Computation of Extreme Values of Time Averaged Observables in Climate Models with Large Deviation Techniques

Abstract

One of the goals of climate science is to characterize the statistics of extreme and potentially dangerous events in the present and future climate. Extreme events like heat waves, droughts, or floods due to persisting rains are characterized by large anomalies of the time average of an observable over a long time. The framework of Donsker–Varadhan large deviation theory could therefore be useful for their analysis. In this paper we discuss how concepts and numerical algorithms developed in the context of with large deviation theory can be applied to study extreme, rare fluctuations of time averages of surface temperatures at regional scale with comprehensive numerical climate models. When performing this type of analysis, unless a rigorous study of the convergence to the large deviation limit is performed, it can be easy to be misled in thinking to have reached the asymptotic regime. In this paper we provide a systematic protocol to study the convergence of large deviation functions tailored for applications to climate problems. Referring to the existing literature on the subject, we provide explicit formulas to compute large deviation functions directly from time series of a deterministic dynamical system that can be applied to climate records, and we describe how to study the convergence. We show how using a rare event algorithm applied to a numerical model can improve the efficiency of the computation of the large deviation functions. As a case study we consider the time averaged European surface temperature obtained with the numerical climate model Plasim. We show how a precise analysis of the convergence leads to the conclusion that the large deviation limit is nor properly reached for time scales shorter than a few years, and is therefore of no practical interest to study midlatitude heat waves. Finally we show how, even in a case like this, rare event algorithms developed to study large deviation functions can be used to improve the statistics of events on time scales shorter than the one needed to reach the large deviation asymptotic regime.

Appendix: Convergence of Direct Estimate of Large Deviation Functions and Statistical Errors

The choice of the size of the time block $\tau _{b}$ and the test of the convergence to the large deviation limit requires computing the autocorrelation time of the process, which sets a lower bound to the values of the averaging time that it makes sense to consider. Figure 2b shows for the first 50 days the autocorrelation function C(t) of the average European surface temperature $T_{s}$ computed from a 1000 years long run. We can see that to a first approximation the function is well described by a double exponential, with a first decay time of about 4 days compatible with the time scale of synoptic variability, followed by a slowly decaying tail that at least in the first part seems to decay exponentially on a time scale of 1 month. The longer time scales inducing the slow decay of the autocorrelation function could be related to the low frequency variability of the atmospheric dynamics, and/or to time scales related to the water vapor cycle in the atmosphere and the land surface processes.

The integral autocorrelation time $\tau _{c}$ is defined as the integral from time lag 0 to $+\infty$ of the autocorrelation function $C(t)=\mathbb {E}\left[ (A(X(t))-\mu )(A(X(0))-\mu )\right] /\sigma ^{2}$. An equivalent expression for $\tau _{c}$ is [3]

$$\begin{aligned} \tau _{c}=\frac{1}{2\sigma ^{2}}\lim _{\tau _{b}\rightarrow +\infty } \frac{1}{\tau _{b}}\int _{0}^{\tau _{b}}\int _{0}^{\tau _{b}} \mathbb {E}\left[ \left( A(X(t))-\mu \right) \left( A(X(s))- \mu \right) \right] dtds. \end{aligned}$$

(21)

Equation 21 gives a better estimator of the autocorrelation time than a simple time integration of the autocorrelation function. In practice what we do is to divide the time series in $N_{b}$ blocks of length $\tau _{b}$, and then we compute the integrals in (21) in each block and approximate the expectation value as a sum of the $N_{b}$ blocks. Figure 9 shows the value of the estimate of the autocorrelation time as a function of $\tau _{b}$. The shaded area represents the 95% confidence interval of the estimate computed as two standard deviations of the sample of estimates over the $N_{b}$ blocks. We can see that the estimate converges to a value of about 7.5 days, but that it is necessary to use a very large value of $\tau _{b}$, of at least 3 years, in order to reach convergence.

