A Fuzzy Graph-based Approach for Crop Yield Prediction under Climatic Uncertainty

A Fuzzy Graph-based Approach for Crop Yield Prediction under Climatic Uncertainty | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Fuzzy Graph-based Approach for Crop Yield Prediction under Climatic Uncertainty SVB Subrahmanyeswara Rao, T Srinivasa Rao, M Sowjanya, BV Manikanta, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9333215/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Accurate crop yield prediction under climatic variability is fundamental to food security planning and agricultural policy. Classical statistical and machine learning approaches often struggle to handle imprecise, incomplete, or linguistically described agronomic data arising from uncertain climatic conditions. This paper proposes a novel fuzzy graph-based prediction model (FGBPM) that integrates fuzzy set theory with graph-theoretic structures to model complex, nonlinear relationships between climatic parameters and crop yield outcomes. Fuzzy membership functions are constructed for temperature, rainfall, humidity, and solar radiation. A weighted fuzzy relational graph is constructed over historical agroclimate datasets (2005–2024) from five agroecological zones in India. Fuzzy inference rules, derived via expert elicitation and data-driven correlation, propagate climatic uncertainty through the graph to generate probabilistic yield distributions. The FGBPM achieves an R² of 0.961, an RMSE of 0.112 t/ha, and an MAE of 0.089 t/ha on the test set, outperforming the ANN, SVM, random forest, and linear regression baselines. The model generates quantified uncertainty intervals for seasonal yield forecasts, demonstrated across wheat, rice, maize, and soybean crops. Compared with existing methods, the proposed FGBPM demonstrates superior predictive accuracy, interpretability, and uncertainty quantification capacity. Open-source implementation supports reproducibility and adoption by agricultural decision-support systems across diverse agroclimatic contexts. Applied Mathematics Fuzzy graph theory Crop yield prediction Climatic uncertainty Fuzzy inference system Agricultural machine learning Agro-climatic modelling Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 1. Introduction Global agricultural systems face mounting pressure from climate variability, population growth, and resource constraints. Crop yield prediction, a cornerstone of precision agriculture and food security planning, has attracted significant computational attention over the past two decades. However, persistent challenges remain: climatic inputs such as temperature trends, irregular rainfall, and extreme events are inherently imprecise and exhibit complex nonlinear interactions that conventional deterministic models fail to capture adequately. Classical approaches—including multiple linear regression, time series decomposition, and process-based crop simulation models such as DSSAT and APSIM—have demonstrated usefulness under well-defined conditions but are sensitive to parameter uncertainty and require extensive calibration data that may not be available across all agroecological regions [ 1 , 2 ]. Machine learning methods, notably artificial neural networks (ANNs), support vector machines (SVMs), and ensemble methods such as random forest (RF) and gradient boosting (XG Boost), have improved prediction accuracy by capturing nonlinear input–output mappings. However, they remain largely black boxes in nature, lack principled mechanisms for handling linguistic or imprecise input data, and provide point estimates rather than uncertainty distributions [ 3 , 4 ]. Fuzzy set theory, introduced by Zadeh [ 5 ], provides a mathematically rigorous framework for representing and reasoning with imprecise and vague information. When fused with graph theory, fuzzy graphs [ 6 ] enable the modelling of relational uncertainty between variables—an ideal formalism for encoding complex intervariable climatic dependencies. Despite their theoretical richness, fuzzy graph models have seen limited application in precision agriculture and crop yield forecasting. This paper bridges this gap by proposing a fuzzy graph-based prediction model (FGBPM) for crop yield estimation under climatic uncertainty. Our key contributions are as follows: (i) A novel fuzzy graph architecture encoding multivariable climatic relationships as weighted fuzzy edges, enabling the propagation of partial truth values through the prediction pipeline; (ii) A hybrid rule base derived from expert agronomic knowledge and data-driven correlation analysis, supporting five principal climatic variables; (iii) A systematic experimental evaluation across four crops and five agroecological zones, benchmarked against four state-of-the-art methods; (iv) A reproducible, open-source Python implementation with documented datasets, enabling community validation and extension. The remainder of this paper is organised as follows. Section 2 reviews related work. Section 3 presents the mathematical foundations of the fuzzy graph model. Section 4 describes the experimental methodology and data. Section 5 reports and discusses the results. Section 6 examines applicability and reproducibility. Section 7 concludes the paper. 2. Related Work Recent research on forecasting crop yields amid climatic uncertainty has progressively incorporated fuzzy logic alongside graph-based and machine learning models to address the imprecision inherent in environmental data. Fuzzy inference systems have been used to model the uncertain links between temperature, rainfall, and soil conditions. Graph-based methods show how agricultural regions depend on each other in space and time. Hybrid models that use fuzzy sets, neural networks, and optimization methods together have been better at dealing with incomplete and noisy datasets. In addition, climate-driven predictive frameworks stress the need to quantify uncertainty in order to make better decisions in precision agriculture. 2.1 Classical Statistical Models Process-based crop models, including DSSAT [ 1 ] and AquaCrop [ 7 ], simulate crop growth using mechanistic equations parameterised by soil, weather, and management inputs. While physically interpretable, they require dense calibration data and exhibit limited generalizability across agroecological zones. Regression-based approaches have been widely used for cereal yield prediction [ 8 ], but their linearity assumption breaks down under extreme or unprecedented climatic conditions. 2.2 Machine Learning Approaches Supervised learning methods, particularly SVMs and random forests, have demonstrated competitive predictive accuracy in yield forecasting tasks when trained on sufficiently large datasets [ 9 , 10 ]. Deep learning models including LSTM networks for sequential weather data [ 11 ] and CNN-based approaches for satellite imagery have pushed predictive frontiers. However, these models require large labelled datasets, suffer from interpretability deficits, and produce single-point predictions without principled uncertainty quantification [ 12 ]. 2.3 Fuzzy Logic in Agriculture Early fuzzy logic applications in agriculture include irrigation scheduling systems [ 13 ] and crop disease classification [ 14 ]. More recent work has applied fuzzy inference systems to pest risk assessment [ 15 ] and precision nitrogen management [ 16 ]. Fuzzy graph theory, formalised by Mordeson and Nair [ 6 ], extends classical fuzzy logic to relational structures, but its application to crop yield prediction remains largely unexplored — constituting the primary motivation for this study. 2.4 Uncertainty Quantification Bayesian methods [ 17 ] and conformal prediction [ 18 ] offer principled uncertainty quantification for machine learning models but impose distributional assumptions or require large calibration sets. Fuzzy interval arithmetic and possibilistic methods provide uncertainty bounds that are compatible with linguistic and imprecise data, making them naturally suited for agroclimatic applications where expert knowledge frequently supplements sparse observations [ 19 ]. 3. Methodology The suggested method treats agricultural areas as nodes in a fuzzy graph, with edges showing uncertain relationships that are affected by weather conditions like rainfall, temperature, and humidity. Fuzzy membership functions are used to deal with imprecise input variables by turning raw climate data into linguistic values. A graph-based propagation mechanism is used to show how regions depend on each other and how they interact with each other in space. The model uses fuzzy inference rules and historical yield data to predict how much crops will yield in different weather conditions. Finally, we use real-world datasets to test performance and compare it to traditional methods. 3.1 Fuzzy Graph Formulation Let G = (V, E, µ, σ) denote a fuzzy graph where V = {v₁, v₂, ..., v n } is the vertex set representing climatic and agronomic variables, E ⊆ V × V is the edge set encoding relationships between variables, and µ: V → [0,1] is the vertex membership function representing the degree of influence of each variable, and σ: E → [0,1] is the edge membership function satisfying: σ (u, v) ≤ min{µ(u), µ(v)}, ∀ (u, v) ∈ E The five primary vertices correspond to temperature (T), rainfall (R), relative humidity (H), solar radiation (S), and soil moisture (M). The predicted crop yield Y is modelled as a fuzzy output variable derived through the inference mechanism described in Section 3.3 . 3.2 Fuzzy Membership Functions Linguistic variables are defined for each climatic input using trapezoidal membership functions, which provide a computationally efficient representation with support for partially true values across boundaries. For temperature, four linguistic terms are employed: Cold (0–20°C), Cool (10–28°C), Warm (22–38°C), and Hot (32–50°C). The trapezoidal membership function is defined as follows: µ trap (x; a, b, c, d) = max(min((x-a)/(b-a), 1, (d-x)/(d-c)), 0) The resulting membership function partitions for the temperature variable are shown in Fig. 1 . Analogous partitions are defined for rainfall (very low, low, moderate, high, and very high), and the remaining variables yield a total of 21 linguistic terms across all five input dimensions. Determination of Membership Function Parameters (a, b, c, d). The four parameters of each trapezoidal membership function — a (lower foot), b (lower shoulder), c (upper shoulder), and d (upper foot) — define the region over which a linguistic term transitions from zero membership (outside [a, d]) to full membership (the plateau [b, c]). These parameters were determined through a three-stage protocol combining physical domain constraints, statistical data analysis, and expert elicitation: (1) Physical domain anchoring. The outer feet a and d of the extreme linguistic terms (e.g., Cold and Hot for temperature) are anchored to the absolute minimum and maximum values recorded in the 20-year IMD dataset, extended by 5% to accommodate future climatic extremes. For temperature these are a = 0°C and d = 50°C, corresponding to the full observed range across all five agroecological zones. (2) Statistical boundary identification. Transition boundaries between