A new way to measure spatial dependencies turns less data into more data

Identifying climate-driven human migration, COVID-19 spread, agricultural trends, and socioeconomic problems in neighboring regions is data-driven—and the more complex the model, the more data is required to understand these spatially distributed phenomena. However, reliable data is often expensive and difficult to obtain, or too scarce to allow accurate predictions.

Maurizio Porfiri, professor of mechanical, aerospace, biomedical, and civil and urban engineering and a member of the Center for Urban Science and Advancement (CUSP) at NYU Tandon School of Engineering, has come up with a new solution based on network and information theory that makes “small data” work big by applying mathematical techniques typically used in time series and spatial operations.

The study, “An information-theoretic approach to studying spatial dependencies in small data sets,” which appeared on the cover of Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, describes how, from a small sample of features at a limited number of locations, observers can make robust inferences about effects, including extrapolating to intermediate or even distant regions that share similar key features.

“Data sets are often poor,” Porfiri explained. “So we took a very basic approach, applying information theory to explore whether the effect in a temporal sense can extend into space, which allows us to work with a very small data set, between 25 and 50 observations.” “We take a single snapshot of the data and draw connections—not on the basis of cause and effect, but on the interaction between individual points—to see if there is some form of underlying collective response in the system.”

The method, developed by Porfiri and collaborator Manuel Ruiz Marín of the Department of Quantitative Methods, Law and Modern Languages, Technical University of Cartagena, Spain, involved:

Combine a given dataset into a small set of acceptable tokens, similar to the way a machine learning system can recognize a face with limited pixel data: chin, cheekbones, forehead, etc.

Applying the principle of information theory to create a non-parametric test (a test that does not assume any underlying model of interaction between locations) to draw associations between events and discover whether uncertainty at a particular location is reduced if one has knowledge about uncertainty at another location.

Professor Maurizio Porfiri, working in his lab at NYU Tandon School of Engineering. Credit: NYU Tandon School of Engineering

Porfiry explained that because the nonparametric approach assumes no underlying structure for the effects between nodes, it gives flexibility in how nodes are connected, or even how the concept of a neighbor is defined.

“Because we abstract this concept of neighbor, we can define it in terms of whatever attribute you like, for example, ideology. Ideologically, California could be a neighbor to New York, even though they don’t share a geography. They might share similar values.”

The team validated the system against two case studies: population migrations in Bangladesh due to sea level rise and automobile deaths in the United States, to derive statistically based insights into the mechanisms of important social and economic problems.

“In the first case, we wanted to see if migration between sites could be predicted by geographic distance or the intensity of flooding of that particular area — whether knowing the proximity of another area or knowing the level of flooding would help predict the magnitude of migration,” Ruiz Marín said.

For the second case, they looked at the spatial distribution of alcohol-related motor vehicle crashes in 1980, 1994, and 2009, comparing states with high rates of such crashes to neighboring states and states with similar legislative ideologies about drinking and driving.

“We found a stronger relationship between countries that share borders than between countries that share legal ideologies regarding alcohol consumption and driving.”

Next, Porfiri and Ruiz-Marín plan to expand their method to analyze spatial and temporal processes, such as gun violence in the United States—a major research project recently funded by the National Science Foundation’s LEAP HI program—or epileptic seizures in the brain. Their work could help us understand when and where gun violence might occur or seizures might start.