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# Assessment of the agro-climatic indices to improve crop yield forecasting

Jordane Mathieu, Filipe Aires

► To cite this version:

Jordane Mathieu, Filipe Aires. Assessment of the agro-climatic indices to improve crop yield forecasting. Agricultural and Forest Meteorology, 2018, 253-254, pp.15-30. 10.1016/j.agrformet.2018.01.031. hal-03880923

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https://hal.science/hal-03880923v1

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# Assessment of the agro-climatic indices to improve crop yield forecasting

Jordane A. Mathieu<sup>a,*</sup>, Filipe Aires<sup>a,b</sup>

<sup>a</sup> Sorbonne Université, Observatory of Paris, PSL Research University, CNRS, LERMA,
75014 Paris, France

<sup>b</sup>Estellus, Paris, France

## Abstract

Weather has a major impact on agriculture. Statistical models have been used to estimate or forecast crop yield from weather information. In this paper, a general statistical framework is developed in order to rank and quantify the information content of weather information. The methodology is tested over the US, for corn yield. The weather sensitivity of different corn production areas is first analyzed. More than fifty agro-climatic indices have been compared. The study shows that variability in yield is more likely to be due to weather variability in medium and low production areas. The two best yield predictors are the temperature and then the Standardized Precipitation - Evapotranspiration Index (SPEI) in July. While corn in the north Eastern regions is not significantly affected by weather variation, on the East coast, the weather-based mixed model is able to explain about 32% (corr=0.57) of the yield variability, but in some particularly weather-sensitive states, such as Virginia, this number can reach 64% (corr=0.80). Two applications are tested: yield estimation at the end of the year for monitoring purpose, and seasonal prediction. It is shown that some agro-climatic indices can significantly improve crop yield modeling when compared to simpler direct weather information (average correlation increase of 0.12 at the US scale).

**Keywords:** weather impact model, statistical model, corn yield, agro-climate indices, agriculture

<footnote>*Corresponding author</footnote>

<footnote>Email address: jordane.mathieu@obspm.fr (Jordane A. Mathieu)</footnote>
## 1. Introduction

How historical, current and future climate affects socio-economic activities has received tremendous attention in areas such as agriculture (Thompson, 1988; Kotlowski, 2007; Kandiannan et al., 2002), energy (Adams et al., 1998; Kaylen and Koroma, 1991), logistics in sales, or tourism (Jewson and Brix, 2005). In agriculture, crop yield is strongly influenced by several factors (e.g., genetics, soil properties, irrigation) but weather is the major uncontrollable factor influencing the development of crops (Taylor and Carlson, 1997). The accurate prediction of crop yields over large areas is critical for the national food supply, prices, farmer plans, or irrigation (Liang, 2004), so quantifying the impact of weather on crop yield is essential.

Many studies have shown the importance of weather information to explain crop yield. Three major modeling approaches have been used to study this relationship: biophysical or crop simulation models (Hoogenboom, 2000; Jones et al., 2003; Kotlowski, 2007); empirical regression models (Kandiannan et al., 2002; Thompson, 1988; Tannura et al., 2008; Lobell et al., 2007), and functional models (Basso et al., 2013; Chipanshi et al., 2015) which are a simplification and/or a combination of the two. In this paper, we focus on empirical regression models: they are less process-oriented than crop models, but they are more data-driven and they require less auxiliary information such as soil properties. They are calibrated on historical data with as little a priori information as possible.

Some agriculture characteristics are sometimes used (e.g., soil properties, grain size) (NASS, May 2016; Asseng et al., 2013) as inputs to the crop yield statistical models. In order to provide the forecasting model information more directly related to crop yield, some agro-climatic indices (i.e. weather-based indices) have been considered as inputs. Agro-climatic indices, especially growing degree-days or some type of water stress indices (e.g. Precipitation-Evaporation), have long been used for crop yield predictions (Lass et al., 1993; Holzkamper et al., 2011). The use of direct weather data became common only in recent
decades, when gridded weather data became a major input for regional yield forecasting (Zhang and Huang, 2011; Moreto and Souza, 2015). Traditionally, agro-climatic indices are obtained from direct weather data in order to better represent the link between weather and crop growth and to facilitate decision making in agriculture (Lepage et al., 2012; Caubel et al., 2015). For instance, Lass et al. (1993), Robertson et al. (2013) or Bornn and Zidek (2012) use Growing Degree Days (GDD); Butler and Huybers (2013) use Growing Degree Days (GDD) and Killing Degree Days; Moreto and Souza (2015), Zhang and Huang (2011) use evapotranspiration; and Moral García et al. (2014) uses various agro-climatic indices to assess the wine suitability in a region.

Some studies relate the changes of agro-climatic indices trend to changes in crop phenology in Eurasia regions (Menzel et al., 2003; Moonen et al., 2002; Genovese et al.; Semenov et al., 2006), or in Sahelian countries (Ben Mohamed et al., 2002), or in the major part of North America (Robeson, 2002; Feng and Hu, 2004; Terando et al., 2011) (the last, with frost days, thermal time, and heat stress index). Torvanger et al. (2004) analyzed the important relationship between yields of potatoes/barley/oats/wheat, and temperature (growing degree days)/precipitation in Norway.

Others studies examined the dependence of agriculture to the weather variability analyzing the time series of several weather indices (Bazgeer et al., 2014). Qian et al. (2010) analyzed a set of agro-climatic indices representing Canadian climatic conditions for field crop production. Qian et al. (2001) used precipitation, maximum temperature, minimum temperature and solar radiation. Bélanger et al. (2000) used water deficit (precipitation - evapotranspiration), effective growing degree-days, and Corn Heat Units (CHU) in order to determine the weather impact on agriculture in Quebec. These indices are considered as the most used weather variables to determine how well suited is a land for cereals growing, according to the Agronomic Interpretation Working Group (1995). Graczyk and Kundzewicz (2016) examined the lengths of the growing season and the frost-free season, the days of occurrence of the last spring frost and first autumn frost, and the annual sums of growing degree-days for three values of
temperature threshold in Poland.

Only a few studies (Trnka et al., 2011; Lalic et al., 2013; Peltonen-Sainio
et al., 2010) tested more than 4 or 5 agro-climatic indices (such as the Free Frost
Period, the Corn Heat Units, the Standardized Precipitation-Evapotranspiration
Index, or the soil moisture), in combination to simple weather variables (tem-
perature and precipitation), in a crop yield forecasting application. Here, we
propose to analyze more than 50 different agro-climatic indices.

Our application focuses on corn yield over the eastern US. The goal of this
paper is: (1) to propose an approach to identify the most important agro-
climatic indices for crop yield forecasting, (2) to identify which US regions are
the most weather-sensitive for crop yield, and most importantly, (3) to measure
the impact of using agro-climatic indices instead of direct weather information
for crop yield forecasting. Section 2 presents the materials and methods. The
sensitivity analysis is presented in Section 3 and the yield forecast improvements
in Section 4.

## 2. Materials and methods

### 2.1. Data

#### 2.1.1. Agricultural data

The corn yield data were collected by the US Department of Agriculture
NASS (National Agricultural Statistics Service) at the county level. A long
historical record is available from 1910 to 2013 but the yield time-series are not
complete for all counties. The eastern US produces more corn than the western
part because of a more favorable climate and topography. The Corn-Belt region
(mainly Iowa, eastern Nebraska, Minnesota, Illinois, and Indiana) is known to be
the biggest corn-producing region in the US. In 2016, these five states produced
more than 60% of the US corn production (18% for Iowa, 11% for Nebraska,
10% for Minnesota, 15% for Illinois, and 6% for Indiana).
2.1.2. The main corn production basins in the US

Corn ranks first in the US grain production. With more than 38 Mha sown
in 2016, it is well ahead of wheat (20 Mha) and soybeans (33 Mha). Maize is
mainly grown in 5 major production areas and these different zones produce
maize under very variable conditions (Fig. 1):

• Zone 1 (Corn Belt): with a humid continental climate, is located in the center of the United States, on rich and deep soils. The area is particularly well suited to maize, which is mostly grown in rotation with soybeans. Since rainfall is abundant, irrigation is not necessary, and dry yields are the highest in the world: 12 t/ha in 2016. Annual precipitation decreases from over 100 cm in the East to less than 60 cm in the West. The amount of rainfall during the five-month growing season is relatively constant (between 43 and 48 cm) (Neild and Newman, 1990). Thus, only 12% of the maize area is irrigated. In 2016, the Corn Belt produced alone 60% of the US maize production.

