# Agri-GNN: A Novel Genotypic-Topological Graph Neural Network Framework Built
on GraphSAGE for Optimized Yield Prediction

Aditya Gupta
William Fremd High School
Palatine, IL
Asheesh Singh
Agronomy Department
Iowa State University

###### Abstract

Agriculture, as the cornerstone of human civilization, constantly seeks to
integrate technology for enhanced productivity and sustainability. This paper
introduces Agri-GNN, a novel Genotypic-Topological Graph Neural Network
Framework tailored to capture the intricate spatial and genotypic interactions
of crops, paving the way for optimized predictions of harvest yields. Agri-GNN
constructs a Graph $\mathcal{G}$ that considers farming plots as nodes, and
then methodically constructs edges between nodes based on spatial and
genotypic similarity, allowing for the aggregation of node information thorugh
a genotypic-topological filter. Graph Neural Networks (GNN), by design,
consider the relationships between data points, enabling them to efficiently
model the interconnected agricultural ecosystem. By harnessing the power of
GNNs, Agri-GNN encapsulates both local and global information from plants,
considering their inherent connections based on spatial proximity and shared
genotypes, allowing stronger predictions to be made than traditional Machine
Learning architectures. Agri-GNN is built from the GraphSAGE architecture,
because of its optimal calibration with large graphs, like those of farming
plots and breeding experiments. Agri-GNN experiments, conducted on a
comprehensive dataset of vegetation indices, time, genotype information, and
location data, demonstrate that Agri-GNN achieves an $R^{2}=.876$ in yield
predictions for farming fields in Iowa. The results show significant
improvement over the baselines and other work in the field. Agri-GNN
represents a blueprint for using advanced graph-based neural architectures to
predict crop yield, providing significant improvements over baselines in the
field.

_The ultimate goal of farming is not the growing of the crops, but the
cultivation and perfection of human beings. —Masanobu Fukuoka_

Key Words: Graph Neural Networks (GNNs), Agricultural Data Integration,
Multimodal Data Fusion, Structured Data Modeling, Adaptive and Modular Design,
Precision Agriculture Enhancement, Complex Interdependencies Modeling

##  1 Introduction

In an era characterized by escalating climate change, which is resulting in
unpredictable weather patterns and increasing environmental stresses, the
agricultural sector faces significant challenges (Anwar et al., 2013).
Unforeseen climatic events such as droughts, floods, and extreme temperatures
are impacting crop yields, highlighting the imperative for advanced, precise,
and resilient crop yield prediction models (Kuwayama et al., 2019). Amidst
this backdrop of climatic uncertainties (Shrestha et al., 2012), the necessity
for accurate and comprehensive crop yield predictions is more acute than ever.
A robust system that can efficiently integrate diverse data types and provide
a detailed and holistic understanding of the agricultural ecosystem is crucial
for mitigating the impacts of climate change on agriculture.

The agricultural ecosystem is inherently complex and interconnected, with
numerous factors playing a pivotal role in determining crop yields.
Traditional Machine Learning models, while powerful, often fall short in
effectively capturing these intricate relationships, as they generally treat
data points as independent entities (Liu et al., 2020). The limited capacity
of these models to handle relational data and their inability to seamlessly
integrate diverse data types such as spatial, temporal, and genetic
information, hamper their effectiveness in providing comprehensive and
accurate crop yield predictions. Moreover, these models tend to be data-
hungry, requiring substantial labeled data for training, which is often a
significant challenge in the agricultural context (Majumdar et al., 2017). On
the contrary, Graph Neural Networks (GNNs) stand out as a more apt choice for
this scenario. GNNs, by design, consider the relationships between data
points, enabling them to efficiently model the interconnected agricultural
ecosystem. They can effectively synthesize diverse data types into a unified
framework, offering a more holistic and nuanced understanding of the factors
influencing crop yields (Zhou et al., 2020). The ability of GNNs to work with
limited labeled data and their flexibility in handling various data modalities
make them a superior choice for developing robust and resilient crop yield
prediction models in the face of climate change.

In light of this, the present study introduces Agri-GNN, a pioneering approach
employing GNNs to offer an inclusive representation of the agricultural
ecosystem. Agri-GNN considers farming plots as nodes, and then constructs
edges between nodes based on spatial and genotypic similarity, allowing for
the aggregation of node information from a refined selection of nodes. This
allows for the model to refine the noise that exists in the dataset and focus
yield prediction efforts for each node on the most similar nodes in terms of
genotypic and spatial similarity. Agri-GNN stands out with its capacity to
amalgamate diverse data modalities into a cohesive framework, adeptly
capturing the complex interaction among genetic, environmental, and spatial
factors (Meng et al., 2018). This innovative model transcends traditional
methodologies by conceptualizing the agricultural ecosystem as a connected
network, where each crop, viewed as an active node, is influenced by its
immediate environment and genetic context.

The employment of the GraphSAGE architecture (Hamilton et al., 2017),
significantly bolsters the effectiveness of Agri-GNN. GraphSAGE is known for
its inductive learning approach, where it leverages node attribute information
to generate representations for data not seen during the training process.
This approach is particularly beneficial for the extensive and heterogeneous
datasets that are commonplace in the field of agriculture. Traditional machine
learning models often struggle with such diverse and expansive data, leading
to suboptimal predictions and insights. However, the GraphSAGE architecture,
with its innovative inductive learning, excels in processing and learning from
large datasets, thereby ensuring the robustness of Agri-GNN.

In agriculture, datasets encompass a wide range of information including
weather conditions, soil types, and genetic information, the ability to
effectively handle and learn from such data is crucial for accurate yield
predictions. The GraphSAGE architecture equips Agri-GNN with this capability,
allowing it to seamlessly integrate and process diverse data types to generate
detailed and reliable yield predictions. This level of granular insight is
important for making informed decisions in agricultural planning and
management, ultimately contributing to enhanced productivity and
sustainability.