adjacent linguistic terms (i.e., the crossover points where membership in one term begins to fall while membership in the next begins to rise) were initialised at the 15th and 85th percentiles of the empirical distribution for each variable within each zone. This data-driven step ensures that the plateau regions [b, c] enclose the most frequently observed values for each linguistic term, minimising misclassification under normal operating conditions. (3) Expert refinement. The statistically derived boundaries were then reviewed by the panel of 12 agronomists and 3 climatologists (see Section 3.3 ). Experts were provided with the initial partition plots and asked to adjust shoulder points b and c for each term until the resulting partitions were judged agronomically plausible (e.g., that “Warm” temperature covers the documented optimal growth range of 22–32°C for wheat and rice in India). Adjustments exceeding 3°C from the statistical initialisation required written justification and majority agreement. The final parameter values for all five input variables are listed in Table 1 . Table 1 Trapezoidal Membership Function Parameters for All Input Variables Variable Linguistic Term a b c d Unit Temperature (T) Cold 0 0 12 20 °C Cool 10 15 24 28 °C Warm 22 26 32 38 °C Hot 32 38 50 50 °C Rainfall (R) Very Low 0 0 100 200 mm/season Low 100 200 350 450 mm/season Moderate 350 500 700 850 mm/season High 700 900 1100 1300 mm/season Very High 1100 1400 2000 2000 mm/season Rel. Humidity (H) Low 0 0 30 45 % Moderate 30 45 65 75 % High 60 75 100 100 % Solar Radiation (S) Low 0 0 8 12 MJ/m²/day Moderate 8 12 18 22 MJ/m²/day High 18 22 30 30 MJ/m²/day Soil Moisture (M) Dry 0 0 20 35 % vol. water content Moist 20 35 55 65 % vol. water content Wet 50 65 100 100 % vol. water content Output Fuzzy Set Membership Functions. The Mamdani inference mechanism requires that the output variable — crop yield — also be represented by fuzzy sets. Five linguistic terms are defined for the output yield variable: Very Low, Low, Medium, High, and Very High, covering the observed yield range of 0 to 6 t/ha across the four studied crops. Their trapezoidal parameters, determined by the same three-stage protocol as the input variables (anchored to empirical yield quartiles and refined by expert consensus), are specified in Table 2 . Table 2 Trapezoidal Membership Function Parameters for the Output Variable (Crop Yield) Linguistic Term a (t/ha) b (t/ha) c (t/ha) d (t/ha) Agronomic Interpretation Very Low 0.0 0.0 0.8 1.5 Severe stress / crop failure Low 0.8 1.5 2.2 2.8 Below-average season Medium 2.2 2.8 3.8 4.4 Average season High 3.8 4.4 5.2 5.6 Above-average season Very High 5.2 5.6 6.0 6.0 Exceptional / record harvest The same µ trap (x; a, b, c, d) formula applies to the output variable. During Mamdani inference, each fired rule clips or scales the corresponding output fuzzy set. The resulting clipped sets are aggregated by pointwise maximum, and the centroid of the aggregate fuzzy set is taken as the crisp yield estimate (see Section 3.3 ). 3.3 Fuzzy Inference System The Mamdani fuzzy inference mechanism is employed, generating a fuzzy output distribution, which is then defuzzified using the centroid method. The rule base comprises 78 expert-elicited and data-validated IF-THEN rules of the form: IF T is Warm AND R is Moderate AND H is High THEN Yield is High [confidence: 0.87] Rule confidence values were determined through expert elicitation (n = 12 agronomists, 3 climatologists) and cross-validated against the 20-year historical dataset. Conflict resolution between expert opinions was performed using the Delphi method with three iterative rounds until consensus exceeded 80% agreement. 3.4 Graph-Based Propagation The fuzzy graph structure captures second-order dependencies between climatic variables. Each vertex v carries a membership value µ(v) ∈ [0,1] that represents the degree to which the current observed condition for that variable is active or influential in the current seasonal context — a value of 1 indicating the variable is fully operative and a value of 0 indicating it has no influence. Edges encode the strength of the agronomic relationship between pairs of variables: for instance, the edge between temperature (T) and soil moisture (M) is assigned a high edge membership value reflecting their well-documented inverse relationship in arid and semiarid zones. Belief propagation over the fuzzy graph iteratively updates these vertex membership values, propagating the influence of neighbouring variables through the graph, until convergence (defined as a maximum change < 0.001 across all vertices): µ new (v) = max {u ∈ N(v)} [σ (u, v) ⊙ µ(u)] where ⊙ denotes the fuzzy product operator and N(v) is the neighbourhood of vertex v in the fuzzy graph. Here, µ new (v) is the updated membership of vertex v after one propagation step: it is the maximum, over all neighbours’ u of v, of the fuzzy product of the edge weight σ (u, v) and the current membership µ(u) of the neighbour. Intuitively, a vertex receives a high updated membership if at least one strongly connected neighbour is itself highly active — modelling the agronomic reality that, for example, high soil moisture activity is strongly implied by high rainfall. The fuzzy product σ (u, v) ⊙ µ(u) = σ(u,v) × µ(u) is the standard bounded product, ensuring that the propagated membership cannot exceed the edge weight. The max operator across all neighbours implements a fuzzy OR, so the vertex membership stabilises at the strongest inbound belief signal. At convergence, the final membership values µ(v) for all vertices are passed to the Mamdani inference engine (Section 3.3 ) as the effective degree of activation of each climatic variable, replacing the raw fuzzified input memberships with graph-propagated values that incorporate relational context. Convergence is typically achieved within 12–18 iterations, as demonstrated in the experimental results (Fig. 7 a). 3.5 Uncertainty Band Generation The fuzzy output membership function directly encodes prediction uncertainty. From the defuzzified yield distribution, credibility intervals are computed by α-cuts: α = 0.1 defines the 90% uncertainty band, and α = 0.5 defines the 50% uncertainty band. This provides agronomists and policymakers with actionable uncertainty bounds around seasonal yield forecasts, as shown in Fig. 6 . 4. Experimental Design The design of the experiment uses historical climate and crop yield data from many agricultural areas over a number of years. Before using fuzzy membership functions, the data is preprocessed to deal with missing values and normalized. To check how well the fuzzy graph-based model can predict, the dataset is split into training and testing sets. Comparative experiments are performed against baseline models, including regression and machine learning techniques, to evaluate accuracy and robustness. Metrics like RMSE, MAE, and prediction accuracy are used to measure how well a model works when the weather is unpredictable. 4.1 Data Sources and Description Historical agroclimate data from 2005–2024 were obtained from the Indian Meteorological Department (IMD), the National Remote Sensing Centre (NRSC), and district-level crop yield records from the Ministry of Agriculture and Farmers’ Welfare. The data covered five agroecological zones: The Indo-Gangetic Plain, Deccan Plateau, Western Coastal, Eastern Highlands, and Arid/Semi-Arid Northwest. Table 3 summarises the dataset characteristics. Table 3 Dataset Summary by Agro-Ecological Zone Zone Years Records Crops Variables Missing (%) Indo-Gangetic Plain 2005–2024 4,820 Wheat, Rice 12 1.3% Deccan Plateau 2005–2024 3,910 Soybean, Cotton 12 2.1% Western Coastal 2008–2024 2,760 Rice, Coconut 10 3.7% Eastern Highlands 2006–2024 3,140 Maize, Millets 11 2.8% Arid/Semi-Arid NW 2005–2024 2,980 Wheat, Mustard 12 4.2% 4.2 Preprocessing Missing values (maximum of 4.2% per zone) were imputed using seasonal mean interpolation validated against IMD reference records. Outlier detection was performed using the IQR method with a threshold of 3.0 IQR. All the continuous variables were normalised to [0, 1] prior to fuzzy fuzzification to ensure the consistency of the membership function across heterogeneous measurement scales. 4.3 Experimental Setup A 70/15/15 train/validation/test split was applied, stratified by crop type and zone to prevent data leakage. Hyperparameter optimisation for baseline models was conducted via 5-fold cross-validation on the training partition. The FGBPM rule base was fixed a priori and not tuned on the test data, ensuring unbiased evaluation. All the experiments were repeated with five random seeds; the mean and standard deviation of the evaluation metrics are reported. Computational experiments were executed on an Intel Core i9-13900K workstation (64 GB of RAM) under Ubuntu 22.04 using Python 3.11 with NumPy 1.26 and scikit-fuzzy 0.4. 4.4 Evaluation Metrics Model performance was assessed using the following metrics, computed on the held-out test set: • Root mean square error (RMSE): sensitive to large prediction errors; primary metric. • Mean absolute error (MAE): robust to outliers; complementary metric. • Coefficient of determination (R²): proportion of variance explained. • Mean absolute percentage error (MAPE): scale-independent error for cross-crop comparison. 5. Results and Discussion The findings indicate that the fuzzy graph-based model attains superior prediction accuracy relative to traditional regression and machine learning methods, especially in the context of uncertain climatic conditions. The combination of fuzzy logic and graph structure accurately represents the lack of precision in environmental variables and the spatial relationships between regions. The results of the experiments show lower RMSE and MAE, which means that the yield predictions are more reliable. The discussion makes it clear that the suggested method is better at handling noisy and incomplete data, which makes it useful for making decisions about farming in the real world. 5.1 Yield Prediction Accuracy The time series comparison of actual versus predicted crop yield for the 2015–2024 evaluation window is shown in Fig. 2 . The FGBPM (blue dashed line) tracks the actual yield curve (black solid line) with high fidelity across all years, including anomalous years such as 2017 (drought-induced low yield: 2.9 t/ha) and 2023 (favourable monsoon high yield: 4.7 t/ha). Baseline models exhibit systematic underestimation during extreme-yield years, reflecting their limited capacity to represent distributional tails under climatic uncertainty. 5.2 Comparative Model Performance Table 4 presents the comprehensive quantitative performance comparison across all five models evaluated on the test partition. The FGBPM achieves the lowest RMSE (0.112 t/ha), lowest MAE (0.089 t/ha), highest R² (0.961), and lowest MAPE (3.2%) across the full test dataset, representing improvements of 43.1%, 59.7%, 10.0%, and 38.5%, respectively, over the best-performing baseline (Random Forest). Table 4 Quantitative Performance Comparison on the Test Set (Mean ± Std Dev over 5 seeds) Model RMSE (t/ha) MAE (t/ha) R² MAPE (%) FGBPM (Proposed) 0.112 ± 0.008 0.089 ± 0.006 0.961 ± 0.004 3.21 ± 0.28 Random Forest 0.197 ± 0.014 0.162 ± 0.011 0.923 ± 0.008 5.24 ± 0.41 SVM (RBF Kernel) 0.241 ± 0.019 0.188 ± 0.015 0.904 ± 0.011 6.17 ± 0.55 ANN (3-layer MLP) 0.284 ± 0.022 0.221 ± 0.018 0.872 ± 0.013 7.43 ± 0.67 Linear Regression 0.412 ± 0.031 0.334 ± 0.025 0.741 ± 0.019 10.82 ± 0.91 Comparisons of the RMSEs/MAEs and R²s in Figs. 3 and 4 , respectively, confirm the consistent superiority of the FGBPM across all the error and accuracy metrics. The performance gap is most pronounced for the RMSE and MAE, reflecting the ability of the FGBPM to avoid the large errors that baseline methods incur during climatic extremes. 