Decadal prediction is a relatively new field in climate research. Skillful prediction of climate from years up to a decade would be beneficial for our society, economy and for a better adaption to a changing climate. Within the large international Coupled Model Intercomparison Project Phase 5 (CMIP5) global decadal predictions of climate key variables like temperature and precipitation have been performed with state-of-the-art Earth system models. In order to validate the prediction skill of the models so-called hindcast experiments are conducted. That means the models are initialized with observations (e.g., in 1961) and then run freely for 10 years and stop at the end of 1970. In 1971, the models are again initialized and start to run for another 10 years and so on. More advanced approaches of initializing every year have followed as well. These hindcasts can be evaluated against observational data to quantify the prediction skill of the models depending on the lead time, which is the time range between the initialization and the forecast datum of interest. In recent years, several studies on decadal predictions have shown the potential of these initialized (global) model runs (e.g., ). However most studies concentrate on regions like the tropical Pacific or North Atlantic and on slowly evolving variables like sea-surface temperature. These regions receive their predictability from large-scale processes like the Atlantic Meridional Overturning Circulation (AMOC) or Pacific Decadal Oscillation (PDO) and thus allow the extraction of predictable signals out of the noise. To be useful for society and climate change adaption, regional climate predictions are required, which should provide skillful forecasts on smaller regions and shorter periods, and they include climate extreme events on populated land areas like the European continent. The European climate is more connected to short-term processes like the North Atlantic Oscillation (NAO), which is to a certain extent predictable on seasonal scales, whereas the decadal predictable signal is weak , which has been shown also for temperature and precipitation in large projects like ENSEMBLES . Further, the complex orography with the Alps in the center contributes to a manifold of general weather situations and hence to a complex climate e.g., World Climate Research Programme Coordinated Regional Downscaling Experiment for Europe (CORDEX-EU),. Nevertheless, the European continent is influenced by the AMOC and thus this process may yield to a certain predictability, although the signal-to-noise ratio is most probably small. Up to now, the prediction skill for Europe is weaker than for regions such as the South Pacific or North Atlantic. have used a regional decadal hindcast ensemble for Europe and detected moderate prediction skill for summer and winter temperature and summer precipitation anomalies within the lead time of 5 years. analyzed the predictability of temperature and precipitation extremes in a global model and found a moderate but significant skill (correlation) for seasonal extremes. They also found skill beyond the first year, but this skill arose from external forcing. Thus, compared initialized climate predictions with uninitialized projections to evaluate the skill gained by initializing and excluding the external forcing. They found that the “impact of initialization is disappointing”.

Another relatively new field in climate research has been established, namely the complex climate network approach. The general idea of climate networks is to consider climate time series, for example, at the grid points of a climate model as nodes of the network and the statistical connection between the time series as links of the network. A link between two arbitrary time series (geolocations) exists if the correlation measure between the time series exceeds a certain threshold.

The climate network community has been very active in recent years. proposed “A new dynamical mechanism for major climate shifts” and explained, e.g., decadal shifts in global mean temperature . discriminated different El Niño types using the network approach, developed a network method to improve El Niño forecasting, and revealed a connection between (paleo-)climate variability and human evolution using recurrence networks, which are similar to the complex climate networks. Generally, it has been shown that climate networks contain useful information for climate applications, e.g., the relation between climate and topography found by , dynamics of the sun activity using visibility graphs , and the prediction of extreme floods .

In this paper, we exploit the idea to use an alternative heat period estimator, based on complex climate networks, and show that its skill is superior to the typical approach of using a fixed temperature threshold for prediction of heat periods on timescales up to a decade.

In Sect.  we introduce the daily maximum temperature data used in this study and the motivation for our approach in Sect. . Section  describes our approach, which includes the preparation of the data, the definition of heat periods, and the construction of time-evolving climate networks. The results for applying the new approach to hindcasts are shown in Sect. . Finally, we give the conclusions and an outlook in Sect. .

We apply the climate network approach to a decadal prediction ensemble generated within the German research project MiKlip (Mittelfristige Klimaprognosen, Decadal Climate Prediction; e.g., ) by the regional COSMO model in CLimate Mode (COSMO-CLM or CCLM) . CCLM has been used in numerous studies recently (e.g., in ). A comprehensive overview can be found here: http://www.clm-community.eu. CCLM has been used to downscale global decadal predictions from the Earth System Model of the Max Planck Institute for Meteorology MPI-ESM,. From a suite of different decadal prediction experiments we have selected the so-called regional baseline 0 ensemble. This ensemble consists of 10 members, each covering the period 1961–2010 for the European region according to CORDEX-EU on a 0.22∘ grid. This ensemble has already been used by .

