Speech, and Signal Processing, 36, 49-62.

Gillard, J. and Zhigljavsky, A. (2013) Stochastic optimization algorithms for Hankel structured low-rank approximation. Unpublished Manuscript. Cardiff School of Mathematics. Cardiff.

See Also

Rssa for an overview of the package, as well as, reconstruct

Examples

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# Decompose co2 series with default parameters
s <- ssa(co2)
# Now make rank 3 approximation using the Cadzow iterations
F <- cadzow(s, rank = 3, tol = 1e-10)
library(lattice)
xyplot(cbind(Original = co2, Cadzow = F), superpose = TRUE)
# All but the first 3 eigenvalues are close to 0
plot(ssa(F))

# Compare with SSA reconstruction
F <- cadzow(s, rank = 3, maxiter = 1, correct = FALSE)
Fr <- reconstruct(s, groups = list(1:3))$F1
print(max(abs(F - Fr)))

# Cadzow with and without weights
set.seed(3)
N <- 60
L <- 30
K <- N - L + 1
alpha <- 0.1

sigma <- 0.1
signal <- cos(2*pi * seq_len(N) / 10)
x <- signal + rnorm(N, sd = sigma)

weights <- rep(alpha, K)
weights[seq(1, K, L)] <- 1
salpha <- ssa(x, L = L,
              column.oblique = "identity",
              row.oblique = weights)
calpha <- cadzow(salpha, rank = 2)

cz <- cadzow(ssa(x, L = L), rank = 2)

print(mean((cz - signal)^2))
print(mean((calpha - signal)^2))

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