We search for the signature of universal properties of extreme events, theoretically predicted for Axiom A flows, in a chaotic and highdimensional dynamical system. We study the convergence of GEV (Generalized Extreme Value) and GP (Generalized Pareto) shape parameter estimates to the theoretical value, which is expressed in terms of the partial information dimensions of the attractor. We consider a twolayer quasigeostrophic atmospheric model of the midlatitudes, adopt two levels of forcing, and analyse the extremes of different types of physical observables (local energy, zonally averaged energy, and globally averaged energy). We find good agreement in the shape parameter estimates with the theory only in the case of more intense forcing, corresponding to a strong chaotic behaviour, for some observables (the local energy at every latitude). Due to the limited (though very large) data size and to the presence of serial correlations, it is difficult to obtain robust statistics of extremes in the case of the other observables. In the case of weak forcing, which leads to weaker chaotic conditions with regime behaviour, we find, unsurprisingly, worse agreement with the theory developed for Axiom A flows.
The investigation of extreme events is extremely relevant for a range of disciplines in mathematical, natural, and social sciences and engineering. Understanding the large fluctuations of the system of interest is of great importance from a theoretical point of view, but also when it comes to assessing the risk associated with low probability and high impact events. In many cases, in order to gauge preparedness and resilience properly, one would like to be able to quantify the return times for events of different intensity and take suitable measures for preventing the expected impacts. Prominent examples are weather and climate extremes, which can have a huge impact on human society and natural ecosystems. The present uncertainty in the future projections of extremes makes their study even more urgent and crucial [
In practical terms, the main goal behind the study of extreme events is to understand the properties of the highest quantiles of the variable of interest. A fundamental drawback comes from the fact that extreme events are rare, so that it is difficult to collect satisfactory statistics from the analysis of a time series of finite length. Additionally, in the absence of a strong mathematical framework, it has its limits to make quantitative statements about the probability of occurrence of events larger than observed. Therefore, statistical inference based on empirical models tends to suffer from the lack of predictive power. A theoretical framework for analysing extreme events is provided by extreme value theory (EVT). After the early contributions by Fisher and Tippett [
The two most important properties of extremes—their rare occurrence and their unusually high or low magnitude—constitute the basis of two popular methods of EVT, the block maxima (BM) and the peakoverthreshold (POT) approaches. The BM approach aims at finding the limiting distribution of maxima
Problems in applying EVT to actual time series result from the fact that, typically, the observed data feature a certain degree of serial correlations [
When performing statistical inference using the BM or POT method (fitting the GEV or GP model, resp., to data), it is crucial to have an appropriate protocol of selection of “good" candidates for extremes [
Classical EVT has been extended and adapted to analyse extremes of observables of chaotic dynamical systems, where the sensitive dependence on initial conditions is fundamentally responsible for generating a de facto stochastic process. The reader is referred to Lucarini et al. [
Several studies dealing with EVT for dynamical systems reveal a link between the statistical properties of the extremes and geometric (and possibly in turn global dynamical) characteristics of the system producing these extremes [
Some preliminary numerical tests show that there is no obvious convergence to the predicted asymptotic shape parameter in lowdimensional cases [
In a previous analysis performed on higherdimensional, intermediate complexity models with
