Reliable data on global temperature trends has been available only since the mid-19th century, and sufficient measurements from weather stations in Germany have been available only since 1881. That is why figures on extreme events such as heavy rainfall or an exceptionally hot once-in-a-century summer have been available only since that time. That is precisely the problem that Marco Oesting, assistant professor of computational statistics, and his doctoral student, Max Thannheimer, are working on. With and despite the sparse data available, they develop mathematical methods that can be used to make long-term predictions about how often certain events occur and what an event that occurs once every 10,000 years looks like.
Extreme value statistics (also known as extreme value theory) is a branch of statistics that deals with the distribution of extreme values. The goal is to model and predict the probability and magnitude of such extreme events.
This field of research advanced considerably after a severe storm surge in the Netherlands in 1953 that claimed the lives of over 1,800 people. Afterwards, the Dutch government set the goal of building dikes systematically rather than based on guesswork. “Dikes that would be flooded only once every 10,000 years were to be built,” explains Marco Oesting. It would be easy to calculate this if records from the past million years were available. “Then we could simply count what happens on average once every 10,000 years.” However, because this data does not exist, Oesting and his research team are using extreme value statistics to develop mathematical methods designed to perform such calculations.
They don’t ask themselves how likely it is that it will rain heavily tomorrow—as with weather forecasts—but rather how often an extreme event will occur over a long period of time and what this event will look like. This question is relevant for long-term protective measures such as the height of dams.
When is an event considered extreme?
But what exactly is an extreme event? For example, at what value is heavy rain considered extreme? “ We try to define it as rain that leads to flooding,” says Thannheimer. In doing so, they take different scenarios such as the amount of rain at a single point on the map or much more widespread rain along an entire river into account in their model.
However, what actually leads to flooding depends not only on the amount of rain that falls at a given time but also on other factors. “We also need meteorological data from the past to know what the soil conditions are like right now. Is the soil saturated? Will the water run off above ground, or can it seep into the ground?” says Oesting. But that would be the next step. “We first build statistical models that simulate rainfall so accurately that we can then incorporate it into these runoff models,” he adds.
“If we know how the rain falls, it’s relatively clear what will happen. Provided we know the soil well enough. Then we can theoretically calculate where every drop of water will be in three days,” says Oesting. The goal of the two scientists is to develop a flexible model and demonstrate that their method works under the specified assumptions. The model is then not limited to rain; it could also be applied to many other processes or events such as storms.
Development of new methods
The models used by the two scientists to describe extreme events are known as Brown–Resnick–Pareto processes. A Brown–Resnick–Pareto process is a mathematical model that describes how extreme values at different locations are related to each other and what extreme events might look like. So not just how high the highest water level of a river will be in a year but rather how much higher the water level can still rise once it is already high.
Conditional simulation means simulating a random system or random variables under a specific condition (i.e., in such a way that only results compatible with a given piece of information or event are generated).
This allows us to model how extreme events are distributed across an area when they occur and how severe they are. “These models are based on the Gaussian normal distribution, which is the standard solution in statistics and has been extensively researched,” explains Thannheimer. “And there’s a huge toolbox that we can access. However, because the scientists work in extreme value statistics, they must combine and apply these tools in ways that have rarely been applied before—for example, for conditional simulation. “We have a large area with lots of points where we want to simulate but very few observations. And then we do a simulation based on the many points, conditional on the few points we have,” says Thannheimer.
Verification in reality
Scientists cannot verify whether their models work in reality. But there are other ways to evaluate them. “We have data from 100 years and want to predict something that happens once every 10,000 years. I could just quote a figure now. That’s my best prediction, but of course I can’t know the number for sure. To be correct, one would also have to specify the likely range of the figure. And the further we go into the future—in other words, the longer the time periods are—the more uncertainties there are,” explains Oesting. That’s why they try to verify these models by pretending that they have only half the data. “Can I then use the model I have to accurately predict what the other half of the data would show?” So these are the kinds of tricks we typically use,” says Oesting.
“We use simulated data to test whether our method would work in theory. We do not specifically simulate rain but rather a random process. We then see how well we can capture this process with our statistical model, whether we can accurately estimate its parameters, and whether we can estimate them better with our more complex model than with a standard model,” says Thannheimer.
