Predicting Winter Wheat Yield with Dual-Year Spectral Fusion, Bayesian Wisdom, and Cross-Environmental Validation

Abstract

Winter wheat is an important grain that plays a crucial role in agricultural production and ensuring food security. Its yield directly impacts the stability and security of the global food supply. The accurate monitoring of grain yield is imperative for precise agricultural management. This study aimed to enhance winter wheat yield predictions with UAV remote sensing and investigate its predictive capability across diverse environments. In this study, RGB and multispectral (MS) data were collected on 6 May 2020 and 10 May 2022 during the grain filling stage of winter wheat. Using the Pearson correlation coefficient method, we identified 34 MS features strongly correlated with yield. Additionally, we identified 24 texture features constructed from three bands of RGB images and a plant height feature, making a total of 59 features. We used seven machine learning algorithms (Cubist, Gaussian process (GP), Gradient Boosting Machine (GBM), Generalized Linear Model (GLM), K-Nearest Neighbors algorithm (KNN), Support Vector Machine (SVM), Random Forest (RF)) and applied recursive feature elimination (RFE) to nine feature types. These included single-sensor features, fused sensor features, single-year data, and fused year data. This process yielded diverse feature combinations, leading to the creation of seven distinct yield prediction models. These individual machine learning models were then amalgamated to formulate a Bayesian Model Averaging (BMA) model. The findings revealed that the Cubist model, based on the 2020 and 2022 dataset, achieved the highest R² at 0.715. Notably, models incorporating both RGB and MS features outperformed those relying solely on either RGB or MS features. The BMA model surpassed individual machine learning models, exhibiting the highest accuracy (R² = 0.725, RMSE = 0.814 t·ha⁻¹, MSE = 0.663 t·ha⁻¹). Additionally, models were developed using one year’s data for training and another year’s data for validation. Cubist and GLM stood out among the seven individual models, delivering strong predictive performance. The BMA model, combining these models, achieved the highest R² of 0.673. This highlights the BMA model’s ability to generalize for multi-year data prediction.

1. Introduction

Winter wheat, a critical global food crop, plays a pivotal role in both agricultural production and food security [1]. With the world’s population rapidly increasing and economic development on the rise, ensuring consistent yield growth for winter wheat has become an urgent agricultural imperative [2]. Consequently, accurate early yield prediction has become a paramount necessity for agricultural progress [2]. Traditional methods for gauging winter wheat yield predominantly hinge on destructive sampling, necessitating the harvesting, threshing, and weighing of sample areas. However, these methods require substantial human, material, and financial resources and do not provide accurate yield information before harvest [3].

Satellite remote sensing has made significant advancements in terms of spatial, temporal, and spectral resolution. However, it comes with limitations such as limited flexibility, high costs, cloud coverage constraints, and lack of automation, which have hindered its widespread adoption [4]. However, the emergence of UAVs has opened up new possibilities in remote sensing. These UAVs provide data with unprecedented spatial, spectral, and temporal resolution, facilitating rapid evolution in the field [5]. The use of images captured by UAVs has become a cost-effective alternative to expensive aircraft and satellite products. UAVs offer the advantage of carrying multiple sensors and are not constrained by issues like repeat cycles. They can undertake multiple missions throughout a crop’s entire growth cycle, capturing data with centimeter-level accuracy [6,7]. UAVs can be equipped with various sensors, including RGB (red-green-blue) sensors and multispectral (MS) sensors. RGB sensors are favored for their cost-effectiveness and high spatial resolution [8]. On the flip side, MS sensors offer excellent spectral resolution, with each sensor being sensitive within a specific spectral range. When using UAVs for crop phenotypic information, the red, near-red, and red-edge bands have proven to be effective [9,10,11]. An example study by Prey [12] used RGB and MS sensors to predict grain yield and demonstrated their effectiveness.

MS features offer valuable insights into estimating aboveground biomass and nitrogen content by enabling the derivation of multiple spectral indices from the five bands of MS data [13,14]. In addition to spectral indices, RGB images also contain texture features that are closely related to vegetation growth. These texture features capture spatial information about the distribution of vegetation and non-vegetation areas in the image and their interactions with the environment. They reflect the visual characteristics of homogeneous phenomena in the image [15]. For example, Yue et al. [16] discovered that integrating RGB spectral indices and image texture metrics at various spatial resolutions improved the R² value for estimating winter wheat biomass. The R² value increased to 0.84, compared to 0.76 when using spectral indices alone. Similarly, Zheng et al. [17] found that combining texture features with spectral indices in stepwise multiple linear regression improved model performance (R² = 0.78). These findings have implications for research aiming to incorporate texture features as input variables in predictive models. To extract plant height (PH) information, a digital surface model (DSM) is typically subtracted from a digital terrain model (DTM) in RGB images. This process yields a canopy height model representing the height difference between the vegetation canopy and the ground, thus providing the plant height feature [18]. Cao et al. [19] demonstrated that integrating the extracted tree height feature into the mangrove classification process improved classification accuracy, underscoring the validity and usefulness of elevation information derived from DSM and DTM.

