We use sophisticated machine-learning techniques on a network of summer temperature and precipitation time series taken from stations throughout Germany for the years from 1960 to 2018. In particular, we consider (normalized) maximized mutual information as the measure of similarity and expand on recent clustering methods for climate modeling by applying a weighted kernel-based

Extremes in temperature have lasting affects on human health (

Modeling extreme events typically involves fitting a generalized extreme value (GEV) probability distribution to these data under the assumption that the parameters of this distribution do not change in time or are “stationary” (

Along with nonstationary modeling, there is the need to create regional models of extremes as policy making and governmental response are most influenced when extreme weather events are predicted on a larger scale. With the introduction of machine-learning techniques such as maximized modularity algorithms (

Recently, spectral clustering methods have been shown to provide reasonable clustering results for climate networks defined by a set of time series (

This paper is outlined as follows. Section 2 is broken into three parts: Sect.2.1 contains details on the chosen subset of data; Sect. 2.2 provides relevant background on clustering methods and a detailed description of the clustering algorithm; and Sect. 2.3 provides relevant background on extreme value modeling and details on the methods used for regional modeling of summer temperature extremes. Section 3 is broken into two parts: Sect. 3.1 contains results on the clustering algorithm applied to summer temperature and precipitation time series taken from the stations across Germany; and Sect. 3.2 outlines the extreme value results on the station level and their incorporation into the regional models formed from the clustering outcome. Section 4 is a brief summary of the paper containing future work motivated by the results in this paper.

The data in this analysis were taken from the Deutscher Wetterdienst Climate Data Center (

DWD data sets used in clustering.

Geographic locations of 68 stations used in regional clustering and modeling of temperature extremes.

The time interval from 1960 to 2018 provides a general representation of all climatology in Germany and a reasonable quantity of extreme values for modeling. For information on station locations used in this analysis, see Fig.

Quality control was performed by the DWD CDC with details provided in

This section discusses some machine-learning tools used to perform regional clustering by viewing the set of time series, corresponding to our subset of weather stations, as a network. The set of time series can be viewed as a weighted network (in graph form: a set of nodes and edges) where each station is a node. The weighted edge between two nodes is calculated by a similarity measure that relies on the corresponding time series (Fig.

Graphical form of the network defined by temperature time series emphasizing the density of the network. Each node represents a station in the network. Each edge represents a positive similarity between two stations. Note that this graph is not complete because low values of similarity

Given a network, the goal of clustering analysis is to find a partition with high intra-cluster similarity and low inter-cluster similarity; however, solving this system is an np-hard (non-polynomial) problem. Relaxations of this problem can be split into two approaches: (1) maximizing modularity through partitioning algorithms and redefined modularity metrics (

The authors in

It is possible that the stable minimum reached is not the global minimum. We run the algorithm for 1000 different initial sets of partitions and choose the cluster associated with the lowest minimized distance. In addition, the number of partitions (clusters) in kernel

Mutual information has become a more common measurement of similarity in recent literature because it is capable of measuring nonlinear relationships between two time series. For two random variables (or time series)

Compressing the time series can result in an obvious loss of information from the system. Following the work of

We use a normalized version of the maximized mutual information (NMMI),

Mutual information may return a large positive value for negatively correlated variables. In this analysis, we cross reference our clustering outcomes from mutual information and correlation. Summer hourly temperature time series (with the daily cycle removed) for all stations in the network are positively correlated. This is expected because Germany is smaller than the average warm air mass. If we attempt to cluster the network by correlation, we find the minimized Euclidean distance plot loses its structure and no longer provides an obvious value for the number of clusters

We remark that mutual information provides a reasonable estimate of similarity in the context of clustering extremes because it compares the probability distributions of the time series to determine their shared information. Recent literature has also introduced measures of similarity defined by the extremal index (EI;

The block maximum method is used in this analysis to model the maximum values of the time series

When modeling real-world data, it is natural to use maximum likelihood estimation (MLE;

Strict independence is not always necessary for convergence of the maxima to the GEV (

Clustering result for the hourly temperature network with similarity defined by NMMI over the time interval from 1960 to 2018.

Stationary assumptions require the location and scale parameters of the maxima to not change in time. We test this assumption using the Mann–Kendall trend test and a mean difference comparison of the time intervals from 1960 to 1990 and from 1991 to 2018 for the temperature maximum at each station where we find evidence of a linear trended location parameter.

Clustering result for the

Clustering result for the daily precipitation amounts network taken from the daily DWD data set over

Example time plots for the 2-year maximum likelihood estimate of the location parameter with standard error bars for weekly maximum temperature values taken from

Finally, we perform a likelihood-ratio test on the stationary and nonstationary model equipped with a linear time-dependent location parameter

The likelihood-ratio test compares the final log-likelihood values

Regional models are created by formulating a mixed GEV distribution from all of the stations in a regional cluster. The mixed GEV distribution is given by

As discussed in Sect.

