# Mathematical Comparison of Drought Trends throughout the United States

According to the National Oceanic and Atmospheric Association, drought ranks second in terms of weather-related economic impact in the U.S., with annual losses approaching nine billion dollars. It also threatens drinking and recreational water supplies, damages local ecosystems, and raises food prices for consumers. Drought can manifest in multiple forms. Meteorological drought occurs when a prolonged lack of precipitation persists in a given area. Hydrological drought stems from meteorological drought and transpires when continued lack of rainfall negatively affects stream/river flow, soil moisture, groundwater recharge, and water levels in reservoirs and other bodies. And agricultural drought happens when available water supplies are unable to meet the hydration needs of crops. Regardless of type, this dangerous phenomenon is undoubtedly impacting the modern world; drought conditions have plagued the Midwest, west, and southeastern U.S. in the last 10 years. In 2013, California experienced its driest year on record.

As a result, researchers continue to analyze drought trends and their subsequent impact. Mathematicians typically use either Markov chains or autoregressive integrated moving average (ARIMA) time series models to study drought in specific locations, such as Maharashtra, India and China’s Laohahe River Basin. However, these existing studies do not account for multiple biomes or locations, and instead tend to focus on only one limited area. During a contributed presentation at the 2018 SIAM Conference on the Life Sciences, currently taking place in Minneapolis, Minn., David Prager of Anderson University compared the Markov chain model with the ARIMA time series approach to determine whether the methods yield similar results. He used the same data set—which included different locations and biomes throughout the U.S.—for both methodologies. “We wanted to compare drought trends over various locations, biomes, and plant communities,” Prager said.

He obtained his data from the United States Drought Monitor, controlled by the National Drought Mitigation Center at the University of Nebraska-Lincoln. The site produces weekly charts with different colors representing varied levels of drought severity. “The upside to this data set is that they categorize drought in a consistent way throughout the U.S.,” Prager said. “This makes it easy to compare one region to another.” However, the monitor has only been active since January 2000, leaving Prager and his team with only 18 years of data with which to work. “In terms of a slow-moving phenomenon like drought, is 18 years of data enough to get accurate estimates of what’s going on in these areas?” Prager asked. He determined that with the exception of a few rare extreme cases, the provided data proved to be sufficient.

David Prager uses data from the United States Drought Monitor to compare the Markov chain model with the ARIMA time series approach when measuring drought in various locations and biomes across the U.S.

Prager began by selecting three cities in South Carolina for his study: Columbia, Walhalla, and Darlington. “Columbia is in the middle of the state, Walhalla is in the mountains, and Darlington is in the Sandhills,” he said. “We chose these three locations to try to get an idea of what’s going to happen for drought conditions over a small geographical area.” He also picked an assortment of national locations representing the major biomes in the U.S.: Sacramento, Calif., Lincoln, Neb., Portland, Ore., Tuscon, Az., and Harrisburg, Penn. Prager gathered data from January 4, 2000 to June 11, 2013 for each of these cities, and set Columbia as the point of comparison. Each week had six possible drought states: $$D_N$$ (no drought) and $$D_0$$ through $$D_4$$.

Prager began with the Markov chains, and employed the standard Markovian assumption that weekly transitions are independent. “The probability of drought for the next week is the same regardless of what’s happened in the past,” he said. He then created a drought transition matrix for each location, eliminating D4 for Columbia because it was not observed during the sample time period. Prager determined that Walhalla and Darlington were both statistically different from Columbia in at least one of the six states of drought. “So even though they’re relatively close in terms of miles, these cities have drastically different drought conditions,” he said. The same was true of all national locations, an unsurprising outcome based on the diverse local results. Next, Prager calculated the mean first passage time to no drought and the mean first passage time to the worst observed drought (both in weeks, and for all locations).

He then moved on to the ARIMA time series assumptions, and examined a separate time series for each drought state and location. Each state—$$D_N$$ through $$D_4$$—has a time series of $$0$$s and $$1$$s. Prager addressed a common quandary of working with time series models: the amount of desired lag. “You want to get the best fit that you possibly can, but for each term you add to your model you lose a degree of freedom,” he said. “In every case, every state, and every location, the best fit is $$p=1$$. This tells us that if you look two weeks back versus one week back, the additional information is not enough to justify losing the degree of freedom.” Prager used the Akaike information criterion to measure the overall quality of his statistical model. Because drought is a slow-moving phenomenon, the autocorrelations are fairly high; this is entirely expected.

Ultimately, the ARIMA time series and Markov chain model produced remarkably similar results of the steady state for this data set. “The most interesting thing in this study is that the two are very similar,” Prager said. “Absolute deviation in most cases is only one or two percent.” Additional conclusions include the realization that different biomes have statistically different drought characteristics, even when locations are geographically close to each other (i.e., the cities in South Carolina). Opportunities for further study include expansion of the number of analyzed locations, an increase in the range of years, and consideration of possible environment and climate factors. “Because climate change is such a hot topic now, it would be interesting to know whether the drought trends have changed,” Prager said.

 Lina Sorg is the associate editor of SIAM News.