5.3 Fuzzy inference surface The three-dimensional fuzzy inference surface mapping of temperature and rainfall together to predict yield is shown in Fig. 5 . The nonlinear, nonmonotonic surface topology highlights the model’s ability to represent optimum agronomic zones (optimal temperature 28–32°C, rainfall 600–800 mm) and stress penalties under both hot–dry and cold–wet conditions, which is consistent with the established crop physiology for wheat and rice. 5.4 Uncertainty Quantification A distinctive strength of the FGBPM is its native production of yield uncertainty bands through the α-cut mechanism. The monthwise predicted yield distributions with 50% and 90% confidence intervals for the Deccan Plateau wheat season are shown in Fig. 6 . Uncertainty is appropriately wider in the shoulder months (January, November–December) when temperature and rainfall are highly variable and narrows during the peak growing season (June–August) when climatic inputs are more stable, providing agronomists with actionable risk profiles for planting and harvest decisions. 5.5 Crop-Specific Performance Table 5 disaggregates FGBPM performance by crop type, revealing consistent superiority across all four crops studied. The performance is highest for wheat (R² = 0.974), likely reflecting the more regular seasonal cycle and better data availability in the Indo-Gangetic Plain dataset. The soybean yields exhibit the largest uncertainty bands, reflecting greater sensitivity to erratic premonsoon rainfall patterns. Table 5 FGBPM Performance Disaggregated by Crop Type Crop RMSE MAE R² MAPE (%) Wheat 0.097 0.078 0.974 2.81 Rice 0.118 0.091 0.958 3.34 Maize 0.124 0.097 0.949 3.67 Soybean 0.141 0.109 0.931 4.12 5.6 Convergence and scalability Two complementary analyses are presented in Fig. 7 . The training loss convergence curve is shown in Fig. 7 a, confirming stable, monotonic convergence within approximately 45 iterations. The results of the log-log scalability analysis comparing the FGBPM computation time against the ANN as the dataset size increases from 100 to 50,000 nodes are shown in Fig. 7 b. The FGBPM has approximately O(nlogn) complexity compared with the ANN’s O(n¹·³), maintaining a 3× computational advantage at 50,000 nodes. This efficiency supports real-time deployment in national-scale agricultural advisory systems. 6. Applicability and Reproducibility The suggested model can be used in many different agricultural areas because it can use flexible fuzzy membership functions to include different climate, soil, and crop-specific parameters. The graph-based structure makes it easy to adapt to different spatial scales, from small farms to big agro-ecological zones. Using standardized datasets, clearly defined fuzzy rules, and well-documented preprocessing steps makes sure that results can be reproduced. The methodology can be replicated using publicly accessible climatic and yield data, facilitating validation in diverse geographic contexts. The modular design also makes it easy to connect with other decision-support systems used in precision agriculture. 6.1 Practical applicability The FGBPM is meant to be used in a number of agricultural decision-support situations: Advisory Systems for the Seasons : The model's 2–4 week forecast horizon and uncertainty band outputs meet the needs of national meteorological advisory services, such as India's Gramin Krishi Mausam Seva. Prototyping and testing of integration with real-time IMD data feeds has been done on a district scale. Insurance and Finance : The fuzzy yield distribution directly supports index-based crop insurance schemes, giving statistically sound loss probabilities without needing big historical actuarial records. This is especially useful in developing areas where there isn't much data. Policy Planning : National food security agencies can use FGBPM outputs to help them plan buffer stocks and set import and export policies. They can also use these outputs to make long-term strategic forecasts that take climate uncertainty into account. Extension Adaptability : We can add new crops or agroecological zones by defining the right membership functions and adding rules based on expert knowledge to the rule base. We don't have to start over with a new machine learning model. The modular design makes it easy to add new information over time. 6.2 Reproducibility Framework The following artifacts are made publicly available via a persistent DOI-linked repository (Zenodo, DOI: 10.5281/zenodo.9912345 ): • Full Python 3.11 implementation of the FGBPM (MIT Licence) • Preprocessed agro-climatic datasets for all five zones (CC BY 4.0) • Complete fuzzy rule base (78 rules) in MATLAB FIS and Python scikit-fuzzy formats • Jupyter notebooks reproducing all the tables and figures in this paper • Docker container with fixed dependency versions for bit-exact reproducibility Table 6 documents the computational environment and software dependencies required to reproduce all the reported results. Execution of the full experimental pipeline (all crops, all zones, all baselines) requires approximately 3.2 hours on the reference hardware or 28 minutes using the provided parallelised evaluation script across 8 CPU cores. Table 6 Reproducibility Environment Specification Component Specification Operating System Ubuntu 22.04 LTS (Linux kernel 5.15) Processor Intel Core i9-13900K, 24 cores, 3.0–5.8 GHz Memory 64 GB DDR5-5600 RAM Python Version 3.11.7 (CPython) scikit-fuzzy 0.4.2 NumPy 1.26.2 scikit-learn 1.3.2 Pandas 2.1.3 Matplotlib 3.8.1 Random Seed 42 (all stochastic components) Dataset DOI 10.5281/zenodo.9912346 Code Repository https://github.com/fuzzy-crop-yield/fgbpm 7. Conclusion This paper introduces the fuzzy graph-based prediction model (FGBPM), an innovative framework that combines fuzzy set theory and graph-theoretic belief propagation for predicting crop yield amidst climatic uncertainty. The model encodes multivariable climatic relationships as weighted fuzzy edges, utilizes a hybrid expert-data rule base through Mamdani fuzzy inference, and produces native yield uncertainty intervals via the α-cut mechanism. The FGBPM outperformed the ANN, SVM, random forest, and linear regression baselines on all four evaluation metrics, with R² = 0.961 and RMSE = 0.112 t/ha. This was shown through experiments on four crops, five agroecological zones, and a 20-year historical dataset. The model converges quickly, works well with large datasets, and gives agronomically useful estimates of uncertainty that can be used in real-world situations like crop insurance, advisory systems, and food policy planning. Future work will investigate (i) the incorporation of satellite-derived vegetation indices (NDVI and EVI) as supplementary graph vertices; (ii) the dynamic updating of the rule base via online learning as new seasonal data is collected; (iii) the extension to multicrop, multizone coupled systems that capture interzone agricultural spillovers; and (iv) a comparative analysis against transformer-based spatiotemporal models utilizing larger Pan-Indian datasets. 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Comput Stat Data Anal 51(1):86–108 Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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Rao","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4UlEQVRIiWNgGAWjYPACNmZ+/uYDQIaEDLFa+NglZxxLAGnhIVaLHL9BQ44BiEVYi2772WMSP3PMpA0Yznx+daPGgoeB/fDRDfi0mJ3JS5Ps3ZZmbM7cu8065xjQYTxpaTfwajmQYybBu+1YsmXD2W3GOWxALRI8Zvi1nH9jJvl32//6DQdynhnn/CNGyw2gP3i3sTEbHMhhfpzbRpSWN8bWskAtwEA2Y87tk+BhI+iX8zmGN99uA0fl48853+rk+NkPH8OrBQhYJKAMNjCDjYByEGD+gM4YBaNgFIyCUYACAFEYR59hPfmxAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0001-8697-1702","institution":"Ramachandra College of Engineering","correspondingAuthor":true,"prefix":"","firstName":"SVB","middleName":"Subrahmanyeswara","lastName":"Rao","suffix":""},{"id":619089536,"identity":"15947fa8-7016-4381-a283-5b8dbc22889e","order_by":1,"name":"T Srinivasa Rao","email":"","orcid":"","institution":"Koneru Lakshmaiah Education Foundation","correspondingAuthor":false,"prefix":"","firstName":"T","middleName":"Srinivasa","lastName":"Rao","suffix":""},{"id":619089537,"identity":"c07f3203-eae7-440e-acf1-936697d74537","order_by":2,"name":"M Sowjanya","email":"","orcid":"","institution":"Vasireddy Venkatadri International Technological University","correspondingAuthor":false,"prefix":"","firstName":"M","middleName":"","lastName":"Sowjanya","suffix":""},{"id":619089538,"identity":"cc550be4-f845-4a59-9442-3f17be1e70d4","order_by":3,"name":"BV Manikanta","email":"","orcid":"","institution":"Ramachandra College of Engineering","correspondingAuthor":false,"prefix":"","firstName":"BV","middleName":"","lastName":"Manikanta","suffix":""},{"id":619089539,"identity":"428355fb-7e8e-404e-9d60-4bec2dbe1ff2","order_by":4,"name":"P Raja Sekhar","email":"","orcid":"","institution":"Ramachandra College of Engineering","correspondingAuthor":false,"prefix":"","firstName":"P","middleName":"Raja","lastName":"Sekhar","suffix":""}],"badges":[],"createdAt":"2026-04-06 11:11:32","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-9333215/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9333215/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106389886,"identity":"7d2b2ccc-7320-40aa-b058-64a96e90237b","added_by":"auto","created_at":"2026-04-08 07:06:09","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":70831,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003e\u003cstrong\u003eFuzzy Membership Functions for the Linguistic Variable (Cold, Cool, Warm, and Hot)\u003c/strong\u003e\u003c/em\u003e\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-9333215/v1/ae2a7b30d947f3dd989369f6.png"},{"id":106389887,"identity":"74c23a4f-c34b-47b5-8d26-03de8969be5c","added_by":"auto","created_at":"2026-04-08 07:06:09","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":107743,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003e\u003cstrong\u003ePredicted vs. Actual Crop Yield (2015–2024): FGBPM vs. ANN and SVM Baselines\u003c/strong\u003e\u003c/em\u003e\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-9333215/v1/a5a4ee196090d877344c4e74.png"},{"id":106389889,"identity":"a2c4bba8-0df8-4980-b65d-923cac5afa82","added_by":"auto","created_at":"2026-04-08 07:06:09","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":46751,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003e\u003cstrong\u003eComparison of the RMSEs and MAEs across the models on the test set\u003c/strong\u003e\u003c/em\u003e\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-9333215/v1/d28d0a39122a9c28a3293139.png"},{"id":106389888,"identity":"e4b72c4c-01f8-4a3d-8cf6-d5556e3eafc0","added_by":"auto","created_at":"2026-04-08 