Machine learning models built based on spectral and structural features are widely used in the development of prediction systems, especially for complex systems with uncertainty [20]. Revill [21] used machine learning with Gaussian process regression to calibrate LAI UAV models using ground data. Barzin [22] used a machine learning approach to estimate the nitrogen (N) content of corn leaves based on UAV MS data and identified Gradient Lifter and Random Forest as the most suitable models for estimating leaf N. Proietti [23] introduced a generalized linear model for a time series that can integrate alternative spectral estimation methods within the same probability-based framework. Li [24] investigated an SVM method for field weed identification involving multiple features using spectral data from soil images. Silva et al. [25] found that the Cubist model is the best multivariate model for predicting sand and clay content from soil VIS-NIR-SWIR spectra. Cao et al. [26] estimated mangrove biomass using the K-Nearest Neighbors (KNN) method, and the results showed an improvement in accuracy with increasing scale. The effectiveness of these seven machine learning methods has been demonstrated in the studies mentioned above. Recursive feature selection as a data dimensionality reduction method can eliminate redundant information and noise interference. It can be applied to filter input features in various prediction models for phenotypic information, such as predicting chlorophyll content [27], predicting winter wheat yield [28], and estimating alpine grassland vegetation cover [29], among others.

Numerous studies have focused on using multi-year data to predict crop phenotypic information, showing promising predictive performance. Wang et al. [30] used four years of hyperspectral data to predict straight-chain starch content in rice. Their results showed that the inclusion of texture variables based on hyperspectral UAV imagery simplified data collection and improved estimation accuracy. Zheng et al. [31] combined two years of UAV MS data with soil hyperspectral data to estimate the nitrogen (N) concentration of rice plants and achieved improved prediction accuracy. Qiao et al. [32] used UAV MS data to estimate leaf area index (LAI) of maize in a two-year experiment conducted in 2021 and 2022. The study found that the depth feature-based LAI estimation model had the highest accuracy when applied to the two-year dataset. Although these studies used multi-year data to construct prediction models, there are currently few research works that use one-year data as a training set and one-year data as a validation set to build prediction models. This poses a challenge for future research in this area.

Furthermore, the non-linear and unsteady properties of UAV spectra make yield prediction difficult, especially when relying only on separate machine learning models. This is because using individual machine learning models to achieve optimal prediction performance does not take into account the structure of the model and parameter settings [33], resulting in uncertainty. Ensemble learning models provide a good way to reduce uncertainty in prediction by combining several good individual machine learning models [34]. Ensemble machine learning algorithms combine multiple weak models to create a stronger and more robust predictive model. Leveraging the strengths of diverse models, these algorithms enhance prediction accuracy and generalization. They have demonstrated impressive performance across various domains. BMA is an ensemble learning model. Unlike other ensemble learning methods, BMA not only gives deterministic weighted averages for the models but also creates prediction distributions to analyze uncertainties linked with deterministic predictions [35]. There are many reports showing that BMA methods can provide better performance than individual machine learning models [36,37]. However, it is less known whether BMA models will improve accuracy and reduce uncertainty in yield prediction.

The aim of this study is to build seven yield prediction models using UAV RGB and MS data from 2020 and 2022. The aim is to evaluate the generalizability and generalization ability of each model. The first step is to extract texture features and plant height features from the RGB data and spectral indices from the MS data. Then, the single- and multi-sensor features are verified using recursive feature elimination (RFE) methods with seven machine learning algorithms. Machine learning models are then created based on the verified features. Finally, one year’s data are used as the training set, while the other year’s data are used as the validation set. The performance of the machine learning models with satisfactory results is evaluated by predicting the yield.

The paper makes several important contributions, outlined below:

Introducing a novel model framework: Combining a RFE method with a machine learning model while combining the results of multiple individual machine learning models to build one ensemble model.

Building Yield Prediction Models: This study built seven yield prediction models using multi-year, multi-sensor data fusion. These models use one year’s data as a training set and another year’s data as a validation set.

Performance of the BMA ensemble model: The BMA ensemble model consistently demonstrated good prediction performance. Even when evaluating unknown datasets, the BMA ensemble model demonstrated a certain level of accuracy.

2. Materials and Methods

2.1. Experimental Area and Design

The experiment took place in the 2019–2020 and 2021–2022 growing seasons at the Chinese Academy of Agricultural Sciences Comprehensive Experimental Base in Xinxiang County, Xinxiang City, Henan Province (113°45′38″E, 35°8′10″N, Figure 1). This region is located in the northern part of Henan Province. The average annual rainfall and temperature are 573 mm and 15.8 °C, respectively.

A total of 180 test plots were established in the 2019–2020 growing season and were subjected to three different irrigation treatments throughout the fertility period: W1 (240 mm), W2 (190 mm), and W3 (145 mm). Each irrigation treatment consisted of 60 plots, and each plot had a row spacing of 20 cm, a length of 8 m, and a width of 1.4 m, resulting in an area of 11.2 m² (Figure 1). Thirty wheat varieties suitable for cultivation in the Huanghuai wheat region were selected as test materials. To ensure the objectivity of the experiment, three replicates of each treatment were performed. The winter wheat was sown at the end of October 2019 and harvested on 3 June 2020. As in the 2019–2020 growing season, a total of 180 test plots were created in the 2021–2022 growing season. During the reproductive period, the wheat plants were subjected to six different irrigation treatments: W1 (300 mm), W2 (240 mm), W3 (180 mm), W4 (120 mm), W5 (60 mm), and W6 (0 mm). A total of 10 varieties of wheat commonly grown on the north China plain were used in the experiment. Each treatment consisted of 30 experimental plots, each 4 m long and 1.4 m wide. Adjacent plots were 0.4 m apart. The winter wheat was sown at the end of October 2021 and harvested on 5 June 2022. During the harvest for both growing seasons 2019–2020 and 2021–2022, wheat grain from each plot was collected separately using a plot combine. The harvested wheat from each plot was then placed in a numbered bag and dried in the laboratory to a constant mass. The winter wheat from each plot was then weighed and the yield was calculated based on the plot area.

