Does the AO index have predictive power regarding extreme cold temperatures in Europe?

With a view to seasonal forecasting of extreme value statistics, we apply the method of Nonstationary extreme value statistics to determine the predictive power of large scale quantities. Regarding winter cold extremes over Europe we find that the monthly mean daily minimum local temperature – which we call a native co-variate in the present context – has a much larger predictive power than the nonlocal monthly mean Arctic Oscillation index. Our results also prompt that the exploitation of both co-variates is not possible from 70 years long data sets. 5

based methodology (Drótos et al., 2015). In particular, seasonal mean temperatures in Northern Europe correlate with the seasonal mean AO index. This is also indicated by the diagram in Fig. 1 (a) showing the daily minimum temperature in Kiev vs the monthly mean AO index (AO) in a scatter plot. We might say that the average temperature depends on negative but not really positive values of AO. Our novel observation is that for extremes this is the other way round: they depend also on positive AO values. Furthermore, the overall dependence of e.g. the location parameter µ(AO) of a "nonstationary GEV distribution" 70 is seemingly nonmonotonic. Therefore, as a minimal model we propose to use a model with quadratic parameter-dependences: Note that we model the shape parameter as a constant, as its inference is known to be more sensitive to data scarcity (Friederichs 75 and Thorarinsdóttir, 2012), but also theory (Holland et al., 2012;Lucarini et al., 2014Lucarini et al., , 2016Bódai, 2017) dictates it, at least under stationary climate, as we explain shortly. We will refer to this model as Model #1 or M1. As shown by Fig. 2 (b), one month looks to be a long enough period (or "block") to yield temperature minima that already conform well to the GEV distribution.
problem for estimating the potential barrier height from escape time data (Bódai, 2018). Like Matlab's gevfit, we perform the rootfinding for ln σ in order to prevent the rootfinder algorithm to select wrong negative values for σ.
We note that the model M1 is almost certainly wrong (apart from the problem of the finite block size of a month used), even if providing a good approximation. We can explain this by considering two aspects of the situation, the second of which to be identified as the one associated with the approximation. Firstly, looking at the dependence of a parameter of the GEV 85 distribution on a co-variate assumes that for any possible fixed value of the co-variate the variable of interest is really distributed according to a GEV distribution. This is actually correct, because 1) the co-variate of any predictive power is some function of the state variables of the considered dynamical system, and 2) sampling the evolution according to some fixed value of it amounts to introducing a Poincaré surface of intersection, yielding a new discrete dynamical system. Actually, already the block maxima of a time-continuous dynamical system are associated with a Poincaré section (Bódai, 2015(Bódai, , 2017. 3) A 90 recent theory (Holland et al., 2012;Lucarini et al., 2014Lucarini et al., , 2016 establishes that observables of dynamical systems feature an extreme value law -or something resembling that with a very good approximation for high-dimensional systems (Bódai, 2017). Interestingly, different Poincaré sectioning surfaces associated with different fixed values of the co-variate should yield sections of the attractor whose geometries are different, yet, their fractal dimensions (Hall and Davies, 1995) and so the shape parameter (Holland et al., 2012;Lucarini et al., 2014;Bódai, 2017) should be the same. Nevertheless, considerable 95 variation of finite-size estimates are possible, even if the pre-asymptotic non-GEV characteristics does not show up (Bódai, 2017;Gálfi et al., 2017). Secondly, unlike the shape parameter, the location and scale parameters should depend in a particular way on the co-variate, which is almost certainly not expressible by elementary functions like it is practical to assume for inference purposes. Yet, some models are better than others, such that they might not even be rejected by suitable statistical tests (Sec. 2.2).

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We rely on further assumptions. There are two sources of nonstationarity that our methodology does not (and probably cannot) take into account. (We note that the term 'nonstationary process' is common to apply to stochastic processes, while deterministic dynamical systems are said to be 'nonautonomous'. Here we adhere to the choice of classical Extreme Value Theory.) -Seasonality: the different months of the winter should have different climatologies (Bódai and Tél, 2012).