Once computed the autocorrelation time, the first step of the direct estimate is to compute the generating function

$$\begin{aligned} \hat{G}(k,\tau _{b},N_{b})=\frac{1}{N_{b}}\sum _{j=1}^{N_{b}}e^{kA_{ \tau _{b}}^{j}},\mathrm {with}\;A_{\tau _{b}}^{j}=\frac{1}{\tau _{b}} \intop _{(j-1)\tau _{b}}^{j\tau _{b}}A(X(t))dt, \end{aligned}$$

(22)

knowing that $\tau _{b}$ will have to be much larger than $\tau _{c}$. When we deal with a discrete time series as the output of a numerical model, where time is discretized in time steps of length $\Delta t$, this means in practice computing

$$\begin{aligned} \hat{G}(k,\tau _{b},N_{b})=\frac{1}{N_{b}}\sum _{j=0}^{N_{b}-1}e^{k \sum _{n=jp+1}^{jp+p}A(X(n\Delta t))\Delta t}, \end{aligned}$$

(23)

where $p=\tau _{b}/\Delta t$. Following [3], a more sophisticated way that makes a better use of the available data would be to compute

$$\begin{aligned} \hat{G}(k,\tau _{b},N_{b})=\frac{1}{2N_{b}}\sum _{j=0}^{2N_{b}-1}e^{k \sum _{n=jp/2+1}^{jp/2+p}A(X(n\Delta t))\Delta t}. \end{aligned}$$

(24)

In (24) the sample mean is computed on $2N_{b}$ blocks overlapping by 50%, as suggested by the Welch estimator of the power spectrum of a random process [46]. Using (24) instead of (23) does not change the results of the estimate or the convergence region, but gives smaller statistical errors where they can be computed. In the following we keep the simpler notation (22) for ease of presentation.

Fig. 9

Convergence with $\tau _{b}$ of autocorrelation time

As discussed in the main text, in a practical application one is constrained by the fixed length T of the time series, and the choice of $\tau _{b}$ and $N_{b}$ has to be considered carefully. The convergence of the estimators has been studied by [36]. In the case of unbounded variables, obtaining a correct estimate is limited by two problems: (1) the artificial linearization of the tails of the functions due to the finite size of the sample and (2) the non-uniform convergence for different values of k.

The linearization effect is an artefact in the estimate of $\hat{G}(k,\tau _{b},N_{b})$ for large values of k which causes the estimate of $\hat{\lambda }(k,\tau _{b},N_{b})$ to become linear in k for any value of k whose module is large enough. This is due to the fact that a sum of exponentials over a finite sample, as the one involved in (22), is dominated for large k by the largest value in the sample, so that $\sum _{j=1}^{N_{b}}e^{kA_{\tau _{b}}^{j}}\approx e^{kA_{\tau _{b}}^{max}}$, with $A_{\tau _{b}}^{max}=\max _{j}\{A_{\tau _{b}}^{j}\}$. Therefore, for a given pair of $\tau _{b}$ and $N_{b}$, for positive k there is an upper critical value $k_{c}^{+}(\tau _{b},N_{b})>0$ for which $\hat{\lambda }(k,\tau _{b},N_{b})\approx kA_{\tau _{b}}^{max}$ for $k>k_{c}^{+}(\tau _{b},N_{b})$. Equivalently for negative k there is a lower critical value $k_{c}^{-}(\tau _{b},N_{b})<0$ for which $\hat{\lambda }(k,\tau _{b},N_{b})\approx kA_{\tau _{b}}^{min}$ for $k<k_{c}^{-}(\tau _{b},N_{b})$. If an observable is bounded, the linear behavior is actually correct. For unbounded variables it is instead an artefact of the finite size of the sample.

Scaling arguments can be provided to estimate $k_{c}^{+}(\tau _{b},N_{b})$ and $k_{c}^{-}(\tau _{b},N_{b})$, as discussed in details in [36]. However, the actual values depend on the underlying probability distribution of the process, and in complex applications they have to be estimated case by case by empirical analysis. A simple way to proceed is to compute the relative contribution of the largest value to the sample mean

$$\begin{aligned} r(k,\tau _{b},N_{b})=\frac{e^{kA_{\tau _{b}}^{max}}}{ \sum _{j=1}^{N_{b}}e^{kA_{\tau _{b}}^{j}}}. \end{aligned}$$

(25)

