Perform the finite rank approximation of the series via Cadzow iterations

1 2 3 4 |

`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 |

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

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.

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 Unpublished Manuscript.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.

`Rssa`

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 | ```
# 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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