In this work, we use a quasigeostrophic (QG) atmospheric model of intermediate complexity, featuring 1056 degrees of freedom, to analyse extremes of different types of observables: local energy (defined at each grid point), zonally averaged energy, and the average value of energy over the midlatitudes. Our main objective is to compare the estimated GEV and GP shape parameters with a shape parameter derived, based on the theory referred to above, from the properties of the attractor and of the measure supported on it along the stable, unstable, and neutral directions. We refer to this as the “theoretical shape parameter.” Thus, we explore numerically the link between the purely statistical properties of extreme events based on EVT and the dynamical properties of the system producing these extremes. We perform simulations applying two different levels of forcing: a strong forcing, producing a highly chaotic behaviour of the system, and a weak forcing, producing a less pronounced chaotic behaviour. The dimensionality of the attractor is much larger in the former than in the latter case. This work goes beyond the previously mentioned studies, based on more simple dynamical systems, in a sense that with our model we can study the convergence for observables being different physical quantities or representing different spatial scales/characteristics of the same physical quantity. Additionally, compared to previous studies also performed on intermediate complexity models, we consider longer time series and a variety of observables. Our model is simple compared to a GCM (General Circulation Model) but contains two of the main processes relevant for midlatitude atmospheric dynamics: baroclinic and barotropic instabilities. Hence, we contribute to bridging the gap between the analysis of extremes in simple and very highdimensional dynamical systems, as in the case of the GCMs used for atmospheric and climate simulations, by using a model that simulates to a certain degree Earthlike atmospheric processes and allows also for computing with feasible computational costs some dynamical system properties, like Lyapunov exponents or KaplanYorke dimensions. The properties of the model have been extensively studied by Schubert and Lucarini [
Based on numerical results (Sebastian Schubert, personal communication), the model is expected to be nonhyperbolic, but one can assume that the chaotic hypothesis applies to it (in the case of sufficiently high forcing levels inducing a strongly chaotic behaviour of the system), and so in analysing the convergence of shape parameter estimates one can take the predicted theoretical shape parameter as reference. We also assume that the symmetry of the model with respect to longitude, which introduces a central direction besides the directions of expansion and contraction in phase space, does not alter the ergodicity of the system at a practical level, and hence the true shape parameter remains uniform.
Although we use an idealised model, our results are transferable to time series obtained from more realistic model simulations or from measurements. By understanding the differences among the analysed observables, we gain insight into the statistical properties of extremes of geophysical observables with different spatial scales. By using two forcing levels, we are able to study the convergence to theoretical shape parameters related to different chaotic systems: one exhibiting fast decaying correlations and another one characterised by slower decaying correlations. These aspects are relevant in the case of geophysical applications where one deals also with time series on several spatial scales and with different degrees of correlations.
The structure of this article is as follows. Section
Let us consider
Under the same conditions, for which the distribution of
From the values of the GEV or GP parameters, it is possible to infer the expected return levels or extreme quantiles. Return levels
In the case of a correlated stationary stochastic process, the same GEV limit laws apply as for i.i.d.r. variables if certain conditions, regarding the decay of serial correlation, are fulfilled [
As mentioned before, several studies on EVT for observables of dynamical systems relate the GEV and GP shape parameters to certain properties of the system itself. In the case of the socalled “distance” observables, one can relate the GEV and GP parameters to basic geometrical properties of the attractor [
While recurrence properties are indeed important for characterising a system, distance observables are not well suited for studying some basic physical properties, such as, in the case of fluids, energy or enstrophy. Hence, Holland et al. [
According to (
We consider a spectral quasigeostrophic (QG) 2layer atmospheric model similar to the one introduced by Phillips [
The model domain is a rectangular channel with latitudinal and longitudinal coordinates
The model is described by the following equations in terms of the barotropic stream function
List of symbols and parameter values for the QG model ((
Variable  Symbol  Unit  Scaling factor  

