Time flies when you're having fun. But what if you could capture that flight of time and analyze it for patterns and trends? That's where time series data comes in. It's like a time capsule, capturing changes and patterns in various phenomena over time. Think of taking hourly temperature measurements for a week - the resulting time series reveals much more than a single snapshot could ever show. 

By collecting repeated measurements over time, researchers study trends, patterns and relationships in various systems - in financial markets, brain signals, or disease epidemics - that are not visible from a single point in time. But here's the catch - traditional statistical tools for analyzing time series mainly rely on pairwise interactions between variables. Real-world events, however, often depend on more than just two variables. As a result, such approaches may not always give us the full picture. 

But what happens when more than two variables come into play? Just like how a group of friends in a pub discusses their vacation plan together, each contributing to create a more detailed plan, multiple variables in climate science, like temperature, humidity, and atmospheric pressure, interact with each other to produce complex weather patterns. Similarly, in the financial market, the prices of different stocks are influenced not only by the overall market trend, but also by the interactions between the stocks themselves. The interplay between groups of more than three variables is usually referred to as higher-order or group interactions. 

 

Based on these intuitions, we have developed a new method to analyze real-world time series data that goes beyond traditional pairwise statistics. Our approach looks at how multiple variables are influencing each other over time, instead of just focusing on pairs. For example, we can detect patterns of synchronization in the stock market, as those observed when multiple financial stocks unexpectedly drop together, revealing hidden trends and patterns that traditional methods can miss. 

 

We first validated our method on computer-generated time series data and found that even when analyzing groups of just three variables, the higher-order approach outperformed traditional pairwise statistics. Our method was able to identify different dynamic phases in simulated data that were only distinguishable through higher-order (group) statistics. 

We also applied our approach to three complex real-world datasets, including brain activity, stock prices, and disease spreading in the 20th-century epidemics in the United States. In brain activity, we found different patterns of neural interactions oscillating between chaos and synchronization. In the stock market, we identified times of crisis and financial stability, and gained insights into the roles of different industrial sectors. In epidemiological data, one might expect that epidemics spread independently over time and space, but with our tool we were able to detect how different diseases interacted with each other over time, such as measles or flu.  

 

Overall, our higher-order approach provides a more complete understanding of the relationships between variables in time series data, helping researchers uncover patterns and trends that may have gone unnoticed. With its ability to detect synchronized patterns and higher-order (group) dependencies among variables, our new method has broad applications across various fields, including finance and healthcare.