By using the GraphSAGE architecture, Agri-GNN is not just limited to data seen
during training. It can generalize and adapt to new data, ensuring that the
model remains relevant and useful as more agricultural data becomes available.
This adaptability is essential in the dynamic field of agriculture, where new
data and insights continuously emerge. The advanced architecture thereby not
only enhances Agri-GNN’s predictive accuracy but also bolsters its longevity
and relevance in the agricultural sector, making it a valuable tool for
tackling the challenges of modern agriculture. Agri-GNN’s modular and scalable
design ensures its adaptability to the fast-paced evolution of the
agricultural sector. This flexibility allows for the effortless integration of
emerging data sources and insights, ensuring the model remains relevant and
effective in a changing landscape (Gandhi and Armstrong, 2016).

Agri-GNN embodies a transformative shift that is taking place in agricultural
modeling, providing a novel perspective that comprehensively addresses the
complexity and interconnectedness of farming systems. By offering a nuanced,
data-driven lens, Agri-GNN stands as a robust tool for navigating the
multifaceted challenges of modern agriculture, particularly in the context of
a changing climate.

##  2 Literature Review

Plant breeding specialists are focused on discovering high-quality genetic
variations that fulfill the needs of farmers, the wider agricultural sector,
and end consumers. One of the key traits often scrutinized is seed yield,
particularly in row crops (Singh et al., 2021c). Traditional ways of assessing
seed yield involve the laborious and time-restricted activity of machine-
harvesting numerous plots when the growing season concludes. This data then
informs decisions about which genetic lines to either advance or discontinue
in breeding programs. This approach is highly resource-intensive, requiring
the harvesting of thousands of test plots each year, thus presenting
operational challenges.

In response to these issues, advancements in technology are being harnessed to
develop more efficient alternatives. An increasing number of scientists and
plant breeders are adopting the use of remote sensing technology, integrated
with machine learning methods. This enables more timely predictions of seed
yield, substantially cutting down on labor and time requirements during the
crucial harvest phase (Li et al., 2022; Yoosefzadeh-Najafabadi et al., 2021;
Chiozza et al., 2021; Shook et al., 2021b; Riera et al., 2021; Guo et al.,
2021; Singh et al., 2021a).

The newly introduced Cyber-Agricultural System (CAS) takes advantage of
cutting-edge continual sensing technology, artificial intelligence, and smart
actuators for enhancing both breeding and production in agriculture Sarkar et
al. (2023). Integral to CAS is the concept of phenotyping, which employs
sophisticated imaging and computational techniques to streamline the gathering
and interpretation of data, thereby facilitating better yield forecasts Singh
et al. (2021b). Numerous studies have honed in on high-throughput phenotyping
through the use of drones, investigating yield predictions in crops such as
cotton, maize, soybean, and wheat Herr et al. (2023). Beyond the 2D data
collected by drones, research has demonstrated the value of canopy
fingerprints, which offer unique insights into the 3D structure of soybean
canopies via point cloud data Young et al. (2023). Despite these advances,
there is still scope for refining models that amalgamate diverse datasets,
including but not limited to soil features and hyperspectral reflectance, for
a more holistic grasp of soybean yields. The soil’s physical and chemical
attributes play a crucial role in nutrient availability, thereby impacting
plant health and growth. Incorporating these soil characteristics could
potentially enhance the precision of yield prediction models.

In recent years, the application of neural networks in crop yield prediction
has moved beyond traditional architectures to more complex models like
Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs)
(Dahikar and Rode, 2014; Gandhi et al., 2016).

Much work in the field takes advantage of remote sensing data, such as
satellite images or NDVI, for yield predictions (You et al., 2017; Nevavuori
et al., 2019; Kim et al., 2019). While these methods have shown promise, they
often struggle to capture the direct relationships between environmental
factors and crop yields. Previous research has also focused on using
environmental factors like temperature and rainfall directly as inputs for
yield prediction models (Cakir et al., 2014; Khaki and Wang, 2019), but the
same failure to capture direct relationships accurately has been seen.

Against this backdrop, Graph Neural Networks (GNNs) offer a significant
advancement by incorporating spatial relationships or neighborhood information
into the prediction models. By adding the spatial context through GNNs, recent
models provide a more nuanced understanding of how localized factors can
impact yield, enhancing prediction accuracy (Fan et al., 2022; Park and Park,
2019; Sajitha et al., 2023). Particularly, Fan et al. (2022) uses a GNN-RNN
based approach that shows promising results, but fails to generalize on graphs
of large size. The GraphSAGE model (Hamilton et al., 2017) stands out as a
notable innovation in leveraging spatial and neighborhood information for more
accurate crop yield predictions. The incorporation of GraphSAGE allows us to
effectively capture localized contextual information, thereby refining our
understanding of how specific factors in localized areas can influence crop
yields. This results in an enhanced level of accuracy in our yield
predictions.

Graph Neural Networks offer a promising avenue for enhancing the state-of-the-
art in crop yield prediction. They facilitate the integration of various types
of data and have the potential to significantly improve the accuracy of
existing models. Future research should focus on leveraging these capabilities
to build models that can generalize well across different conditions and
scales.

##  3 Background

In this section, we provide background information on Graph Neural Networks
and GraphSAGE. We also provide a background on the data features used in the
creation of Agri-GNN. This background information is necessary for
understanding Agri-GNN.

###  3.1 Graph Neural Networks

Graphs are a robust and versatile means of capturing relationships among
entities, known as nodes, and their interconnections, termed edges. Graph
Neural Networks elevate this representational power by extending classical
neural network architectures to operate directly on graph-structured data.
These networks are particularly effective at generating meaningful node-level
or graph-level embeddings, representations that can be subsequently used in
various downstream tasks such as node classification, link prediction, and
community detection. For an exhaustive review of the techniques and methods
employed in GNNs, we direct the reader to seminal survey papers by Battaglia
et al. (2018) and Chami et al. (2021).

A GNN can be represented as $f(A,X;W)\rightarrow y$, where $y$ denotes the set
of predicted outcomes (e.g., plot yield predictions), $A$ is an $n\times n$
adjacency matrix encapsulating the graph structure, $X$ is an $n\times p$
feature matrix detailing node attributes, and $W$ represents the trainable
weight parameters of the network. In this function, $A$ and $X$ serve as
inputs, and the network $f$ is parameterized by $W$.

