Decomposes a series into components by diagonalising its lag-covariance
matrix, then reconstructs each one. With window (embedding dimension) M
and series length N, Vautard, Yiou & Ghil use the Toeplitz estimate of the
lag-covariance matrix,
c_j = sum_{t=1}^{N-j} z_t z_{t+j} / (N - j) and
C_ij = c_{|i-j|}, on the centred series z = y - mean(y).
This is the estimator they recommend over the trajectory-matrix
(Broomhead-King) form because it is unbiased term by term and keeps C
Toeplitz, so its eigenvectors stay close to sines and cosines for a
quasi-periodic signal. C is symmetric and is diagonalised by Jacobi
rotations, eigenvalues sorted decreasing.
Value
List with estimate (leading variance fraction),
eigenvalues, variance_fraction, reconstructed,
leading_fraction, pair_fraction, total_variance,
trace_check, reconstruction_error, c0,
mean, n, window, n_lagged, method.
Details
The k-th principal component is the projection of the delay vectors on
eigenvector k, a^k_t = sum_j z_{t+j-1} E_jk for
t = 1..N-M+1, and the reconstructed component is recovered by
diagonal averaging of a^k_t E_jk over the delay index with the
boundary-corrected normalisation. The reconstructed components sum
exactly to the centred series, reported as
reconstruction_error.
Since trace(C) = M c_0 = M var(y), the eigenvalues sum to M times
the series variance; trace_check reports that identity. The
Toeplitz estimate is not guaranteed positive semi-definite – each
c_j carries its own divisor N - j – so a few trailing
eigenvalues may come out slightly negative and an individual variance
fraction may exceed 1. That is a property of the estimator Vautard,
Yiou & Ghil chose, not an error.
References
Vautard, R., Yiou, P. & Ghil, M. (1992). Singular-spectrum analysis: a toolkit for short, noisy chaotic signals. Physica D 58(1-4), 95-126. doi:10.1016/0167-2789(92)90103-T
Examples
Singsd(y = c(1, 2, 3, 4, 5, 6, 7, 8), window = 5L)
#> $estimate
#> [1] 0.5102578
#>
#> $eigenvalues
#> [1] 13.3942680 10.2974813 1.0358520 0.9236687 0.5987300
#>
#> $variance_fraction
#> [1] 0.51025783 0.39228500 0.03946103 0.03518738 0.02280876
#>
#> $reconstructed
#> [1] -0.8510674 -0.9494330 -0.7142292 -0.2137613 0.2137613 0.7142292 0.9494330
#> [8] 0.8510674
#>
#> $leading_fraction
#> [1] 0.5102578
#>
#> $pair_fraction
#> [1] 0.9025428
#>
#> $total_variance
#> [1] 26.25
#>
#> $trace_check
#> [1] 2.131628e-14
#>
#> $reconstruction_error
#> [1] 2.220446e-15
#>
#> $c0
#> [1] 5.25
#>
#> $mean
#> [1] 4.5
#>
#> $n
#> [1] 8
#>
#> $window
#> [1] 5
#>
#> $n_lagged
#> [1] 4
#>
#> $method
#> [1] "Singular spectrum analysis, Toeplitz lag covariance (Vautard, Yiou & Ghil 1992)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"