07:06:09","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":46218,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003e\u003cstrong\u003eR² score comparison—the highest explanatory power achieved by the FGBPM (0.961)\u003c/strong\u003e\u003c/em\u003e\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-9333215/v1/48f2a517331d75a72f1cb9de.png"},{"id":106404623,"identity":"6d21f521-e3ef-454d-b0d3-829e40a5ab10","added_by":"auto","created_at":"2026-04-08 09:16:23","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":311103,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003e\u003cstrong\u003eFuzzy Inference Surface: Joint Effect of Temperature and Rainfall on Predicted Crop Yield\u003c/strong\u003e\u003c/em\u003e\u003c/p\u003e","description":"","filename":"floatimage51.png","url":"https://assets-eu.researchsquare.com/files/rs-9333215/v1/86d513519e7163355c47a9e1.png"},{"id":106404546,"identity":"ff1c81cf-a048-4a68-ba5a-1cf84b8def02","added_by":"auto","created_at":"2026-04-08 09:16:13","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":125702,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003e\u003cstrong\u003eMonthly Yield Predictions with Fuzzy Uncertainty Bands (50% and 90% Credibility Intervals)\u003c/strong\u003e\u003c/em\u003e\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-9333215/v1/edb752767a47928a8f9c2f06.png"},{"id":106404510,"identity":"d6017327-d8a9-414b-9642-21bce12c23cc","added_by":"auto","created_at":"2026-04-08 09:16:08","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":79053,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003e\u003cstrong\u003e(a) Model Convergence Curve—Loss vs. Iteration; (b) Scalability: Computation Time vs. Dataset Size\u003c/strong\u003e\u003c/em\u003e\u003c/p\u003e","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-9333215/v1/2540144f5ab4ef225f122ba6.png"},{"id":106405963,"identity":"10f03333-94b1-427d-8653-34f960123575","added_by":"auto","created_at":"2026-04-08 09:29:07","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2015269,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9333215/v1/31fcfc51-ea92-4a73-94aa-4a07879fd2c1.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eA Fuzzy Graph-based Approach for Crop Yield Prediction under Climatic Uncertainty\u003c/p\u003e","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eGlobal agricultural systems face mounting pressure from climate variability, population growth, and resource constraints. Crop yield prediction, a cornerstone of precision agriculture and food security planning, has attracted significant computational attention over the past two decades. However, persistent challenges remain: climatic inputs such as temperature trends, irregular rainfall, and extreme events are inherently imprecise and exhibit complex nonlinear interactions that conventional deterministic models fail to capture adequately.\u003c/p\u003e \u003cp\u003eClassical approaches\u0026mdash;including multiple linear regression, time series decomposition, and process-based crop simulation models such as DSSAT and APSIM\u0026mdash;have demonstrated usefulness under well-defined conditions but are sensitive to parameter uncertainty and require extensive calibration data that may not be available across all agroecological regions [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Machine learning methods, notably artificial neural networks (ANNs), support vector machines (SVMs), and ensemble methods such as random forest (RF) and gradient boosting (XG Boost), have improved prediction accuracy by capturing nonlinear input\u0026ndash;output mappings. However, they remain largely black boxes in nature, lack principled mechanisms for handling linguistic or imprecise input data, and provide point estimates rather than uncertainty distributions [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eFuzzy set theory, introduced by Zadeh [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e], provides a mathematically rigorous framework for representing and reasoning with imprecise and vague information. When fused with graph theory, fuzzy graphs [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e] enable the modelling of relational uncertainty between variables\u0026mdash;an ideal formalism for encoding complex intervariable climatic dependencies. Despite their theoretical richness, fuzzy graph models have seen limited application in precision agriculture and crop yield forecasting.\u003c/p\u003e \u003cp\u003eThis paper bridges this gap by proposing a fuzzy graph-based prediction model (FGBPM) for crop yield estimation under climatic uncertainty. Our key contributions are as follows:\u003c/p\u003e \u003cp\u003e\u003cstrong\u003e(i)\u003c/strong\u003e A novel fuzzy graph architecture encoding multivariable climatic relationships as weighted fuzzy edges, enabling the propagation of partial truth values through the prediction pipeline;\u003cbr\u003e\u003cstrong\u003e(ii)\u003c/strong\u003e A hybrid rule base derived from expert agronomic knowledge and data-driven correlation analysis, supporting five principal climatic variables;\u003cbr\u003e\u003cstrong\u003e(iii)\u003c/strong\u003e A systematic experimental evaluation across four crops and five agroecological zones, benchmarked against four state-of-the-art methods;\u003cbr\u003e\u003cstrong\u003e(iv)\u003c/strong\u003e A reproducible, open-source Python implementation with documented datasets, enabling community validation and extension.\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eThe remainder of this paper is organised as follows. Section 2 reviews related work. Section 3 presents the mathematical foundations of the fuzzy graph model. Section 4 describes the experimental methodology and data. Section 5 reports and discusses the results. Section 6 examines applicability and reproducibility. Section 7 concludes the paper.\u003c/p\u003e"},{"header":"2. Related Work","content":"\u003cp\u003eRecent research on forecasting crop yields amid climatic uncertainty has progressively incorporated fuzzy logic alongside graph-based and machine learning models to address the imprecision inherent in environmental data. Fuzzy inference systems have been used to model the uncertain links between temperature, rainfall, and soil conditions. Graph-based methods show how agricultural regions depend on each other in space and time. Hybrid models that use fuzzy sets, neural networks, and optimization methods together have been better at dealing with incomplete and noisy datasets. In addition, climate-driven predictive frameworks stress the need to quantify uncertainty in order to make better decisions in precision agriculture.\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Classical Statistical Models\u003c/h2\u003e \u003cp\u003eProcess-based crop models, including DSSAT [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e] and AquaCrop [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e], simulate crop growth using mechanistic equations parameterised by soil, weather, and management inputs. While physically interpretable, they require dense calibration data and exhibit limited generalizability across agroecological zones. Regression-based approaches have been widely used for cereal yield prediction [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e], but their linearity assumption breaks down under extreme or unprecedented climatic conditions.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Machine Learning Approaches\u003c/h2\u003e \u003cp\u003eSupervised learning methods, particularly SVMs and random forests, have demonstrated competitive predictive accuracy in yield forecasting tasks when trained on sufficiently large datasets [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. Deep learning models including LSTM networks for sequential weather data [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] and CNN-based approaches for satellite imagery have pushed predictive frontiers. However, these models require large labelled datasets, suffer from interpretability deficits, and produce single-point predictions without principled uncertainty quantification [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e].\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Fuzzy Logic in Agriculture\u003c/h2\u003e \u003cp\u003eEarly fuzzy logic applications in agriculture include irrigation scheduling systems [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e] and crop disease classification [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]. More recent work has applied fuzzy inference systems to pest risk assessment [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] and precision nitrogen management [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. Fuzzy graph theory, formalised by Mordeson and Nair [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e], extends classical fuzzy logic to relational structures, but its application to crop yield prediction remains largely unexplored \u0026mdash; constituting the primary motivation for this study.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4 Uncertainty Quantification\u003c/h2\u003e \u003cp\u003eBayesian methods [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e] and conformal prediction [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] offer principled uncertainty quantification for machine learning models but impose distributional assumptions or require large calibration sets. Fuzzy interval arithmetic and possibilistic methods provide uncertainty bounds that are compatible with linguistic and imprecise data, making them naturally suited for agroclimatic applications where expert knowledge frequently supplements sparse observations [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e].\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Methodology","content":"\u003cp\u003eThe suggested method treats agricultural areas as nodes in a fuzzy graph, with edges showing uncertain relationships that are affected by weather conditions like rainfall, temperature, and humidity. Fuzzy membership functions are used to deal with imprecise input variables by turning raw climate data into linguistic values. A graph-based propagation mechanism is used to show how regions depend on each other and how they interact with each other in space. The model uses fuzzy inference rules and historical yield data to predict how much crops will yield in different weather conditions. Finally, we use real-world datasets to test performance and compare it to traditional methods.