A Twisted Path to Equation-Free Prediction

Gabriel Popkin

Complex natural systems defy standard mathematical analysis, so one ecologist is throwing out the equations.


Sometimes ecological data just don’t make sense. The sockeye salmon that spawn in British Columbia’s Fraser River offer a prime example. Scientists have tracked the fishery there since 1948, through numerous upswings and downswings. At first, population numbers seemed inversely correlated with ocean temperatures: The northern Pacific Ocean surface warms and then cools again every few decades, and in the early years of tracking, fish numbers seemed to rise when sea surface temperature fell. To biologists this seemed reasonable, since salmon thrive in cold waters. Represented as an equation, the population-temperature relationship also gave fishery managers a basis for setting catch limits so the salmon population did not crash.
But in the mid-1970s something strange happened: Ocean temperatures and fish numbers went out of sync. The tight correlation that scientists thought they had found between the two variables now seemed illusory, and the salmon population appeared to fluctuate randomly.
Trying to manage a major fishery with such a primitive understanding of its biology seems like folly to George Sugihara, an ecologist at the Scripps Institution of Oceanography in San Diego. But he and his colleagues now think they have solved the mystery of the Fraser River salmon. Their crucial insight? Throw out the equations.
Sugihara’s team has developed an approach based on chaos theory that they call “empirical dynamic modeling,” which makes no assumptions about salmon biology and uses only raw data as input. In designing it, the scientists found that sea surface temperature can in fact help predict population fluctuations, even though the two are not correlated in a simple way. Empirical dynamic modeling, Sugihara said, can reveal hidden causal relationships that lurk in the complex systems that abound in nature.
Sugihara and his colleagues are now putting their insight to use. Earlier this year they reported in the Proceedings of the National Academy of Sciences (PNAS) that their method predicted the 2014 Fraser River salmon run more precisely than any other method. Sugihara’s technique predicted a run of between 4.5 million and 9.1 million fish, while the Pacific Salmon Commission’s models predicted anywhere from 6.9 million to 20 million fish — a forecast so broad as to be of little benefit to, for instance, a fisher wanting to know how many boats to deploy in the coming season. The final count was around 8.8 million.
This success built on an earlier result Sugihara and his colleagues had achieved with Pacific sardines, and they’re working with scientists at the National Oceanographic and Atmospheric Administration (NOAA) to apply the methods to Gulf and Atlantic menhaden. Leading ecologists hope Sugihara’s methods can provide the field with some much-needed predictive power, and not just for marine fisheries but for many other ecosystems. Don DeAngelis, an ecologist with the U.S. Geological Survey in Miami, calls it “a huge theoretical breakthrough.”
Sugihara and others are now starting to apply his methods not just in ecology but in finance, neuroscience and even genetics. These fields all involve complex, constantly changing phenomena that are difficult or impossible to predict using the equation-based models that have dominated science for the past 300 years. For such systems, DeAngelis said, empirical dynamic modeling “may very well be the future.”

A New Set of Coordinates
The roots of empirical dynamic modeling go back more than 30 years. In the late 1970s, the Dutch mathematician Floris Takens was studying chaos theory, which had begun to emerge in the 1960s as scientists recognized that many of nature’s complex phenomena seem to defy prediction. In chaotic systems, small perturbations can have large and seemingly unpredictable effects, as in the archetypical example of a butterfly’s flapping wings influencing the weather thousands of miles away.
Takens helped find order in the chaos. Along with the physicist David Ruelle, he developed the notion of a “strange attractor” — a set of points in a coordinate system made of the variables that influence a system, around which the system’s state, plotted over time, swirls like a ball of yarn.
In many natural systems, however, the number of relevant variables that make up the coordinate system is immense. The factors that determine the weather in a certain place at a certain time are almost limitless, and some of these can be very hard to measure — the air pressure three miles above the North Pole, for example.
But let’s say you could consistently and accurately measure one variable, such as the temperature in New York City. Takens found a way to use present and past measurements of that one variable to capture all the information in the system. The method involves creating an alternate coordinate system from those past measurements; in other words, one coordinate axis might be the temperature in Times Square today, a second axis might be the temperature yesterday, a third the temperature two days ago, and so on. Takens showed that the full state of a chaotic system can, in theory at least, be embedded in a time series of a single variable. He published his “embedding theorem” in 1981.
The theorem “caused a big hullabaloo,” said Timothy Sauer, a mathematician at George Mason University in Fairfax, Va., who has extended the original theorem so it can be applied more generally.
The next step was for people to use it in the real world, but the messiness of nature had a way of impinging on the purity of Takens’ math. Despite the fact that weather provided much of the initial impetus for chaos theory, it rebuffed efforts at prediction, because too many constantly changing factors are involved, and no one variable can truly capture them all. Sauer said that Takens’ theorem can be most effectively applied to systems in which the number of influencing factors is relatively small.

Sugihara learned about Takens’ theorem as a Princeton graduate student working with Robert May, a physicist by training who switched to ecology in the early 1970s. May specialized in simple and elegant theoretical studies, including one proving that the population of even a single species could fluctuate chaotically. Sugihara became interested in seeing if he could build on May’s advances using real-world data. In 1986, a few years after earning his doctorate, Sugihara moved to Scripps to get his hands on plankton data that a researcher there had collected in the 1920s and ’30s. “It’s an amazing data set,” Sugihara said. “I knew there was some way to get good information out of it.”
Based on the plankton data as well as work on measles and chicken-pox cases by other researchers, Sugihara and May published a paper in Nature in 1990 showing how Takens’ theorem could help make short-term predictions of some nonlinear systems. The essence of the method involves identifying points in a system’s attractor graph that are close to the point representing the system’s present state. For one or two time steps, one can then predict that the system will evolve similarly to how it did in the past. The paper has since been cited more than 1,000 times by scientists all over the disciplinary map. The paper also prompted Sugihara to make a mid-career foray into finance, as firms were very interested in forecasting stock prices using methods similar to those he had applied in ecology.
In 2002, Sugihara returned to science. He had unfinished business: convincing the world that ecosystems, though complex and chaotic, could be predicted, and that managers could use those predictions to do their jobs better. “I feel like I have a mission,” he said, “to get people to understand how this all works — to begin to embrace natural systems as they are as opposed to as we hope they would be.”