r
s i +r =
k =1
a k s i +r − k .
If r is minimal, it is called the rank of the signal SN . General form of the signal: sn = k Pmk (n)ρn k cos(2πωk n + φk ) ∈ R or sn = k Pmk (n)µn k ∈ C, where µk ∈ C. Problems: Extraction of the signal Signal forecasting Parameter estimation
4/22
Nina Golyandina
On the choice of parameters in SSA
Signal extraction by SSA
F N = S N + R N , F N = ( f0 , . . . , fN − 1 ) Let SN be a signal of rank r . Scheme: f f ... f
7/22
Nina Golyandina
On the choice of parameters in SSA
Model time series
Examples: fn = sn + rn , sn = b n cos(2πω n), n = 1, . . . , N , (1) rn = c , (2) rn = σεn , √ (3) rn = (σεn + c )/ 2, (4) rn = σηn . Here εn is a white gaussian noise with variance 1, ηn is the autoregressive process of order 1 (red noise) with parameter α and variance Dηn = 1, that is, ηn = αηn−1 + n , where n has variance 1 − α2 . Set c = σ = 0.1, α = 0.5, b = 1, ω = 1/10. The parameters of noise (=residual) are chosen to obtain the same signal-to-noise ratio (SNR).
2/22
Nina Golyandina
On the choice of parameters in SSA
Basic SSA algorithm
Time series FN = (f0 , . . . , fN −1 ) Parameters: window length L; 1 < L < N ; K = N − L + 1. Decomposition 1 Embedding Series F ↔ L × K trajectory matrix X f0 f1 . . . fK − 1 f p p p p p p fK 1 p p X = pp — Hankel matrix p p p pp pp p f L− 1 f L . . . f N − 1 2 SVD d √ √ T XXT ⇒ (λm , Um ) ⇒ X = λm U m V m , V m = X T U m / λm .
0 1
K −1
T FN − − →X= . . . . . . . .. L fL−1 fL ...
f1
f2 ...
fK
f N −1
SVD:(µi ,Ui ,Vi ), Pr − − − − − − − − − − → . − . . r
s0 s1 s1 ... sK −1 s2 ... sK
H − →S= . . .
sL−1 sL ... sN −1
. . . . ..
T −1 − − → SN . − . .

Short notation of the SSA-extraction algorithm: SN = T −1 HPr T FN Cadzow iterations: (m) FN = T −1 (HPr )m T FN
(s)
+ δerror
(projector)
, SN = SN + δerror .
(signal)
Questions: Convergence of the errors to 0 as the time series length N tends to infinity? Conditions on the signal SN and the residual RN ? Conditions on window lengths L = L(N )? Conventional technique: the formal expansion with respect to the perturbation level. Difficulty: the residual RN is not small, i.e. RN does not tend to zero. The paper Nekrutkin (SII, 2010): theory of perturbation for convergence as N → ∞. This talk: a general view on convergence, dependence of errors on L; numerical experiments.

Para, . . . , Ur ) Pr is the projector on Lr T T S= r i =1 Ui (X Ui ) = Pr X L is a window length. r is the signal rank.
6/22 Nina Golyandina On the choice of parameters in SSA
Structure of the experiments
Types of residuals: Pure noise (white or red) Deterministic residuals Combined residuals Behavior of errors: is common for pure-noise residuals, is specific for deterministic residual, intermediate situation for combined residuals. Rate of convergence as N → ∞: quite different for fixed L and L ∼ αN /2. Features of interest: Estimation of the projector Extraction of the signal Forecasting Parameter estimation
On the choice of parameters in Singular Spectrum Analysis and related subspace-based methods
Nina Golyandina
St.Petersburg State University Mathematical Department
m=1
Reconstruction 1 Grouping X = XI1 + . . . + XIc , where XI =
2
√
m∈I
T λm U m V m = m∈I
T Um Um X
Diagonal averaging Matrix XI Hankel matrix XI ↔ rec.series F (I ) (I1 ) Output: F = F + . . . + F (Ic ) . Results: eigenvalues (λi ), eigenvectors EOF (Ui ), factor vectors/principal components (Vi ), reconstructed components RC (F (Ij ) ).
We observe: FN = SN + RN , where RN is a perturbation (we call it the residual).
We apply SSA: SN = T −1 HPr T FN , where Pr is the orthogonal projector on the estimated signal subspace Lr = span(U1 , . . . , Ur ). We obtain: Pr = Pr
UK-China workshop on singular spectrum analysis and its applications 20 September 2010, Cardiff University, UK
1/22
Nina Golyandina
On the choice of parameters in SSA
Outline
Consider Singular Spectrum Analysis (SSA) and subspace-based methods in signal processing. Describe common and specific features of these methods Consider different kinds of problems: signal reconstruction, forecasting and parameter estimation. Provide general recommendations on the choice of parameters Demonstrate that the error behavior depends on the type of residuals, deterministic or stochastic, and whether the noise is white or red. Discuss convergence rate The analysis is based on known theoretical results and extensive computer simulations. “On the choice of parameters in Singular Spectrum Analysis and related subspace-based methods” Journal: “Statistics and Its Interface”, Special Issue: Theory and Practice in Singular Spectrum Analysis of Time Series. 2010