• Zone 2: in the Northern Great Plains (North and South Dakota), non-irrigated corn, in rotation with soybean, competes with HRS (Hard Red Spring) wheat. The yield was 9,3 t/ha in 2016, and this area produced 9% of the US maize.

• Zone 3 (the central plains): Nebraska, Kansas, Colorado are the second largest producing area, with much of the maize irrigated from a 450,000 km² ground-water (Ogallala aquifer) that extends on 8 states. It is located east of the Rocky Mountain Range, with a dry continental climate and produces a quarter of the US maize production. Corn competes with HRW wheat (Hard Red winter) and soybeans, which are also irrigated. The level of the groundwater table is nevertheless decreasing (-77 cm on average between 1980 and 1995), which could eventually pose irrigation problems. In 2016, this area produced 17% of the maize in the US and had a yield of 9.8 t/ha.

• Zone 4: maize is also grown along the Mississippi River (Lower Mississippi River Basin), where crops are irrigated with moderation. They include SRW (Soft Red winter), soybean, sorghum, and cotton. In 2016, this area yielded 3% of the maize of the US and had a yield of 9.6 t/ha.
<figure><img src="image_3.png" /><figcaption>Figure 1: Top: Major corn production basins in the United States (Leveau and Peigney, 2008) (Source: USDA, 2002 Agricultural Census): (1) The Corn Belt, (2) The Northern Great Plains, (3) The central plains, (4) The Mississippi River, and (5) The East coast. Green and orange colors indicate respectively non-irrigated and irrigated areas.</figcaption></figure>

<caption>Bottom: corn production (left) and yield (right) in 2013. Counties left blank are not available for this year.</caption>

• Zone 5: finally, the East Coast produces about 3.4% of maize with an average yield of 6.2 t/ha (all numbers come from the US Department of Agriculture, National Agricultural Statistics Service). Irrigation is limited.

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134

Irrigation is very important in some of the western regions. According to the US Geological Survey in 2005, “the majority of withdrawals (83%) and irrigated acres (74%) were in the 17 western states”. These regions should be discarded in order to obtain a clearer link between precipitation and corn yield. Therefore, all federal states located west of 103° West and some other isolated states (for instance Nebraska and Texas) have not been considered in
the study. Finally, the 28 States considered in the study are: Illinois, Indiana, Iowa, Kentucky, Michigan, Minnesota, Missouri, Ohio, and Wisconsin (Zone 1); North Dakota, and South Dakota (Zone 2); Kansas, Nebraska, and Oklahoma (Zone 3); Arkansas, Louisiana, Mississippi, and Tennessee (Zone 4); New Jersey, Delaware, Maryland, Virginia, West Virginia, Pennsylvania, North Carolina, South Carolina, Georgia, Alabama (Zone 5).

The analysis has been realized at the level of each production zone, and also at the US level (considering the 28 states listed above).

Fig. 2 illustrates the natural dispersion and variability of the yield anomalies for the five production zones. In zone 1, there is a lower dispersion (suggesting a lower weather sensitivity and low inter-annual variability), but many negative extreme values (suggesting a higher sensitivity to adverse weather conditions). Areas 2, 3 and 5 have very similar natural variability. Zone 4 has a low dispersion and few extreme values.

<figure><img src="image_4.png" /><figcaption>Figure 2: Box-plot of yield anomalies (in %) for the five production zones. It summarizes the variability of the yield anomalies for the five production zones. The standard deviation of the statistics is also provided.</figcaption></figure>
### 2.1.3. Direct meteorological data

Temperature (monthly mean, daily min, daily max and daily mean) and cumulative precipitation data were collected for the period 1979-2013, from the ERA-Interim re-analysis of the European Center for Medium-Range Weather Forecasts (Uppala et al., 2005). We will focus on these 35 years. The US territorial organization and its subdivision into counties are used to project data, from its original 75 km×75 km regular grid into county-level data. To perform the spatial interpolation, we used the closest pixel technique. It is crude and could be improved (bi-linear interpolation for instance) but in our case, this had no impact: the monthly temperatures and precipitations are so smooth spatially that neighbor pixels are very correlated and any interpolation technique gives a similar answer.

### 2.1.4. Agro-climatic indices and soil moisture

Agro-climatic variables are derived from monthly and daily temperature and rainfall data in order to generate more meaningful agronomic variables. The agro-climatic variables are supposed to better represent the temperature or water impact on a crop. Thermal indices include cumulative measures (such as Growing Degree Days, Corn Heat Units, or Cold Degree Days), threshold measures (e.g., number of days for which the temperature is above 30°C), or date-driven measures (such as the first fall frost date, the growing season length, or the date of the growing season start/end). Water indices include cumulative precipitation on different periods of time (growing season, summer month), crop evapotranspiration, or water balance (henceforth named P-ET). Formulas used to calculate these indices are available in Appendix, Tab. A.1. Acronym definitions are also given in this table. Standardized Precipitation-Evapotranspiration Index (SPEI) comes from Beguería et al. (2014) and have been calculated with the FAO-56 Penman-Monteith estimation of potential evapotranspiration<fnref n="1" />. As

<footnote n="1">¹For more information on the calculation details of SPEI, see http://sac.csic.es/spei/database.html.</footnote>
SPEI, Soil Moisture (SM) is also an alternative to precipitation and should better take into account the quantity of water actually available for crops. Soil moisture data come from ECMWF, with a 75 km × 75 km spatial resolution. Soil moisture refers to the ECMWF variable “volumetric soil water layer 2 (7-28 cm)” in $m^3/m^3$ (Albergel et al., 2012; Balsamo et al., 2012).

## 2.2. Crop yield statistical forecast

A scheme of the dataset and the methods used in the study is presented in Fig. 3.

### 2.2.1. Mixed-Effects model

A statistical model based on a Mixed-Effects (ME) model has been developed in Mathieu and Aires (2016). ME models are regression models able to take some group-information into account (for instance, geographic group). This improves the estimation of the variability within and between groups of data. The generalization quality of a model refers to its ability to give a good estimation on new data.

In statistics, the term “effect” is related to the response of the model to an input perturbation. Effects are said to be “fixed” when they are constant for a whole population. Estimating these fixed effects is the traditional domain of regression modeling. Random effects, by comparison, are group-dependent random variables, meaning that the effects are varying from one type of population to another, and we model these variations with a statistical distribution. Thanks to the random effects, this model adapts its behavior and therefore its estimation, for each group. Here, inputs are climate variables (direct or agro-climatic indices). Outputs are yield anomalies (percentage) with respect to a long-term trend which is estimated from historical data (Mathieu and Aires, 2016). Groups are the US county classification (about 2000 counties are considered here). Model parameters are derived from historical data and then, the model can be applied on new data. ME models are based on the idea that each
data belongs to a particular county, and the ME model takes into account the particularities of each county (Pinheiro and Bates, 2009; West et al., 2007).

206

Let us introduce some notations: the corn grain yield for year $t$ is noted $y(t)$, and the long-term trend value by $\bar{y}(t)$. This trend represents the slow evolution of the corn yield (mostly from changes in agricultural practices). The corn yield anomaly $a(t)$ is then defined as a percentage variation around the trend:

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$$a(t) = \frac{y(t) - \bar{y}(t)}{\bar{y}(t)} \qquad (1)$$

If $a(t) > 0$, the yield on that year is higher than a regular year, and if $a(t) < 0$, the yield is lower. For instance, an anomaly $a(t) = 0.25$ means that the corn yield for year $t$ is 25% higher than the annual trend.

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214

In this paper, several linear mixed effects model will be used with different sets of inputs. The inputs will be specified in due course. All mixed models used here for corn yield anomalies forecasting are fitted with fixed effects and uncorrelated random effects for all weather inputs.

