Geology Reference
In-Depth Information
Robust (0.04)
Robust (0.1)
Classical
99%
95%
90%
AR(1)
99%
95%
90%
AR(1)
99%
95%
90%
AR(1)
ρ
est
= 0.8790
ρ
est
= 0.8300
ρ
est
= 0.9207
10
2
0.050
0.055
10
1
Actual
ρ
= 0.90
10
0
0
0.02
0.04
0.06
0.08
0.1
Frequency (1/n)
Figure 4.20
Classical and robust red noise modeling of the “signal+red noise” time series from Figure 4.17. The
actual red noise in the time series has ρ = 0.90, but the classical red noise model overestimates ρ = 0.9207. The robust
red noise models calculated using the SSA-MTM Toolkit with 0.1 and 0.04 median filter lengths and linear fitting
result in underestimated ρ = 0.8300 and ρ = 0.8790. Alternative log fitting (not shown) results in ρ = 0.8600 for a
median filter of length 0.1, and ρ = 0.9100 for a shorter median filter of length 0.04, which reduces the influence of
the two signal frequencies on the modeled noise for f ≤ 0.04.
But also of significance is that the estimated ρ = 0.9207 from the lag - 1 of the
signal + red noise autocorrelation function is larger than the actual red noise
component of the time series (ρ = 0.9). Thus, the AR(1) model, also known as
the “classical red noise model,” is biased: R(f ) overestimates noise at low
frequencies, and to a lesser degree, underestimates noise at high frequencies.
The bias from estimating the AR(1) null model from a time series con-
taining both signal and noise was recognized by Mann and Lees (1996) who
introduced “robust red noise” estimation. This procedure removes narrow
band spectral components prior to AR(1) modeling on the premise that
such components represent signal. Two methods were used to identify
narrow band signal components: a harmonic F-test to detect individual lines
(see Section 4.3.6.2) and median smoothing of the data spectrum to reject
outliers presumed to represent signal.
Comparison of “classical” versus “robust” red noise modeling (Figure 4.20)
shows that the former overestimates the noise, but the latter has adjustable
parameters (median filter length, and linear vs. log fitting) that can be
searched for the ρ closest to true ρ. Of course in practical applications, true ρ
is unknown. Thus, a procedure is needed to identify the most correctly
estimated ρ which has yet to be developed. Meyers (2012) has described an
alternative “lowess” smoothing of the spectrum in robust AR(1) modeling to
optimize ρ estimation. These classical and robust AR(1) models, despite the
shortcomings, are the conventional null models presently used in cyclostratig-
raphy. Figure 4.21 demonstrates application of both models on the Arguis