2.2. UAV Spectral Data Acquisition

The MS data were collected using a DJI M210 drone manufactured by Shenzhen DJI Technology Co., Ltd., located in Shenzhen, China. The M210 was equipped with a Micasense RedEdgeMX MS camera. Additionally, the RGB data were captured using a DJI Phantom 4 Pro with an RGB sensor (Figure 2). The RedEdge MX sensor utilizes five bands of spectral imaging technology, including red, green, blue, near-infrared and red-edge. The red, green, and blue bands have center wavelengths of 668 nm, 560 nm, and 475 nm, respectively, while the near-infrared band has a center wavelength of 840 nm, and the red-edge band has a center wavelength of 717 nm. All channels have a resolution of 1280 × 960 and deliver high quality images with a wide 47.2° field of view. On the other hand, the DJI Phantom 4 Pro is equipped with a high-quality RGB sensor that can capture detailed, clear and crisp images. The sensor has a resolution of 20 megapixels.

On 6 May 2020 and 10 May 2022, two UAVs were used for aerial photography during the grain filling stage. The flights took place between 11:00 a.m. and 2:00 p.m., and this time was chosen to ensure optimal lighting conditions in clear and cloudless weather. Both UAVs maintained a constant altitude of 30 m throughout the missions. Directional overlap was set to 85%, and side overlap was set to 80%. To ensure data accuracy, a global navigation satellite system (GNSS) with millimeter precision was used for each sensor.

2.3. Pre-Processing of UAV Images

Figure 2 illustrates the data acquisition process. In this study, the UAV captured MS and RGB images during the flowering period, which were subsequently imported into the software (version 4.0.8) (Shenzhen DJI Technology Co., Ltd., Shenzhen, China) for image alignment and processing. The software first generated a sparse point cloud of the flight area based on the UAV images and position data. This sparse point cloud was used to create a spatial grid of the flight area and integrate ground control points (GCPs) to provide accurate spatial coordinate information. This approach resulted in sparse point clouds with precise locations as well as information about the surface geometry and spatial texture of the flight area. The MS images were calibrated using known reflectance to convert the DN values to reflectance. The processed data were then used to create a high-resolution digital orthophotos map (DOM), digital surface models (DSM) and digital terrain models (DTM) of the flight area. Finally, the processed images were exported in TIFF image format. ArcMap 10.5 software (Environmental Systems Research Institute, Inc., Redlands, CA, USA) was then used to divide the MS HD digital orthophotos into plots, generating corresponding shapefile files for 180 separate areas with unique IDs. For each region, the spectral reflectance information of the corresponding ID region was identified and extracted. To reduce the influence of edge effects on the image, the shapefile was created by excluding the image edge regions. Using the raster calculator function, we performed DSM–DTM calculations to obtain plant height information from the digital surface model. In addition, the cropped RGB images were imported into ENVI software (version 5.3) (Exelis Visual Information Solutions, Inc., Boulder, CO, USA) to extract texture features. The window size for texture extraction was 7 × 7. These features included mean (ME), variance (VA), homogeneity (HO), contrast (CO), dissimilarity (DI), entropy (EN), second moment (SE) and correlation (COR). All means were extracted according to each ID and used as features of the corresponding region.

2.4. Spectral Features

Based on previous studies, we computed 34 yield-sensitive MS features from the spectral reflectance of the MS. These features were then used as inputs for the yield prediction models. Furthermore, we extracted eight texture and plant height features from the RGB images and utilized them as input features for the yield prediction model.

Table 1. Information about the MS and RGB features.