Another verification method used to check whether the model fits is based on statistical indicators. “For example, they indicate how strongly the rainfall in Stuttgart and the rainfall in Reutlingen are related to each other. This means examining all heavy rainfall events in Reutlingen and determining how often it rains in Stuttgart at the same time. And then we can check whether the model accurately reflects that,” says Oesting.
The forecaster’s dilemma
In this case, “correct” means that it looks similar to what the data suggests. That’s because the less data we have, the less reliable the conclusions we can draw from it. “We can only try to make the model fit the limited data we have as well as possible,” says Oesting. “The trick is to check this and present the probabilities in such a way that it becomes clear how uncertain we actually are.”
In the extreme value community, this uncertainty is also referred to as the forecaster’s dilemma. It describes a fundamental problem that arises when forecasts are evaluated on the basis of rare events. Forecasts of extreme events are particularly interesting, but statistically speaking, they hardly ever occur. That is why it is difficult to assess whether the model or the forecast was good.
“We could take the easy way out and build a model that predicts something bad always happens. When it actually happens, those who predicted it are celebrated, in a manner of speaking: ‘I’ve been saying for 10 years that there will be a terrible flood.’ But the real trick is predicting the probabilities,” explains Oesting.
Models are of interest to geologists and hydrologists
The models developed by the two scientists are also of interest to other researchers. “ “The model we have developed has the advantage of allowing us to simulate rainfall events that are extreme in the sense that they lead to flooding. This is also of interest to other scientists, especially geologists and hydrologists,” says Thannheimer, who recently saw a presentation on a similar topic at a conference in the United States.
“Scientists have also studied what extreme rainfall looks like—but more from a geological and hydrological perspective. However, they did not have the ability to simulate this particular extreme case of rain. We could help each other out,” says Thannheimer. But insurance companies and local authorities are also interested in such models for calculating risks or planning protective measures.
Accounting for climate change
The extent to which climate change influences these extreme weather events is being investigated in another project led by Oesting. ClimXtreme is a joint project involving several research institutions that are investigating various aspects such as the effect of extreme events on water supply or the extent of damage. Oesting and his doctoral student Carolin Forster are also developing statistical methods to describe these effects. “It’s essentially about adapting the models we already have a little better,” says Oesting. “Ideally, we want to model extreme events as accurately as possible under variable climatic and geographical conditions. But there are so many different aspects that need to be taken into account and so many steps that still need to be taken,” he adds.
Manuela Mild | SimTech Science Communication
Read more
Legrand, P. Naveau & M. Oesting (2025). Evaluation of binary classifiers for asymptotically dependent and independent extremes. Journal of the American Statistical Association 120(551), 1558–1568.
Forster & M. Oesting (2025). Non-stationary max-stable models with an application to heavy rainfall data. Extremes 28, 523–556.
Thannheimer (2024, joint work with M. Oesting). Bayesian inference for functional extreme events defined via partially unobserved processes. Oberwolfach Reports 21(3), 2189-2191
Engelke, R. de Fondeville & M. Oesting (2019). Extremal Behavior of Aggregated Data with an Application to Downscaling. Biometrika 106(1), 127-144.
About the scientists
Marco Oesting studied mathematics at the University of Goettingen, where he received his doctorate in extreme value modeling. After postdoctoral positions at the University of Mannheim, INRA in Paris, and the University of Twente, he worked as an academic councilor at the University of Siegen, where he also completed his habilitation. Since 2020, he has been a SimTech Junior Professor at the Institute of Stochastics and Applications at the University of Stuttgart, where he now heads the Computational Statistics working group. His research focuses on extreme value theory and statistics, spatial statistics, simulation of stochastic processes, and random fields as well as the statistical modeling of extreme events in climate and environmental sciences. At SimTech, he works on the PN 5-10 project. He also organizes the “Mathe Macht!” lecture series, where mathematics students can learn about career opportunities in companies at an early stage.
Max Thannheimer completed both his bachelor’s and master’s degrees in mathematics at the University of Stuttgart and is now pursuing his doctorate under Marco Oesting in extreme value statistics, a field he already explored in his master’s thesis. This introduced him to a field he had never encountered before and that immediately sparked his interest. He finds basic research fascinating because it allows him to revisit and apply concepts that have existed in conventional statistics for over 200 years. He finds it especially exciting that extreme value statistics is still a small and relatively unknown field; that means he often meets the leading experts in this area at conferences. He even went to a baseball game with them.
