Several studies have shown that the global food production would have to double by 2050 to meet the needs of rising population and diet shifts (Bruinsma, 2009; Tilman et al., 2011; OECD and Food and Agriculture Organization of the United Nations, 2012). However, a prior study found that the current growth rates in yield for the major cereals grown across the globe are insufficient to achieve this target (Ray et al., 2013). According to the Fifth Assessment Report (AR5) of the Intergovernmental Panel on Climate Change (IPCC), surface temperature is projected to rise over the twenty-first century under all assessed emission scenarios with a high degree of likelihood of an increase in the intensity and duration of heat waves and extreme precipitation events in many regions (IPCC, 2013b). This is expected to cause a significant decline in the global crop production (Gourdji et al., 2013; Deryng et al., 2014), thus making the world more food insecure in future. Figures 1A,B are graphics taken from the IPCC AR5 Working Group 2 report on Food Security and Food Production Systems (IPCC, 2013a) which provide a summary of results from several studies on the impact of climate change on yields for four major crops grown in different regions of the world. An overwhelming majority of these studies show a declining trend in yields over the historical period 1960–2013 (shown in Figure 1A), with several of them also projecting major declines in future across different regions of the globe, especially toward the end of the twenty-first century (shown in Figure 1B). The threat to food security from climate change is a critical issue for a number of businesses like food and beverage, retail, agriculture, insurance, biofuels, transportation and weather analytics. With the world population expected to hit 9 Billion by 2050, governments across the globe need to be well-equipped to deal with supply shocks in major cereals.

Statistical models and, more recently, tools from machine learning have been used to model crop yield variability using weather indices as inputs. Previous studies have shown the importance of growing season-averaged temperature and precipitation in explaining crop yield variability (Schlenker and Roberts, 2009; Lobell and Burke, 2010; Lobell and Field, 2011; Lobell et al., 2011b; Urban et al., 2012; Osborne and Wheeler, 2013; Moore and Lobell, 2014; Ray et al., 2015). However, extreme weather events from the recent past, like the droughts in Russia in 2010-2011 and in United States (U.S.) in 2012 and their impact on the regional crop production and global commodity markets has clearly made the case to also consider weather extremes in crop yield modeling (Otto et al., 2012). Winter Wheat, for example, has been shown to be particularly susceptible to freezing temperatures during Fall and to heat stress during grain filling and stem elongation (Tack et al., 2015). This vulnerability to extreme temperatures is believed to be the reason behind a decline in wheat yields across Europe (Brisson et al., 2010). As per a different study (Schauberger et al., 2017), each day above 30°C causes a decline in maize and soybean yields by upto 6% under rainfed conditions. Similarly, the interannual variation in rainfall also has a crucial role to play in crop growth. Although a few studies did consider extreme weather indices in their analysis, their scope was either restricted to measuring conditional relationship with yields (Troy et al., 2015) or the extreme event types considered were limited (Lobell and Burke, 2010; Lesk et al., 2016). Non-linear and threshold-type relationships have been shown to exist between yields and weather indices (Schlenker and Roberts, 2009; Lobell et al., 2011a; Troy et al., 2015). However, most of the previous studies have modeled this nonlinearity using regression models with quadratic terms for mean weather indices without appropriate justification. Understanding the exact relationship between weather outcomes and yield is essential given that a prior study reported a significant stagnation and declines in yield for major cereal crops on more than a quarter of global croplands (Ray et al., 2012).

This study addresses the following two research questions:

Using linear and nonlinear measures for pairwise dependence along with a suite of linear and nonlinear regression models, this study tries to understand the nature of the crop yield-weather relationship with the hypotheses that extreme weather indices have a statistically significant information content and that nonlinear regression models capture yield variability better than linear approaches.