Data Type	Features	Formulas	References	Applications
MS	Normalized difference vegetation index	$NDVI=(NIR-R)/(NIR+R)$	[38]	Agriculture. Vegetation
	Normalized difference red-edge	$NDRE=(NIR-RE)/(NIR+RE)$	[38]	Vegetation
	Blue NDVI	$\mathrm{BNDVI}=(\mathrm{NIR}-\mathrm{B})/(\mathrm{NIR}+\mathrm{B})$	[38]	Vegetation
	Green NDVI	$GNDVI=(NIR-G)/(NIR+G)$	[38]	Vegetation
	Blue-wide dynamic range vegetation index	$\mathrm{BWDRVI}=(0.1\mathrm{NIR}-\mathrm{B})/(0.1\mathrm{NIR}+\mathrm{B})$	[39]	Vegetation
	Canopy chlorophyll content index	$\begin{array}{l}CCCI=(NIR-RE)/(NIR+RE)/\\ (NIR-R)/(NIR+R)\end{array}$	[38]	Agriculture. Vegetation
	Coloration index	$CI=(R-B)/R$	[38]	Vegetation
	Green ratio vegetation index	$GRVI=NIR/G$	[40]	Vegetation
	Red-green ratio	$RGR=R/G$	[41]	Vegetation
	Red-edge ratio index 1	$RRI1=NIR/RE$	[42]	Remote sensing
	Red-edge ratio index 2	$RRI2=RE/R$	[42]	Remote sensing
	Soil and atmospherically resistant vegetation	$SARVI=2.5(NIR-R)/(1+NIR+6R-7.5B)$	[43]	Soil, Vegetation
	Adjusted transformed soil-adjusted vegetation index	$\begin{array}{l}ATSAVI=1.22(NIR-1.22R-1.22)/\\ (1.22NIR+R-1.22\ast 0.03+0.08\ast (1+{1.22}^2))\end{array}$	[44]	Soil, Vegetation
	Chlorophyll index green	$CI\mathrm{g}=(NIR/G)-1$	[45]	Vegetation
	Chlorophyll index red-edge	$CI\mathrm{re}=(NIR/RE)-1$	[38]	Vegetation
	Ideal vegetation index	$IVI=(NIR-0.03)/(1.22R)$	[46]	Vegetation
	Difference vegetation index	$DVI=NIR-R$	[38]	Vegetation
	Iron oxide	$IO=R/B$	[47]	Geology
	Weighted difference Vegetation index	$WDVI=NIR-1.22R$	[42]	Vegetation
	Transformed vegetation index	$TVI=\sqrt{NDVI+0.5}$	[48]	Vegetation
	Wide dynamic range Vegetation index	$\mathrm{WDRVI}=(0.1NIR-R)/(0.1NIR+R)$	[49]	Biomass, LAI
	Transformed NDVI	$TNDVI=\sqrt{(NIR-R)/(NIR+R+0.5)}$	[50]	Vegetation
	Soil-adjusted vegetation index	$SAVI=1.5\times (NIR-R)/(NIR+R+0.16)$	[38]	Soil, Vegetation
	Green difference vegetation index	$GDVI=NIR-G$	[51]	Vegetation
	Enhanced vegetation index	$EVI=2.4(NIR-R)/(NIR+R+1)$	[48]	Vegetation
	Green leaf index	$GLI=(2G-R-B)/(2G+R+B)$	[48]	Agriculture. Vegetation
	Green atmospherically resistant vegetation index	$\begin{array}{l}GARI=(NIR-(G-(B-R)))/\\ (NIR-(G+(B-R)))\end{array}$	[48]	Vegetation
	Green soil adjusted vegetation index	$GSAVI=1.5\times (NIR-G)/(NIR+G+0.5)$	[52]	Soil, Vegetation
	Norm G	$N\mathrm{orm}G=G/(NIR+R+G)$	[53]	Vegetation
	Norm NIR	$N\mathrm{orm}NIR=NIR/(R+G+NIR)$	[53]	Vegetation
	Norm R	$N\mathrm{orm}R=R/(R+G+NIR)$	[53]	Vegetation
	Normalized green-red difference index	$NGRDI=(G-R)/(G+R)$	[49]	Vegetation
	Redness index	$RI=(R-G)/(R+G)$	[48]	Agriculture
	Shape index	$IF=(2R-G-B)/(G-B)$	[54]	Vegetation
RGB	Gray-level co-occurrence matrix	ME, HO, DI, EN, SE, VA, CO, COR	[38]	Vegetation
	Plant height	$PH=DSM-DTM$	/	Agriculture. Vegetation

/—empirical visible vegetation index, ME—mean, HO—homogeneity, DI—dissimilarity, EN—entropy, SE—second moment, VA—variance, CO—contrast, COR—correlation.

shows the basic information of these features.

2.5. Model Framework

Figure 2 illustrates the modeling framework and processing flow in this study. The proposed modeling framework is based on a feature selection method. First, input features are selected by applying a RFE method to different individual models. This process aims to identify the best combination of input features and build yield prediction models using a 5-fold cross-validation method. Seven machine learning algorithms were used to evaluate the feasibility and generalizability of the modeling framework. For every input feature combination, RFE was carried out with the seven models. The best feature combination for each model was chosen, yielding a model with strong performance in yield prediction. To further validate the models, this study employs two validation approaches. In validation approach A, we validate the 2022 data by training it on the 2020 data. In validation approach B, we validate the 2022 data by training it on the 2020 data as well. Identifying the training and validation set samples, the RFE method could not be screened. Both models predicting 2020 and 2022 used the RFE method to obtain the best input feature combination. The results of these prediction models for 2020 and 2022 were then used as input features for additional research. We used these input features to create seven individual machine learning models. To enhance prediction performance, we built BMA models by combining the predictions from the seven individual machine learning models. This allowed us to assess the prediction performance of the ensemble learning models and their capability to handle unseen datasets.

RFE [55] is a common method for selecting the most relevant subset of features. It works by iteratively removing features with lower importance or weight from the initial feature set. The process includes training a model, ranking features by importance, and iteratively eliminating less important features. The model is then retrained and the importance rankings are updated. This iterative process continues until a certain number of features or a predefined stopping criterion is reached [56].