One of the distinguishing characteristics of GNNs lies in their ability to
accumulate and propagate information across nodes (Hamilton et al., 2017). A
node’s feature vector is updated iteratively based on the feature vectors of
its neighboring nodes. The depth or number of layers, $\ell$, in the GNN
controls which neighbors are involved in these updates. Specifically, the
final representation of a node only includes information from nodes that are
at most $\ell$-hop distance away. This scope of nodes and edges involved in
the computation of a node’s representation is termed its computational graph,
formally defined as $G=(A,V)$. Here, $A$ is the adjacency matrix for the
subgraph, and $X_{v}$ is the feature matrix for nodes within the $\ell$-hop
neighborhood of the node $v$.

###  3.2 GraphSAGE

GraphSAGE (Graph Sample and Aggregation) is a pioneering extension to general
Graph Neural Networks and was specifically designed to address challenges such
as scalability and inductive learning on large graphs (Hamilton et al., 2017).
Unlike traditional GNN architectures that require the entire graph to be
loaded into memory for training, GraphSAGE leverages a sampling strategy to
extract localized subgraphs, thereby allowing for mini-batch training on
large-scale graphs. The key innovation in GraphSAGE is its novel aggregation
mechanism, which uses parameterized functions to aggregate information from a
node’s neighbors. These functions can be as simple as taking an average or as
complex as employing a neural network for the aggregation process.

GraphSAGE is expressed as $f_{\text{SAGE}}(A,X;W)\rightarrow y$, where
$f_{\text{SAGE}}$ represents the GraphSAGE model, $y$ is the output (such as
node embeddings or graph-level predictions), $A$ is the adjacency matrix, $X$
is the feature matrix, and $W$ are the trainable parameters. Like generic
GNNs, GraphSAGE accumulates and combines information from a node’s
neighborhood to update its feature representation. However, GraphSAGE can
generalize to unseen nodes during inference by leveraging learned aggregation
functions, making it particularly valuable for evolving graphs where the node
set can change over time. It has been employed effectively in diverse
applications such as social network analysis, recommendation systems, and even
in specialized fields like computational biology and agronomy, showcasing its
adaptability and efficiency(Xiao et al., 2019).

In formal terms, the $l$-th layer of GraphSAGE is defined. The aggregated
embedding from neighboring counties, denoted $\mathbf{a}_{c,t}^{(l)}$, is
calculated using the function $g_{l}$ applied to the embeddings
$\mathbf{z}_{c^{\prime},t}^{(l-1)}$ for all neighboring counties $c^{\prime}$
of county $c$, represented as:

$\mathbf{a}_{c,t}^{(l)}=g_{l}(\\{\mathbf{z}_{c^{\prime},t}^{(l-1)},\forall
c^{\prime}\in\mathcal{N}(c)\\})$

Here, $\mathcal{N}(c)=\\{c^{\prime},\forall A_{c,c^{\prime}}=1\\}$ denotes the
set of neighboring counties for $c$.

The embedding for the $l$-th layer, $\mathbf{z}_{c,t}^{(l)}$, is then obtained
by applying a non-linear function $\sigma$ to the product of a weight matrix
$\mathbf{W}^{(l)}$ and the concatenation of the last layer’s embedding
$\mathbf{z}_{c,t}^{(l-1)}$ and $\mathbf{a}_{c,t}^{(l)}$:

$\mathbf{z}_{c,t}^{(l)}=\sigma(\mathbf{W}^{(l)}\cdot(\mathbf{z}_{c,t}^{(l-1)},\mathbf{a}_{c,t}^{(l)}))$

Where $\mathbf{z}_{c,t}^{(0)}=h_{c,t}$ as per a previous equation, and $l$
belongs to the set $\\{0,1,...,L\\}$.

The aggregation function for the $l$-th layer, $g_{l}(\cdot)$, can be a mean,
pooling, or graph convolution (GCN) function.

In this process, $\mathbf{a}_{c,t}^{(l)}$ is first concatenated with
$\mathbf{z}_{c,t}^{(l-1)}$, and then transformed using the weight matrix
$\mathbf{W}^{(l)}$. The non-linear function $\sigma(\cdot)$ is applied to this
product to obtain the final embedding for the $l$-th layer.

###  3.3 Vegetation Indices

Vegetation indices are essential metrics used in the field of remote sensing
phenology to quantify vegetation cover, assess plant health, and (in our
study) to estimate crop yields. These indices leverage the spectral data
gathered by electromagnetic radiation, which measure various wavelengths of
light absorbed and reflected by plants. A fundamental understanding of how
these wavelengths interact with vegetation is crucial for interpreting these
indices. Specifically, the pigments in plant leaves, such as chlorophyll,
absorb wavelengths in the visible spectrum, particularly the red light.
Conversely, leaves reflect a significant amount of near-infrared (NIR) light,
which is not visible to the human eye. The indices used in the construction of
Agri-GNN are available in Appendix A.

One of the most commonly used vegetation indices is the Normalized Difference
Vegetation Index (NDVI). It is calculated using the formula
$\text{NDVI}=\frac{(NIR-RED)}{(NIR+RED)}$, where $NIR$ represents the near-
infrared reflectance and $RED$ is the reflectance in the red part of the
spectrum. The NDVI value ranges between -1 and 1, with higher values typically
indicating healthier vegetation and lower values signifying sparse or stressed
vegetation. This index is invaluable for various applications, ranging from
environmental monitoring to precision agriculture. For Agri-GNN, vegetation
indices like NDVI can serve as informative node attributes in agronomic
graphs, enhancing the model’s ability to make accurate and meaningful
predictions in agricultural settings(Bannari et al., 1995).

##  4 Methods

Our proposed framework, Agri-GNN, aims to provide a comprehensive solution to
crop yield prediction by leveraging the power of Graph Neural Networks (GNNs).
The methods section goes over the various stages involved in the design,
construction, and validation of Agri-GNN.

###  4.1 Graph Construction

To effectively utilize GNNs for agricultural prediction, we first represent
the agricultural data as a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$,
where $\mathcal{V}$ denotes the set of nodes and $\mathcal{E}$ denotes the set
of edges.