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The data used to calibrate a mixed model fall into one of $m$ distinct counties $i = 1, ..., m$. For each county $i$, there is a population of $n_i$ samples ($n_i$ is at most equal to 35 years). To account for the counties in the impact model, the response $j$ in county $i$ is written as:

220

$$a_{ij} = X_i\beta + Z_i b_i + \varepsilon_{ij} \qquad (2)$$

222

for $i \in [1; m]$ and $j \in [1; n_i]$

where $a_{ij}$ is the corn yield anomaly (Eq. 1), $X_i$ and $Z_i$ are two matrices of predictors (i.e., weather anomalies) for county $i$. Predictors in $X_i$ have a fixed effect on yield anomalies, whatever the county, so the so-called “fixed effects” $\beta$ is constant. On the contrary, predictors in $Z_i$ have an effect that is different for each county, so $b_i$ - the so-called “random-effects” - are dependent on county $i$. The term $\varepsilon_{ij}$ is the measurement or process errors (within county variability). They are supposed to be independent and normally distributed.

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### 2.2.2. Input selection process

The complexity of the weather/crop relationships requires as much information as possible in the inputs of the impact model, but the number of available crop data limits the complexity of the model that can be used. An input selection is necessary to reduce the overfitting issues. A model overfits when it works well on training data but has poor predictive performance on new data. For instance, overfitting occurs when a model is excessively complex, such as having too many parameters or too many inputs, and not enough data to calibrate it.

A forward induction method is used to obtain a hierarchy of the most important weather inputs (Draper and Smith, 1981): the first selected weather input is the one that shares the greatest covariance with corn anomalies. The second input is chosen among all the predictors (except the first selected input), as the one that, combined with the first selected input, defines a 2-input linear model that shares the greatest covariance with corn anomalies. This process is repeated until enough inputs are selected. This method identifies the inputs that are more correlated to corn yield but it also ensures the selection of the most complementary inputs. Then a cross validation process helps define the total number of useful inputs: if adding another input does not improve the generalization quality of the model, the inputs addition is stopped at this stage. If necessary, the correlation matrix between the first selected variables can be used to assist decision-making. For instance, if a variable share a relatively important correlation with the previous selected variables, it can be neglected. Besides, it is more rigorous to use several criteria to select the good variables (here we use the CORrelation coefficient (COR), the Root Mean Squared Error (RMSE), and the Akaike Information Criterion (AIC)) and sometimes, the different criteria do not give the same cross-validation results.

Much care is required when using space-time data because data of spatially close pixels are often highly correlated. For cross-validation, the learning and validation sets should be independent and the cross-validation method in presence of spatial auto-correlation needs to be adapted (Le Rest, 2013) to avoid
<figure><img src="image_5.png" /><figcaption>Figure 3: Scheme representing the datasets and the methods used in the study.</figcaption></figure>

non-reliable results and false conclusions. To solve this issue, our analysis uses independent years for the learning and the validation data sets. To define a learning set, 28 years are randomly chosen (80% of the 35 available years) and the remaining 7 years are used for the test set.

## 3. Weather sensitivity analysis

Before modeling the impact of weather on corn yield, a sensitivity analysis can be performed to better understand our application.

### 3.1. *Best predictors classification, depending on the production areas*

Fifty-eight potential weather predictors are available on our database (Tab. A.1). This section compares and quantifies the links between the various possible weather predictors with the yield anomalies according to the production basins. The best predictors are therefore ranked for each zone. To quantify the robustness of this selection, we consider several quality criteria: (1) correlation with yield anomalies, (2) RMSE, and (3) AIC criterion. Tab. 1 summarizes the first
274 6 selected predictors (from left to right) according to the COR criterion. Similar tables are given for RMSE and AIC in appendices Tabs. A.2 and A.3.

<table><caption>Table 1: The six first selected predictors (successively) for the five production zones, according to the correlation criterion. The numbers indicate the correlation between yield anomalies and the output of a linear model constructed with the predictor above and the preceding predictors. See Tab. A.1 for the signification of the predictors. Predictors that do not appear in parentheses will be used as a model inputs, later in the article.</caption><thead><tr><th>Zone</th><th colspan="6">→ Successive selected predictors →</th></tr></thead><tbody><tr><td>1</td><td>DDaugust<br>0.38</td><td>SPEIjuly<br>0.47</td><td>OCA<br>0.50</td><td>SMapril<br>0.52</td><td>SPEIjune<br>0.54</td><td>(CHU)<br>0.55</td></tr><tr><td>2</td><td>OCA<br>0.28</td><td>Tseptember<br>0.33</td><td>SPEIjuly<br>0.36</td><td>Tjuly<br>0.40</td><td>P-ET<br>0.41</td><td>DDaugust<br>0.42</td></tr><tr><td>3</td><td>Tjuly<br>0.44</td><td>DDaugust<br>0.46</td><td>SPEIjune<br>0.49</td><td>DDOctober<br>0.50</td><td>Toctober<br>0.52</td><td>(OCA)<br>0.53</td></tr><tr><td>4</td><td>DDjuly<br>0.50</td><td>Paugust<br>0.53</td><td>SPEIjune<br>0.57</td><td>TCD<br>0.59</td><td>Tapril<br>0.61</td><td>(Tseptember)<br>0.62</td></tr><tr><td>5</td><td>SMjuly<br>0.42</td><td>SPEIjune<br>0.50</td><td>SPEIjuly<br>0.52</td><td>Tjuly<br>0.55</td><td>SMapril<br>0.57</td><td>DDaugust<br>0.58</td></tr></tbody></table>

276 For zones 1, 2 and 3, the three quality criteria (COR, RMSE and AIC) selected the same first 10 predictors. In the case of zones 4 and 5, the selected variables are very similar but are sometimes not chosen in the same order (it is known that such selection procedures can slightly vary depending on the quality criteria, in particular for the order of predictors). However, these strong similarities confirm the robustness of these results.

282 In the following paragraphs, we interpret the selected predictors in agro-nomic terms. The information provided by this classification is of two types: on the one hand, they give information on the months of importance, and on the other hand, they indicate which of temperature or water supply is of greater importance for the yield.

284 Selection for zone 1 underlines the importance of the accumulated degree-days in August, DDaugust (variable negatively correlated with yield anomalies) and
then water in July. In general, due to the accumulating water demand of grow-
ing biomass during the previous growing period, the lowest soil water contents
is reached during August. It is a limited irrigated area so the natural water
supply is important during the warm period. The SPEI in July (SPEIjuly) is
selected as the second most important (positively correlated) weather indicator.

Corn is highly sensitive to moisture in July and August because it is the period
of grain formation. During moisture stress, especially at low relative humidity,
high temperatures can desiccate silks and damage or kill pollen. The moisture
stress during this stage may cause up to 50% yield reduction (Wiatrak, 2010).

It is an area where the number of days with $T > 30°C$ is often low (only 7% of
the counties have an OCA higher than five days from 1979 to 2013). However,
the frequency of temperatures above $30°C$ (OCA) is selected as 3rd predictor,
so this seems to have a significant impact on the yield. Thus, data show that
heat stress is also important, as highlighted in Schlenker and Roberts (2009).

Moreover, it is difficult to state which heat or moisture stress is the most im-
portant for Zone 1, because these two stresses interact with each other.

Zone 2 is located in the north, where the number of days with $T > 30°C$ is low.
For this area too, the parameter OCA seems to explain a big part of the vari-
ability of yield anomalies (negative correlation, but low compared to the other
correlations of the 1st selected predictors in the other zones). The two states in
zone 2 are among those with the latest harvest dates (8oct-19nov on average).

The September temperature (positively correlated) can therefore influence the
yield. Next, the importance of water in July is underlined with the selection
of SPEI (and not precipitation) and temperature in July (the month in which
corn is growing the most). Note that on this predictors selection, SPEI and
soil moisture (SM) often appear before precipitation: for Zone 4, it seems that
evapotranspiration is less limiting; this is logical because these agro-climatic
indices better represent the water available to the plant than the precipitation.

