BOUNDARY CONDITIONS FOR SYMMETRIC BANDED TOEPLITZ MATRICES: AN APPLICATION TO TIME SERIES ANALYSIS Alessandra Luati Dip. Scienze Statistiche University of Bologna Tommaso Proietti S.E.F. e ME. Q. University of Rome “Tor Vergata” ????? D UE GIORNI DI ALGEBRA LINEARE NUMERICA ,B OLOGNA , 6-7 MARZO 2008
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BOUNDARY CONDITIONS FOR SYMMETRIC
BANDED TOEPLITZ MATRICES: AN APPLICATION
TO TIME SERIES ANALYSIS
Alessandra LuatiDip. Scienze StatisticheUniversity of Bologna
Tommaso ProiettiS.E.F. e ME. Q.
University of Rome “Tor Vergata”
? ? ? ? ?
DUE GIORNI DI ALGEBRA LINEARE NUMERICA, BOLOGNA, 6-7 MARZO 2008
This talk concerns the spectral properties of matrices associated with linear
filters for the estimation of the underlying trend of a time series.
These matrices are finite approximations of infinite symmetric banded Toeplitz
(SBT) operators subject to boundary conditions.
The interest lies in the fact that the eigenvectors can be interpreted as the latent
components of a time series that the filter smooths through the eigenvalues.
In this study, analytical results on the eigenvalues and eigenvectors of matrices
associated with trend filters are derived by interpreting the latter as perturbations of
matrices belonging to algebras with known spectral properties, such as the
circulant and the reflecting τ11.
The results allow to design new estimators based on cut off eigenvalues, which
are less variable and almost equally biased as the original estimators.
Signal extraction of a time series
Time series additive models
yt = µt + εt, t = 1, . . . , n
yt observed time series
µt trend component or signal, smooth function of time
εt irregulars or noise, zero mean stationary stochastic process.
The aim is to estimate µt using the available observations.
Smoothing methods like local polynomial regression may serve to this purpose.
Local polynomial regression methods
The basic assumption is that µt locally approximated by a p-degree polynomial
function of the time distance j between yt and neighboring yt+j
On the matrix H (Bini and Capovani, 1983, Proposition 2.2).
H =n∑
j=1
cjTj−1ψϕ
where
Tψϕ =
ψ 1 0 · · · 0
1 0 1. . . 0
0 1. . .
. . . 0...
. . .. . . 0 1
0 . . . 0 1 ϕ
with ψ,ϕ = 0,1,−1, and c is the solution of the upper triangular system Qc = hwhere h′ is the first row of H and Q is the matrix whose j-th column equals the
first column of Tj−1ψϕ .
Eigenvalues
Theorem Let S be an n× n smoothing matrix associated with the symmetric filter
{w−h, ..., w0, ..., wh}, and let H be the corresponding matrix in τ11. Hence,
∀λ ∈ σ(S), ∃i ∈ {1, 2, .., n} such that
|λ− ξi| ≤ δH
where
ξi =h+1∑
j=1
(2 cos
(i− 1)πn
)j−1
wj−1 +
bh−j−12 c∑
q=0
(−1)q+1(j)q
(q + 1)!(j + 2q + 1)wj+2q+1
and δH = ‖S−H‖2.
Note that ‖A‖2 =√
ρ(A′A) and ρ(A) is the spectral radius of A.
Figure 1: Transfer function of the symmetric Henderson filter, h = 6, ν ∈ [0, π](line) and eigenvalues of the associated reflecting matrix H (crosses), n = 51.
0 0.5 1 1.5 2 2.5 3 3.5-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Figure 2: Absolute eigenvalue distributions of H (crosses) with asymmetric