The null-hypotheses

A hypothesis test is conducted for each of the three orbital parameters where the null-hypotheses are

H^p₀, glacial terminations are independent of the phase of precession H^e₀, glacial terminations are independent of the phase of eccentricity H^θ₀, glacial terminations are independent of the phase of obliquity

To proceed it is necessary to translate the H^p,e,θ₀ into PDFs of nonlinear coherence. In estimating these PDFs, one could assume that the phase between orbital variations and the terminations is uniformly distributed. Although this uniform assumption would be simple, it is also most likely incorrect in that it requires the interval be-tween consecutive terminations to be uniformly distributed bebe-tween zero and infinity.

Instead, it appears that ice-sheets have an intrinsic timescale associated with growth and collapse [e.g. Imbrie et al., 1993;Marshall and Clark, 2002], and a null-hypothesis is developed with incorporates a plausible timescale for glacial variability.

The simple stochastic glacial model introduced by Wunsch [2003b] is used as the basis for estimating the PDF associated with the H₀. The stochastic model postulates a random walk in ice-volume,

Vt = Vt−1+aηt +b, (4.3)

if Vt <0, Vt = 0, if Vt > ξ, Vt = 0,

where V is ice-volume, t is a discrete time measured in one KY intervals, ηt is a zero-mean white noise process, and a, b, and ξ are constants. The noise term, η, is independent and normally distributed so that the expected variance, < (aη)² >, is a² where a is chosen as two. So that no particular sequence of termination times is made more likely, the initial ice-volume is set to a random value between 0 andξwith uniform probability.

For a simple random walk, the variance of the timeseries is expected to grow linearly in time at a rate controlled bya. However, Eq. 4.4 incorporates two threshold conditions which modify the behavior of this random walk. First, a barrier is imposed at zero ice-volume so that values of Vt which are less than zero are reset to zero.

0 100 200 300 400 500 600 700 0

20 40 60 80 100

time (ky)

ice−volume

Figure 4-4: A realization of the random walk model of ice-volume (Eq. 4.4). The selected parameterizations are a noise amplitude of a=2, a drift of b=1, and an ice-volume threshold ofξ=100. Terminations are triggered once the threshold (indicated by the horizontal red line) is crossed. All units are normalized.

Second, a collapse threshold is imposed at ξ above which ice-volume is also reset to zero. The nth crossing of this threshold is identified with the triggering time, T_n. Wunsch [2003b] added an additional stochastic term to the collapse threshold, but here ξ is simply taken to be constant at a value of 100. A drift term, not in the original model, ofb = 1 is included to provide a bias toward accumulation. A similar bias is observed in the histogram of the rates of change of EOF1 shown in Figure 4-3. Without this bias the distribution of termination intervals is similar to a Poisson process where long intervals of more than 200KY are common. The positive bias towards accumulation makes terminations occur at a more regular rate; if there was no noise, the terminations would be periodic at 100KY. A realization of the model output is shown in Figure 4-4.

The threshold crossing statistics of V_t could be computed analytically using the theory of Brownian motion with reflecting and absorbing barriers [e.g. Feller, 1957], but such an approach is not pursued here. Rather, it suffices to derive the pertinent statistics associated with Vt using a Monte Carlo approach. To obtain a single real-ization of the nonlinear coherence associated with H0, first a sequence of eight glacial termination are generated using the stochastic glacial model, in analogy with the values listed in Table 4.1. Next, the relative phasing between the terminations and each orbital parameter is calculated using Eq. 4.1. Finally, the nonlinear coherence is computed from each set of eight phases using Eq. 4.2. Fifty-thousand Monte Carlo

0 0.5 1

Figure 4-5: From left to right are the null-hypothesis PDFs associated with preces-sion, obliquity, and eccentricity. The vertical bar indicates the estimated nonlinear coherence, and shading indicates theα= 0.05 rejection region for the null-hypothesis of no connection between the parameter and the timing of the terminations. The critical value (v) and nonlinear coherence (c) with EOF1 are (v = 0.6,c = 0.43) for precession, (v = 0.6,c= 0.7) for obliquity, and (v = 0.84, c = 0.66) for eccentricity.

Only obliquity has a nonlinear coherence greater than its critical value so that the obliquity null-hypothesis alone is rejected.

realizations of nonlinear coherence are used.

The PDFs associated with the H^p,e,θ₀ are estimated by sorting fifty-thousand Monte Carlo realizations of the nonlinear coherence for each orbital parameter into twenty bins centered on {0.025,0.05...0.975} and then normalizing the histogram area to one. Critical values are estimated by finding the nonlinear coherence above which 5%

of the Monte Carlo realizations reside. The resulting PDFs and critical values are shown in Figure 4-5 and tabulated in Table 4.2. Only the obliquity null-hypothesis, H^θ₀, can be rejected as the obliquity nonlinear coherence (0.70) is greater than the obliquity critical value (0.60). Thus a significant coupling exists between obliquity and the glacial terminations at the 5% level. In fact, the likelihood of obtaining such an obliquity nonlinear coherence by chance alone is one in a hundred. Furthermore, it is estimated that the power of the obliquity test is 0.58; see Appendix 4.6.2 for further discussion of how the power of the test was calculated and how it should be interpreted.

c v β precession, 0.43 0.60 — obliquity, 0.70 0.60 0.42 eccentricity, 0.66 0.84 —

Table 4.2: Summary results of the hypothesis test. From left to right, columns refer to the nonlinear coherence between orbital variations and the terminations (c); the nonlinear coherence critical values (v) for an α = 0.05 significance level; and the probability of making a Type II error, β.