####  4.1.1 Node Representation

Each node $v_{i}\in\mathcal{V}$ corresponds to a specific plot and is
associated with a feature vector $\mathbf{x}_{i}$. This feature vector
encapsulates all given data information:

Vegetation Indices ($\mathbf{r}_{i}$): The variable $r_{i}$ is derived from
remote sensing imagery and encapsulates various spectral indices, including
but not limited to reflectance values, that are crucial for assessing
vegetation health and vitality. These spectral indices often make use of
different reflectance bands, such as near-infrared and red bands, to capture
detailed information about plant life. These indices serve as a valuable
source of information, particularly when integrated with other types of data
like genotypic and soil information. The computational procedure to obtain
$r_{i}$ often involves sophisticated algorithms that account for atmospheric
corrections and other sources of noise in the imagery. The specific formulas
and methodologies used to determine $r_{i}$ are detailed in Appendix A.

Genotypic Data ($\mathbf{g}_{i}$): The variable $g_{i}$ encapsulates the
genetic information associated with a specific crop or plant. This genotypic
data serves as a foundational element in the realm of Cyber Agricultural
Systems. Utilizing advanced imaging techniques like hyperspectral imaging,
along with computational methods such as machine learning algorithms, the
acquisition and analysis of $g_{i}$ have been significantly streamlined. These
advancements not only ease the process of data collection but also enable more
accurate and comprehensive genetic profiling. Such in-depth genotypic
information is invaluable for understanding plant characteristics, disease
resistance, and yield potential, thereby playing a crucial role in the
development of precision agriculture strategies and sustainable farming
practices Singh et al. (2021b).

Weather Data ($\mathbf{w}_{i}$): The variable $w_{i}$ encompasses an array of
meteorological factors related to a specific agricultural plot, capturing
elements such as temperature, precipitation, humidity, wind speed, and solar
radiation. These weather conditions are collected through a variety of
methods, including on-site weather stations, remote sensors, and even
satellite data. The comprehensive nature of $w_{i}$ allows it to serve as a
vital input for crop health monitoring systems and predictive yield models.
For instance, high-resolution temporal data on factors like soil moisture and
air temperature can be instrumental in predicting potential stress events for
crops, such as drought or frost risks. Furthermore, when integrated with other
data types like genotypic and vegetation indices, $w_{i}$ contributes to
creating a multifaceted, dynamic model of the agricultural
environment(Mansfield et al. (2005)).

The final node feature vector is a concatenation of these features:

$\mathbf{x}_{i}=[\mathbf{r}_{i},\mathbf{g}_{i},\mathbf{w}_{i}]$ (1)

####  4.1.2 Edge Representation in Agri-GNN

Edges play an indispensable role in graph-based neural network architectures,
encoding vital relationships between nodes. Within the Agri-GNN architecture,
the edge set $\mathcal{E}$ is meticulously constructed to encapsulate the
intricate relationships between agricultural plots. This is achieved by
harnessing both spatial and genotypic attributes.

There are two edge constructions that are undertaken.
$\mathcal{E}_{\text{spatial}}$ encompasses the edges that are created through
spatial proximity. Given the geographical coordinates $\mathbf{c}(v_{i})$
associated with a node $v_{i}$, the pairwise distance to another node $v_{j}$
having coordinates $\mathbf{c}(v_{j})$ can be defined as:

$d(v_{i},v_{j})=\|\mathbf{c}(v_{i})-\mathbf{c}(v_{j})\|$ (2)

For every node $v_{i}$, edges are constructed to nodes that fall within the
bottom 3% of all pairwise distances, thereby ensuring that the model captures
localized environmental intricacies and dependencies:

$e_{ij}=\begin{cases}1&\text{if }d(v_{i},v_{j})\in\text{bottom 3\% of
distances for }v_{i}\\\ 0&\text{otherwise}\end{cases}$ (3)

The second set of edges constructed is represented by
$\mathcal{E}_{\text{genotypic}}$. The genotypic data serves as a repository of
the genetic characteristics of agricultural plots, offering a window into
inherent traits and susceptibilities. Let $g(v_{i})$ represent the genotypic
data for node $v_{i}$. An edge is formed between nodes $v_{i}$ and $v_{j}$ if
their genotypic attributes resonate:

$e_{ij}=\begin{cases}1&\text{if }g(v_{i})=g(v_{j})\\\
0&\text{otherwise}\end{cases}$ (4)

The culmination of the edge formation process results in the edge set
$\mathcal{E}$, which is a fusion of edges derived from both spatial proximity
and genotypic similarity:

$\mathcal{E}=\mathcal{E}_{\text{spatial}}\cup\mathcal{E}_{\text{genotypic}}$
(5)

By harmonizing spatial and genotypic data, Agri-GNN crafts a robust and
nuanced representation of the agricultural milieu, establishing itself as an
efficacious tool for diverse applications in the realm of agriculture.

###  4.2 Graph Neural Network Architecture

For our crop yield prediction task, we introduce the Agri-GNN model, an
adaptation of the GraphSAGE (Graph Sample and Aggregation) architecture
discussed in section 3. The model operates on graph $\mathcal{G}$, designed
for efficient aggregation and propagation of information across the graph.

AgriGNN has four GraphSAGE convolutional layers to process and refine node
features, ensuring an effective representation for downstream tasks. The model
architecture is explained in detail in this section.

In the initial layer of the Architecture, whose primary objective is
transitioning the input node features to an intermediate representation using
hidden channels. The transformation for node $i$ in this layer is shown by
equation (6).

$\mathbf{h}_{i}^{(1)}=\sigma\left(\mathbf{W}_{\text{init}}\cdot\mathbf{x}_{i}+\mathbf{b}_{\text{init}}\right)$
(6)

In equation (6), $\mathbf{x}_{i}$ stands for the initial input features of
node $i$, while $\mathbf{W}_{\text{init}}$ and $\mathbf{b}_{\text{init}}$
denote the weight matrix and bias vector of this layer, respectively. The
function $\sigma$ represents the activation function. The Rectified Linear
Unit (ReLU) is used.

