Time Series Analysis for Cyclostratigraphy - Rock Magnetic Cyclostratigraphy - page 79

series) at frequency f are combined to estimate the mean modulus, or
amplitude μ(f ), for which μ(f ) 2 is a “signal” power estimate with 2 dof.
Subtraction of μ(f ) from the K individual eigencoefficients at frequency f is
used to estimate a residual modulus μ res (f ), for which μ res (f ) 2 is a “noise” power
estimate with ν(f ) - 2 dof. The ratio, F = μ(f ) 2 /μ res (f ) 2 , is a signal-to-noise ratio
of variances with an F-distribution of 2 and ν(f ) - 2 dof. It is applied as a con-
ventional F-test: when an F value exceeds the F-distribution at a specified
1 - α level, a harmonic line may be inferred that is significantly above the level
of the noise.
Figure 4.23 shows the results of the harmonic F-test of the “signal + red
noise” test series, which is an extremely noisy time series with a signal
with a variance of 1 (constrained to frequencies 0.050 and 0.055) and red
noise (ρ = 0.9) with a variance of 25. The signal is actually not visually
discernable in the time series (Figure 4.17a); it is notable that the spectral
analysis techniques demonstrated thus far have been able to detect it. The
reason for this is that over the entire length of the time series the signal
maintains perfect repeatability with constant (“coherent”) phase. The
noise, on the other hand, has irregular phasing with variance at any given
frequency that can “cancel out” when evaluated over the entire series
length. The question is whether in the presence of the strong noise in this
test time series, F-testing will be able to identify the signal frequencies as
harmonic lines.
At very low frequencies (f<0.04), the noise dominates the spectrum
(Figures 4.23a and b); nonetheless the two signal frequencies are clearly
distinguished. Over 90% of the power occurs from f=[0,0.1], and the
F-testing should therefore focus on this range. The dof used in the F-testing
is based on ν(f ) - 2 (Figures 4.23c and d) where ν(f ) was previously shown
in Figure 4.17c. A useful by-product of the F-test procedure is an estimate
of the amplitude spectrum (Figure 4.23e and f ), which indicates that the
red noise has added variance to both frequencies (amplitudes slightly over
1.0, the true amplitude). Several of the noise-only frequencies indicate pos-
sible coherent phase behavior, registering F-values with significance levels
in excess of 99% (Figure 4.23j), but the highest significance levels occur at
the two signal frequencies at 0.050 and 0.055. In practice, a spectrum with
N/2 + 1 frequencies will register α(N/2 + 1) F-tests exceeding the 1 - α sig-
nificance level. Here, with (4N)/2 + 1 = 4097 frequencies, and α = 0.01, we
would expect ~41 F-tests at 0.99 and higher; 39 such F-tests are recorded in
Figure 4.23i.
In summary, the harmonic F-test detected the two lines in the signal + red
noise test time series. However, other high F-values (37 in total) generated by
the noise produced numerous “false positives” that could have been (incor-
rectly) interpreted as lines. Our interpretation was possible based on clues
from the power spectrum indicating elevated power at the line frequencies,
and a priori knowledge of the existence of the lines. However, for time series
with unknown spectral content, which is always the case in cyclostratigraphy,
F-testing must be undertaken with great caution. Thomson (2009) advises