Ko E ti2 Si2 k7 E Xi Si

k10 = E x(i) S(i)-2, k11 = E t(i) x(i) S(i) ¿=¿1+1 ¿=¿1+1

To estimate the change-points in time, a brute-force search over all pairs of candidate points is performed because gradient techniques are inapplicable owing to the non-differentiability with respect to t1 and ¿2:

Because the number of pairs of search points grows with the data size as — 1)/2, it is advisable to use computational measures to keep computing costs low (Section 4.5). A positive by-product of the brute-force search is that the solution is a global optimum. Because the candidate points t1 and t2 are from the set {i(i)}™=1, the solution is as "coarse" as the spacing. This may be a problem when the spacing (at around t1 and t2) is larger than the standard errors, seti and se^. However, in climatological applications this likely occurs only when we wish to quantify a climate transition using an archive with a hiatus located at around the place of a transition change-point.

Because in practice the variability S(i) is unknown, an iterative estimation procedure via the residuals e(i) is indicated (Section 4.1.1).

4.2.1.2 Example: Northern Hemisphere Glaciation

Application of the ramp model to the marine ¿1SO record ODP 846 shows that the Northern Hemisphere Glaciation was a slow climate transition (Fig. 4.6). Whereas the t1 estimate of around 2.5 Ma before present is in general agreement with the climatological literature (Shack-leton et al. 1984; Haug et al. 1999, 2005), the t2 estimate (3.7 Ma) is about 0.5 Ma earlier than what was previously thought. (Mudelsee and Raymo (2005) analysed a total of 45 ¿18O records using the ramp and found an average t2 of ~ 3.6 Ma; the inter-record variation is likely caused by contrasting temperature trends.) As outliers (defined in the paper as more than 3S(i) away from the ramp fit), two prominent glaciation peaks (termed M2-MG2) appear at around 3.3 Ma (Fig. 4.6). These findings are robust against the estimation uncertainties (Fig. 4.6).

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