Probabilities on the vine

Galaxies, stock prices, wind directions, whatever: The model that keeps Professor Thomas Nagler busy allows ‘random dependencies’ to be calculated. “Wherever things behave randomly but are somehow interrelated, you can calculate something with what are known as copula models,” the mathematician and statistician says.

Born and bred in Munich, Nagler has been preoccupied with models of stochastic dependencies since writing his master’s thesis in mathematics at the Technical University of Munich (TUM). He did his doctorate on the same subject at the same seat of learning. In the process, he says, he “slipped into statistics”, a discipline he had encountered during a semester abroad at Leuven in Belgium. In the role of Assistant Professor, he next moved to Leiden University in the Netherlands, becoming part of a major mathematical statistics research group and winning a prestigious teaching award. After a year-long stint at Delft University of Technology, he took up his current position at LMU in April 2022.

Nagler’s research area is “a mixture of mathematics and statistics”, as he puts it, ranging from theoretical work to practical software implementation. He continues to devote considerable energy to copula models: models that can identify a functional relationship between the marginal distribution functions of different random variables and their common probability distribution.

“One simple example would be to compare, say, BMW’s stock with the Audi share. Mostly, both stocks will trend up or down on the same days, because if conditions are good for the automotive sector, they are good for both shares.” So a certain dependency exists, he explains, albeit a stochastic dependency. “You can’t say that if the one share goes up 2 percent, the other one will go up 3 percent! No, sometimes it will rise by 3, sometimes by 1, sometimes by 0.5 percentage points. In other words, a measure of randomness is involved.”

Lessons learned from the crisis

The copula models first rose to prominence – or rather, notoriety – in financial mathematics: A New York Times article published during the Lehmann Brothers crisis called them ‘The formula that killed Wall Street’. “That was a special copula model known as a Gaussian copula model,” Nagler notes. “It had been assumed that loans accumulated in banks were subject to a mild dependency.” But what market watchers failed to take into account was that this dependency is much more pronounced in crash scenarios than in booms or periods of calm. Nagler sees this as the real strength of these models: They let us simulate and analyze ‘stress test scenarios’ on the basis of data from past events. “What would we do in the middle of a crash? It turned out that we have to model these scenarios more flexibly and more precisely, and that we need to weigh up asymmetries. This is where the vine copula models came into play – an area I have researched in very great depth.”

Vine copulas – named after their graphical visualization which, at least in rudimentary cases, looks like a vine – can be used to model high-dimensional problems by observing the various dependencies in pairs. That, Nagler adds, makes things much simpler, because you do not have to analyze all ten dimensions at once, with all their interconnected overlaps.

“In academia and in industry alike, there have been lots of cooperative ventures to date,” Nagler says. One project with an automotive supplier centered around ‘automated driving’. Projects involving astronomers touched on dependencies between the properties of galaxies. Working with environmental engineers, Nagler calculated how wind directions and angles change relative to each other over time. Going forward, the mathematician envisages numerous opportunities for cooperation within LMU, too – in the context of econometry, for example.

Students of everything from astronomy to informatics

Besides investigating copula models, Nagler is also Principal Investigator at the Munich Center for Machine Learning. “I am especially interested in quantifying uncertainties in predictions,” he states. The fundamental question is this: “If you want to make predictions about time series, how do you correctly model dependencies between different points in time?”

Excited as he is about his research, Professor Nagler is also deeply committed to teaching. In his capacity as a mathematician, he is now teaching for the first time at an institute of statistics in Munich: “Adapting my own teaching to the students in front of me is always a fascinating challenge.” He already has experience with teaching a heterogeneous body of students who read subjects from astronomy to flight engineering to informatics. At LMU, Nagler concentrates primarily on delivering a basic mathematical education. “This will be very important later in their studies, even if the students themselves do not know that yet.” And that, he says, is the advantage of being both a mathematician and a statistician: “As a mathematician, I can teach with the knowledge of what will be vital in statistics later on.”