In addition to mean weather indices like growing season-averaged maximum and minimum temperature and growing season-averaged precipitation, this study also considered extreme weather indices, as defined by the CCI/CLIVAR/JCOMM Expert team on Climate Change Detection and Indices (ETCCDI) (Karl et al., 1999), as predictors in the regression models. Table 1 provides the list of mean and extreme weather indices along with their definitions. The crop considered for this study was Corn (Maize), a major agricultural input to food production. The U.S. is the largest producer and exporter of this crop with 36% of the world's production (Schlenker and Roberts, 2009). The majority of the U.S. corn production takes place in the midwest region (also known as the “Corn Belt”). The county of Cerro Gordo situated in the state of Iowa in the U.S. midwest was chosen as the area of interest for this study. Yearly values for corn yield (measured in bushels/acre) were collected over a 76-years period starting from 1940 to 2015 from the NASS portal of the USDA (USDA, 2010) for this county. The time series of corn yield over this period, shown in Figure 2, has a strong positive trend due to advancements in farming technology over the years. In order to account for this trend, the year corresponding to the yield was used as one of the predictors in the regression model.

Data for three weather variables: daily maximum temperature (T_max) in °C, daily minimum temperature (T_min) in °C and daily precipitation (Precip) in mm were collected for the period of interest for three weather stations within the county from the Global Historical Climate Network (GHCN) daily database (Menne et al., 2012) using the Climate Data online portal of the National Oceanic and Atmospheric Administration (NOAA) (NOAA, 2018). The county-averaged time series of weather was created by taking an average of the daily data from the three stations, as shown in Figure 3. May 10^th and Oct 20^th were chosen as the start (sowing) and end (harvesting) dates for the growing season and were kept constant over the entire period of interest. Any fluctuations in weather occurring outside the growing period were assumed to have no impact on crop growth. The predictor and response variables were normalized prior to their use by subtracting the mean and dividing by their standard deviation.

Figure 6 shows the bias-corrected estimates for λ^ and ρ between corn yield and each of the 11 mean and extreme weather indices. The shaded areas in blue and red represent the 90% confidence bounds (5% and 95% quantiles) for the bias-corrected estimates generated using jackknife resampling. For some indices like GSP, Cold Wave index and Longest Wet Spell, the gap between the λ^ and ρ is narrow. This shows that the variation of yield with respect to these indices is mostly linear in nature. Mean weather indices like GDD, GST_max and GST_min and extreme weather indices like Summer days, Longest Dry Spell and prcp95p have a strong nonlinear relationship with yield even though the absolute value of their linear dependence is weak. It is interesting to note that the information contained in certain extreme weather indices like Summer days, Heat Wave index and Longest Wet Spell is more than that contained in mean weather indices, thus making the case for their inclusion as predictors in regression models.

The results obtained here indicate a high degree of susceptibility of crop yield to extreme weather, thereby conforming with the key insights from past research (Lobell et al., 2011b, 2013). Many of the previous studies did not include extreme weather indices in their regression models for multiple reasons. The most common being the lack of availability of daily weather data (Lobell et al., 2011b). Also, some of these studies assessed the impact of climate change on crop yield using temperature and precipitation derived from Global Circulation Models (GCMs). The outputs from the current generation of GCMs, however, are usually not thought to be credible at the spatiotemporal resolutions required to directly capture the effect of weather extremes on crop yield. Including extreme weather indices is crucial as they capture the variability of weather within the growing season which is not taken into account in mean weather indices. For example, the same average growing season temperature may arise from two very different seasons, one with little temperature variation and the other with wide fluctuations in temperature. A growing season with widely varying temperatures can result in an increased exposure to extreme conditions, which may critically impact the yields. The insights from this work also agree with those from a different study on the negative impact of temperatures on crop yield (Zhao et al., 2017), which state that with each °C increase in global mean temperature, the global maize yield would reduce by about 7.4% (without any consideration of adaptation strategies or effects of CO₂ fertilization).