The regional baseline 0 ensemble (based on the global MPI-ESM model) has been initialized every 10 years (1961, 1971, 1981, 1991, 2001). Within a decade the CCLM model runs freely, except for the prescription of the atmospheric boundary conditions by the global MPI-ESM model.

More details on the development of the ensemble and the initialization can be found in , , and .

In the study presented here we use daily maximum near-surface temperatures from the CCLM model and from the E-OBS v8.0 gridded climatology for the European continent. The E-OBS data basically are measurements interpolated to a regular latitude–longitude grid.

For our comparison, we use the so-called Prudence regions http://prudence.dmi.dk/, namely British Isles, Iberian Peninsula, France, Central Europe, Scandinavia, Alps, the Mediterranean, and Eastern Europe (shown in Fig. ).

Generally, heat periods are maximum temperature values persisting for several days and occurring on spatially expanded regions. This means that many temperature time series (grid points) behave in a “cooperative mode” (see, e.g., ). This cooperative state can be described by the link strength, i.e., essentially the correlation between time series, of a climate network. Thus, the link strength of a climate network could turn out to be the better heat period estimator for model data, because it is independent of the typically critical thresholds used in classical extreme value detection.

The standard estimator for heat periods according to the World Meteorological Organization (WMO) is that the daily maximum temperature is 5K above the 1961–1990 mean maximum temperature at five consecutive days at least . Thus, the standard method to compare the prediction skill of heat periods between observations and model would be to count the heat periods, e.g., for each year in an observational reference data set and similarly in the model data, both according to the WMO definition (cf. Fig. ).

A crucial problem of the standard estimator for model predictions is the inherent static threshold used to detect heat periods. Although this threshold can be adapted to the model climatology (as we do it in Sect. ) the problem is that it is still likely that the model slightly undershoots or otherwise slightly misses the threshold if a heat period according to the definition at the very beginning of this section occurs, assuming the model exhibits at least some predictive skill.

To account for this situation in decadal predictions we propose a new method, based on complex climate networks, to detect heat periods, which is independent of a fixed temperature threshold (see Sect. ). Again we want to emphasize that no new method for the detection of heat periods is needed, if past observational data or short-term forecasts are used. The WMO-based definition works well. However, the increased uncertainty in decadal predictions requires new methods to handle climate extremes like heat periods.

The following schematic examples in Figs. a–c and illustrate why the complex network approach is able to detect heat periods without using a temperature threshold. The black curves represent (artificially generated) daily maximum temperature model data. Further we assume that one heat period has actually occurred in Fig. a–c persisting for 15days from day 11 to day 25. Accordingly the black curves show different possible model results if the model exhibits predictive skill to detect a signal out of the noise.

Figure a depicts that using the standard approach the model correctly detects one heat period above the threshold. In Fig. b the model detects a signal, but this signal is too weak to cross the threshold; thus no heat period would have been detected and the model underestimates the number of heat periods. Overestimation of the number of heat periods happens in Fig. c, where the model detects two heat periods (5 days above the threshold at the edges and below the threshold in between). Now, the key point for our motivation is that a heat period constitutes an event in space and time; thus in a certain region, many time series would look like the ones in Fig. . The link strength of a network would be given by the correlation between these coherent time series. Since the signals in Fig. a–c look quite similar, the link strength of the network would thus be very similar in all three cases. Whereas the standard approach would correctly estimate the heat period in only one case (Fig. a), the networks' link strength would correctly estimate it in all three cases, given a proper relation between link strength and heat periods.

To test the relation in principle, we created 100 artificial time series (Gaussian noise) and included successively 0–9 heat periods. Figure a shows such a time series with three artificial heat periods indicated by the dashed lines. In a following step, we calculated the mean correlation (link strength) between these 100 coherent time series dependent on the number of included heat periods depicted in Fig. b. As can be seen, more heat periods are connected with a larger link strength. This simplified test shows that a proper relation between link strength and heat periods could exist. Note that Fig. b is not a calibration curve for real data, because we simply used Gaussian noise to create the time series.

It is clear that the argumentation above concerning the link strength as a heat period estimator is quite simplistic, but it elucidates our approach and the main idea.

Our hypothesis is that complex network measures may be better estimators for climate extremes than standard measures like absolute threshold exceedances.