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-Climate change: just like the seasonal cycle, 20th century human activity imposes an external forcing and is known to cause climate change, i.e., probability distributions and so expectations, etc., do change (Drótos et al., 2015).
Our assumption regarding both types of nonstationarity is that while the unconditional temperature distribution is more significantly affected, the probabilities conditioned on the co-variates are more robust in the considered situation. Furthermore, we believe that even if even the conditional probabilities were somewhat affected, our conclusion on the predictive power of 110 the co-variates as we define it here is rather robust to this. Otherwise, the effect due to seasonality would be a conservative with "brackets" around it (σ(AO)) in the range of data availability.  It stands to reason that we have a better co-variate than the monthly mean AO index in the monthly mean temperature, or 120 the monthly mean daily minimum temperature, at the same location, as it is 1) the same physical quantity and 2) pertains to the same location. We can call it a "native" co-variate. We rerun the inference using the same model as (1)-(3) but with the monthly mean daily minimum temperature T as the co-variate, to be referred to as M2, and present the resounding results in Fig. 1 (b) on top of the scatter plot. While using the AO there is just a small gain possible: σ s = 4.63 > σ(t) = 4.41, using T we can, potentially, gain much more: σ s = 4.63 > σ(t) = 2.39.

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As we found -at least for Kiev -that both AO and T have predictive power, we will also examine a quadratic model, M3, featuring both as co-variates: where e.g. AO * (t) = AO(t) − AO 0 . However, with M3 over M2 we make no gain on this occasion, as σ(t) = 2.68; what is more, we have something like the opposite of a synergistic effect: the deterioration of the performance when trying to exploit co-variates that are "skilled" on their own. No doubt this is an overfitting problem, having too many parameters of the model 135 to infer from too little data. To alleviate this problem, we attempt to remove some parameters. Experimenting with different possibilities (Matlab's mle did not return confidence intervals when including the shift parameters e.g. AO 0 ), we found the best result σ(t) = 2.29 with enforcing σ 2,AO = 0, σ 2,T = 0 (M4). We should have a better idea of the systematic gains of M4 over M3 when we repeat this exercise for other gridpoints.
To do this we downloaded E-OBS gridded data from www.ecad.eu/download/ensembles/download.php, namely, daily mini- is based on daily "observational" (gridded E-OBS) data and Extreme Value Theory only; no seasonal forecast systems are evaluated. Our main conclusions -an encouraging one, on the one hand, and a negative result, on the other -are the following: -A native co-variate will perform far better: the predictive power of the local monthly mean daily minimum temperature is many times better in Europe than the nonlocal monthly mean AO index.

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-Trying to exploit both co-variates leads to deteriorating predictability as opposed to a synergistic effect, likely due to overfitting.
Even fairly unsophisticated quadratic models turn out to feature too many parameters to be inferred from a fairly short observational record. This data scarcity should be somewhat alleviated by "pooling" correlated data from neighbouring gridpoints, assuming a smoothness of the spatial dependence of the parameters of the models. So-called Max Stable processes are com-195 monly used to model spatial extremes, see e.g. (Padoan et al., 2010;Ribatet, 2017), which model can be extended to feature nonstationarity (Huser and Genton, 2016). We will explore this avenue in future work.
Another way of dealing with data scarcity is to consider more coarse properties -with respect to either the probability density or its co-variate-dependence (or both  In some gridpoints and for some model we were able to reject the GEV model. Therefore, we are prompted to check in the future if monthly temperature minima in circulation models, for which abundant data can be generated, also defy the GEV 205 distribution. In our analysis we relied on the assumption that probabilities conditioned on the co-variates considered are not affected much either by seasonal and long-term nonstationarity as a forced response, or by significant internal variability on multidecadal time scales. We also relied on theory that under a stationary climate the true shape parameter is constant. Nevertheless, a nonparametric statistical test has been recently devised to test the hypothesis of the constancy of the shape parameter (de Haan   210 and Zhou, 2017). Clearly, it is meaningless to assume some dependence and carry out inference if the hypothesis of constancy cannot be rejected. It is also known that the estimation of the shape parameter of a stationary process is rather sensitive to data shortage (Friederichs and Thorarinsdóttir, 2012).