Table 2 compares the performance of linear and nonlinear regression models based on metrics like R² and RMSE. For the linear models, PCR and Ridge regression were found to have R² values of 0.89 and 0.88, respectively and RMSE values of 0.32 and 0.33, respectively. Nonlinear regression methods like SVR and Random Forest were found to have slightly better performance. R² values were 0.90 and 0.93 for SVR and Random Forest, respectively with the corresponding RMSE values being 0.32 and 0.25. Overall, Random Forest regression was found to have the best R² and RMSE. This could be attributed to its robustness to data with multicollinearity and for being adept at capturing non-linear interactions. The existence of nonlinear relationships between crop yield and weather indices is not newfound and have been conformed by multiple studies in the past (Schlenker and Roberts, 2009; Lobell et al., 2011a).

Results from this study could help researchers interested in understanding the impact of environmental factors on crop production. Mechanistic crop simulation models have been traditionally used to model crop growth and yield and to understand patterns of crop yield response to climate change. However, gaps exist in our understanding of crop growth and development processes. One example being the effect of extreme temperatures on crop growth. Asseng et al. (2013) simulated climate change impacts on future global wheat yields and concluded that a greater proportion of the uncertainty was due to variations among mechanistic crop models than to variations among downscaled climate models. Insights from this study could contribute toward a better understanding of the relevant predictors in crop yield modeling and improve our existing knowledge on the precise nature of crop-weather relationship.

Future studies should focus on expanding the scope of this study in terms of the number of crops considered and the spatial extent of the study. When performing this analysis for a broader region, care should be taken to include effects, such as spatial autocorrelation of environmental variables. The presence/absence of irrigation has been shown to negate some of the effects of extreme heat stress on crop growth (Siebert et al., 2017) and hence, should also be considered. There are several limitations of this study. First, the way in which some of the weather indices are computed can have a sizeable impact on the results. A separate analysis was performed to test the sensitivity of some of the extreme weather indices to the specific value of thresholds, as shown in Figures S1, S2. With a couple of indices as test cases (Summer Days and pth percentile precipitation), it was found that the value of threshold used can have a huge impact on the value of the index. This is a problem that has also been acknowledged in previous studies. According to Tack et al. (2015), when calculating growing degree days, including information on the distribution of temperature within each day provides a statistically significant improvement in capturing yield variability. For this particular study, data on intraday variability in temperature was not available and therefore not used. Second, different crop growth stages have different sensitivities to an extreme event. Although this study did include extreme weather indices, it did not consider the specific crop growth stage affected by it. Third, this study included only temperature and precipitation-based indices. However, other environmental factors like relative humidity, ozone and CO₂ concentration have also been shown to affect yield.

Changes in the mean and extreme weather pose a major risk to governments and businesses all across the globe. With corn as a test case, the aim of this study was to come up with a systematic approach to understand the nature of the crop yield-weather relationship and determine if extreme weather indices are relevant for yield modeling. Using Mutual Information as a metric for pairwise dependence, it can be concluded that the yield-weather relationship is indeed nonlinear. The information contained in certain extreme weather indices like Summer days, Heat Wave index, Longest Dry spell and Longest Wet Spell was found to be greater than or equal to that contained in the mean weather indices, thus making a case for their inclusion as predictors in crop yield modeling. The results also suggest that Mutual Information can be a better metric for covariate selection over Pearson correlation coefficient as it gives a measure of the overall relationship (linear and nonlinear) between the predictor and response variables. Using a combination of mean and extreme weather indices as inputs, the nonlinear regression models were found to have a slightly better fit than the linear models, with the Random Forest regression giving the best fit and least error on the test set. Future studies should focus on expanding the scope of this analysis, both in terms of the spatial scale and number of crops considered. The implications of this work are important for researchers, businesses and government agencies and especially for platforms like NASA Earth Exchange which facilitate the generation and dissemination of impacts relevant weather data and indices using a multitude of satellite-derived datasets and model outputs.

The data that support the findings of this study are openly available at https://www.ncdc.noaa.gov/cdo-web/ and https://quickstats.nass.usda.gov/.

VK, TV, SG, and AG contributed to the conceptualization of this study. VK, TV, and AG contributed to the methodology. VK led the preparation of the manuscript with guidance from AG. VK, TV, and AG edited the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.