A salient feature of the Agri-GNN is its intermediary layers, which not only
collate features from neighboring nodes but also incorporate skip connections
to preserve the essence of the original node’s features. The aggregation of
features from neighboring nodes in these layers is depicted in equation (7).

$\mathbf{m}_{i}^{(l)}=\text{AGGREGATE}\left(\\{\mathbf{h}_{j}^{(l-1)}\,\forall
j\in\text{N}(i)\\}\right)$ (7)

Subsequent to this aggregation, the features undergo a transformation, as
expressed in equation (8).

$\mathbf{h}_{i}^{(l)}=\sigma\left(\mathbf{W}^{(l)}\cdot\text{CONCAT}(\mathbf{h}_{i}^{(l-1)},\mathbf{m}_{i}^{(l)})+\mathbf{b}^{(l)}\right)$
(8)

Here, $\text{N}(i)$ represents the neighboring nodes of node $i$, and
$\mathbf{W}^{(l)}$ and $\mathbf{b}^{(l)}$ signify the weight matrix and bias
vector for layer $l$, respectively.

The architecture culminates in the final layer, a pivotal component that
produces the model’s refined output. This layer mirrors the operations of the
intermediary layers in aggregating neighboring node features, but
distinguishes itself by excluding the addition of original node features. The
aggregation of features in this layer is portrayed in equation (9).

$\mathbf{m}_{i}^{(4)}=\text{AGGREGATE}\left(\\{\mathbf{h}_{j}^{(3)}\,\forall
j\in\text{N}(i)\\}\right)$ (9)

The subsequent transformation, harnessing the aggregated features to yield the
final output, is described in equation (10).

$\mathbf{h}_{i}^{(4)}=\sigma\left(\mathbf{W}^{(4)}\cdot\text{CONCAT}(\mathbf{h}_{i}^{(3)},\mathbf{m}_{i}^{(4)})+\mathbf{b}^{(4)}\right)$
(10)

To ensure stability in convergence and enhance generalization, each hidden
layer is succeeded by a batch normalization step. After normalization, dropout
regularization with a rate of $p=0.5$ is employed to combat overfitting, as
described by equation (11):

$\mathbf{h}_{i}^{(l)}=\text{Dropout}(\mathbf{h}_{i}^{(l)},p=0.5)$ (11)

The final output of the model is a prediction of the yield of the given
node(s).

The final Agri-GNN model is designed to take in initial node features
$\mathbf{x}_{i}$ and produce an output $\mathbf{o}_{i}$ for each node,
representing the predicted yield. The model is summarized as:

$\mathbf{o}_{i}=\text{{Agri-GNN}}(\mathbf{x}_{i};\mathcal{G},\Theta)$ (12)

Here, $\Theta$ stands for the set of all learnable parameters within the
model.

The model’s performance is evaluated using a Mean Squared Error (MSE) loss
between the predicted crop yields $\mathbf{o}_{i}$ and the actual yields.
Optimization is carried out via the Adam optimizer, and hyperparameters such
as learning rates are fine-tuned for optimal performance.

Agri-GNN’s architecture is summarized in Figure 1.

Figure 1: Summary of Model Architecture

##  5 Applications and Experimental Results

The Agri-GNN framework is now applied to plot fields in Ames, Iowa. Agri-GNN’s
performance is compared to various baseline yield prediction models to
accurately gauge its potential.

###  5.1 Data Collection and Processing

Data on soil attributes, hyperspectral reflectance, seed yield, and weather
were collected systematically over two consecutive years, 2020 and 2021. The
data collection covered both Preliminary Yield Trials (PYT) and Advanced Yield
Trials (AYT), and each year involved multiple trial locations.

####  5.1.1 Trial Design

The Preliminary Yield Trials (PYT) were designed using a row-column
configuration. In this layout, each plot had a width of 1.52m and a length of
2.13m. Plots within each row were interspaced by 0.91m. The Advanced Yield
Trials (AYT) followed a similar row-column design, but with each plot
measuring 1.52m in width and 5.18m in length. The interspacing between plots
remained consistent at 0.91m for both trial types.

####  5.1.2 Soil Data Collection

Digital soil mapping techniques were used to identify ten specific soil
attributes, supplementing the collection of hyperspectral data. Soil cores
were extracted down to a depth of 15 cm following a 25m grid sampling pattern,
using specialized soil probes. Digital soil maps were then generated with
3mx3m pixel resolution using the Cubist regression machine learning
algorithmKhaledian and Miller (2020).

For each plot, boundaries were outlined using polygon shape files, and the
average value of each soil feature was calculated. To improve data
reliability, a 3x3 moving mean was computed for each soil attribute within the
plot, and this smoothed value was then used for more detailed analyses. The
assessed soil features encompassed Calcium (Ca), Cation Exchange Capacity
(CEC), Potassium (K), Magnesium (Mg), Organic Matter (OM), Phosphorus (P1),
Percent Hydrogen (Ph), and proportions of Clay, Sand, and Silt.

####  5.1.3 Hyperspectral Reflectance Data

Hyperspectral reflectance data for each plot was captured using a Thorlabs
CCS200 spectrometer, based in Newton, NJ. The methodology adheres to the
system outlined in the study by Bai et al. (2016). Spectral data was collected
annually at three distinct time points: T1, T2, and T3, captured sequentially,
covering wavelengths from 200 nm to 1000 nm, as illustrated in Figure 2.

Figure 2: Reflectance plot showcasing wavelengths from 400 nm to 1000 nm. The
depicted red, green, and blue bands are illustrative of the visible spectrum
wavelengths.

Particular emphasis is placed on the T3 timepoint, which has been identified
as having superior feature importance values according to preliminary data
assessments. As vegetation nears physiological maturity, the correlation
between hyperspectral reflectance data and crop yield becomes more
significant.

Fifty-two vegetation indices were calculated based on the collected
hyperspectral reflectance values, following the methodology detailed in the
study by Li et al. (2022). A comprehensive list of these indices is available
in Appendix A. The distribution of the collected data across the four fields
is summarized in Table 1.