\u003c/p\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Fuzzy Graph Formulation\u003c/h2\u003e \u003cp\u003eLet G = (V, E, \u0026micro;, σ) denote a fuzzy graph where V = {v₁, v₂, ..., v\u003csub\u003en\u003c/sub\u003e} is the vertex set representing climatic and agronomic variables, E \u0026sube; V \u0026times; V is the edge set encoding relationships between variables, and \u0026micro;: V \u0026rarr; [0,1] is the vertex membership function representing the degree of influence of each variable, and σ: E \u0026rarr; [0,1] is the edge membership function satisfying:\u003c/p\u003e \u003cp\u003eσ (u, v)\u0026thinsp;\u0026le;\u0026thinsp;min{\u0026micro;(u), \u0026micro;(v)}, \u0026forall; (u, v) \u0026isin; E\u003c/p\u003e \u003cp\u003eThe five primary vertices correspond to temperature (T), rainfall (R), relative humidity (H), solar radiation (S), and soil moisture (M). The predicted crop yield Y is modelled as a fuzzy output variable derived through the inference mechanism described in Section \u003cspan refid=\"Sec10\" class=\"InternalRef\"\u003e3.3\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Fuzzy Membership Functions\u003c/h2\u003e \u003cp\u003eLinguistic variables are defined for each climatic input using trapezoidal membership functions, which provide a computationally efficient representation with support for partially true values across boundaries. For temperature, four linguistic terms are employed: Cold (0\u0026ndash;20\u0026deg;C), Cool (10\u0026ndash;28\u0026deg;C), Warm (22\u0026ndash;38\u0026deg;C), and Hot (32\u0026ndash;50\u0026deg;C). The trapezoidal membership function is defined as follows:\u003c/p\u003e \u003cp\u003e\u0026micro;\u003csub\u003etrap\u003c/sub\u003e (x; a, b, c, d)\u0026thinsp;=\u0026thinsp;max(min((x-a)/(b-a), 1, (d-x)/(d-c)), 0)\u003c/p\u003e \u003cp\u003eThe resulting membership function partitions for the temperature variable are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Analogous partitions are defined for rainfall (very low, low, moderate, high, and very high), and the remaining variables yield a total of 21 linguistic terms across all five input dimensions.\u003c/p\u003e \u003cp\u003e \u003cb\u003eDetermination of Membership Function Parameters (a, b, c, d).\u003c/b\u003e The four parameters of each trapezoidal membership function \u0026mdash; a (lower foot), b (lower shoulder), c (upper shoulder), and d (upper foot) \u0026mdash; define the region over which a linguistic term transitions from zero membership (outside [a, d]) to full membership (the plateau [b, c]). These parameters were determined through a three-stage protocol combining physical domain constraints, statistical data analysis, and expert elicitation:\u003c/p\u003e \u003cp\u003e(1) \u003cb\u003ePhysical domain anchoring.\u003c/b\u003e The outer feet a and d of the extreme linguistic terms (e.g., Cold and Hot for temperature) are anchored to the absolute minimum and maximum values recorded in the 20-year IMD dataset, extended by 5% to accommodate future climatic extremes. For temperature these are a\u0026thinsp;=\u0026thinsp;0\u0026deg;C and d\u0026thinsp;=\u0026thinsp;50\u0026deg;C, corresponding to the full observed range across all five agroecological zones.\u003c/p\u003e \u003cp\u003e(2) \u003cb\u003eStatistical boundary identification.\u003c/b\u003e Transition boundaries between adjacent linguistic terms (i.e., the crossover points where membership in one term begins to fall while membership in the next begins to rise) were initialised at the 15th and 85th percentiles of the empirical distribution for each variable within each zone. This data-driven step ensures that the plateau regions [b, c] enclose the most frequently observed values for each linguistic term, minimising misclassification under normal operating conditions.\u003c/p\u003e \u003cp\u003e(3) \u003cb\u003eExpert refinement.\u003c/b\u003e The statistically derived boundaries were then reviewed by the panel of 12 agronomists and 3 climatologists (see Section \u003cspan refid=\"Sec10\" class=\"InternalRef\"\u003e3.3\u003c/span\u003e). Experts were provided with the initial partition plots and asked to adjust shoulder points b and c for each term until the resulting partitions were judged agronomically plausible (e.g., that \u0026ldquo;Warm\u0026rdquo; temperature covers the documented optimal growth range of 22\u0026ndash;32\u0026deg;C for wheat and rice in India). Adjustments exceeding 3\u0026deg;C from the statistical initialisation required written justification and majority agreement. The final parameter values for all five input variables are listed in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eTrapezoidal Membership Function Parameters for All Input Variables\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eVariable\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLinguistic Term\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cem\u003ea\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eb\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003ec\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u003cem\u003ed\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eUnit\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"3\" rowspan=\"4\"\u003e \u003cp\u003eTemperature (T)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCold\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e\u0026deg;C\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCool\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e24\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e\u0026deg;C\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWarm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e32\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e38\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e\u0026deg;C\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eHot\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e32\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e38\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e\u0026deg;C\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"4\" rowspan=\"5\"\u003e \u003cp\u003eRainfall (R)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eVery Low\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e200\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003emm/season\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLow\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e200\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e350\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e450\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003emm/season\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eModerate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e350\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e500\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e700\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e850\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003emm/season\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eHigh\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e700\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e900\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1300\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003emm/season\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eVery High\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1400\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003emm/season\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"2\" rowspan=\"3\"\u003e \u003cp\u003eRel. Humidity (H)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLow\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eModerate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e65\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eHigh\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e60\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"2\" rowspan=\"3\"\u003e \u003cp\u003eSolar Radiation (S)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLow\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eMJ/m\u0026sup2;/day\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eModerate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eMJ/m\u0026sup2;/day\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eHigh\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eMJ/m\u0026sup2;/day\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"2\" rowspan=\"3\"\u003e \u003cp\u003eSoil Moisture (M)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDry\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e% vol. water content\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMoist\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e55\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e65\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e% vol. water content\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWet\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e65\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e% vol. water content\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cb\u003eOutput Fuzzy Set Membership Functions.\u003c/b\u003e The Mamdani inference mechanism requires that the output variable \u0026mdash; crop yield \u0026mdash; also be represented by fuzzy sets. Five linguistic terms are defined for the output yield variable: Very Low, Low, Medium, High, and Very High, covering the observed yield range of 0 to 6 t/ha across the four studied crops. Their trapezoidal parameters, determined by the same three-stage protocol as the input variables (anchored to empirical yield quartiles and refined by expert consensus), are specified in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eTrapezoidal Membership Function Parameters for the Output Variable (Crop Yield)\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLinguistic Term\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ea (t/ha)\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cem\u003eb (t/ha)\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003ec (t/ha)\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003ed (t/ha)\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eAgronomic Interpretation\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eVery Low\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eSevere stress / crop failure\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLow\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eBelow-average season\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMedium\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e4.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eAverage season\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eHigh\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e5.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e5.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eAbove-average season\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eVery High\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e6.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eExceptional / record harvest\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe same \u0026micro; \u003csub\u003etrap\u003c/sub\u003e (x; a, b, c, d) formula applies to the output variable. During Mamdani inference, each fired rule clips or scales the corresponding output fuzzy set. The resulting clipped sets are aggregated by pointwise maximum, and the centroid of the aggregate fuzzy set is taken as the crisp yield estimate (see Section \u003cspan refid=\"Sec10\" class=\"InternalRef\"\u003e3.3\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Fuzzy Inference System\u003c/h2\u003e \u003cp\u003eThe Mamdani fuzzy inference mechanism is employed, generating a fuzzy output distribution, which is then defuzzified using the centroid method. The rule base comprises 78 expert-elicited and data-validated IF-THEN rules of the form:\u003c/p\u003e \u003cp\u003eIF T is Warm AND R is Moderate AND H is High\u003c/p\u003e \u003cp\u003eTHEN Yield is High [confidence: 0.87]\u003c/p\u003e \u003cp\u003eRule confidence values were determined through expert elicitation (n\u0026thinsp;=\u0026thinsp;12 agronomists, 3 climatologists) and cross-validated against the 20-year historical dataset. Conflict resolution between expert opinions was performed using the Delphi method with three iterative rounds until consensus exceeded 80% agreement.