For Zone 3, the two most important predictors are related to temperature, which
is consistent for an area that requests moderate irrigation (highly irrigated re-
gions have been removed from our analysis (Section 2.1.1)). The average tem-
perature in July (Tjuly) is selected first (anti-correlated): July is the flowering period of corn. A high temperature in July affects the photosynthesis that becomes less effective. Heat stress leads to morphological damage and hence reduced yield (Vollenweider and Günthardt-Goerg, 2005). Corn yield may also be reduced due to high air temperatures (30°C and higher) during pollination (Sánchez et al., 2014). Here too, although the analysis of data seems to give more importance to heat stress than water stress (because more heat-related predictors are selected than water-related ones), it is difficult to state the predominance of one over the other as their impacts are closely related to each other. The months of July and August are still selected for this zone. Depending on the zones, either temperatures (T) or degree-days (DD) is chosen, but there is no overall predominance of T on DD (for July, it is rather T which is selected, whereas for August it is DD).

Selection for zone 4 underlines the importance of temperature in July (with a high correlation of -0.5) then variables related to the water supply for the months of June and August.

Finally, for zone 5, three water variables of June and July are selected first (that is consistent because it is an area with little irrigation), and the three of them are negatively correlated. The SPEI in June (SPEIjune) is selected as the second weather indicator (positively correlated). Corn is actively growing in June and the water supply is essential. When June moisture stress occurs, the availability and uptake of nutrients are reduced and stress-weakened plants are more susceptible to disease and insect damage. The temperature in July also seems to be important. The selection of SPEIjuly after that of the soil moisture in July (SMjuly) is curious because they are two very similar variables (correlation of 0.48). However, the p-value for the nullity test of the parameter related to SPEIjuly is very close to zero, so this selection of SPEIjuly is significant.

These results show that water and heat variations during summer months have an important impact on corn yield, with a positive impact of water supply and a negative impact of heat. The low information of some agro-climatic
indices could be surprising (e.g. the date of First Autumnal Frost (FAF) or the accumulated Corn Heat Units (CHU)). It appears that their values are very similar across the US so they do not discriminate enough between the different counties. Furthermore, their times series do not show important variations from one year to another. Even if they are biologically related to corn yield, they do not explain the monthly and inter-annual corn yield anomaly variations. Therefore, such indices do not appear appropriate for our application, but conclusions could be different for other parts of the world or for other crops.

## 3.2. Comparison of the weather sensitivity of the different zones

We now want to compare the weather sensitivity of the different production areas using the predictors classification made in Section 3.1. Since the areas are different in size, they do not contain the same number of samples. The AIC criterion is highly influenced by the number of samples, so it cannot be used to compare the different zones. We focus on the RMSE criterion, and the COR criterion (the adjusted $R^2$ which is less influenced by the sample size - and therefore better suited to model comparisons - provides exactly the same ranking as COR for the five zones).

To compare the five areas, the idea is to use a model (more sophisticated than the simplistic linear model used to classify predictors) with the selected predictors of the different zones, in order to estimate year-end yield. Linear mixed effects models are chosen (Section 2.2.1). The first step in this modeling approach is to know how many predictors one keeps for each zone. This step is performed by cross-validation (Fig. 4). It is important to follow the parsimony principle to reduce the risk of over-learning. Fig. 4 shows the evolution of the COR (blue) and RMSE (orange) criteria in test samples (i.e., generalization statistics) when increasing the number of predictors. When the generalization correlation does not increase significantly anymore, or the RMSE no longer decreases significantly when considering another predictor, the addition of predictors to the list of the model inputs is stopped. The cross-validation indicates that, the first five predictors are needed for zones 1, 3, and 4; for zones 2 and
380 5, the first six predictors are required.

382 not appear in parentheses). When the decision indicated by COR is not exactly

384 the same as that indicated by RMSE, it is the matrix of cross-correlations of
inputs that helps make the final decision. For example, in zone 3, the 6th in-
put OCA being relatively correlated with the 1st and 2nd input (the average

386 temperature in July (Tjuly) and DDaugust) (corr = 0.78 for Tjuly and 0.54 for DDaugust), it was decided not to keep it. Similarly for zone 5, inputs 7

388 (BCHU: Beginning of the Corn Heat Units) is relatively correlated with the pre-
vious inputs, so the selection was stopped at input 6. Moreover, the correlation

390 does not improve much afterwards. Therefore, the selected inputs are relatively uncorrelated. Finally, we have one model by zone, which use the inputs of

392 Tab. 1.
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>Successive selected predictors</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 4: COR and RMSE statistics during the input selection. Only test results are provided. These measurements are obtained using a cross validation based on 200 different training/test sets. These results help us choosing the number of inputs needed for each zone. The two last figures compare the COR and RMSE/sd (the RMSE divided by the standard deviation of each production zone) curves for the five production zones.</figcaption></figure>
In order to compare the different zones, it is better to divide the RMSE by the standard deviation of each zone. In Fig. 4 (bottom right), the COR curves of the five zones are compared to each other. Since the predictors differ on each zone, there is no common abscissa axis. According to the COR criteria, the least to the most weather-sensitive areas are:

$$ \underbrace{\text{Zone 2} < \text{Zone 3} \approx \text{Zone 1}}_{\text{the three most productive basins}} < \underbrace{\text{Zone 4} < \text{Zone 5}}_{\text{the least productive}} $$

The sub-figure with superimposed curves of RMSE divided by the standard deviation of each zone gives similar results (Fig. 4, bottom left). The areas that produce the most are the least weather-sensitive, or those that produce least are more sensitive to weather variations. In addition to the soil suitability for the corn culture, growers have selected areas that are not too much affected by weather, or areas that can be irrigated. These regions are the most productive.

<table><caption>Table 2: Correlation and RMSE statistics in the training and testing datasets, between model and yield anomalies, according to production zones. Standard deviations of the statistics are also provided, quantifying the sensitivity of COR and RMSE to 200 different training/testing sets.</caption><thead><tr><th rowspan="2">Zones</th><th colspan="2">Learning</th><th colspan="2">Testing</th></tr><tr><th>COR</th><th>RMSE</th><th>COR</th><th>RMSE</th></tr></thead><tbody><tr><td>1</td><td>0.62 ±0.029</td><td>0.15 ±0.004</td><td>0.45 ±0.158</td><td>0.16 ±0.026</td></tr><tr><td>2</td><td>0.50 ±0.019</td><td>0.22 ±0.005</td><td>0.32 ±0.153</td><td>0.24 ±0.033</td></tr><tr><td>3</td><td>0.58 ±0.022</td><td>0.21 ±0.005</td><td>0.45 ±0.177</td><td>0.23 ±0.025</td></tr><tr><td>4</td><td>0.64 ±0.021</td><td>0.16 ±0.002</td><td>0.53 ±0.134</td><td>0.16 ±0.013</td></tr><tr><td>5</td><td>0.69 ±0.011</td><td>0.18 ±0.002</td><td>0.54 ±0.095</td><td>0.20 ±0.009</td></tr></tbody></table>

Tab. 2 summarizes the correlation and RMSE criteria in the learning and testing sets, between model and yield anomalies, for the five production areas. Zones 2 and 3 are the least sensitive to weather variations and the least well predicted by the mixed effects model.
The standard deviations are provided too. For RMSEs, the standard devia-
tions are small, illustrating the stability of the RMSE estimates. The standard
deviations are higher for COR, in test, showing the difficulty of obtaining a ro-
bust estimate. The conditional quantile plots showing the quality of the forecast
for the five zones are provided in Fig. 5. The average curve of the scatter-plot
and four quantiles illustrate the dispersion of the errors. Fig. 5 represents the
joint distribution of forecasts and observations of yield anomalies. The condi-
tional distribution of the observation given each of the forecasts are represented
in terms of selected quantiles, in comparison to the 1:1 diagonal line represent-
ing a perfect forecast. Here, it can be seen that the forecasts exhibit a small
degree of over-estimation (the conditional medians of the observed anomalies
are almost always smaller than the forecasts). The forecasts are globally un-
biased, except for the left tails of zones 1 and 2. The histograms in the lower
part of the panels represent the forecast distribution. It can be seen that the
forecasts in zones 1, 2, 3 are more refined, with more extreme anomalies being
forecasted, especially on the left tail.

Correlation of yield anomalies can sometimes be misleading: if a production
zone is less weather-sensitive, the natural variability can be small. Zone 2 is
the least correlated but we saw in Fig. 2 that it is the zone with the highest
variability, in particular with positive outliers that are difficult to explain. It is
very possible that the low correlation in zone 2 is caused by the data outliers.
Zone 1 appears well retrieved on average: despite a reduced corn yield variabil-
ity (Fig. 2), it has many negative extremes probably related to meteorological
adverse conditions. This ranking clearly depends on the list of the weather pre-
dictors used in this study. Another model might be selected if another list of
predictors was used (Burnham and Anderson, 2002).

