Boosting Ensembles for Statistics of Tails at Conditionally Optimal Advance Split Times

TL;DR

The paper investigates how to efficiently sample extreme climate events using rare-event sampling with ensemble boosting by optimizing the advance split time (AST). It formalizes conditionally optimal advance split time (COAST) and develops two tail-estimators, MoCTail and PoPTail, to aggregate conditional tails from boosted ensembles toward climatological tail estimates. Through a two-layer quasigeostrophic model with a passive tracer, it shows that optimal ASTs (1–3 eddy turnover times) substantially improve tail estimates over direct DNS, with threshold-free criteria based on thresholded entropy (TE) and expected improvement (EI) offering robust AST selection. The work provides practical guidance for designing efficient sampling strategies in climate extremes and suggests online, adaptive AST optimization as a promising direction for future rare-event algorithms.

Abstract

Climate science needs more efficient ways to study high-impact, low-probability extreme events, which are rare by definition and costly to simulate in large numbers. Rare event sampling (RES) and ensemble boosting use small perturbations to turn moderate events into a severe ones, which otherwise might not come for many more simulation-years, and thus enhance sample size. But the viability of this approach hinges on two open questions: (1) are boosted events representative of the yet-unrealized events? (2) How does this depend on the specific form of perturbation, i.e., timing and structure? Timing in particular is crucial for sudden, transient events like precipitation. In this work, we formulate a concrete optimization problem for the advance split time (AST) hyperparameter, and study it on an idealized but physically informative model system: passive tracer fluctuations in a turbulent channel, which captures key elements of midlatitude storm track dynamics. Three major questions guide our investigation: (1) Can RES methods, in particular ``ensemble boosting'' equipped with a probability estimator and ``trying-early adaptive multilevel splitting'', accurately and efficiently sample extreme events? (2) What is the optimal AST, and how does it depend on the event definition, in particular the target location and surrounding flow conditions? (3) Can the AST be optimized ``online'' while running RES? Our answers support RES as a viable method: (1) RES can meaningfully improve tail estimation, using (2) an optimal AST of 1-3 eddy turnover timescales depending on location. (3) A ``thresholded entropy'' statistic is a good proxy for AST optimality, bypassing the tedious threshold-setting that often hinders RES methods. Our work clarifies aspects of the response function of transient extreme events to perturbations, giving a guide for designing efficient, reliable sampling strategies.

Figures (15)