The Gaussian Process (GP) [57] is a non-parametric machine learning technique that creates probabilistic models between input and output variables. It employs Bayesian principles to learn from training data and predict new input values, offering uncertainty estimates for the predictions. GP is particularly useful for capturing complex relationships between variables and can handle various types of data. It offers flexibility in feature selection and can be adapted to different problem domains and datasets. GP is also robust to missing values and outliers, making it a reliable choice in noisy environments.

Gradient Boosting Machine (GBM) [3] is a powerful machine learning algorithm used primarily for regression problems. It works by iteratively training multiple weak learners, each focusing on the residual errors of the previous learner, thereby gradually improving the overall model performance. GBM excels at handling diverse data types and features, allowing it to effectively capture complex non-linear relationships. It also has automatic feature selection capabilities and adapts well to different problem domains and datasets. In addition, GBM is robust to noise as it can effectively deal with missing values and outliers.

The Generalized Linear Model (GLM) [58] is a statistical learning method that extends linear regression models to cover a wider range of data types and application scenarios. By introducing a link function and a family of distributions, GLM enables non-linear modeling of the relationship between input and output.

The K-Nearest Neighbors algorithm (KNN) [57] is a classic and simple supervised learning method used for regression problems. It predicts new samples by finding the nearest neighbors among the training samples using a distance metric. During training, KNN stores the feature vectors and corresponding category labels of the training examples. In the prediction phase, the algorithm calculates the distance between each new sample and the training samples. It then selects the K nearest neighbors and calculates their average as the prediction result.

Support Vector Machine (SVM) [3] aims to construct an optimal hyperplane in a high-dimensional feature space to maximise the separation between samples of different categories. SVM is known for its ability to generalize and its robustness, making it suitable for handling complex data structures and high-dimensional feature spaces.

Random Forest (RF) [3] is a commonly used ensemble learning algorithm for regression problems. RF builds multiple decision trees by randomly selecting training data and feature subsets and then combines their predictions to enhance accuracy and generalization. It performs well with high-dimensional data and large datasets, providing good predictive performance and generalization capability.

Cubist [59] is a machine learning algorithm designed for regression tasks using tree models and rule extraction techniques. The goal is to generate models with high prediction accuracy and interpretability. Cubist divides the input feature space into various subspaces by creating a rule set and generates a linear regression model for each subspace. It processes datasets with a large number of features and complex relationships, automatically performs feature selection and models interaction features. The models created by Cubist are easy to interpret and understand and provide insights for decision making based on the data.

BMA [60] is a model ensemble approach that combines various statistical models using Bayesian statistical theory. This method considers uncertainty and model diversity, leading to more accurate and robust predictions. The BMA algorithm is implemented in R Language (version 4.3.1).

Let y be the predicted yield, $D=[d_1,d_2,\dots ,d_r]$ is the observed yield, and $f=[f_1,f_2,\dots ,f_k]$ represents the model space consisting of GP, GBM, SVM, RF, GLM, KNN and Cubist. According to the law of total probability, the posterior probability y of a simulated variable in the BMA can be expressed as [61]:

$$p(y|D)={\displaystyle \sum _{k=1}^{k}p(f_k|D)\cdot p_k(y|f_k,D)}$$

(1)

where $p(y|D)$ is the posterior probability of the predicted sequence $f_k$, which reflects the degree of coincidence between $f_k$ and the observed yield. $p(f_k|D)$ is the posterior probability of the k-th model $f_k$ given the measured data $w_k$ (In other words, $w_k$ is the weight of each model). $p_k(y|f_k,D)$ is the posterior distribution of the predicted y given the model $f_k$ and data D.

The predicted values of the BMA are obtained from the weighted average of each data-driven model. Assuming that the predicted and measured values of each model follow a normal distribution, the prediction of the BMA model can be expressed as:

$$E(y|D)={\displaystyle \sum _{k=1}^{k}p(f_k|D)}\cdot E[p_k(y|f_k,D)]={\displaystyle \sum _{k=1}^k{w}_k{f}_k}$$

(2)

$$\mathrm{Var}[y|D]={\displaystyle \sum _{k=1}^k{w}_k}(f_k-{\displaystyle \sum _{k=1}^k{w}_k}{f}_k{)}^2+{\displaystyle \sum _{k=1}^k{w}_k}{\sigma }_k{ }^2$$

(3)

where ${\sigma }_k{ }^2$ is the variance of the BMA predictor.

2.6. Parameters for Model Accuracy Evaluation

In this study, Pearson’s correlation coefficient was used to evaluate the correlation between the input features and the yield. To evaluate the predictive performance of the prediction model, 3 parameters were selected: R² (determination coefficient), RMSE (root mean squared error), and MSE (mean squared error). A higher value of R² and lower values of RMSE and MSE indicate better accuracy and better prediction performance. The formulas for these evaluation metrics are as follows:

$${\rho }_{a,b}=\frac{Cov(a,b)}{\sqrt{Var(a)Var(b)}}$$

(4)

where $p_{a,b}$ represents the Pearson’s correlation coefficient, Cov(a,b) represents the covariance of $\mathrm{a}$ and $\mathrm{b}$, Var(a) is the variance of $\mathrm{a}$, and Var(b) is the variance of $\mathrm{b}$.

$$R^2=1-\frac{{\displaystyle {\sum }_{i=1}^n{({\stackrel{⌢}{y}}_i-y_i)}^2}}{{\displaystyle {\sum }_{i=1}^n{(y_i-\overline{y})}^2}}$$

(5)

$$RMSE=\sqrt{\frac{{\displaystyle {\sum }_{i=1}^n{({\widehat{y}}_i-y_i)}^2}}{N}}$$

(6)

$$MSE=\frac{{\displaystyle {\sum }_{i=1}^n{(\widehat{y}-y_i)}^2}}{N}$$

(7)

where $y_i$ is the observed value, $\widehat{y_i}$ is the predicted value, $\overline{y}$ is the mean of the observed values, and N is the sample size.

