Linear and nonlinear coherence

As defined within the main text of the chapter [see Eq 4.2], the nonlinear coherence is

where φn is the phase of a periodic or quasi-periodic signal relative the nth event identified in another signal. N is the total number of phase estimates, and|.|indicates

the magnitude of a complex quantity. Eq 4.2 has a form similar to standard linear coherence estimates [e.g. Priestley, 1984]. For comparison purposes, consider the linear coherence estimate,

Here φn is the phase between nth set of harmonics, where the harmonics must be of equal frequency. Note that unlike the nonlinear coherence, the linear coherence weights eachφnterm according to the amplitude associated with each of the harmon-ics, here denoted as an and bn. In the case that both an and bn are constant (a and b need not be equal), the linear coherence estimate reduces to the same form as the nonlinear coherence estimate.

The maximum value of both cand c⁰ is one, occurring when φ is a constant; the minimum value is zero, occurring when the sum of the vectors exactly cancel. Even when the true coherence is zero, however, it is unlikely for the estimated coherence to be zero, given a finite number of observations. While it is possible to correct for this bias, it is easier to account for this effect when determining significance levels, and the latter approach is adopted here. Table 4.2 lists the estimated nonlinear coherence of each orbital parameter with the glacial terminations.

The similarity between the nonlinear and linear coherence estimates suggests that they will have similar distributions. Amos and Koopman [1963] have derived an ex-pression for the probability density function (PDF) associated the linear coherence estimates under the assumptions that the phase estimates are independent and uni-formly distributed. Figure 4-8 shows how the cumulative density functions (CDFs) for linear coherence varies with increasing degrees of freedom (DOF), i.e. increas-ing independent phase estimates. The CDF of the nonlinear coherence is estimated using a Monte Carlo approach. A single Monte Carlo realization is made by com-puting the nonlinear coherence ofN randomly selected phases distributed uniformly on the interval−180^◦ to 180^◦. A histogram of nonlinear coherence is generated from 10⁵ Monte Carlo realizations whose area is normalized to one. The cumulative sum of this histogram of nonlinear coherence the provides an estimate of the nonlinear coherence CDF and results are shown in Figure 4-8 along with the linear CDFs.

A small systematic offset exists between the linear and nonlinear coherence CDFs but which diminishes with increasing DOF. This discrepancy presumably arises

be-0 0.2 0.4 0.6 0.8 1 0

0.2 0.4 0.6 0.8 1

coherence

cumulative distribution

Figure 4-8: The cumulative density function (CDF) for linear coherence (black) from the analytical results of Amos and Koopman [1963] and nonlinear coherence (red) estimated using a Monte Carlo method as described in the text. From left to right the pairs of linear and nonlinear CDFs are for 32, 16, 8, 4, and 2 degrees of freedom.

Results assume the phase is uniformly distributed. The nonlinear coherence CDFs tend to be shifted toward lower values, but for greater degrees of freedom this dif-ference becomes small. The vertical dashed line indicates the 95% confidence level above which the null-hypothesis of a uniform phase distribution can be rejected.

2 4 6 8 10 12 14 16 18 20

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

degrees of freedom

95% confidence level

Figure 4-9: The 95% confidence level for linear coherence (black) and nonlinear coher-ence (red) as a function of the degrees of freedom (DOF). For large DOF the differcoher-ence between the difference between linear and nonlinear coherence becomes negligible.

cause of the weighting terms included in estimating c⁰ (Eq 4.5) but which are not present in the expression for c (Eq. 4.2). Figure 4-9 shows the 95% confidence level plotted as a function of DOF for the linear and nonlinear coherence where the nonlin-ear coherence confidence level is estimated using the Monte Carlo approach previously described. At eight DOF there is only a 0.01 difference between the linear and non-linear 95% confidence levels. At sixteen DOF this difference is less than 0.005. This results suggests that for sufficiently large DOF, the analytical results of Amos and Koopman [1963] can be used to judge the significance of nonlinear coherence esti-mates, assuming uniformly distributed and independent phases. However, because the nonlinear coherence estimates made here have only eight degrees of freedom, Monte Carlo techniques are used to estimate the PDF of nonlinear coherence.

The linear coherence is typically computed over a range of frequencies. One of the useful features of such a calculation is that one obtains a sense of the behavior of the coherence statistic. For instance, one would generally be hesitant to conclude much from a coherence which has a large variance and shows little structure. As another example, if one thought two records were unrelated and yet found what appeared to be significant coherence, it would be prudent to perhaps explore the coherence estimate using more or less DOF, perhaps a different record interval, or data from other sources. Similarly, to get a sense for the behavior of the nonlinear coherence estimate it is useful to calculate the nonlinear coherence of the glacial terminations with a range of periodic signals with differing frequencies.

The timing of the glacial terminations are listed in Table 4.1, and the timing of the nth termination will be designated asTn. The periodic signal can be written as

x(t) = cos(wt+α),

wherewis the circular frequency andαis a phase constant. It is possible to compute the nonlinear coherence by calculating the phase at the terminations times,

φ_n =wT_n+α, and substituting into Eq. 4.2.

Figure 4-10 shows the spectrum of 400 nonlinear coherence estimates computed between Tn and a periodic signals with frequencies ranging from 1/10KY to 1/2KY with a 1/1000KY bandwidth spacing. This range of frequencies is selected because

at frequencies above 1/10KY there is no known orbital pacing of the glacial cycles and 1/2KY is the Nyquist frequency associated with a one KY sampling interval.

The level below which 95% of the nonlinear coherence estimates fall is 0.58, in good agreement with the Monte Carlo and analytically derived 95% confidence levels. This correspondence between multiple estimates supports the accuracy of the confidence level estimates and indicates that the T_n are not an unusual case.

The distribution of the spectrum of nonlinear coherence estimates provides a basis for judging the significance of the orbital nonlinear coherence estimates. The orbital nonlinear coherence between the Tn and precession is 0.43, 0.66 for eccentricity, and 0.70 for obliquity. Figure 4-10 shows that 100 out of 400 nonlinear coherence esti-mates exceed 0.43, 10 out of 400 exceed 0.66, and only 3 out of 400 exceed 0.70.

Thus precession appears to be unrelated to the terminations, the significance of the eccentricity nonlinear coherence is somewhat ambiguous, and obliquity appears to be highly coherent. The significance of each of these nonlinear coherence results is more formally considered in section 4.5.

There are some similarities between the assessment of phase coupling presented here, and the paper byRahmstorf [2003] which calls attention to how the Dansgaard-Oeschger events are spaced by roughly integer multiples of 1500 years. At some level both studies seek to quantify the regularity in reoccurrence times of events. There are, however, two important differences. First,Rahmstorf [2003] never quantitatively assesses the likelihood of finding the regularity in the Dansgaard-Oeschger events as a function of chance. Second, the regularity of the glacial cycles are compared with the long-term variations in modes of insolation forcing, while the regularity in the Dansgaard-Oeschger events are not associated with any known forcing. The latter situation is not unlike finding a very large nonlinear coherence between terminations and some arbitrary periodic signal and attempting to ascertain its physical significance

— a much more demanding task.