• Figure 1: Schematic summarizing the ensemble boosting and tail estimation procedure, using a simple Langevin dynamics with a potential that is quadratic for $x\in(-0.25,0.25)$---the blue-shaded region---and logarithmic outside this range. Appendix A specifies the system completely. The position variable $X(t)$ exhibits intermittent, transient extremes (a.i) and power law tails $\mathbb{P}\{|X|>|x|\}\sim|x|^{-3.1}$ (a.ii). We set a threshold for severity (horizontal black dashed line) at roughly the minimum probability estimable from a relatively short (duration 1600) timeseries (see the black empirical PDF in a.ii and the black empirical CCDFs in (b,c,d).iii, as compared with the true PDF and CCDF in gray). We then identify the peaks over the threshold (marked by vertical black dashed lines in a.i), and perturb the simulation in advance of these peaks. Three choices of advance split time (AST) are shown in rows b,c,d, marked by vertical red lines, each resulting in "boosted" peak ensembles, shown as red curves in (b,c,d).(i,ii) and summarized by complementary CDFs (CCDFs) shown in light red in (b,c,d).(iii). Combining these conditional CCDFs together using the "MoCTail" estimator introduced later in Eq. \ref{['eq:moctail']} gives the dark red dashed line, which is meant to approximate the ground truth (gray line) better than the short DNS alone can do, including by going to higher values of $x$. The intermediate AST (c) is best among the three for this task, and our goal is to formulate and characterize this optimal AST more generally.
• Figure 2: Snapshots of the QG system configuration in the upper layer. Contours indicate the anomaly streamfunction $\psi$, which varies over a non-dimensional range of approximately $\pm18$, dashed contours indicating negative anomalies. Colors indicate (a) tracer concentration $c$, (b) zonal wind velocity $u=U-\partial_y\psi$, where $U=1$ is the basic background shear, and (c) meridional velocity $v=\partial_x\psi$. The timestamps increase from left to right, and come from the long DNS. The small square represents an example target region in which to sample extremes of the local tracer concentration, in this case centered at $x_0=\frac{1}{2}L,y_0=\frac{26}{64}L$ and extending $\pm\ell=\frac{2}{64}L$ in both meridional and zonal directions. This same region is the target used in the following results, and we consistently refer to the domain coordinates in fractions of 64 across all figures.
• Figure 3: Hovmöller diagrams of anomalies (departures from time-means) of zonal-mean concentration (a.i) and zonal-mean zonal wind (b.i). Contours indicate zonal-mean streamfunction anomaly (range $\pm10$, negatives values dashed). Column (ii) shows bottom topography, which directly affects the lower layer only, but indirectly sets the preferred jet positions in the upper layer as well. For the same quantities, column (iii) shows the zonal and time mean and column (iv) shows the zonal mean of the temporal standard deviation. The Hovmöller diagrams give context to the snapshots of $u$ from Fig. \ref{['fig:snapshots']}b, which come from times (i) 3300, when the upper and lower jets are both shifted south; (ii) 3400, when the jets are unusually far apart; and (iii) 3500, when the jets are unusually close together. These intermittent, discrete shifts in jet location happen every $\sim100$ days, which we call the "jet meandering timescale". During a typical 100-day timespan of stationary jet, the fields shown oscillate roughly 10 times; hence we assign the eddy turnover timescale a nominal value of 10 days.
• Figure 4: Summary statistics of latitude-dependent climatological tail distributions of local tracer concentrations, also called "intensities", which are denoted $R$ and defined as the average concentration $c$ over a box $(x,y)\in(x_0,y_0)+[-\ell,\ell]^2$. $x_0=\frac{1}{2}L$ and $\ell=\frac{1}{32}L$ are fixed, while $y_0$ varies across the midlatitudes from $\frac{10}{64}L$ to $\frac{54}{64}L$. Panel (a) shows the lower-layer topography in this same range of middle latitudes, (b) shows the mean intensity $\langle R\rangle(y_0)$, after subtracting a nominal trend of $\frac{y_0}{L}$ to reveal a finer-scale structure that resembles the underlying topography, and (c) shows the standard deviation of intensity $\sqrt{\langle R^2\rangle-\langle R\rangle^2}$. Dashed curves in (b) and (c) indicate the mean and standard deviation, respectively, of the concentration field $c$ without box-averaging. Panels (d,e,f) summarize the distribution of intensities $R^*$ via the parameters of the generalized Pareto distributions (GPD), inferred by the peaks-over-threshold fitting procedure (see section \ref{['sec:target_variable']} for details). The threshold is set to the $(\frac{1}{2})^5$-complementary quantile, denoted $\mu[(\frac{1}{2})^5]$ and shown in (d) with linear trend removed. Panels (e, f) display the estimated (scale, shape) parameters ($\sigma,\xi$).
• Figure 5: Probability distributions of local tracer concentrations at latitude $y_0=\frac{26}{64}L$ and averaged over a box of half-width $\ell=\frac{2}{64}L$. (a) The full PDF of intensity $R$. (b) The CCDF (tail integral) of intensity $R$, restricted to $R>\mu[\frac{1}{2}]$. (c) Further zoomed-in CCDF of the severity $R^*$ (peaks of $R$ over $\mu[(\frac{1}{2})^5]$). In all three panels, solid black and red lines represent estimates from long and short DNS, respectively, with shaded 90% confidence intervals obtained by repeating the inference $64$ times, once for each possible longitudinal rotation of the dataset. Error bars become degenerate at levels experienced by $<5\%$ of longitudes. Black dashed lines show the mean over all longitudinal rotations, our best estimate of ground truth. The gray line in (b,c) represents the GPD fit to $R^*$ with $\mu=0.52$, $\sigma=0.06$, and $\xi=-0.31$, and this is a much better fit to the severities in (c) which makes sense given they are defined in terms of peaks.