parest function

Estimate periods from (set of) eigenvectors

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

## S3 method for class '1d.ssa'
parestimate(x, groups, method = c("esprit", "pairs"),
            subspace = c("column", "row"),
            normalize.roots = NULL,
            dimensions = NULL,
            solve.method = c("ls", "tls"),
            ...,
            drop = TRUE)
## S3 method for class 'toeplitz.ssa'
parestimate(x, groups, method = c("esprit", "pairs"),
            subspace = c("column", "row"),
            normalize.roots = NULL,
            dimensions = NULL,
            solve.method = c("ls", "tls"),
            ...,
            drop = TRUE)
## S3 method for class 'mssa'
parestimate(x, groups, method = c("esprit", "pairs"),
            subspace = c("column", "row"),
            normalize.roots = NULL,
            dimensions = NULL,
            solve.method = c("ls", "tls"),
            ...,
            drop = TRUE)
## S3 method for class 'cssa'
parestimate(x, groups, method = c("esprit", "pairs"),
            subspace = c("column", "row"),
            normalize.roots = NULL,
            dimensions = NULL,
            solve.method = c("ls", "tls"),
            ...,
            drop = TRUE)
## S3 method for class 'nd.ssa'
parestimate(x, groups,
            method = c("esprit"),
            subspace = c("column", "row"),
            normalize.roots = NULL,
            dimensions = NULL,
            solve.method = c("ls", "tls"),
            pairing.method = c("diag", "memp"),
            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)
• dimensions: a vector of dimension indices to perform ESPRIT along. 'NULL' means all dimensions.
• method: For 1D-SSA, Toeplitz SSA, and MSSA: parameter estimation method, 'esprit' for 1D-ESPRIT (Algorithm 3.3 in Golyandina et al (2018)), 'pairs' for rough estimation based on pair of eigenvectors (Algorithm 3.4 in Golyandina et al (2018)). For nD-SSA: parameter estimation method. For now only 'esprit' is supported (Algorithm 5.6 in Golyandina et al (2018)).
• solve.method: approximate matrix equation solving method, 'ls' for least-squares, 'tls' for total-least-squares.
• pairing.method: method for esprit roots pairing, 'diag' for `2D-ESPRIT diagonalization', 'memp' for ``MEMP with an improved pairing step'
• 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 nD-ESPRIT, coefficient(s) in convex linear combination of shifted matrices. The length of beta should be ndim - 1, where ndim is the number of independent dimensions. If only one value is passed, it is expanded to a geometric progression.

Returns

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 nD-SSA, a list of objects of S3-class fdimpars.nd'. Each object is named list of n 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.

Details

See Sections 3.1 and 5.3 in Golyandina et al (2018) for full details.

Briefly, the time series is assumed to satisfy the model

$$x_n = \sum_k{C_k\mu_k^n}$$

for complex $\mu_k$ or, alternatively,

$$x_n = \sum_k{A_k \rho_k^n \sin(2\pi\omega_k n + \phi_k)}.$$

The return value are the estimated moduli and arguments of complex $\mu_k$, more precisely, $\rho_k$ ('moduli') and c("$T_k =\n$", "$1/\\omega_k$") ('periods').

For images, the model

$$x_{ij}=\sum_k C_k \lambda_k^i \mu_k^j$$

is considered.

Also print' and plot' methods are implemented for classes fdimpars.1d' and fdimpars.nd'.

References

Golyandina N., Korobeynikov A., Zhigljavsky A. (2018): Singular Spectrum Analysis with R. Use R!. Springer, Berlin, Heidelberg.

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.

Shlemov A, Golyandina N (2014) Shaped extensions of Singular Spectrum Analysis. In: 21st international symposium on mathematical theory of networks and systems, July 7--11, 2014. Groningen, The Netherlands, pp 1813--1820.

See Also

Rssa for an overview of the package, as well as, ssa, lrr,

Examples

# Decompose 'co2' series with default parameters
s <- ssa(co2, neig = 20)
# Estimate the periods from 2nd and 3rd eigenvectors using 'pairs' method
print(parestimate(s, groups = list(c(2, 3)), method = "pairs"))
# Estimate the peroids from 2nd, 3rd, 5th and 6th eigenvectors using ESPRIT
pe <- parestimate(s, groups = list(c(2, 3, 5, 6)), method = "esprit")
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 background)
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"))