Fig. 2: (N)BLE applied to prototype synthetic examples.
Dark- and light-orange shadings represent 16% to 84% and 1% to 99% percentile credibility bands (CBs) for the BLE, respectively; the same applies in green for the NBLE. a, d, g, j The results of the (N)BLE applied to the four test sets. The red, dotted vertical lines in (a–c, j–l) indicate the approximate times when noise-induced tipping (N-tipping) into a flickering regime and bifurcation-induced tipping (B-tipping) take place, respectively. The example datasets are analysed within windows of size Nw = 2 ⋅ 103 points with a shift of 100 points per window. The estimates' shift in time is due to the rolling window approach and ascribing the estimates to the last point of each window. Ascribing them to the midpoint of each window makes unbiased estimates match the true values almost perfectly (cf. SI S14, Fig. S12). In the examples (d, g, j), the data of each window are linearly detrended to account for the non-stationary trends in the mean. b At time t = 1386.8, the N-tipping causes artificial drift slope peaks with the width of one rolling time window as indicated by the grey-shaded area. b, c, e, f The BLE \(\hat{\zeta }\) and \(\hat{\sigma }\) are unbiased for the Markovian examples. h, i, k, l The BLE yields strongly biased estimates which work as qualitative leading indicators (due to correct trends), similar to AR1 \({\hat{\rho }}_{1}\) and STD \(\tilde{\sigma }\), apart from the noise level estimates in vicinity of the bifurcation point in (i). The BLE bias in (l) increases because of the increasing influence of the hidden process y due to multiplicative coupling via x. The strong BLE bias is perfectly compensated by the NBLE estimates. It mirrors the constant noise before the bifurcation in (i) and also works under the imperfect model parameterisation in (k, l).