Table 1: Number of datapoints in each of the four fields. Field No. | Number of observations
---|---
1 | 770
2 | 912
3 | 800
4 | 679

It is interesting to note that many of the vegetation indices show a strong
correlation with each other, especially when considering their underlying
mathematical formulations. As depicted in Figure 3, certain pairs of indices
exhibit notably high correlation values, suggesting that they might be
capturing similar information about the vegetation. This redundancy could be
attributed to the fact that many vegetation indices are derived from the same
spectral bands, primarily the red and near-infrared (NIR) regions, which are
known to be indicative of plant health and vigor. However, while two indices
might be highly correlated, they might still provide unique insights to
different vegetation properties, and thus, all of the vegetation indices were
kept as features in the construction of the dataset.

Figure 3: Correlation of selected vegetation indices is shown.

####  5.1.4 Yield Data

Seed yield data at the plot level was collected using an Almaco small plot
combine, headquartered in Nevada, IA. For consistency, all yield data was
normalized to a moisture content of 13% and then converted to kilograms per
hectare (Kg/ha). The data collection process was organized in blocks,
mirroring the layout established by the breeding program. This arrangement
grouped the plots based on their genetic lineage and corresponding maturity
groups.

Prior to the computation of vegetation indices, a rigorous data preprocessing
phase was undertaken to omit anomalies and outliers. The steps encapsulated:

  1. 1.

Omission of all observations for band values falling below 400 nm due to
detected anomalies in these bands’ readings.

  2. 2.

Exclusion of datapoints with negative hyperspectral values within the 400 nm
to 1000 nm band range.

  3. 3.

Removal of datapoints showcasing negative seed yield values.

###  5.2 Graph Construction

Once the dataset was pruned to retain only the relevant columns, we started
the task of graph construction, as explained in detail in Section 4.1. The
nodes of the graph represented individual data points from the dataset, while
the edges encoded two primary relationships: spatial proximity and genotype
similarity.

The geospatial coordinates (‘Latitude‘ and ‘Longitude‘) of each data point
facilitated the computation of pairwise distances between them. By harnessing
this distance matrix, we established a threshold—specifically the 3rd
percentile of non-zero distances—and used it as a criterion to draw edges
between nodes. In essence, if the spatial distance between two nodes was less
than this threshold, an edge was drawn between them.

Beyond spatial relationships, our graph also recognized the significance of
genetic similarities between data points. This was achieved by drawing edges
between nodes that shared the same genotype, as denoted by the ‘Population‘
column. By adopting this strategy, our graph was enriched with edges that
encapsulated intrinsic genetic relationships, which has an influence on
agricultural yield (Shook et al., 2021a)

To facilitate subsequent graph-based deep learning, we represented our graph
using the PyTorch Geometric framework. This involved a series of
transformations. Firstly, the combined spatial and genotype edges were
aggregated. Secondly, the dataset was processed to handle missing values, by
imputing them with column-wise means, and categorical columns were one-hot
encoded. The final graph representation incorporated node features (derived
from the processed dataset), the aggregated edges, and the target label
(‘Yield‘).

The resulting graph, thus constructed, served as the foundation for our Agri-
GNN experiments.

###  5.3 Neural Architecture

The model was developed using the PyTorch framework (Paszke et al., 2017). For
the specific needs of graph-based neural networks, we turned to the PyTorch
Geometric (PyG) library (Fey and Lenssen (2019)). This library is a
comprehensive toolkit, providing a range of architectures and tools tailored
for graph neural networks.

Our model, Agri-GNN, is an augmentation of the conventional GraphSAGE
architecture Hamilton et al. (2017). It features four GraphSAGE convolutional
layers. The initial layer transitions input features to hidden channels. The
two intermediary layers, enhanced with skip connections, amplify the model’s
capacity to discern both rudimentary and advanced patterns in the data. The
final layer is designed to yield the model’s output. To ensure the stability
and efficiency of training, batch normalization is applied following each
convolutional layer. Furthermore, to mitigate the risk of overfitting, dropout
regularization is integrated after each batch normalization, with a rate of
$0.5$.

The dimensionality of the input features was determined dynamically based on
the dataset. The model was trained for $500$ epochs, and was monitored to
gauge the model’s performance accurately and take measures against potential
overfitting.

An $80-20$ split was utilized, where $80\%$ of the farm nodes were randomly
chosen to be part of the training dataset,

###  5.4 Hyperparameter Tuning

Hyperparameter tuning was critical to ensuring we have the optimal model. We
conducted an exhaustive exploration of various hyperparameter combinations to
pinpoint the most conducive setting for our model. The hyperparameters that we
varied are summarized in Table 2.

Hyperparameter | Values Explored
---|---
Learning Rates | $0.001,0.005,0.01,0.02$
Hidden Channels | $32,64,128$
Dropout Rates | $0.3,0.5,0.7$
Table 2: Hyperparameters and their explored values

The best hyperparameters can be seen in Table 3.

Hyperparameter | Best Value
---|---
Learning Rate | $0.02$
Hidden Channels | $32$
Dropout Rate | $0.3$
Table 3: Optimal hyperparameters after tuning

##  6 Model Performance and Results

We now show the results of Agri-GNN on the described dataset. The results are
compared with two baselines on the same dataset: a K-Nearest Neighbors Model
and an Ensemble Machine Learning model (Chattopadhyay et al., 2023).

We first visualize the embeddings derived from our graph neural network model.
To better understand the spatial distribution of these embeddings, we employed
the t-Distributed Stochastic Neighbor Embedding (t-SNE) algorithm for
dimensionality reduction (Hinton and Roweis (2002)). The embeddings, initially
generated by the forward method of our model, were reduced to two dimensions
using t-SNE. As shown in Figure 4, each point represents a node in the graph,
and the spatial arrangement captures the similarity between node embeddings.
Distinct clusters and patterns in the visualization indicate nodes with
similar features within the graph. The embeddings can be seen in Figure 4.

Figure 4: t-SNE visualization of graph node embeddings.

From Figure 4, we see that multiple distinct graphs may possibly be formed
from the data, particularly when subsets of the data are not related in any
spatial or genotypic way. Agri-GNN is designed to be able to learn the
inherent patterns in each of these subgraphs, while further increasing the
accuracy of the main base model.

