# Estimate periods from (set of) eigenvectors

### Description

Function to estimate the parameters (frequencies and rates) given a set of SSA eigenvectors.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 | ```
## S3 method for class '1d.ssa'
parestimate(x, groups, method = c("pairs", "esprit-ls", "esprit-tls"),
subspace = c("column", "row"),
normalize.roots = NULL,
..., drop = TRUE)
## S3 method for class 'toeplitz.ssa'
parestimate(x, groups, method = c("pairs", "esprit-ls", "esprit-tls"),
subspace = c("column", "row"),
normalize.roots = NULL,
..., drop = TRUE)
## S3 method for class 'mssa'
parestimate(x, groups, method = c("pairs", "esprit-ls", "esprit-tls"),
subspace = c("column", "row"),
normalize.roots = NULL,
..., drop = TRUE)
## S3 method for class '2d.ssa'
parestimate(x, groups,
method = c("esprit-diag-ls", "esprit-diag-tls",
"esprit-memp-ls", "esprit-memp-tls"),
subspace = c("column", "row"),
normalize.roots = NULL,
...,
beta = 8,
drop = TRUE)
``` |

### Arguments

`x` |
SSA object |

`groups` |
list of indices of eigenvectors to estimate from |

`...` |
further arguments passed to 'decompose' routine, if necessary |

`drop` |
logical, if 'TRUE' then the result is coerced to lowest dimension, when possible (length of 'groups' is one) |

`method` |
estimation method. For 1dSSA: 'pairs' for rough estimation based on pair of eigenvectors, 'esprit-ls' for least-squares 1D-ESPRIT and 'esprit-tls' for total-least-squares 1D-ESPRIT. For 2dSSA: 'esprit-diag-ls' or 'esprit-diag-tls' for ‘Two-Dimensional ESPRIT Method’ (2D-ESPRIT diagonalization) and 'esprit-memp-ls' or ‘esprit-memp-tls’ for ‘MEMP with an improved pairing step’. Suffix '-ls' or '-tls' means approximate matrix equation solve method, as in 1D-ESPRIT case. |

`subspace` |
which subspace will be used for parameter estimation |

`normalize.roots` |
logical vector or 'NULL', force signal roots to lie on unit circle. 'NULL' means automatic selection: normalize iff circular topology OR Toeplitz SSA used |

`beta` |
In 2D-ESPRIT, coefficient in convex linear combination of shifted matrices. |

### Details

The time series is assumed to satisfy the model

*
x_n = ∑_k{C_kμ_k^n}
*

for complex *μ_k* or, alternatively,

*
x_n = ∑_k{A_k ρ_k^n \sin(2πω_k n + φ_k)}.
*

The return value are the estimated moduli and arguments of complex
*μ_k*, more precisely, *ρ_k* ('moduli') and *T_k =
1/ω_k* ('periods').

For images, the model

*
x_{ij}=∑_k C_k λ_k^i μ_k^j
*

is considered.

Also ‘print’ and ‘plot’ methods are implemented for classes ‘fdimpars.1d’ and ‘fdimpars.2d’.

### Value

For 1D-SSA (and Toeplitz), a list of objects of S3-class ‘fdimpars.1d’. Each object is a list with 5 components:

- roots
complex roots of minimal LRR characteristic polynomial

- periods
periods of dumped sinusoids

- frequencies
frequencies of dumped sinusoids

- moduli
moduli of roots

- rates
rates of exponential trend (

`rates == log(moduli)`

)

For 'method' = 'pairs' all moduli are set equal to 1 and all rates equal to 0.

For 2D-SSA, a list of objects of S3-class ‘fdimpars.2d’. Each object is unnamed list of two ‘fdimpars.1d’ objects, each for corresponding spatial coordinate.

In all cases elements of the list have the same names as elements of
`groups`

. If group is unnamed, corresponding component gets name
‘Fn’, where ‘n’ is its index in `groups`

list.

If 'drop = TRUE' and length of 'groups' is one, then corresponding list of estimated parameters is returned.

### References

Golyandina, N., Zhigljavsky, A. (2013): *Singular Spectrum
Analysis for time series*. Springer Briefs in Statistics. Springer.

Roy, R., Kailath, T., (1989): *ESPRIT: estimation of signal parameters via
rotational invariance techniques*. IEEE Trans. Acoust. 37, 984–995.

Rouquette, S., Najim, M. (2001): *Estimation of frequencies and damping factors by two-
dimensional esprit type methods*. IEEE Transactions on Signal Processing 49(1), 237–245.

Wang, Y., Chan, J-W., Liu, Zh. (2005): *Comments on “estimation of frequencies and
damping factors by two-dimensional esprit type methods”*.
IEEE Transactions on Signal Processing 53(8), 3348–3349.

### See Also

`Rssa`

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

,
`lrr`

,

### Examples

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 | ```
# Decompose 'co2' series with default parameters
s <- ssa(co2, neig = 20)
# Estimate the periods from 2nd and 3rd eigenvectors using default
# 'pairs' method
print(parestimate(s, groups = list(c(2, 3))))
# Estimate the peroids from 2nd, 3rd, 5th and 6th eigenvectors using ESPRIT
pe <- parestimate(s, groups = list(c(2, 3, 5, 6)), method = "esprit-ls")
print(pe)
plot(pe)
# Artificial image for 2D SSA
mx <- outer(1:50, 1:50,
function(i, j) sin(2*pi * i/17) * cos(2*pi * j/7) + exp(i/25 - j/20)) +
rnorm(50^2, sd = 0.1)
# Decompose 'mx' with default parameters
s <- ssa(mx, kind = "2d-ssa")
# Estimate parameters
pe <- parestimate(s, groups = list(1:5))
print(pe)
plot(pe, col = c("green", "red", "blue"))
# Real example: Mars photo
data(Mars)
# Decompose only Mars image (without backgroud)
s <- ssa(Mars, mask = Mars != 0, wmask = circle(50), kind = "2d-ssa")
# Reconstruct and plot texture pattern
plot(reconstruct(s, groups = list(c(13,14, 17, 18))))
# Estimate pattern parameters
pe <- parestimate(s, groups = list(c(13,14, 17, 18)))
print(pe)
plot(pe, col = c("green", "red", "blue", "black"))
``` |