3. Results

3.1. Yield Distribution

Figure 3 shows the winter wheat yields for all plots in 2020 and 2022. In 2020, the average yield for winter wheat was 6.55 t·ha⁻¹. Among the three water treatments, the yields followed a descending order: W1 > W2 > W3. The W1 treatment achieved the highest yield at 10.44 t·ha⁻¹. In 2022, the average yield for winter wheat increased to 8.21 t·ha⁻¹. The pattern across the six water treatments was W1 > W2 > W3 > W4 > W5 > W6. The W6 treatment recorded the lowest yield at 5.63 t·ha⁻¹. This trend supports the hypothesis that increased irrigation correlates with higher yields. The yield data for the plots exhibited clear color distributions, emphasizing notable differences between the treatments. Additionally, there was a distinct separation observed within the datasets.

3.2. Spectral Feature Selection

In this study, a total of 25 RGB features and 34 MS features were acquired from the relevant data sources. We conducted a Pearson correlation analysis on the 59 spectral features obtained for the years 2020 and 2022 to assess their relationships with yield.

As shown in Figure 4a, it is evident that the majority of spectral features exhibit robust and statistically significant correlations with yield for the year 2020. MS features exhibit a stronger correlation with yield than the RGB features. NGRDI and RI show the highest correlation, with an absolute value reaching 0.67. Figure 4b illustrates that all the spectral features exhibited pronounced and statistically significant correlations with yield in 2022. Interestingly, in that year, the correlation between the RGB features and yield was stronger than that of MS features. The correlation reached an absolute value of 0.74. When comparing the data from the two years in relation to yield, the spectral features showed a more robust correlation in 2022.

Although many spectral features showed significant and strong correlations with yield, the large number of features created in this study required a feature selection process. To streamline these for model building, we employed the RFE method. Therefore, we applied the RFE method to filter the RGB and MS features, MS features, and RGB features in the 2020, 2022, and 2020 and 2022 datasets using the seven machine learning algorithms. The RMSE regression loss of each algorithm was calculated. Figure 5 illustrates the relationship between the number of input features (on the x-axis) and the RMSE regression loss (on the y-axis). Different lines represent the variation across the folds. A lower RMSE indicates a higher prediction accuracy for the model. The optimal number of input features for each of the seven machine learning algorithms is detailed in

Table 2. Optimal number of features to combine using recursive feature selection.

Year	Sensor Type	GP	GBM	SVM	RF	GLM	KNN	Cubist
2020	RGB	16	17	22	17	7	12	17
	MS	30	32	22	30	16	20	8
	RGB and MS	42	18	50	43	10	45	47
2022	RGB	25	6	12	13	13	23	22
	MS	34	17	22	34	9	10	23
	RGB and MS	22	40	30	50	17	25	18
2020 and 2022	RGB	25	14	18	16	15	9	23
	MS	32	15	12	34	25	20	23
	RGB and MS	51	46	56	40	21	41	50

.

The range of RMSE with increasing number of input features is narrow for all models except for the GLM model. In summary, the year 2022 had the lowest RMSE regression loss, ranging from 0.52 to 0.83 t·ha⁻¹, indicating the best model fit. The dataset for 2020 and 2022 followed, with RMSE ranging from 0.8 to 1.5 t·ha⁻¹. When RGB features were used as inputs, the Cubist model achieved the lowest RMSE of 0.52 t·ha⁻¹ in 2022 with 18 features. Additionally, the Cubist model exhibited the minimum RMSE value in 2020 and 2022. For MS features, the RMSE in 2022 varied from 0.65 to 0.78 t·ha⁻¹. The GLM model achieved the lowest RMSE of 0.619 t·ha⁻¹ with nine features. When RGB and MS features were evaluated, the Cubist model again excelled, with an RMSE of 0.520 t·ha⁻¹ with 18 features. Both the SVM and Cubist models exhibited their lowest RMSE values when 50 features were used as input for the 2020 and, 2020 and 2022 datasets.

3.3. Analysis of the Model Accuracy

The machine learning models underwent feature filtering using seven RFE methods for each of the nine datasets. BMA was employed to ensemble the seven machine learning models. The accuracy achieved by each model is shown in

Table 3. Prediction accuracy of each machine learning model based on the best combination of features.