4. Yield forecast improvement using agro-climatic indices

The use of agro-climatic indices as inputs can be complicated because they
are sometimes difficult to obtain over a large spatial and temporal extent. In
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>Figure 5: Conditional quantile plot of observations and forecasts of yield anomalies for each zone. It represents the joint distribution of forecasts and observations of yield anomalies. The conditional distribution of the observation given each of the forecasts are represented in terms of selected quantiles, in comparison to the 1:1 diagonal line representing a perfect forecast. The histograms in the lower parts of the panels represent the forecast distribution.</figcaption></figure>
particular, it can be difficult to obtain them as a climate model output. It is therefore legitimate to ask (and quantify) whether the use of these indices brings a real improvement compared to simpler weather variables. In this section, the corn yield model is no longer separated according to the production areas. It will be easier in this way to measure the impact of the use of the agro-climatic indices. Furthermore, this will help reducing the over-fitting issues, because more data is available for the general model, and this will lead to even more robust results.

In this section, two configurations will be compared in order to measure the impact of using the agro-climatic indices in addition to the direct weather information:

• “Input configuration (W)” will refer to direct Weather indicators, namely the monthly temperature and precipitation. It will use the linear mixed effects model built in (Mathieu and Aires, 2016). Inputs of this model are provided in Tab. 3 (first line).

• “Input configuration (I)” will refer to agro-climatic Indices, in addition to the direct weather indicators. Another linear mixed model trained nationwide is also used here. Inputs of this second model are also provided in Tab. 3 (second line). Inputs are in the order of their selection for the two configurations (W) and (I).

Tjuly and T/DDaugust are both on the two selections. In July, SPEIjuly has been selected for (I) instead of the accumulated precipitations in July (Pjuly) for (W); this is not surprising since the SPEI variable is more related to the water actually available for the crop. In June, water balance appears to be more informative than monthly mean temperature.

Two types of applications are considered here: (1) the “monitoring mode” requires the full knowledge of weather indicators (i.e., all the predictors until the harvest), and the estimation of the crop yield can be performed only once the growing season is concluded. This mode aims at analyzing the weather-sensitivity of the final crop yield, obtaining an independent crop yield assess-
<table><caption>Table 3: Weather predictors selected (listed following the selection order) for the two input configurations: with direct weather data only (W), or when adding agro-climatic indices (I). We also give the correlation $COR_{cum}$ between yield anomaly observations and the forecast of the mixed models.</caption><thead><tr><th>(W)</th><th>Tjuly</th><th>Taug</th><th>Pjuly</th><th>Tjune</th><th>Tmay</th><th>Paug</th></tr></thead><tbody><tr><td><i>COR</i><sub>cum</sub></td><td>0.40</td><td>0.42</td><td>0.43</td><td>0.44</td><td>0.44</td><td>0.45</td></tr><tr><td>(I)</td><td>Tjuly</td><td>SPEIjuly</td><td>SPEIjune</td><td>DDaug</td><td>DDapril</td><td></td></tr><tr><td><i>COR</i><sub>cum</sub></td><td>0.40</td><td>0.45</td><td>0.50</td><td>0.51</td><td>0.53</td><td></td></tr></tbody></table>

ment at the end of the season (the weather-sensitive part). (2) The “seasonal forecasting mode” intends to estimate the final crop yield during the growing season, before the harvest. Application of this mode is straightforward: help farmers, big cooperatives (often owned by farmers) or even traders for the management of stocks. Government can also use these forecasts for government assistance in poor crop years.

### 4.1. Improvement on the monitoring mode

We first analyze the monitoring mode at the US scale, then we focus on a particular district.

#### 4.1.1. At the US scale

We compare here the forecast ability for configurations (W) and (I) at the US scale. Tab. 4 provides the correlation and RMSE between observed and predicted corn yield anomalies (calculated over the test samples, to test the model generalization ability). Thirty five years are available in the dataset, and each year contains a similar number of data. To define a learning set, 28 years are randomly chosen (80% of the 35 available years) and the remaining 7 years are used for the test set. Several quality measures are saved for this test set. This method is repeated 100 times to analyze the model on different test sets and obtain more robust results. Global quality measures are provided, as the average of the 100 values.

The percentage of yield variance explained by weather inputs rises from 20%
(Corr=0.45) to 28% (Corr=0.53) from configurations (W) to (I), showing the significative impact of using agro-climatic indices in addition to direct weather information. These levels are realistic: for instance, Ray et al. (2015) show that globally, climate variations explain a third of global crop yield variability. Using well-selected agro-climatic indices instead of simple weather data allows reaching this number.

<table><caption>Table 4: Correlation and RMSE (in test cases) between observed and predicted corn yield anomaly, for both configurations (W) and (I), at the US scale (half east of the US). These results come from an adapted cross-validation and Monte-Carlo processes (Section 2.2) using a learning and test sets gathering 60 000 observations.</caption><thead><tr><th></th><th>Config. (W)</th><th>Config. (I)</th></tr><tr><th></th><th>direct weather data</th><th>agro-climatic indices</th></tr></thead><tbody><tr><th>COR</th><td>0.45</td><td>0.53</td></tr><tr><th>RMSE</th><td>0.197</td><td>0.181</td></tr></tbody></table>

The previous Tab. 4 provides statistics at the US scale, but it is possible to analyze the spatial heterogeneity of the weather sensitivity: Fig. 6 (respectively Fig. 7) represents the correlation maps (respectively, the RMSE maps) between corn yield anomaly and weather-based prediction, at the district scale, and for both configurations (W) and (I). From (W) to (I), correlation rises for 185 districts (over the 210 ones), with an average correlation increase of 0.12. Regions benefiting most from (W) to (I) are located on the East, Central and north East. South and Kansas districts have not improved; agro-climatic indices used in (I) and chosen at the US scale, do not seem adapted for this state. On the contrary, districts that are already well predicted with (W) (those with a correlation above 50%) benefit significantly from the agro-climatic indices of (I). In general, for both RMSE and COR, the improvement is lower for less well estimated districts.
<figure><img src="image_18.png" /><figcaption>Figure 6: Top: Correlation maps (corn yield anomaly and weather-based prediction, at the district scale) for configurations (W) and (I). Bottom: Difference between Corr(I) and Corr(W).</figcaption></figure>

4.1.2. *At the district scale*

We focus here on the Virginia state to see the (W)-to-(I) improvement in
terms of time series. This state is one of the most weather-sensitive states for
corn yield. In Fig. 8, the yield forecast anomaly with respect to a long-term trend
(left) and the yield forecast (right) are represented for the two configurations
(W) and (I). Overall variations are well represented for both configurations, but
larger errors for large/extreme amplitudes are much reduced under configuration
(I), for instance in 1980, 1983 or in 2012. For this state, the agro-climatic
indices significantly improve the forecast quality: correlation between observed
and predicted yield anomalies increases by 0.13.
<figure><img src="image_19.png" /><figcaption>Figure 7: Top: RMSE maps (corn yield anomaly and weather-based prediction, at the district scale) for configurations (W) and (I). Bottom: Difference between RMSE(W) and RMSE(I).</figcaption></figure>

## 4.2. Improvement on the seasonal forecasting mode

For the seasonal forecast application, new input selections have been made from May to August and for both configurations (W) and (I): the selection is done among all the predictors available before or during the forecasting month. For instance, for the June prediction, all weather predictors from January to June can be selected as input of the June-forecasting-model. The different inputs (obtained after a cross-validation, as before) are given in Appendix (Tab. A.4).

### 4.2.1. At the US scale

The maps in Fig. 9 (respectively, Fig. 10), represents the correlations (respectively, the RMSE) between observed and predicted yield anomalies, from
<figure><img src="image_20.png" /><figcaption>Figure 8: Monitoring mode over Virginia (“central” agricultural district) for yield anomaly forecast (left) and yield forecast (right). Correlation between observed corn yield anomaly time-series and predicted one (in test samples) for configurations (W) and (I) are also provided.</figcaption></figure>

May to August and for both configurations (W) and (I). There is a clear improvement in the seasonal forecast of yield anomalies, for May and July. There is also a clear advantage in considering the configuration (I) compared to (W): at the end of July, the average (across all districts) correlation between observations and predictions is 0.41 for (W) and 0.50 for (I).