Stream function 




Temperature 

K 


Velocity 




Energy 






Parameter  Symbol  Dimensional value  Nondimensional value  Scaling factor 


Forced meridional temperature difference 

133 & 40 K  0.188 & 0.0564 

Ekman friction 


0.022 

Eddymomentum diffusivity 




Eddyheat diffusivity 




Thermal damping 


0.011 

Stability parameter 


0.0329 & 0.0247 

Coriolis parameter 


1 

Beta ( 


0.509 

Aspect ratio 

0.6896  0.6896  — 
Meridional length 




Zonal length 




Specific gas constant 


2 

Vertical pressure difference 

500 hPa  1 

Time scale 


1 

Length scale 


1 

The vertical velocity is set to 0 at the top level,
Using hydrostatic approximation [
We substitute (
The spectral output of the model is transformed into the grid point space using the Fast Fourier Transform resulting in
We obtain the zonally averaged energy
Although we record the model output, as stated above, every 5.5 hours, we save only the maximum values over one month in the case of strong forcing and over three months in the case of weak forcing. We estimate the GEV and GP parameters based on block maxima and threshold exceedances obtained from the monthly, respectively, 3monthly, maxima series. Such an operation has no effect on the subsequent GEV analysis. Instead, it might modestly impact the GP analysis, as some abovethreshold events might be lost, because they could be masked by a larger event occurring within the same 1month or 3month period. Nonetheless, since we consider very high thresholds and an extremely low fraction of events, the risk of losing information is negligible. The GEV and GP parameters are inferred by maximum likelihood estimation (MLE), as described by Coles [
As mentioned before, the simulations are performed using two different configurations, where the value of the parameter
Before presenting the results related to the statistics of extreme events, we outline some general statistical properties of the analysed observables. As emphasised in Sections
By taking the ergodic hypothesis, we estimate the autocorrelation coefficient for the local energy according to (
Statistical properties of the total energy for
Figures
In what follows, we present the results of the EVT analysis starting with the local observables. We first discuss the convergence of the shape parameter for GEV and GP, then the convergence of the GP modified scale parameter (to be introduced below), and, at the end, the convergence of return levels. Taking advantage of the fact that statistics are uniform in the zonal direction, we concatenate the monthly maxima series for every second longitude one after the other in the
The theory discussed in Section
GEV and GP shape parameters as well as bias and precision estimates in the case of the local observables, for
First we assess the uniformity for the local observables. Figure
To assess the goodness of fit, we perform a onesample KolmogorovSmirnovtest (KStest) [
Figure
We perform another test to check whether the GP distribution is a good approximation for the distribution of threshold exceedances based on our data and consider the GP modified scale parameter. The GP scale parameter depends on the chosen threshold according to
GP modified scale parameter estimates in the case of the local observables, for
Having practical applications in mind, the BM and POT methods aim at obtaining statistical estimates of either return levels or expected return periods, for even unobserved extreme events. Figures
Return levels for
If the GEV distribution is an adequate model for extreme events for a certain block size, one expects return levels with a certain return period not to change much anymore with increasing block size. Figures
After having discussed in detail the convergence in the case of the local observables, we proceed with the results for the zonally averaged observables. Figure
Same as Figure
Our observation that the estimated shape parameters depend strongly on the considered latitude has to do with the effect of serial correlation on the convergence to the limiting distribution. We obtain weak autocorrelations, fast convergence to
We present now the analysis of extremes of the average midlatitude observable. Figure
GEV and GP shape parameter as well as bias and precision estimates in the case of the average midlatitude observable, for
In short, our numerical results do allow for conclusions regarding the universality of extremes, as predicted by the theory presented in Section
Before analysing the extreme events for weak forcing, we discuss some statistical (and dynamical) properties of our observables, which influence directly the statistics of extremes. Figure
Statistical properties of the total energy for
In contrast to the case of strong forcing, the zonal averages of the local energy observables show remarkable deviations from a Gaussian behaviour, even more than the PDFs of the local energy observables (Figures
For the analysis of extreme events, we use a similar procedure as in the case of strong forcing (
In the case of weak forcing, the theoretical shape parameter is −0.03, shown by the grey horizontal line in Figure
Shape parameter for
In the case of the zonally averaged and average midlatitude observables, we cannot detect any convergence. This is an expected result, considering the statistical and dynamical characteristics of our data and the fact that the length of the time series is in this case even shorter than for the local observables. As an effect of the “shoulders” in the PDFs, we obtain very uncertain estimates even for large block sizes, and the KStests reject the hypothesis of a GEV model in these cases. The shape parameter estimates have a large latitudinal spread due to the varying form of PDFs with different latitudes. Except for the differences between the GEV (Figures
In this paper, we have studied the convergence of statistically estimated GEV and GP shape parameters to the theoretical shape parameter, which, following the mathematical findings reported in [