		2020			2022			2020 and 2022
		RGB	MS	RGB and MS	RGB	MS	RGB and MS	RGB	MS	RGB and MS
GP	R2	0.505	0.493	0.509	0.660	0.532	0.663	0.626	0.631	0.657
	RMSE/(t·ha−1)	1.141	1.150	1.116	0.541	0.641	0.544	0.942	0.941	0.905
	MSE/(t·ha−1)	1.302	1.323	1.247	0.293	0.411	0.296	0.887	0.885	0.819
GBM	R2	0.497	0.531	0.538	0.610	0.562	0.657	0.618	0.655	0.700
	RMSE/(t·ha−1)	1.126	1.105	1.087	0.574	0.641	0.550	0.951	0.911	0.842
	MSE/(t·ha−1)	1.268	1.221	1.182	0.329	0.411	0.303	0.904	0.830	0.709
SVM	R2	0.532	0.512	0.597	0.643	0.517	0.658	0.616	0.612	0.707
	RMSE/(t·ha−1)	1.101	1.129	1.022	0.560	0.660	0.549	0.963	0.965	0.834
	MSE/(t·ha−1)	1.212	1.275	1.044	0.314	0.436	0.301	0.927	0.931	0.696
RF	R2	0.537	0.505	0.554	0.641	0.530	0.661	0.620	0.642	0.695
	RMSE/(t·ha−1)	1.097	1.145	1.072	0.558	0.635	0.550	0.952	0.931	0.857
	MSE/(t·ha−1)	1.203	1.311	1.149	0.311	0.403	0.303	0.906	0.867	0.734
GLM	R2	0.494	0.445	0.508	0.634	0.561	0.638	0.654	0.660	0.671
	RMSE/(t·ha−1)	1.17	1.206	1.124	0.567	0.619	0.578	0.922	0.902	0.888
	MSE/(t·ha−1)	1.369	1.454	1.263	0.321	0.383	0.334	0.850	0.814	0.789
KNN	R2	0.446	0.531	0.566	0.612	0.495	0.647	0.492	0.627	0.673
	RMSE/(t·ha−1)	1.196	1.131	1.076	0.582	0.680	0.565	1.109	0.947	0.885
	MSE/(t·ha−1)	1.430	1.279	1.158	0.339	0.462	0.319	1.230	0.897	0.783
Cubist	R2	0.498	0.516	0.595	0.683	0.571	0.703	0.649	0.663	0.715
	RMSE/(t·ha−1)	1.132	1.106	1.029	0.524	0.637	0.520	0.910	0.900	0.831
	MSE/(t·ha−1)	1.281	1.223	1.059	0.275	0.406	0.270	0.828	0.810	0.691
BMA	R2	0.539	0.535	0.600	0.712	0.616	0.713	0.685	0.681	0.725
	RMSE/(t·ha−1)	1.084	1.084	1.008	0.508	0.592	0.508	0.882	0.877	0.814
	MSE/(t·ha−1)	1.175	1.177	1.017	0.258	0.351	0.258	0.778	0.768	0.663

. Evaluating the accuracy of the eight models under nine input combinations, some conclusions can be drawn.

Among the seven individual machine learning models, the 2020 SVM model performed best, reaching an R² of 0.597 with RGB and MS as input features. Notably, the 2022 and, 2020 and 2022 models, both employing the Cubist algorithm, excelled with significant R² of 0.703 and 0.715, respectively, when utilizing RGB and MS features. This underscores the superior performance of the Cubist model for winter wheat yield analysis. Models with diverse input features achieved the highest R². The R² values for the 2020 model ranged from 0.445 to 0.597, while those for the 2022 model varied from 0.495 to 0.703. Additionally, the R² for the 2020 and 2022 model consistently exceeded 0.6 in most cases, underscoring its robust performance across various scenarios. Importantly, the ensemble model’s accuracy consistently surpassed that of the individual 2020 and 2022 models.

The BMA prediction model was created by combining predictions from seven separate machine learning models, each developed independently. The model’s results, including R² values, are presented in Table 3. Our analysis revealed that BMA models developed with varying input features consistently exhibited superior accuracy when compared to the individual machine learning models. Notably, the highest accuracy was achieved in the 2020 and 2022 model, boasting an impressive R² of 0.725, with RMSE and MSE values of 0.815 t·ha⁻¹ and 0.663 t·ha⁻¹, respectively. Compared to the top-performing individual machine learning models with diverse input features, the BMA model showed a significant R² improvement, ranging from 0.002 to 0.045. This robustly underscores the practical significance and superiority of the BMA model.

3.4. Model Generalizability Validation Analysis

To further validate the model’s performance, this study employed two distinct validation approaches, labeled A and B. Because the training and validation sets were already established, we could not use the RFE method to find the best input combination for the prediction model. Instead, the optimal input feature combination of each individual model was determined through the RFE method. Consequently, the prediction results obtained from the 2020 and 2022 dataset served as the input features for constructing the yield prediction model.

The accuracy of the seven individual machine learning models is presented in

Table 4. Accuracy of models based on validation approaches A and B.