To test for time invariance of the clusters, we apply the algorithm to the intervals from 1960 to 2000, from 1970 to 2010, and from 1980 to 2018. Resulting clusters are equivalent for all three time intervals.

For further investigation, we consider clusters obtained by maximizing the mutual information between pairs of stations represented by their corresponding daily precipitation amounts. Daily precipitation amounts for this analysis are taken from the same hourly recording data set as the temperature data. Precipitation data in this set are only available for the time period from 1995 to 2018. All available data over the period from 1995 to 2018 are used to show the robustness of the regional clusters; however, these data are not used in the final regional cluster analysis of temperature extremes. Hourly precipitation values are summed over each 24 h period and the resulting time series is used to compute the MMI matrix. Clustering results suggest the network of daily precipitation amounts is regionally equivalent to that of hourly temperature (Fig.

For added measure, we consider the daily precipitation amounts taken from the daily DWD CDC set which have values over the same time period (1960–2018) as our temperature data set.

Results from our clustering algorithm show preference for four (

Scatter plot illustrating the difference in mean summer temperature taken over the period from 1960 to 1990 (

For the remaining analysis, we define an extreme (or maximum) value of the temperature time series as the maximum taken over data blocks of 7 d (168 data points). We find independence over almost all blocks at the

Confidence intervals for transformed regional models.

Return level plots comparing the parametric mixed GEV distribution with confidence intervals and actual mixed temperature maxima for the

Recall from Sect.

Nonstationary generalized extreme value model overlay of 1960 and 2018 weekly temperature extremes for the

Results from these tests provide justification for considering nonstationary modeling of the weekly summer temperature extremes. We use maximum likelihood estimation of the log-likelihood function equipped with a linear trended location parameter to generate the nonstationary GEV model for every station in the data set. We compare these results with the stationary model using the likelihood ratio test, and we find that the nonstationary model provides a significant (

We form regional GEV models using the mixed GEV distribution described by Eq. (14). Stations located at high altitude, including Zugspitze and (the mountains near) Görlitz, were removed from the regional model. These stations have location parameter values that are further than 2 standard deviations away from the regional location parameter. This deviation causes issues with the modality and accuracy of regional temperature extreme probability estimation. When creating regional models of extremes, care needs to be taken regarding the homogeneity of the region in terms of the shape parameter. This is especially important for regional models of extremes where local phenomena are observed. We remark that the mixed GEV distribution has been used to describe regional dynamics with unknown local phenomena through likelihood estimation (

To determine how well the regional models represent the pooled time series, we first transform the time series of maxima for each station into a stationary Gumbel distribution using Eq. (13) and perform the Anderson–Darling goodness of fit test on the transformed maxima under the null hypothesis that these data follow a stationary Gumbel distribution with

We use the MLE parameters

Regional tail probability estimates (in units of occurrences per year) for weekly temperature extremes above

In this analysis, we expand on the work of

We use the clusters we obtain from our algorithm to create regional models of the weekly summer temperature extremes. We find significant increasing linear trends in the location parameter of these models and a preference for nonstationary modeling for all stations in the network. Regional nonstationary models are created by defining a mixed nonstationary GEV distribution from likelihood results. Return time plots are used to validate the results. Regional distribution models reveal an increase in the probability of observing a weekly summer temperature maximum above

In future work, we will consider more complicated networks defined by both temperature and precipitation values, simultaneously. We plan to use these results to form joint probability densities of temperature and precipitation extremes. Viewing the data in this way will allow us to make conclusions about the conditional extreme probabilities. We are also interested in determining whether these clusters are seasonally dependent or consistent throughout the year.

The similarity matrix

Kernel

DWD station IDs and locations.

Continued.

Location trend results for the hourly temperature time series.

MLE nonstationary models for temperature maxima.
scale

The hourly temperature and precipitation data sets and the daily precipitation data set are freely available from the DWD CDC website:

The majority of the work for this study was performed by MC with heavy guidance from HK. Both authors worked closely together on this analysis and held regular discussions on all aspects of this paper.

The authors declare that they have no conflict of interest.

The authors thank Kevin Bassler for helpful discussions and the Deutscher Wetterdienst Climate Data Center for the hourly and daily data sets used in this analysis. The authors are also grateful to Jakob Zscheischler and the anonymous reviewer for their helpful comments and suggestions.

This paper was edited by William Hsieh and reviewed by Jakob Zscheischler and one anonymous referee.