Jökull - 01.01.2014, Page 20

W. Menke and H. Menke
Figure 4. (a) The i = 1 (black) and i = N (grey) members of a set of N = 100 time series u(i)0 whose
shapes evolve systematically with i. (b) The corresponding noisy time series u(i) (s.n.r.=3). (c) Reconstructed
versions of (a) without weights fail to capture the systemic evolution. (d) Reconstructed versions of (a) with
weights correctly capture the systemic evolution. (e)-(f) Weights v(1, i) and v(N, i), respectively, used to com-
pute the reconstructions in (d). In order to emphasize the accuracy of the reconstruction, the target time series
were omitted from their own reconstructions. – (a) Sú fyrsta (i = 1, svört) og síðasta af N =100 tilbúnum
tímaröðum; lögun merkisins er látin breytast á kerfisbundinn hátt með hækkandi i. (b) Hér hefur suði (r =
3) verið bætt við tímaraðirnar. (c) Eftir einfalda úrvinnslu gagnanna líkjast allar endurheimtu tímaraðirnar
meðaltali þeirra. (d) Ef þær nágranna-tímaraðir sem líkastar eru hverri tímaröð um sig eru látnar vega þyngst
(með stuðlum úr jöfnu (7) eins og sýnt er í (e) og (f)) í úrvinnslunni fyrir hana, má ná því að fylgja hinum
kerfisbundnu breytingum eftir.
the out-member averaging is disadvantageous; limit-
ing the averaging to a subset of similarly shaped time
series is more sensible. In such cases, we propose
using a weighted average, with the weights monoton-
ically increasing with the maximum value cestij of the
cross-correlation. Such averaging also allows simi-
larly shaped time series to be stacked to produce an
estimate ū(i) that has reduced noise (Figure 4):
τwij =
N−2∑
k=1
w
(k)
ij τ
(k)
ij
/N−2∑
k=1
w
(k)
ij and
ū(i) =
N∑
k=1
v
(i)
k u
(i)(t− τwik)
/ N∑
k=1
v
(i)
k
w
(k)
ij = f
[
1/2(cestik +c
est
kj )
]
and v
(i)
k = f
[
cestik
]
(7)
Here f(c) is a monotonically increasing function of
c. We use the function f(c) = exp(c/s) where s
is an empirically-chosen scale factor. This type of
stacking is similar to the non-local means denoising
method (Buades et al., 2005; Buades, 2010; Bonar
and Sacchi, 2012), except that the optimal time delay
is itself being computed as part of the process. The
weighted stack has the additional benefit of rejecting
outlier time series (e.g. data with uncharacteristically
high s.n.r.). We therefore suggest its routine use, even
in cases where no evolution of the deterministic pulse
is expected.
20 JÖKULL No. 64, 2014