By fixing an arbitrary upper threshold for $r(k,\tau _{b},N_{b})$, one finds an estimate for the value of $k_{c}^{+}(\tau _{b},N_{b})$ (and an equivalent procedure gives a value for $k_{c}^{-}(\tau _{b},N_{b})$ ). Figure 10a shows $r(k,\tau _{b},N_{b})$ as a function of k for different values of $\tau _{b}$ for which there is actual convergence to the large deviation limit. Figure 10b shows the estimate of $k_{c}^{+}(\tau _{b},N_{b})$ as a function of $\tau _{b}$, obtained taking a threshold of 50% for $r(k,\tau _{b},N_{b})$. We can see that there is a large difference in $k_{c}^{+}(\tau _{b},N_{b})$ if taking a value of $\tau _{b}$ of about 1 year or 3-4 years. However, the estimate for lower values of $\tau _{b}$ is extremely unstable, showing that if proper convergence in time is not reached, also the convergence of the statistical estimator itself is not well behaved. For $\tau _{b}$ larger than 3 years the estimate of $k_{c}^{+}(\tau _{b},N_{b})$ stabilizes around a value of 5 $K^{-1}years^{-1}$. We have therefore taken $\tau _{b}=3$ years and $k_{c}^{+}(\tau _{b},N_{b})=5\,K^{-1}years^{-1}.$ A similar analysis gives $k_{c}^{-}(\tau _{b},N_{b})=-2.5\,K^{-1}years^{-1}$.

Fig. 10

a Contribution of the largest value in the sample to the estimate of the generating function as a function of k for different values of $\tau _{b}$. b Estimate of upper convergence limit as a function of $\tau _{b}$, taking as threshold a 50% contribution from the largest value in the sample

Once identified the convergence region, one can compute statistical errors in half of it, following [36]. The error on the generating function can be naturally estimated as

$$\begin{aligned} {{\text {err}}}[\hat{G}(k,\tau _{b},N_{b})]= \sqrt{\mathrm {var}(\hat{G}(k,\tau _{b},N_{b}))/N_{b}}, \end{aligned}$$

(26)

where $\mathrm {var}(\hat{G}(k,\tau _{b},N_{b}))$ is the empirical variance associated with the sample mean replacing the expectation value. An estimate of the associated error on $\hat{\lambda }(k,\tau _{b},N_{b})$ can be computed by taking a Taylor expansion of the estimator [33, 36]

$$\begin{aligned} {{\text {err}}}[\hat{\lambda }(k,\tau _{b},N_{b})]= \frac{{{\text {err}}}[\hat{G}(k,\tau _{b},N_{b})]}{\hat{G}(k,\tau _{b},N_{b})}. \end{aligned}$$

(27)

The statistical error on $\hat{a}(k,\tau _{b},N_{b})$ can be estimated by

$$\begin{aligned} {{\text {err}}}[\hat{a}(k,\tau _{b},N_{b})]= \sqrt{\frac{{{\text {err}}}[\hat{H}(k, \tau _{b},N_{b})]^{2}}{\left( \hat{G}(k,\tau _{b},N_{b}) \right) ^{2}}+\frac{\left( \hat{H}(k,\tau _{b},N_{b}) \right) ^{2}{{\text {err}}}[\hat{G}(k, \tau _{b},N_{b})]^{2}}{\left( \hat{G}(k,\tau _{b},N_{b})\right) ^{4}}}, \end{aligned}$$

(28)

where $\hat{H}(k,\tau _{b},N_{b})=\sum _{j=1}^{N_{b}}A_{\tau _{b}}^{j}e^{kA_{\tau _{b}}^{j}}$ and ${{\text {err}}}[\hat{H}(k,\tau _{b},N_{b})]$ is computed as ${{\text {err}}}[\hat{G}(k,\tau _{b},N_{b})]$. This formula is obtained assuming that $\hat{H}(k,\tau _{b},N_{b})$ and $\hat{G}(k,\tau _{b},N_{b})$ are independent [36]. The error on $\hat{I}(\hat{a}(k,\tau _{b},N_{b}),\tau _{b},N_{b})$ can then be estimated as

$$\begin{aligned} {{\text {err}}}[\hat{I}(\hat{a}(k,\tau _{b},N_{b}), \tau _{b},N_{b})]=\sqrt{k{{^{2}\text {err}}}[ \hat{a}(k,\tau _{b},N_{b})+{{\text {err}}}[ \hat{\lambda }(k,\tau _{b},N_{b})}. \end{aligned}$$

(29)

again assuming independence between $\hat{a}(k,\tau _{b},N_{b})$ and $\hat{\lambda }(k,\tau _{b},N_{b})$.