The maps in the right-hand column illustrate the districts whose forecasts have increased or decreased with the addition of agro-climatic indices. The results show a monotonous increase in the quality of the seasonal forecast when approaching the harvest. The improvement in June is less significant: in June, the average correlation between observations and predictions is 0.23 for (W) and 0.30 for (I). (I) provides information almost everywhere, particularly in the northern US, where (W) appears to be ineffective. We obtain with (I) a slightly more spatially homogeneous forecast quality. The configuration (I) is always better than (W), whatever the month. No information is provided in May by (W), whereas knowledge is not zero for (I). More information can already be provided in June for the south of the east coast. Finally, for both configurations, most of the information is provided by the end of July. Maps for RMSE give
similar results.
<figure><img src="image_21.png" /><figcaption>Figure 9: Correlation maps between observed and predicted yield anomalies, of the seasonal forecast, from May to August and for both configurations (W) and (I). Maps for August have already been shown in Fig 6. The maps in the right-hand column illustrate the correlation difference (I)-(W).</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>Figure 10: RMSE maps between observed and predicted yield anomalies, of the seasonal forecast, from May to August and for both configurations (W) and (I). Maps for August have already been shown in Fig 6. The maps in the right-hand column illustrate the RMSE difference (W)-(I).</figcaption></figure>
### 4.2.2. At the district scale

In this section, we quantify the information provided by weather inputs from May to August to predict corn yield using a mixed model, over the Virginia state. Tab. 5 summarizes the forecast ability (in terms of correlation between observed and predicted yield-anomalies or yield time series) using weather inputs available from May to August among all inputs of configurations (W) and (I).

<table><caption>Table 5: Correlations between observed and predicted yield-anomalies (left) or yield (right) time series, over Virginia (central district). Statistics are provided for configurations (W) and (I), under seasonal forecasting mode, from May to August. Results for the NASS model (August and September report) are also provided for comparison purpose.</caption><thead><tr><th rowspan="2"></th><th colspan="4">Yield Anomaly</th><th colspan="4">Yield</th></tr><tr><th>May</th><th>June</th><th>July</th><th>Aug.</th><th>May</th><th>June</th><th>July</th><th>Aug.</th></tr></thead><tbody><tr><th>(W)</th><td>0.00</td><td>0.35</td><td>0.67</td><td>0.67</td><td>0.38</td><td>0.53</td><td>0.72</td><td>0.73</td></tr><tr><th>(I)</th><td>0.10</td><td>0.52</td><td>0.76</td><td>0.80</td><td>0.39</td><td>0.63</td><td>0.79</td><td>0.82</td></tr><tr><th>NASS</th><td>-</td><td>-</td><td>-</td><td>-</td><td>na</td><td>na</td><td>0.90</td><td>0.95</td></tr></tbody></table>

As shown in Tab. 5, configuration (W) provides less information before June (COR≤0.35) whereas a correlation of 0.52 can be obtained with input configuration (I) at the end of June. These numbers underline the importance of information provided by SPEIjune in (I): the other two inputs (the average temperature in June (Tjune) and April (Tapril) for (W), or Tjune and the accumulated degree-days in April (DDapril) for (I) - Tab. A.4) are very similar. Improvement from June to July, or from July to August is very impressive on configuration (W): roughly 0.5 from June to July, and less from July to August.

In July, 91% of the maximal value (those observed in August) is still reached for configuration (W), and 95% for configuration (I). So, almost all information is available already at the end of July. Once again, weather variation in July appears to be very important for corn growth, and with the agro-climatic indices, it is possible to obtain some exploitable information at the end of July.

Regarding yield forecast (Tab. 5, right), the differences between (W) and (I)
are less important in May than for yield anomaly forecast because both of the configurations use the same trend to calculate yield from yield anomaly. Using the (I) inputs, the mixed model shows that in Virginia, and in July, 62% of the yield anomaly variations are explained by weather variations (Corr=0.79).

A NASS crop model has been developed by the US Department of Agriculture (NASS, May 2016). The NASS currently uses two major survey techniques for crop yield forecasting and estimation. In the first one, statistically sampled farmers are asked to report their final harvested yield or their best evaluation of potential yield based on current conditions during the forecast season. The second one is named the “Objective Yield Surveys” and utilizes plant counts and fruit measurements from random plots in selected fields. The overall NASS model does not use any weather data: it is based on more direct yield information. It is therefore natural that its performance is better than the weather based model. With less than 5 weather predictors, the ME (I) model proposed here is much simpler than NASS, and less expensive to put in place. Furthermore, our model could be extended to other parts of the world where it would be difficult to organize the two necessary in situ surveys for the NASS model. It is also interesting to note that it is possible to have some relevant information on June or July when NASS survey is available only in mid-September. It is conceivable to combine the information of the configurations (I) and (W) with those of the NASS surveys to obtain better information.

These results are illustrated further in Fig. 11: it illustrates the predicted time series of corn yield anomaly in Virginia, for configurations (W) and (I) when we use the selection inputs for May (top left), then June (top right), then July (bottom left), and finally August - i.e., the same inputs as for monitoring mode (bottom right). Correlations between observed and predicted time series are also provided. In June, no information on yield anomalies is provided by (W), and little by (I). As early as June, even if the yield and production anomalies are far from being well estimated, the variations of the predictions are correct. In July, the negative amplitudes of the predictions are much better for (I) than
596 for (W), especially in the years 1983 and 1989. Only small improvements are visible from July to August.

<figure><img src="image_23.png" /><figcaption>Figure 11: Seasonal forecast of yield anomaly in Virginia. For each month of forecast, both input configurations (W) and (I) are compared.</figcaption></figure>
**Heat stress and water availability** - Many studies analyzed the impact of heat stress and water deficit on corn yield, and emphasized the importance of these predictors too. Kaushal et al. (2016) analyzed the effects of heat stress on various food crops and their responses. Stan and Näescu (1997) and Aguilar et al. (2007) studied the influence of drought on several maize hybrids. Çakir R. (2014), Stone et al. (1997) and Soltani et al. (2013) determined the effect of water deficit stress on corn. Aslma et al. (2013) studies showed that drought stress have deleterious effects on the seedling establishment, vegetative growth, photosynthesis, root growth, pollination and grain formation in maize crop. Besides, Hawkins et al. (2013) found a significant reduction in French maize yield for each day with a maximum temperature above 32°C. Similar studies refer to African maize (Lobell et al., 2011). Deryng et al. (2014) quantified at the global scale, the impacts of extreme heat stress on maize yield, and Lobell et al. (2014) suggested that agronomic changes tend to translate improved drought tolerance of plants but not to decreasing drought sensitivity of yields. Moreover, Bishop et al. (2016) showed that the pollinator dependency of crops can be modified by heat stress, and suggest that insect pollination may become more important in crop production as the probability of heat waves increases. Most of these studies analyzing the impact of heat stress or water deficit on corn yield do support our selection of weather predictors.

It should be noted, however, that the heat impact might depend also on the soil water storage capacity (that is a location-specific information). For a soil with a low heat storage capacity, the heat can have a prominent effect on the crop.

**Weather predictors interaction** - Drought and heat stress often (but not always) occur in combination, especially in July/August, leading to the stronger responses in these months. In the literature, many articles have studied this point: Kebede et al. (2012) showed that the combined effect of heat and severe moisture deficit stress had a much greater impact for non-irrigated
corn than the independent effect. Similar studies analyzed the independent and combined effects of high temperature and drought stress on plant yield (Grig-orova et al., 2011; Fábián et al., 2008; Prasad et al., 2011).

Other parameters influence the crop yield such as soil type, fertilizers, seeds type, diseases, etc. Part of this variability is taken into account in the long-term trend that was modeled using a mixed-effect model. Unlike soil type or crop variety, weather has a short-term impact on crop yield, and explains part of the inter-annual variance. Such differences between counties are already taken into account thanks to the group information in the mixed effects model, but this additional information could probably improve further our results. This could be the subject of a forthcoming study.