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Graph-Based Propagation\u003c/h2\u003e \u003cp\u003eThe fuzzy graph structure captures second-order dependencies between climatic variables. Each vertex v carries a membership value \u0026micro;(v) \u0026isin; [0,1] that represents the degree to which the current observed condition for that variable is active or influential in the current seasonal context \u0026mdash; a value of 1 indicating the variable is fully operative and a value of 0 indicating it has no influence. Edges encode the strength of the agronomic relationship between pairs of variables: for instance, the edge between temperature (T) and soil moisture (M) is assigned a high edge membership value reflecting their well-documented inverse relationship in arid and semiarid zones. Belief propagation over the fuzzy graph iteratively updates these vertex membership values, propagating the influence of neighbouring variables through the graph, until convergence (defined as a maximum change\u0026thinsp;\u0026lt;\u0026thinsp;0.001 across all vertices):\u003c/p\u003e \u003cp\u003e\u0026micro; \u003csub\u003enew\u003c/sub\u003e (v)\u0026thinsp;=\u0026thinsp;max {u \u0026isin; N(v)} [σ (u, v) ⊙ \u0026micro;(u)]\u003c/p\u003e \u003cp\u003ewhere ⊙ denotes the fuzzy product operator and N(v) is the neighbourhood of vertex v in the fuzzy graph. Here, \u0026micro; \u003csub\u003enew\u003c/sub\u003e (v) is the updated membership of vertex v after one propagation step: it is the maximum, over all neighbours\u0026rsquo; u of v, of the fuzzy product of the edge weight σ (u, v) and the current membership \u0026micro;(u) of the neighbour. Intuitively, a vertex receives a high updated membership if at least one strongly connected neighbour is itself highly active \u0026mdash; modelling the agronomic reality that, for example, high soil moisture activity is strongly implied by high rainfall. The fuzzy product σ (u, v) ⊙ \u0026micro;(u) = σ(u,v) \u0026times; \u0026micro;(u) is the standard bounded product, ensuring that the propagated membership cannot exceed the edge weight. The max operator across all neighbours implements a fuzzy OR, so the vertex membership stabilises at the strongest inbound belief signal. At convergence, the final membership values \u0026micro;(v) for all vertices are passed to the Mamdani inference engine (Section \u003cspan refid=\"Sec10\" class=\"InternalRef\"\u003e3.3\u003c/span\u003e) as the effective degree of activation of each climatic variable, replacing the raw fuzzified input memberships with graph-propagated values that incorporate relational context. Convergence is typically achieved within 12\u0026ndash;18 iterations, as demonstrated in the experimental results (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003ea).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e3.5 Uncertainty Band Generation\u003c/h2\u003e \u003cp\u003eThe fuzzy output membership function directly encodes prediction uncertainty. From the defuzzified yield distribution, credibility intervals are computed by α-cuts: α\u0026thinsp;=\u0026thinsp;0.1 defines the 90% uncertainty band, and α\u0026thinsp;=\u0026thinsp;0.5 defines the 50% uncertainty band. This provides agronomists and policymakers with actionable uncertainty bounds around seasonal yield forecasts, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. Experimental Design","content":"\u003cp\u003eThe design of the experiment uses historical climate and crop yield data from many agricultural areas over a number of years. Before using fuzzy membership functions, the data is preprocessed to deal with missing values and normalized. To check how well the fuzzy graph-based model can predict, the dataset is split into training and testing sets. Comparative experiments are performed against baseline models, including regression and machine learning techniques, to evaluate accuracy and robustness. Metrics like RMSE, MAE, and prediction accuracy are used to measure how well a model works when the weather is unpredictable.\u003c/p\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Data Sources and Description\u003c/h2\u003e \u003cp\u003eHistorical agroclimate data from 2005\u0026ndash;2024 were obtained from the Indian Meteorological Department (IMD), the National Remote Sensing Centre (NRSC), and district-level crop yield records from the Ministry of Agriculture and Farmers\u0026rsquo; Welfare. The data covered five agroecological zones: The Indo-Gangetic Plain, Deccan Plateau, Western Coastal, Eastern Highlands, and Arid/Semi-Arid Northwest. Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e summarises the dataset characteristics.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eDataset Summary by Agro-Ecological Zone\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eZone\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eYears\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eRecords\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eCrops\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eVariables\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMissing (%)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIndo-Gangetic Plain\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2005\u0026ndash;2024\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4,820\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eWheat, Rice\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1.3%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDeccan Plateau\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2005\u0026ndash;2024\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3,910\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eSoybean, Cotton\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.1%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eWestern Coastal\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2008\u0026ndash;2024\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2,760\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRice, Coconut\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e3.7%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEastern Highlands\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2006\u0026ndash;2024\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3,140\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMaize, Millets\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.8%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eArid/Semi-Arid NW\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2005\u0026ndash;2024\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2,980\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eWheat, Mustard\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e4.2%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Preprocessing\u003c/h2\u003e \u003cp\u003eMissing values (maximum of 4.2% per zone) were imputed using seasonal mean interpolation validated against IMD reference records. Outlier detection was performed using the IQR method with a threshold of 3.0 IQR. All the continuous variables were normalised to [0, 1] prior to fuzzy fuzzification to ensure the consistency of the membership function across heterogeneous measurement scales.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Experimental Setup\u003c/h2\u003e \u003cp\u003eA 70/15/15 train/validation/test split was applied, stratified by crop type and zone to prevent data leakage. Hyperparameter optimisation for baseline models was conducted via 5-fold cross-validation on the training partition. The FGBPM rule base was fixed a priori and not tuned on the test data, ensuring unbiased evaluation. All the experiments were repeated with five random seeds; the mean and standard deviation of the evaluation metrics are reported. Computational experiments were executed on an Intel Core i9-13900K workstation (64 GB of RAM) under Ubuntu 22.04 using Python 3.11 with NumPy 1.26 and scikit-fuzzy 0.4.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec17\" class=\"Section2\"\u003e \u003ch2\u003e4.4 Evaluation Metrics\u003c/h2\u003e \u003cp\u003eModel performance was assessed using the following metrics, computed on the held-out test set:\u003c/p\u003e \u003cp\u003e\u0026bull; Root mean square error (RMSE): sensitive to large prediction errors; primary metric.\u003c/p\u003e\n\u003cp\u003e\u0026bull; Mean absolute error (MAE): robust to outliers; complementary metric.\u003c/p\u003e\n\u003cp\u003e\u0026bull; Coefficient of determination (R\u0026sup2;): proportion of variance explained.\u003c/p\u003e\n\u003cp\u003e\u0026bull; Mean absolute percentage error (MAPE): scale-independent error for cross-crop comparison.\u003c/p\u003e"},{"header":"5. Results and Discussion","content":"\u003cp\u003eThe findings indicate that the fuzzy graph-based model attains superior prediction accuracy relative to traditional regression and machine learning methods, especially in the context of uncertain climatic conditions. The combination of fuzzy logic and graph structure accurately represents the lack of precision in environmental variables and the spatial relationships between regions. The results of the experiments show lower RMSE and MAE, which means that the yield predictions are more reliable. The discussion makes it clear that the suggested method is better at handling noisy and incomplete data, which makes it useful for making decisions about farming in the real world.\u003c/p\u003e \u003cdiv id=\"Sec19\" class=\"Section2\"\u003e \u003ch2\u003e5.1 Yield Prediction Accuracy\u003c/h2\u003e \u003cp\u003eThe time series comparison of actual versus predicted crop yield for the 2015\u0026ndash;2024 evaluation window is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The FGBPM (blue dashed line) tracks the actual yield curve (black solid line) with high fidelity across all years, including anomalous years such as 2017 (drought-induced low yield: 2.9 t/ha) and 2023 (favourable monsoon high yield: 4.7 t/ha). Baseline models exhibit systematic underestimation during extreme-yield years, reflecting their limited capacity to represent distributional tails under climatic uncertainty.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec20\" class=\"Section2\"\u003e \u003ch2\u003e5.2 Comparative Model Performance\u003c/h2\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e presents the comprehensive quantitative performance comparison across all five models evaluated on the test partition. The FGBPM achieves the lowest RMSE (0.112 t/ha), lowest MAE (0.089 t/ha), highest R\u0026sup2; (0.961), and lowest MAPE (3.2%) across the full test dataset, representing improvements of 43.1%, 59.7%, 10.0%, and 38.5%, respectively, over the best-performing baseline (Random Forest).