Figure  depicts the number of observed heat periods (solid line) and the corresponding link strength (dashed line) retrieved from the complex evolving network, both from E-OBS data for France (Prudence region 3), and shows that the link strength Wτ is a suitable estimator of heat periods. It shows that the network contains climate information in the sense that the dynamics of the link strength Wτ is similar to the dynamics of heat periods, both based on the same data. So, the link strength can here be considered as an estimator for heat periods, which is comparable to the standard heat period estimator. Jumps between the decades occur as the running mean filter is only applied within the decades (see Sect. ). The corresponding figures for the seven other Prudence regions can be found in the supplementary material.

As an example, Prudence region 8 (Eastern Europe) is a region where the network method performs better than the standard approach (Figs.  and ). Figure  shows the E-OBS number of heat periods o (black) and the CCLM ensemble mean number of heat periods m (blue) for Eastern Europe together with the interquartile range (25th and 75th percentiles), and Fig.  shows again the E-OBS number of heat periods now compared to the CCLM link strength. Comparing the absolute mean differences, denoted as M in the two figures, reveals that our network approach enhances the skill in four decades, namely 1970s, 1980s, 1990s, and 2000s. Especially the 1970s, 1980s, and 1990s show a clear improvement and our network approach better reflects the low-frequency dynamics of the heat periods. The 2000s seem to be off in both model cases, the number of heat periods, and the link strength, which indicates a failed model initialization.

In order to see how the prediction skill of the standard as well as the network heat period estimators vary with the considered region, we performed the same analysis as above for the eight Prudence regions in Europe and for the 1960s, 1970s, 1980s, 1990s, and 2000s. The corresponding figures for the other regions can be found in the Supplement.

To summarize the results we calculated the absolute mean differences (Eqs. 6 and 7) for all the Prudence regions, see Fig. . Blue colors in the panels stand for low values (high skill) whereas red colors depict high values (low skill) in the absolute mean difference.

The left panel of Fig.  shows how the network method performs using only E-OBS data similar to Fig. . Therefore we estimated the number of heat periods in the E-OBS data and the link strength (from the complex network) of the E-OBS data and accordingly calculated the differences (after normalization, cf. Eq. ) between these two estimators. As can be seen blue colors dominate the plot, i.e., low differences and hence high skill. Thus, this reference test shows that the link strength is coupled to the number of heat periods in maximum daily summer temperature data and so can be used as an alternative, possibly better, heat period estimator. There are some exceptions like the 1990s and 2000s of Prudence region 6. Further investigation on the reasons of these cases has to be performed.

The middle panel of Fig.  shows how well the standard method performs in predicting heat periods using E-OBS observations and CCLM model data. The right panel indicates the performance of the new network method in estimating heat periods using E-OBS and CCLM data. In contrast to the relatively low values in the left panel, the values in the middle and right panels on the one hand are higher for many decades and Prudence regions. On the other hand, the visual impression is that the absolute differences of the right panel are slightly smaller than those of the middle panel, especially during the 1990s and 2000s.

Thus, we can conclude with Fig.  that the method works in principle but that the uncertainties in the model simulations lead to increased differences between observations and model simulations. In addition, the link strength seems to work better than the standard approach for the model simulations with respect to the observations.

To quantify this last statement, we calculated the difference between the middle and right panels of Fig.  (see Fig. ). This basically shows which method performs better regarding the eight regions (columns) and five decades (rows). Blue color in Fig.  indicates that the network approach performs better (Mdr(ψ) < Mdr(μ)) and red color stands for a better performance of the standard approach (Mdr(ψ) > Mdr(μ)). White boxes in Fig.  denote a tie between the methods in the case of too small differences (|Mdr(ψ)-Mdr(μ)| ≤ 0.05). The matrix of Fig.  shows that the network method is clearly superior in three regions (5, 7, 8) and slightly superior in two regions (4, 6), the standard approach is superior in two regions (1, 3), and in region 2 we observed a tie, i.e., no clear result.