The performance of Agri-GNN was assessed using standard regression metrics, as
presented in table 4:

Metric | Agri-GNN Value
---|---
Root Mean Squared Error (RMSE) | $4.565$
Mean Absolute Error (MAE) | $3.590$
Coefficient of Determination ($R^{2}$) | $0.876$

Table 4: Performance metrics of the Agri-GNN

model

Agri-GNN achieves a Root Mean Squared Error (RMSE) of $4.565$ on this dataset.
The Mean Absolute Error (MAE) is $3.590$. The model has an $R^{2}$ value of
$0.876$. These results show significant improvement over the baselines used.

The K-Nearest Neighbor (K-NN) Algorithm is used as the first baseline model.
The K-NN model predicts the yield of a node based on the average yield of its
$k$ nearest neighbors based on latitude and longitude. This model is optimized
by performing a grid search to find the best value of $k$ from a range of 1 to
20. The optimized K-NN model used $k=18$ as the number of neighbors for making
predictions.

The performance metrics of the optimized K-NN model are presented in Table 5.
The model achieved a Root Mean Squared Error (RMSE) of $12.93$, a Mean
Absolute Error (MAE) of $10.33$, and a Coefficient of Determination ($R^{2}$)
of $0.026$. The Agri-GNN model shows superior performance to the baseline. The
metrics signify a substantial enhancement over the baseline K-NN model,
demonstrating the effectiveness of Agri-GNN in yield prediction.

Metric | Baseline K-NN Value
---|---
Root Mean Squared Error (RMSE) | $12.93$
Mean Absolute Error (MAE) | $10.33$
Coefficient of Determination ($R^{2}$) | $0.026$
Table 5: Performance metrics of K-NN baseline model ($k=18$

Agri-GNN’s prediction error and $R^{2}$ shows significant potential when
compared to recent work on a similar dataset (Chattopadhyay et al., 2023),
where the highest $R^{2}$ value achieved was less than $.8$. Such an
enhancement underscores the potential benefits of graph-based neural networks,
especially when handling datasets with inherent relational structures.

In Figure 5, we present a comparison between the Actual and Predicted Yields.
This graphical representation provides a comprehensive insight into the
accuracy and precision of our model’s predictions. An interesting observation
from Figure 5 is the remarkable accuracy of the model’s predictions for yields
ranging between 50 and 90. The data points in this range cluster closely
around the line of perfect agreement, suggesting that the model is
particularly adept at predicting yields within this interval. This is
noteworthy as it indicates that our model is not only reliable in general but
also exhibits enhanced performance when predicting yields in this specific
range.

Figure 5: The Actual vs. Predicted Yield of our predictions. The red line
signifies a perfect model. The model achieves an $R^{2}$ value of $0.876$

###  6.1 Scalability and Robustness

In the previous section, an application of Agri-GNN is outlined, focusing on
agricultural yield prediction in farms located in Ames, Iowa. The results
demonstrate the model’s capability to efficiently and accurately predict
yields based on geographical coordinates. This section delves into the
scalability and robustness of the Agri-GNN framework, discussing its potential
application on a broader scale, both nationally and internationally.

Agri-GNN is designed with scalability in mind, allowing for seamless
integration and deployment in diverse farming contexts across the world. The
model’s adaptability to various geographical and environmental conditions
ensures consistent and reliable yield predictions irrespective of the
location. The use of cloud computing resources and distributed computing
frameworks enhances the scalability of Agri-GNN, enabling real-time processing
and analysis of large datasets encompassing numerous farms across different
geographical locations.

Robustness is at the core of the Agri-GNN framework. The model employs
advanced machine learning algorithms and techniques to ensure stability and
reliability in yield predictions, even in the face of data variability and
uncertainties. The robust nature of Agri-GNN ensures uninterrupted and
consistent performance, bolstering the confidence of farmers and stakeholders
in the accuracy and reliability of the yield predictions provided by the
model. Moreover, the continuous learning and adaptation capabilities of Agri-
GNN further enhance its robustness, ensuring it remains at the forefront of
agricultural yield prediction technology. Unfortunately, Agri-GNN is optimized
to work best with large datasets where ample information from previous
experiments is available. In many farming contexts, there is not enough
information to be able to train $Agri-GNN$ to have reliable performance
(Wiseman et al., 2019). Further research should explore how Agri-GNN can be
optimized to perform well in instances where a lack of ample farming data may
be a problem.

##  7 Conclusion

In an ever-evolving agricultural landscape marked by climatic uncertainties,
the pressing need for accurate and holistic crop yield predictions has never
been greater. This study introduced Agri-GNN, a pioneering approach that
harnesses the power of Graph Neural Networks to provide a comprehensive
representation of the agricultural ecosystem. Unlike traditional models that
often operate in silos, Agri-GNN’s strength lies in its ability to synthesize
diverse data modalities into a unified framework, capturing the intricate
interplay between genetic, environmental, and spatial factors.

Agri-GNN transcends conventional methodologies by viewing the agricultural
ecosystem as an interconnected network, where crops aren’t just passive
entities but active nodes influenced by their immediate surroundings and
broader contexts. This perspective, combined with GNN’s superior data
processing capabilities, enables Agri-GNN to deliver predictions that are both
granular and holistic.

Furthermore, Agri-GNN’s modular design ensures its relevance in a rapidly
changing agricultural sector, allowing for seamless integration of new data
sources and insights. Its precision agriculture approach not only aids in
enhancing productivity but also paves the way for sustainable practices that
respect both economic and environmental considerations.

Our applications of the Agri-GNN framework’s capabilities to the plot fields
of Ames, Iowa show promising results, even obtaining better performance than
the results obtained in Chattopadhyay et al. (2023). Agri-GNN’s performance
metrics, including $RMSE$, $MAE$, and $R^{2}$ highlighted its proficiency in
yield prediction. Notably, the model showcased significant improvements over
existing models, reaffirming the potential of graph-based neural networks in
agricultural applications. The t-SNE visualizations further provided insights
into the model’s embeddings, reinforcing the cohesive and interconnected
nature of the data.