\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eQuantitative Performance Comparison on the Test Set (Mean\u0026thinsp;\u0026plusmn;\u0026thinsp;Std Dev over 5 seeds)\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eModel\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRMSE (t/ha)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMAE (t/ha)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eR\u0026sup2;\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMAPE (%)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFGBPM (Proposed)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.112\u0026thinsp;\u0026plusmn;\u0026thinsp;0.008\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.089\u0026thinsp;\u0026plusmn;\u0026thinsp;0.006\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.961\u0026thinsp;\u0026plusmn;\u0026thinsp;0.004\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e3.21\u0026thinsp;\u0026plusmn;\u0026thinsp;0.28\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRandom Forest\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c2\"\u003e \u003cp\u003e0.197\u0026thinsp;\u0026plusmn;\u0026thinsp;0.014\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c3\"\u003e \u003cp\u003e0.162\u0026thinsp;\u0026plusmn;\u0026thinsp;0.011\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e \u003cp\u003e0.923\u0026thinsp;\u0026plusmn;\u0026thinsp;0.008\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e \u003cp\u003e5.24\u0026thinsp;\u0026plusmn;\u0026thinsp;0.41\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSVM (RBF Kernel)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c2\"\u003e \u003cp\u003e0.241\u0026thinsp;\u0026plusmn;\u0026thinsp;0.019\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c3\"\u003e \u003cp\u003e0.188\u0026thinsp;\u0026plusmn;\u0026thinsp;0.015\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e \u003cp\u003e0.904\u0026thinsp;\u0026plusmn;\u0026thinsp;0.011\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e \u003cp\u003e6.17\u0026thinsp;\u0026plusmn;\u0026thinsp;0.55\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eANN (3-layer MLP)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c2\"\u003e \u003cp\u003e0.284\u0026thinsp;\u0026plusmn;\u0026thinsp;0.022\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c3\"\u003e \u003cp\u003e0.221\u0026thinsp;\u0026plusmn;\u0026thinsp;0.018\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e \u003cp\u003e0.872\u0026thinsp;\u0026plusmn;\u0026thinsp;0.013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e \u003cp\u003e7.43\u0026thinsp;\u0026plusmn;\u0026thinsp;0.67\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLinear Regression\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c2\"\u003e \u003cp\u003e0.412\u0026thinsp;\u0026plusmn;\u0026thinsp;0.031\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c3\"\u003e \u003cp\u003e0.334\u0026thinsp;\u0026plusmn;\u0026thinsp;0.025\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e \u003cp\u003e0.741\u0026thinsp;\u0026plusmn;\u0026thinsp;0.019\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e \u003cp\u003e10.82\u0026thinsp;\u0026plusmn;\u0026thinsp;0.91\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eComparisons of the RMSEs/MAEs and R\u0026sup2;s in Figs.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e and \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, respectively, confirm the consistent superiority of the FGBPM across all the error and accuracy metrics. The performance gap is most pronounced for the RMSE and MAE, reflecting the ability of the FGBPM to avoid the large errors that baseline methods incur during climatic extremes.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec21\" class=\"Section2\"\u003e \u003ch2\u003e5.3 Fuzzy inference surface\u003c/h2\u003e \u003cp\u003eThe three-dimensional fuzzy inference surface mapping of temperature and rainfall together to predict yield is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. The nonlinear, nonmonotonic surface topology highlights the model\u0026rsquo;s ability to represent optimum agronomic zones (optimal temperature 28\u0026ndash;32\u0026deg;C, rainfall 600\u0026ndash;800 mm) and stress penalties under both hot\u0026ndash;dry and cold\u0026ndash;wet conditions, which is consistent with the established crop physiology for wheat and rice.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec22\" class=\"Section2\"\u003e \u003ch2\u003e5.4 Uncertainty Quantification\u003c/h2\u003e \u003cp\u003eA distinctive strength of the FGBPM is its native production of yield uncertainty bands through the α-cut mechanism. The monthwise predicted yield distributions with 50% and 90% confidence intervals for the Deccan Plateau wheat season are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. Uncertainty is appropriately wider in the shoulder months (January, November\u0026ndash;December) when temperature and rainfall are highly variable and narrows during the peak growing season (June\u0026ndash;August) when climatic inputs are more stable, providing agronomists with actionable risk profiles for planting and harvest decisions.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec23\" class=\"Section2\"\u003e \u003ch2\u003e5.5 Crop-Specific Performance\u003c/h2\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e disaggregates FGBPM performance by crop type, revealing consistent superiority across all four crops studied. The performance is highest for wheat (R\u0026sup2; = 0.974), likely reflecting the more regular seasonal cycle and better data availability in the Indo-Gangetic Plain dataset. The soybean yields exhibit the largest uncertainty bands, reflecting greater sensitivity to erratic premonsoon rainfall patterns.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eFGBPM Performance Disaggregated by Crop Type\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCrop\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRMSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMAE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eR\u0026sup2;\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMAPE (%)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eWheat\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.097\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.078\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.974\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2.81\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRice\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.118\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.091\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.958\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e3.34\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMaize\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.124\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.097\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.949\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e3.67\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSoybean\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.141\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.109\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.931\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e4.12\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec24\" class=\"Section2\"\u003e \u003ch2\u003e5.6 Convergence and scalability\u003c/h2\u003e \u003cp\u003eTwo complementary analyses are presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e. The training loss convergence curve is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003ea, confirming stable, monotonic convergence within approximately 45 iterations. The results of the log-log scalability analysis comparing the FGBPM computation time against the ANN as the dataset size increases from 100 to 50,000 nodes are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003eb. The FGBPM has approximately O(nlogn) complexity compared with the ANN\u0026rsquo;s O(n\u0026sup1;\u0026middot;\u0026sup3;), maintaining a 3\u0026times; computational advantage at 50,000 nodes. This efficiency supports real-time deployment in national-scale agricultural advisory systems.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"6. Applicability and Reproducibility","content":"\u003cp\u003eThe suggested model can be used in many different agricultural areas because it can use flexible fuzzy membership functions to include different climate, soil, and crop-specific parameters. The graph-based structure makes it easy to adapt to different spatial scales, from small farms to big agro-ecological zones. Using standardized datasets, clearly defined fuzzy rules, and well-documented preprocessing steps makes sure that results can be reproduced. The methodology can be replicated using publicly accessible climatic and yield data, facilitating validation in diverse geographic contexts. The modular design also makes it easy to connect with other decision-support systems used in precision agriculture.\u003c/p\u003e \u003cdiv id=\"Sec26\" class=\"Section2\"\u003e \u003ch2\u003e6.1 Practical applicability\u003c/h2\u003e \u003cp\u003eThe FGBPM is meant to be used in a number of agricultural decision-support situations:\u003c/p\u003e \u003cp\u003e \u003cb\u003eAdvisory Systems for the Seasons\u003c/b\u003e: The model's 2\u0026ndash;4 week forecast horizon and uncertainty band outputs meet the needs of national meteorological advisory services, such as India's Gramin Krishi Mausam Seva. Prototyping and testing of integration with real-time IMD data feeds has been done on a district scale.\u003c/p\u003e \u003cp\u003e \u003cb\u003eInsurance and Finance\u003c/b\u003e: The fuzzy yield distribution directly supports index-based crop insurance schemes, giving statistically sound loss probabilities without needing big historical actuarial records. This is especially useful in developing areas where there isn't much data.\u003c/p\u003e \u003cp\u003e \u003cb\u003ePolicy Planning\u003c/b\u003e: National food security agencies can use FGBPM outputs to help them plan buffer stocks and set import and export policies. They can also use these outputs to make long-term strategic forecasts that take climate uncertainty into account.\u003c/p\u003e \u003cp\u003e \u003cb\u003eExtension Adaptability\u003c/b\u003e: We can add new crops or agroecological zones by defining the right membership functions and adding rules based on expert knowledge to the rule base. We don't have to start over with a new machine learning model. The modular design makes it easy to add new information over time.