**Sensitivity of the production zones** - The first part of this study involved, among other things, locating the corn production areas that were most sensitive to weather. Regions in the south and east are the most sensitive: here, the weather accounts for more than 28% of the maize yield variability. They are the regions that actually produce the least amount of corn. In other regions, less than 20% of the variability in yield is related to weather. These numbers may seem weak, but they remain consistent with recent research on this subject: for example Ray et al. (2015) showed that on a global scale, weather information could explain 1/3 of the yield variability.

**6. Conclusion**

This study focuses on the annual yield variations related to weather conditions. Climate is both a resource and a constraint for agriculture so it is important to have a clear understanding of its impact on crop yield. Weather information can be exploited to develop impact models that estimate or forecast crop yield. We introduced a rigorous statistical methodology in order to rank the most effective weather-based information and quantify their impact for crop yield forecasting. The sensitivity analysis showed that temperature
and precipitation information are required. Our methodology allows measuring
the weather-sensitivity of the agricultural regions: the first part of the study
found, using a classification of weather predictors, that zone 2 (north) and zone 3
(west) were less weather-sensitive relative to the other zones we evaluated. This
is confirmed by the second part of the article that showed that they are less
well-predicted.

There are variations from one production area to another, but the similarities are sufficiently important to make a general model in the second part of this study, allowing more robust results, given the limited data available to parameterize several local models. The statistical models manage to estimate a significant part of the crop yield variability (64% in Virginia), especially if the inputs of the model are optimized, using agro-climatic indices. It is shown that agro-climatic indices (such as SPEI) are more effective than more direct weather information for yield forecasting. On the other hand, the sophisticated agro-climatic indices do not seem to be very informative: simple predictors work well. The SPEI index is not too sophisticated, nor as simple as rainfall; it seems to be a good compromise, providing more direct information for the plant but in a simple and efficient way. This type of model can be used to exploit indirect weather information in order to predict corn yield as a monitoring tool, and in seasonal forecasting mode (during the growing season, and from the end of July).

# 7. Perspectives

Our work can be extended in several ways. First, it seems conceivable to
consider a different statistical model for each production area if longer time
series were available. The NOAA ESRL database could provide the necessary
datasets over a wider period (1948 to present).

Secondly, the regression model is performing a satisfactory prediction of the
values around the mean. But for the most extreme years (either high or low) it
is consistently underpredicting. Forecasting these extreme yield losses or gains
requires a different method of analysis: it requires a different set of tools, it
has been treated for instance in Ben-Ari et al. (2016) (using a mono-variable
approach), Mathieu and Aires (in review) (using a neural network classification)
or in Schlenker and Roberts (2009).

Besides, it is very possible that the low correlation in some production zone
is caused by the data outliers. Robust regression method has been applied in
yield forecasting studies to deal with outlier problems (Rousseeuw and Leroy,
2005; Newlands et al., 2014). A future study could consider this new approach
to improve the forecasting models. Outliers can be extreme yield losses too.

Finally, climate forecasting (for the 20 to 50 next years) is a possible ex-
tension of this work. Improving our understanding of the influence of climate
on agricultural production is needed to cope with expected future changes in
temperature and precipitation. Our work allows the selection of agro-climatic
indices that need to be obtained from the Global Climate Model (GCM) simu-
lations, in order to use them as inputs of our impact models. This will improve
our knowledge on the climate impact on agriculture, allows developing efficient
impact models, and therefore enable a better weather-risk management. A fu-
ture study will therefore focus on, using the model on climatic runs to analyze
the impact of climate change on agriculture as practiced today, and to facilitate
the development of adaptation strategies. In this case, the long-term trend be-
comes important. This long-term trend in corn yield is not only the result of
climate change (Lobell et al., 2003), but rather the result of agriculture practice
changes (fertilization, tools, irrigation practices, genetics on crop seeds, etc.).
So, an external trend yield for the next decades would be necessary for any kind
of climate forecast, whatever the model (statistical or vegetation).

**Acknowledgments.** This work was supported by the PSL Research University.
The authors are grateful to the National Agricultural Statistics Service (NASS)
of the United State Department of Agriculture for the quality of their survey and studies used in this work. We thank the European Center for Medium-Range Weather Forecasts for providing meteorological data. This study shows that it is important to increase interactions and collaborations between agronomist, climate and statistics scientists: we appreciate the interesting exchange with two agronomists A. Perrier and A. Tuzet. Finally, we deeply thank two anonymous reviewers and the editor who handled this manuscript, for their useful comments.

## Appendix A.

Tab. A.1 provides the list of agro-climatic indices used in this study, their abbreviation and the formulas used to calculate them.

Tabs. A.2 and A.3 provide the first ten weather predictors that are selected according to the AIC (Tab. A.2) and RMSE (Tab. A.3) criteria. The first predictor is written on the left and the tenth on the right.