The crucial question is whether this result indicates that the network method performs significantly better than the standard approach or not. However, testing for statistical significance bears serious problems. There are so many factors involved in the analysis, i.e., the models themselves, the downscaling, the ensemble, the initialization, the different regions, the filtering, etc. that any null hypothesis would be not well-posed and any test would be questionable. This issue is discussed in detail in a 2013 paper entitled “Testing ensembles of climate change scenarios for `statistical significance”' by climate statistics instances of Hans von Storch and Francis Zwiers , who claim that “a statistical null hypothesis may not be a well-posed problem…” and “Even if statistical testing were completely appropriate, the dependency of the power of statistical tests on the sample size n remains a limitation on interpretation.” and finally “propose to employ instead a simple descriptive approach for characterising the information in an ensemble…”. Although we totally agree with the argumentation by that a “classical” significance test would most probably fail in our analysis, we think that alternative significance tests, based on bootstrapping or surrogate data, could definitely help obtain a better interpretation of the results. Thus, we construct the following significance test based on surrogate data to answer the following question: what is the probability of getting a rank matrix like the one in Fig.  by chance?

First, we have to define what is the possibly “significant” characteristic of the matrix in Fig. . It is, as we concluded above, that the network method is superior in five regions. Thus the question is the following: what is the probability to observe at least five regions in which we have in each at least one blue matrix element more than a red one by chance? Accordingly we constructed matrices like in Fig.  by randomly coloring 20 matrix elements blue and 20 red. Afterwards we colored 8 matrix elements white as in Fig. . Finally, we repeated this surrogate procedure 1000 times and counted the cases (regions) where the blue matrix elements dominate. Table shows the probabilities that blue matrix elements dominate in n regions. Since we have 16 blue elements and 16 red, it is sure that blue dominates in n=1 region, and it is impossible to dominate in n=7 and n=8 regions. As can be seen from Table  the probability of dominating in n=5 regions by chance is only about 5 %; thus the results of our network approach have to be stated significant. Due to the symmetry of the test, the same argumentation is valid for red matrix elements. Dominating in n=2 regions, as achieved by the standard approach (Fig. ), can be realized easily by chance with a probability of approx. 99 %. Page 1 in the Supplement shows an example of 12 of these randomly generated matrices, where one matrix, depicted by a black frame, fulfills the “significance” criterion.

We presented a novel approach examining heat periods using a complex network analysis. We have investigated the predictability of the slow dynamics of the occurrence of heat periods in Europe based on daily maximum near-surface temperature data.

We found that the network approach is superior (significance is ≈ 5 %) to the standard approach in estimating heat periods in Europe, hence highlighting the potential of network methods to improve the skill estimation in decadal prediction experiments. Picking up our hypothesis and simplified argumentation from Sect. , the crucial point why we detect heat periods with the network link strength is that heat periods are cooperative events in space and time. Thus, the link strength can be used as an estimator of heat periods. The drawback of the standard approach is most probably the inflexible threshold for the detection of heat periods (cf. Fig. ). If the climate model contains the signal of a heat period, but with a slightly too small amplitude, the threshold will not be crossed and no heat period will be detected. In contrast, the complex climate network does not depend on such fixed thresholds and can use this information, which makes it the more robust estimator of heat periods.

The general prediction skill of climate in Europe using standard measures is still moderate. In this sense our work adds new aspects to our previous study and also the work of , who found a strong variation of skill with region and decade. In essence, we found regions and decades in Europe where our climate model output, or more specifically the used network estimator, follows the slowly evolving dynamics of observed heat periods. We also found regions and decades in which the network estimator is not able to represent the observational reference. Understanding of this variability in prediction skill is one of the future challenges of decadal predictions.

Concluding, our approach shows that the complex climate networks approach yields meaningful climate information and has the potential to improve skill measures within the framework of climate prediction. It is the first time that such network techniques have been used in climate predictions. Since climate or decadal predictions aim to predict natural variability on the order of years, suitable statistics are needed. Natural variability on the order of years evolves highly dynamically and often nonlinearly. Thus, the complex climate networks could bear the potential to be very useful in climate predictions. Our approach, which is even based on the most simple network measure, the node degree (or as we used it the link strength), yields optimistic results. So, we think that our analysis could be the starting point for using the complex networks in climate predictions. Using other measures and/or multivariate data could turn out to be the better way of analyzing predictions of natural variability years ahead than using methods from short- or medium-range forecasting. Further, from the network perspective it would be interesting to analyze other network measures like clustering, similarities, or path lengths and how they are connected to climate evolution. The incorporation of other relevant variables like precipitation, wind, or soil moisture into the network is an appealing aspect. From a physical or climatological point of view it is important to understand why the network measures are able to represent climate dynamics, which could also contribute to a better understanding of the sources of decadal predictability. Thus, the incorporation and investigation of processes like the AMOC, PDO, or NAO together with complex networks and climate prediction might be an option for the future.

E-OBS can be downloaded via the website www.ecad.eu.