In summary, Agri-GNN represents a paradigm shift in agricultural modeling. By
capturing the complexity and interconnectedness of farming systems, it offers
a fresh lens through which to view and address the multifaceted challenges of
modern agriculture. As we stand at the crossroads of traditional practices and
technological innovation, Agri-GNN serves as a beacon, guiding the way towards
informed, sustainable, and resilient agricultural futures.

##  8 Acknowledgments

The authors thank Joscif Raigne, Dr. Baskar Ganapathysubramanian, and Dr.
Soumik Sarkar for their invaluable feedback on the draft manuscript.

The authors thank Joscif Raigne for his help in the construction of spatial
data for the field experiments.

The authors thank Shannon Denna and Christopher Grattoni for their support.

The authors thank staff and student members of SinghSoybean group at Iowa
State University, particularly Brian Scott, Will Doepke, Jennifer Hicks, Ryan
Dunn, and Sam Blair for their assistance with field experiments and
phenotyping.

This work was supported by the Iowa Soybean Association, North Central Soybean
Research Program, USDA CRIS project IOW04714, AI Institute for Resilient
Agriculture (USDA-NIFA #2021-647021-35329), COALESCE: COntext Aware LEarning
for Sustainable CybEr-Agricultural Systems (CPS Frontier #1954556), Smart
Integrated Farm Network for Rural Agricultural Communities (SIRAC) (NSF S & CC
#1952045), RF Baker Center for Plant Breeding, and Plant Sciences Institute.

##  9 Author Contributions

Aditya Gupta conceptualized and designed the proposed model, developed the
methodology, conducted the data analysis, constructed and implemented the
Agri-GNN, provided visualization of the results, and took primary
responsibility for writing, revising, and finalizing this paper. All authors
have read, reviewed, and agreed to the published version of the manuscript.

##  10 Conflict of Interest

The authors declare no conflict of interest.

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##  Appendix A Table of Vegetation Indices

Table 6: Summary of the 52 vegetation indices. Here Tx denote the
hyperspectral reflectance value at x nm.

Full form | Spectral Index/Ratio | Formula
---|---|---
Curvature index | Cl | $T675\times T690/T683^{2}$
Chlorophyll Index red-edge | Clre | $T750/T710-1$
| Datt1 | $(T850-T710)/(T850-T680)$
| Datt4 | $T672/(T550\times T708)$
| Datt6 | $T860/(T550\times T708)$
Double difference index | DDI | $(T749-T720)-(T701-T672)$
Double peak index | DPI | $(T688+T710)/T697^{2}$
Gitelson2 |  | $(T750-T800)/(T695-T740)-1$
Green normalized difference vegetation index | GNDVI | $(T750-T550)/(T750+T550)$
Modified chlorphyll absorption ratio index | MCARI | $[(T700-T670)-0.2(T700-T550)](T700/T670)$
| MCARI3 | $[(T750-T710)-0.2(T750-T550)](T750/T715)$
Modified normalized difference | MND1 | $(T800-T680)/(T800+T680-2\times T445)$
| MND2 | $(T750-T705)/(T750+T705-2\times T445)$
Modified simple ratio | mSR | $(T800-T445)/(T680-T445)$
Modified simple ratio 2 | mSR2 | $(T750/T705-1)/(\sqrt{T750/T705+1})$
MERIS terrestrail cholrophyll index | MTCI | $(T754-T709)/(T709-T681)$
Modified traingular vegetation index 1 | MTVI1 | $1.2[1.2(T800-T550)-2.5(T670-T550)]$
Normalized difference 550/531 | ND1 | $(T550-T531)/(T550+T531)$
Normalized difference 682/553 | ND2 | $(T682-T553)/(T682+T553)$
Normalized difference chlorophyll | NDchl | $(T925-T710)/(T925+T710)$
Normalized difference red edge | NDRE | $(T790-T720)/(T790+T720)$
Normalized difference vegetation index | NDVI1 | $(T750-T650)/(T750+T650)$
| NDVI2 | $(T750-T550)/(T750+T550)$
| NDVI3 | $(T750-T710)/(T750+T710)$
Normalized pigment cholrophyll index | NPCL | $(T680-T430)/(T680+T430)$
Normalized difference pigment index | NPQI | $(T415-T435)/(T415+T435)$
Optimized soil-adjusted vegetation index | OSAVI | $(1+0.16)(T800-T670)(T800+T670-0.16)$
Plant biochemical index | PBI | $T810/T560$
Plant pigment ratio | PPR | $(T550-T450)/(T550+T450)$
Physiological reference index | PRI | $(T550-T530)/(T550+T530)$
Pigment-specific normalized difference | PSNDb1 | $(T800-T650)/(T800+T650)$
| PSNDc1 | $(T800-T500)/(T800+T500)$
| PSNDc2 | $(T800-T470)/(T800+T470)$
Plant senescence reflectance index | PSRI | $(T678-T500)/T750$
Pigment-specific simple ratio | PSSRc1 | $T[800]/T[500]$
| PSSRc2 | $T[800]/T[740]$
Photosynthetic vigor ratio | PVR | $T([550]-T[650])/(T[550]+T[650])$
Plant water index | PWI | $T970/T900$
Renormalized difference vegetation index | RDVI | $(T800-T670)/\sqrt{(T800+T670)}$
Red-edge stress vegatation index | RVSI | $((T718+T748)/2)-T733$
Soil-adjusted vegatation index | SAVI | $1.16((T800-T670)/(T800+T670+0.16))$
Structure intensive pigment index | SIPI | $(T800-T445)/(T800+T680)$
Simple ratio | SR1 | $T430/T680$
| SR2 | $T440/T740$
| SR3 | $T550/T672$
| SR4 | $T550/T750$
Disease -water stress index 4 | DSWI-4 | $T550/T680$
Simple ratio pigment index | SRPI | $T430/T680$
Transformed chlorophyll absorption ratio | TCARI | $3((T700-T670)-0.2(T700-T550)(T700/T670))$
Traingular cholrophyll index | TCI | $1.2(T700-T550)-1.5(T670-T550)\times\sqrt{T700/T670}$
Triangular vegetation index | TVI | $0.5(120(T750-T550)-200(T670-T550))$
Water band index | WBI | $T970/T902$