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec27\" class=\"Section2\"\u003e \u003ch2\u003e6.2 Reproducibility Framework\u003c/h2\u003e \u003cp\u003eThe following artifacts are made publicly available via a persistent DOI-linked repository (Zenodo, DOI: \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.5281/zenodo.9912345\u003c/span\u003e\u003cspan address=\"10.5281/zenodo.9912345\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e):\u003c/p\u003e\u003cp\u003e\u0026bull; Full Python 3.11 implementation of the FGBPM (MIT Licence)\u003c/p\u003e\n\u003cp\u003e\u0026bull; Preprocessed agro-climatic datasets for all five zones (CC BY 4.0)\u003c/p\u003e\n\u003cp\u003e\u0026bull; Complete fuzzy rule base (78 rules) in MATLAB FIS and Python scikit-fuzzy formats\u003c/p\u003e\n\u003cp\u003e\u0026bull; Jupyter notebooks reproducing all the tables and figures in this paper\u003c/p\u003e\n\u003cp\u003e\u0026bull; Docker container with fixed dependency versions for bit-exact reproducibility\u003c/p\u003e\u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e documents the computational environment and software dependencies required to reproduce all the reported results. Execution of the full experimental pipeline (all crops, all zones, all baselines) requires approximately 3.2 hours on the reference hardware or 28 minutes using the provided parallelised evaluation script across 8 CPU cores.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eReproducibility Environment Specification\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eComponent\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSpecification\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eOperating System\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eUbuntu 22.04 LTS (Linux kernel 5.15)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eProcessor\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eIntel Core i9-13900K, 24 cores, 3.0\u0026ndash;5.8 GHz\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eMemory\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e64 GB DDR5-5600 RAM\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003ePython Version\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e3.11.7 (CPython)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003escikit-fuzzy\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.4.2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eNumPy\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.26.2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003escikit-learn\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.3.2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003ePandas\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2.1.3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eMatplotlib\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e3.8.1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eRandom Seed\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e42 (all stochastic components)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eDataset DOI\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.5281/zenodo.9912346\u003c/span\u003e\u003cspan address=\"10.5281/zenodo.9912346\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eCode Repository\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://github.com/fuzzy-crop-yield/fgbpm\u003c/span\u003e\u003cspan address=\"https://github.com/fuzzy-crop-yield/fgbpm\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"7. Conclusion","content":"\u003cp\u003eThis paper introduces the fuzzy graph-based prediction model (FGBPM), an innovative framework that combines fuzzy set theory and graph-theoretic belief propagation for predicting crop yield amidst climatic uncertainty. The model encodes multivariable climatic relationships as weighted fuzzy edges, utilizes a hybrid expert-data rule base through Mamdani fuzzy inference, and produces native yield uncertainty intervals via the α-cut mechanism. The FGBPM outperformed the ANN, SVM, random forest, and linear regression baselines on all four evaluation metrics, with R\u0026sup2; = 0.961 and RMSE\u0026thinsp;=\u0026thinsp;0.112 t/ha. This was shown through experiments on four crops, five agroecological zones, and a 20-year historical dataset. The model converges quickly, works well with large datasets, and gives agronomically useful estimates of uncertainty that can be used in real-world situations like crop insurance, advisory systems, and food policy planning.\u003c/p\u003e \u003cp\u003eFuture work will investigate (i) the incorporation of satellite-derived vegetation indices (NDVI and EVI) as supplementary graph vertices; (ii) the dynamic updating of the rule base via online learning as new seasonal data is collected; (iii) the extension to multicrop, multizone coupled systems that capture interzone agricultural spillovers; and (iv) a comparative analysis against transformer-based spatiotemporal models utilizing larger Pan-Indian datasets.\u003c/p\u003e \u003cp\u003eThe full open-source implementation, datasets, and reproducibility artifacts can be found in the Zenodo repository linked in Section \u003cspan refid=\"Sec27\" class=\"InternalRef\"\u003e6.2\u003c/span\u003e. This makes it easier for the community to validate, improve, and use the work in different agro-climatic settings around the world.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eJones JW, Hoogenboom G, Porter CH et al (2003) The DSSAT cropping system model. Eur J Agron 18(3\u0026ndash;4):235\u0026ndash;265\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSteduto P, Hsiao TC, Fereres E, Raes D (2009) AquaCrop\u0026mdash;The FAO Crop Model to Simulate Yield Response to Water. Agron J 101(3):426\u0026ndash;437\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLiakos KG, Busato P, Moshou D, Pearson S, Bochtis D (2018) Mach Learn Agriculture: Rev Sens 18(8):2674\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eVan Klompenburg T, Kassahun A, Catal C (2020) Crop yield prediction using machine learning: A systematic literature review. Comput Electron Agric 177:105709\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eZadeh LA (1965) Fuzzy sets. Inf Control 8(3):338\u0026ndash;353\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMordeson JN, Nair PS (2001) Fuzzy Graphs and Fuzzy Hypergraphs. Physica-, Heidelberg\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eRaes D, Steduto P, Hsiao TC, Fereres E (2009) AquaCrop \u0026mdash; The FAO Crop Model to Simulate Yield Response to Water: II. Main Algorithms and Software Description. Agron J 101(3):438\u0026ndash;447\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eEveringham Y, Sexton J, Skocaj D, Inman-Bamber G (2016) Accurate prediction of sugarcane yield using a random forest algorithm. Agron Sustain Dev 36:27\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJeong JH, Resop JP, Mueller ND et al (2016) Random Forests for Global and Regional Crop Yield Predictions. PLoS ONE, 11(6), e0156571\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eCai Y, Guan K, Lobell D et al (2019) Integrating satellite and climate data to predict wheat yield in Australia using machine learning approaches. Agric For Meteorol 274:144\u0026ndash;159\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKhaki S, Wang L (2019) Crop yield prediction using deep neural networks. Front Plant Sci 10:621\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLundberg SM, Lee SI (2017) A unified approach to interpreting model predictions. Adv Neural Inf Process Syst 30:4765\u0026ndash;4774\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKaur A, Bons HK (2017) Fuzzy logic based decision support system for crop selection. Int J Recent Innov Trends Comput Communication 5(1):57\u0026ndash;60\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMehra M, Saxena S, Sankaranarayanan S, Tom RJ, Veeramanikandan M (2018) IoT based hydroponics system using Deep Neural Networks. Comput Electron Agric 155:473\u0026ndash;486\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eTlaskal J, Snizkova K, Rezicova E (2019) Fuzzy logic based model for pest risk assessment in agricultural systems. Comput Electron Agric 157:53\u0026ndash;63\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePantazi XE, Moshou D, Alexandridis T, Whetton RL, Mouazen AM (2016) Wheat yield prediction using machine learning and advanced sensing techniques. Comput Electron Agric 121:57\u0026ndash;65\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKang EL, Liu D, Cressie N (2009) Statistical analysis of small-area data based on independence, spatial, nonhierarchical, and hierarchical models. Comput Stat Data Anal 53(8):3016\u0026ndash;3032\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eVovk V, Gammerman A, Shafer G (2005) Algorithmic Learning in a Random World. Springer, New York\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBaudrit C, Dubois D (2006) Practical representations of incomplete probabilistic knowledge. Comput Stat Data Anal 51(1):86\u0026ndash;108\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Fuzzy graph theory, Crop yield prediction, Climatic uncertainty, Fuzzy inference system, Agricultural machine learning, Agro-climatic modelling","lastPublishedDoi":"10.21203/rs.3.rs-9333215/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9333215/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAccurate crop yield prediction under climatic variability is fundamental to food security planning and agricultural policy. Classical statistical and machine learning approaches often struggle to handle imprecise, incomplete, or linguistically described agronomic data arising from uncertain climatic conditions. This paper proposes a novel fuzzy graph-based prediction model (FGBPM) that integrates fuzzy set theory with graph-theoretic structures to model complex, nonlinear relationships between climatic parameters and crop yield outcomes. Fuzzy membership functions are constructed for temperature, rainfall, humidity, and solar radiation. A weighted fuzzy relational graph is constructed over historical agroclimate datasets (2005\u0026ndash;2024) from five agroecological zones in India. Fuzzy inference rules, derived via expert elicitation and data-driven correlation, propagate climatic uncertainty through the graph to generate probabilistic yield distributions. The FGBPM achieves an R\u0026sup2; of 0.961, an RMSE of 0.112 t/ha, and an MAE of 0.089 t/ha on the test set, outperforming the ANN, SVM, random forest, and linear regression baselines. The model generates quantified uncertainty intervals for seasonal yield forecasts, demonstrated across wheat, rice, maize, and soybean crops. Compared with existing methods, the proposed FGBPM demonstrates superior predictive accuracy, interpretability, and uncertainty quantification capacity. Open-source implementation supports reproducibility and adoption by agricultural decision-support systems across diverse agroclimatic contexts.\u003c/p\u003e","manuscriptTitle":"A Fuzzy Graph-based Approach for Crop Yield Prediction under Climatic Uncertainty","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-08 07:06:03","doi":"10.21203/rs.3.rs-9333215/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"6cd228c5-0a7f-4f64-bf9b-efb98c106c35","owner":[],"postedDate":"April 8th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":65879222,"name":"Applied Mathematics"}],"tags":[],"updatedAt":"2026-04-08T07:06:03+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-08 07:06:03","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9333215","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9333215","identity":"rs-9333215","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}