Finally, Tab.A.4 gives the input selections for the seasonal forecast application: for the May selection, the list of potential predictors contains only predictors that can be calculated by the end of May (similarly for June, July and August).
<table><caption>Table A.1: Agro-climatic indices used in this study. For more details, see Côté and Ouranos (2012). Usual planting and harvesting dates by state in the US are available in (USDA, 2010).</caption><thead><tr><th>Variables frequently used</th><th>Units</th></tr></thead><tbody><tr><td>P<sub>j</sub></td><td>Total precipitation of day j</td></tr><tr><td>Pmonth</td><td>Cumulated Precip. for month "month"</td></tr><tr><td>Tave<sub>j</sub></td><td>average temperature for day j<br>Tave<sub>j</sub> = <sup>Tmin<sub>j</sub> + Tmax<sub>j</sub></sup>&frasl;<sub>2</sub></td></tr><tr><td>Tbase</td><td>baseline growing temperature</td></tr><tr><td>Tfrost</td><td>frost temperature threshold</td></tr><tr><td>Tmax<sub>j</sub></td><td>maximum temperature for day j</td></tr><tr><td>Tmin<sub>j</sub></td><td>minimum temperature for day j</td></tr><tr><td>Tmonth</td><td>Average temperature for month "month"</td></tr><tr><td colspan="2">Agro-climatic indices name and formula</td></tr><tr><td>AAT</td><td>Average Annual Temperature<br>AAT = min{Tmin<sub>j</sub>}, j=1, ..., 365</td></tr><tr><td>BCHU</td><td>Beginning of Corn Heat Units<br>BCHU = min{j|TMM5<sub>j</sub> > 12.8°C}</td></tr><tr><td>BGS</td><td>Date of Beginning of Growing Season<br>BGS = min{j|TMMP5<sub>j</sub> > 5.5°C}<br>j=1, ..., 365</td></tr><tr><td>CHU</td><td>Cumulated Corn Heat Units<br>CHU = &Sigma;<sup>ECHU</sup><sub>j=BCHU</sub> CHU<sub>j</sub><br>CHU<sub>j</sub> = 1/2(Ymax<sub>j</sub> + Ymin<sub>j</sub>)<br>Ymax<sub>j</sub> = <span style="font-size: 2em; vertical-align: middle;">{</span> 0.33(Tmax<sub>j</sub> - 10) - 0.084(Tmax<sub>j</sub> - 10)<sup>2</sup> if Tmax<sub>j</sub> > 30°C<br>0 if Tmax<sub>j</sub> ≤ 10°C<br>Ymin<sub>j</sub> = <span style="font-size: 2em; vertical-align: middle;">{</span> 1.8(Tmin<sub>j</sub> - 4.44) if Tmin<sub>j</sub> > 4.44°C<br>0 if Tmin<sub>j</sub> ≤ 4.44°C<br>DD</td></tr><tr><td>DD</td><td>Total degree-days from April to October<br>DD = &Sigma;<sup>304</sup><sub>j=91</sub> DD<sub>j</sub><br>DD<sub>j</sub> = max{0, (Tave<sub>j</sub> - Tbase)}</td></tr><tr><td>DDgs</td><td>Total Degree-days in the Growing Season<br>DDgs = &Sigma;<sup>EGS</sup><sub>j=BGS</sub> DD<sub>j</sub></td></tr><tr><td>ECHU</td><td>End of Corn Heat Units<br>ECHU = min{j|Tmin<sub>j</sub> ≤ -2°C}<br>j=213, ..., 365</td></tr><tr><td>EGS</td><td>Date of end of growing season<br>EGS = max{j|TMMP5<sub>j</sub> > 5.5°C}<br>j=1, ..., 365</td></tr><tr><td>FAF</td><td>Date of First Autumnal Frost<br>FAF<sub>Tfrost</sub> = min{j|Tmin<sub>j</sub> ≤ Tfrost}<br>j=213, ..., 365</td></tr><tr><td>FFP</td><td>Free Frost Period<br>FFP<sub>Tfrost</sub> = FAF<sub>Tfrost</sub> - LSF<sub>Tfrost</sub></td></tr><tr><td>LGS</td><td>Length of growing season<br>LGS = EGS - SGS</td></tr><tr><td>LSF</td><td>Date of Last Spring Frost<br>LSF<sub>Tfrost</sub> = max{j|Tmin<sub>j</sub> ≤ Tfrost}<br>j=1, ..., 212</td></tr><tr><td>OCA</td><td>Frequency of temperatures above 30°C<br>OCA<sub>TSUP30</sub> = &Sigma;<sup>365</sup><sub>j=1</sub> TSUP30<sub>j</sub><br>TSUP30<sub>j</sub> = <span style="font-size: 2em; vertical-align: middle;">{</span> 1 if Tmax<sub>j</sub> > 30°C<br>0 if Tmax<sub>j</sub> ≤ 30°C</td></tr><tr><td>Pgs</td><td>Precipitation during the growing season<br>Pgs = &Sigma;<sup>EGS</sup><sub>j=BGS</sub> P<sub>j</sub></td></tr><tr><td>TCD</td><td>Total Cooling Degree Days<br>TCD = &Sigma;<sup>FPE-1</sup><sub>j=213</sub> TCD<sub>j</sub><br>TCD<sub>j</sub> = <span style="font-size: 2em; vertical-align: middle;">{</span> 0 if j = 212<br>max{0, TCD<sub>j-1</sub> + DF<sub>j</sub>} else<br>FPE = min{j|Tmin<sub>j</sub> ≤ -10°C and j > 212}<br>DF<sub>j</sub> = CD<sub>j</sub> - DD<sub>j</sub><br>CD<sub>j</sub> = <span style="font-size: 2em; vertical-align: middle;">{</span> 0 if Tave<sub>j</sub> ≥ 5°C<br>|Tave<sub>j</sub> - 5| if Tave<sub>j</sub> < 5°C<br>0 if Tave<sub>j</sub> ≤ 5°C<br>DD<sub>j</sub> = <span style="font-size: 2em; vertical-align: middle;">{</span> Tave<sub>j</sub> - 5 if Tave<sub>j</sub> > 5°C</td></tr><tr><td>TDD</td><td>Cumulated Degree Days during frost period<br>TDD = &Sigma;<sup>Ewin</sup><sub>Bwin</sub> max{0, Tave<sub>j</sub>}<br>Bwin = min{j|Tmin<sub>j</sub> ≤ -15°C}<br>Ewin = max{j|Tmin<sub>j</sub> ≤ -15°C}</td></tr><tr><td>TMM5<sub>j</sub></td><td>moving average of daily mean temperature<br>during 5 days<br>1/5(Tave<sub>j-4</sub> + Tave<sub>j-3</sub> + Tave<sub>j-2</sub> + ... + Tave<sub>j-1</sub> + Tave<sub>j</sub>)</td></tr><tr><td>TMMP5<sub>j</sub></td><td>weighted moving average of daily mean temperature during 5 days<br>1/16(Tave<sub>j-4</sub> + 4.Tave<sub>j-3</sub> + ... + 6.Tave<sub>j-2</sub> + 4.Tave<sub>j-1</sub> + Tave<sub>j</sub>)</td></tr></tbody></table>
<table><caption>Table A.2: The ten first weather predictors successively selected for the five zones, according to the AIC criteria.</caption><thead><tr><th rowspan="2">Zone</th><th colspan="10">→ Successive selected inputs →</th></tr><tr><th>DDaugust</th><th>SPEIjuly</th><th>OCA</th><th>SMapril</th><th>SPEIjune</th><th>CHU</th><th>DDjuly</th><th>SPEIaugust</th><th>DDmarch</th><th>DDgs</th></tr></thead><tbody><tr><td>1</td><td>OCA</td><td>Tseptember</td><td>SPEIjuly</td><td>Tjuly</td><td>P-ET</td><td>DDaugust</td><td>Paug-oct</td><td>DDapril</td><td>SMapril</td><td>Pmarch</td></tr><tr><td>2</td><td>Tjuly</td><td>DDaugust</td><td>SPEIjune</td><td>Toctober</td><td>DDoctober</td><td>OCA</td><td>Tseptember</td><td>LGS</td><td>Poctober</td><td>Taugust</td></tr><tr><td>3</td><td>DDjuly</td><td>Paugust</td><td>SPEIjune</td><td>Paug-oct</td><td>DDseptember</td><td>AAT</td><td>Toctober</td><td>LSF</td><td>SMaugust</td><td>Papril</td></tr><tr><td>4</td><td>SMjuly</td><td>Tjuly</td><td>SPEIjune</td><td>SPEIjuly</td><td>SMapril</td><td>DDaugust</td><td>Tseptember</td><td>FFP</td><td>Papril</td><td>SPEIaugust</td></tr><tr><td>5</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>Table A.3: The ten first weather predictors successively selected for the five zones, according to the RMSE criteria.</caption><thead><tr><th rowspan="2">Zone</th><th colspan="10">→ Successive selected inputs →</th></tr><tr><th>DDaugust</th><th>SPEIjuly</th><th>OCA</th><th>SMapril</th><th>SPEIjune</th><th>CHU</th><th>DDjuly</th><th>SPEIaugust</th><th>DDmarch</th><th>DDgs</th></tr></thead><tbody><tr><td>1</td><td>OCA</td><td>Tseptember</td><td>SPEIjuly</td><td>Tjuly</td><td>P-ET</td><td>DDaugust</td><td>Paug-oct</td><td>DDapril</td><td>SMapril</td><td>Pmarch</td></tr><tr><td>2</td><td>Tjuly</td><td>DDaugust</td><td>SPEIjune</td><td>Toctober</td><td>DDoctober</td><td>OCA</td><td>LGS</td><td>Tseptember</td><td>Poctober</td><td>Taugust</td></tr><tr><td>3</td><td>DDjuly</td><td>TCD</td><td>SPEIjune</td><td>Paugust</td><td>Tapril</td><td>Tseptember</td><td>Paug-oct</td><td>Toctober</td><td>AAT</td><td>SMaugust</td></tr><tr><td>4</td><td>SMjuly</td><td>SPEIjune</td><td>SPEIjuly</td><td>Tjuly</td><td>SMapril</td><td>DDaugust</td><td>B-CHU</td><td>LSF</td><td>AAT</td><td>Papril</td></tr><tr><td>5</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>Table A.4: List of the inputs (in their order of selection) for the different months of the seasonal forecast, and for both configurations (W) and (I).</caption><thead><tr><th>Month</th><th colspan="7">→ Successive selected inputs →</th></tr></thead><tbody><tr><td>May</td><td colspan="7">Tapril</td></tr><tr><td>June<br>(W)</td><td>Pjune</td><td>Tjune</td><td>Tapril</td><td></td><td></td><td></td><td></td></tr><tr><td>July</td><td>Tjuly</td><td>Pjune</td><td>Tapril</td><td>Pjuly</td><td>Tjune</td><td></td><td></td></tr><tr><td>August</td><td>Tjuly</td><td>Taugust</td><td>Pjuly</td><td>Tjune</td><td>Tmay</td><td>Paugust</td><td></td></tr><tr><td>May</td><td>DDapril</td><td>SPEImay</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>June<br>(I)</td><td>SPEIjune</td><td>Tjune</td><td>DDapril</td><td></td><td></td><td></td><td></td></tr><tr><td>July</td><td>Tjuly</td><td>SPEIjuly</td><td>SPEIjune</td><td>DDapril</td><td></td><td></td><td></td></tr><tr><td>August</td><td>Tjuly</td><td>SPEIjuly</td><td>SPEIjune</td><td>DDaugust</td><td>DDapril</td><td></td><td></td></tr></tbody></table>
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