We start the discussion of our results with the strong forcing regime. In this case, we observe a roughly monotonic increase of the estimated GEV (GP) shape parameters towards the theoretical value
Despite the predicted universal asymptotic properties of extremes, if we consider a certain block size (threshold), we find that the shape parameter estimates are different among the observables and latitudes. Thus, in view of finitesize estimates, extremes show rather diverse properties. The speed of convergence to the asymptotic level is not universal. The local observables exhibit highfrequency fluctuations, as an effect of boundary fluxes, and, at the same time, the fastest convergence of the shape parameter estimates to the theoretical value. Since the energy is transported mostly along the zonal direction by the zonal mean flow, by averaging along a latitudinal band the highest frequencies are filtered out, and fluctuations with lower frequencies become stronger. In the case of the zonally averaged observables, we obtain weak autocorrelations and fast convergence to
We assume that the extremely slow convergence has to do mainly with the fact that
Our conclusions regarding the convergence of the estimated shape parameter to
In the case of weak forcing, temporal and spatial correlations are very strong due to a regime behaviour of our system, which exhibits two welldefined regimes: a more unstable one with stronger fluctuations and a less unstable one with reduced fluctuations. Due to such a regime behaviour the statistics of extreme events is “contaminated”: if the block size (threshold) is not large (high) enough, we select events from both regimes, whereas if it is large (high) enough, only extremes from the more unstable regime are selected. This induces nonmonotonic changes of the estimated shape parameters by increasing the block size (threshold) and leads to the appearance of positive, that is, nonphysical, or very low shape parameter estimates. In the case of the local observables, the estimated shape parameters seem to converge at almost every latitude to a value which is lower (≈−0.06) than the theoretical shape parameter (
Our results show that with increasing block size or threshold the shape parameters of the GEV and GP distributions are becoming more and more similar, according to the asymptotic equivalence of the two models [
We use the KolmogorovSmirnov test (KStest) to verify the fit of the GEV (GP) distribution to the distribution of extremes, selected as block maxima (threshold exceedances). Our results show that the KStest is merely an indicator of the fit quality and does not show whether the convergence to the correct GEV (GP) distribution is reached or not. The KStest suggests a good fit to the GEV (GP) distribution even in the cases when the distance between the estimated and the asymptotic shape parameter is substantial and even if no convergence can be detected. The misleading property of
Concluding, we would like to emphasise some key messages one can get from our results:
Indeed, we have been able to find the signature of the universal properties of the extremes of physical observables in strongly chaotic dynamical systems, as predicted in the case of Axiom A systems. Nonetheless, given the availability of very long yet finite time series, we have been able to find more convincing results (yet with a relatively large uncertainty) only for specific observables, because in the case of observables featuring serial correlations it is extremely hard to collect robust statistics of extremes.
We have observed that in the case of strong forcing the estimate of the shape parameter increases monotonically towards its asymptotic value for stricter and stricter criteria of selection of extremes. This corresponds to the fact that we manage to collect more detailed information on the local properties of the attractor and of the measure supported on it near the point of absolute maximum of the observable, and thus we explore all the dimensions of the attractor.
We also remark that agreement of the results with the theory of extremes of observables of dynamical systems developed in the context of Axiom A flows cannot be found in the case of the weakly chaotic flow featuring regime behaviour and strong spatial and temporal correlations, as these features suggest strong deviations from the conditions behind the chaotic hypothesis. Note that conceptually analogous results had been reported in a simple model in [
We note that the predicted and estimated shape parameters are extremely small so that the statistics of extremes is virtually indistinguishable, up to ultralong return periods, from what would be predicted by a Gumbel distribution (
We conclude by noting that in some cases of great practical relevance one finds results in contradiction with the basic tenets of the theory of extremes of dynamical systems, suggesting that one should never find block maxima distributed according to the Fréchet distribution, which allows for arbitrarily large extremes. The precipitation, as opposed to geophysical fields like temperature or pressure, is a nonsmooth intermittent field with multifractal properties in space and time [
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors would like to thank Sebastian Schubert, Christian Franzke, Maida Zahid, and Richard Blender for useful discussions. The authors are indebted to Sebastian Schubert for his support in performing some simulations and providing the code for computing the Lyapunov Exponents and KaplanYorke dimensions. Valerio Lucarini acknowledges the many exchanges on these topics with Davide Faranda, Antonio Speranza, and Renato Vitolo. Valerio Lucarini also acknowledges support received from the Sfb/Transregio Project TRR181 and from the StGERC Project NAMASTE (Grant No. 257106). Valerio Lucarini and Tamás Bódai are grateful for support from the CRESCENDO Project (Grant no. 641816). Vera Melinda Gálfi acknowledges funding from the International Max Planck Research School on Earth System Modelling (IMPRSESM).