Scientific Data Has Become So Complex, We Have to Invent New Math to Deal With It

Simon DeDeo, a research fellow in applied mathematics and complex systems at the Santa Fe Institute, had a problem. He was collaborating on a new project analyzing 300 years’ worth of data from the archives of London’s Old Bailey, the…

Simon DeDeo, a research fellow in applied mathematics and complex systems at the Santa Fe Institute, had a problem. He was collaborating on a new project analyzing 300 years’ worth of data from the archives of London’s Old Bailey, the central criminal court of England and Wales. Granted, there was clean data in the usual straightforward Excel spreadsheet format, including such variables as indictment, verdict, and sentence for each case. But there were also full court transcripts, containing some 10 million words recorded during just under 200,000 trials.
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“How the hell do you analyze that data?” DeDeo wondered. It wasn’t the size of the data set that was daunting; by big data standards, the size was quite manageable. It was the sheer complexity and lack of formal structure that posed a problem. This “big data” looked nothing like the kinds of traditional data sets the former physicist would have encountered earlier in his career, when the research paradigm involved forming a hypothesis, deciding precisely what one wished to measure, then building an apparatus to make that measurement as accurately as possible.
“In physics, you typically have one kind of data and you know the system really well,” said DeDeo. “Now we have this new multimodal data [gleaned] from biological systems and human social systems, and the data is gathered before we even have a hypothesis.” The data is there in all its messy, multi-dimensional glory, waiting to be queried, but how does one know which questions to ask when the scientific method has been turned on its head?

DeDeo is not the only researcher grappling with these challenges. Across every discipline, data sets are getting bigger and more complex, whether one is dealing with medical records, genomic sequencing, neural networks in the brain, astrophysics, historical archives, or social networks. Alessandro Vespignani, a physicist at Northeastern University who specializes in harnessing the power of social networking to model disease outbreaks, stock market behavior, collective social dynamics, and election outcomes, has collected many terabytes of data from social networks such as Twitter, nearly all of it raw and unstructured. “We didn’t define the conditions of the experiments, so we don’t know what we are capturing,” he said.
Gunnar Carlsson, a mathematician at Stanford University, uses topological data analysis to find structure in complex, unstructured data sets. Image: Peter DaSilva for Quanta Magazine
Today’s big data is noisy, unstructured, and dynamic rather than static. It may also be corrupted or incomplete. “We think of data as being comprised of vectors – a string of numbers and coordinates,” said Jesse Johnson, a mathematician at Oklahoma State University. But data from Twitter or Facebook, or the trial archives of the Old Bailey, look nothing like that, which means researchers need new mathematical tools in order to glean useful information from the data sets. “Either you need a more sophisticated way to translate it into vectors, or you need to come up with a more generalized way of analyzing it,” Johnson said.
Vespignani uses a wide range of mathematical tools and techniques to make sense of his data, including text recognition. He sifts through millions of tweets looking for the most relevant words to whatever system he is trying to model. DeDeo adopted a similar approach for the Old Bailey archives project. His solution was to reduce his initial data set of 100,000 words by grouping them into 1,000 categories, using key words and their synonyms. “Now you’ve turned the trial into a point in a 1,000-dimensional space that tells you how much the trial is about friendship, or trust, or clothing,” he explained.
Scientists like DeDeo and Vespignani make good use of this piecemeal approach to big data analysis, but Yale University mathematician Ronald Coifman says that what is really needed is the big data equivalent of a Newtonian revolution, on par with the 17th century invention of calculus, which he believes is already underway. It is not sufficient, he argues, to simply collect and store massive amounts of data; they must be intelligently curated, and that requires a global framework. “We have all the pieces of the puzzle — now how do we actually assemble them so we can see the big picture?” he said. “You may have a very simplistic model at the tiny local scale, but calculus lets you take a lot of simple models and integrate them into one big picture.” Similarly, Coifman believes that modern mathematics — notably geometry — can help identify the underlying global structure of big datasets. A data set might be organized by geography or climate, for example, each of which will produce a very differently shaped map.
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