Feature	Feature Category	Metrics	GP	GBM	SVM	RF	Cubist	KNN	GLM	BMA
Validation Approach A	RGB and MS	R2	0.543	0.468	0.039	0.350	0.634	0.432	0.594	0.673
		RMSE/(t·ha−1)	0.639	0.909	1.406	0.780	0.593	0.762	0.617	0.560
		MSE/(t·ha−1)	0.408	0.390	1.978	0.608	0.352	0.580	0.381	0.313
	MS	R2	0.444	0.389	0.016	0.197	0.543	0.331	0.561	0.586
		RMSE/(t·ha−1)	0.700	0.811	1.763	0.964	0.678	0.935	0.642	0.640
		MSE/(t·ha−1)	0.491	0.659	3.109	0.929	0.459	0.874	0.412	0.410
	RGB	R2	0.526	0.497	0.266	0.341	0.568	0.379	0.595	0.651
		RMSE/(t·ha−1)	0.681	1.013	1.391	0.950	0.654	1.462	0.620	0.594
		MSE/(t·ha−1)	0.463	1.026	1.934	0.902	0.428	2.136	0.384	0.353
Validation Approach B	RGB and MS	R2	0.478	0.306	0.005	0.235	0.567	0.342	0.499	0.569
		RMSE/(t·ha−1)	1.149	1.404	1.932	1.470	1.056	1.389	1.137	1.044
		MSE/(t·ha−1)	1.321	1.970	3.735	2.161	1.116	1.928	1.293	1.089
	MS	R2	0.463	0.162	0.003	0.003	0.512	1.156	0.489	0.525
		RMSE/(t·ha−1)	1.165	1.478	1.900	1.938	1.118	1.537	1.140	1.096
		MSE/(t·ha−1)	1.358	2.183	3.609	3.755	1.250	2.363	1.300	1.202
	RGB	R2	0.451	0.355	0.154	0.375	0.498	0.313	0.488	0.510
		RMSE/(t·ha−1)	1.267	1.537	1.776	1.490	1.165	1.939	1.203	1.170
		MSE/(t·ha−1)	1.605	2.361	3.156	2.220	1.358	3.760	1.448	1.368

. SVM, RF, and KNN did not perform well in terms of accuracy. Their R² values were below 0.48, and the RMSE exceeded 0.7 t·ha⁻¹, with a maximum RMSE of 1.939 t·ha⁻¹. The GBM model displayed some predictive capability when constructing the yield prediction model based on RGB features in validation approach A (R² = 0.497, RMSE = 1.013 t·ha⁻¹, MSE = 1.026 t·ha⁻¹). In contrast, all Cubist and GLM models demonstrated reliable predictive performance, with R² values exceeding 0.48. Among them, the Cubist model exhibited the best predictive performance (R² = 0.634, RMSE = 0.593 t·ha⁻¹, MSE = 0.352 t·ha⁻¹). Overall, the models constructed using validation approach A generally exhibited higher accuracy than those using validation approach B. It was deduced that an R² value below 0.48 may not provide reliable predictive performance. Only individual machine learning models with an R² greater than 0.48 were used to build the BMA predictive models. Their accuracies are summarized in Table 4. Notably, the BMA models consistently outperformed any individual machine learning model. The highest prediction accuracy was achieved using validation approach A (R² = 0.673, RMSE = 0.560 t·ha⁻¹, MSE = 0.313 t·ha⁻¹). Furthermore, the R² values of the BMA models fell within the range of 0.51–0.673, while the RMSE values ranged from 0.313 to 1.170 t·ha⁻¹, confirming the reliability of their predictive performance.

4. Discussion

4.1. Multi-Sensor Features

The selection and fusion of features from MS and RGB sensors offer significant advantages in scientific research. MS sensors capture data from various spectral bands, providing rich spectral features [62]. Conversely, RGB sensors offer high-resolution image information [63]. Combining features extracted from these two sensor types, we leverage their strengths to create a more accurate representation. Feature fusion amalgamates the advantages of MS and RGB sensors, enhancing prediction accuracy and stability in regression models. Most models in this study showed improved accuracy when RGB and MS features were fused, aligning with previous findings [38]. Feature fusion also addresses limitations of individual sensors [64]. These limitations include insensitivity to specific spectral bands in MS sensors and susceptibility to lighting conditions and color distortion in RGB sensors [65]. Combining features, we compensate for these shortcomings. This approach improves adaptability to different scenarios and data variations. Using fused multi-sensor features from 2020 and 2022, each model demonstrated better adaptability compared to single-sensor features, with slightly higher R² values. Additionally, feature fusion expanded the dataset, increasing the number of input features and training samples, which improved generalization ability and stability. It is important to note that this study exclusively utilized features from two sensors. Future research can explore incorporating thermal infrared, hyperspectral, and other sensor features. This exploration aims to further enhance predictive capabilities and conduct more comprehensive investigations.

4.2. Advantages of the RFE Approach Based on Multiple Individual Models

Multi-model RFE is a valuable strategy for achieving more accurate and robust results in model building and prediction. In this approach, the evaluation of feature importance using multiple base models plays a crucial role [66]. The base models chosen in this study encompass linear models, tree models, and support vector machines. Each of these models provides distinct indicators of feature importance. Through the utilization of multiple base models, we obtain a comprehensive and diverse evaluation of feature importance. Different individual models possess the capability to capture various types of feature relationships and non-linear associations. This diversity leads to a more accurate ranking of feature importance. In this study, the amalgamation of feature importance from multiple base models helps mitigate biases or errors that may arise from a single model. This approach ultimately enhances the stability and reliability of feature selection. This is in alignment with previous findings [67]. Throughout the process of RFE, we iteratively remove features contributing less to the model’s prediction. This gradual reduction of the feature space allows us to obtain a more concise feature combination. This aids in the reduction of feature dimensionality, minimizes model complexity, and improves model interpretability [68]. The iterative selection of the best feature combination offers flexibility in the number of features included. This flexibility is observed in the feature selection methods based on different base models (Table 2). This approach significantly enhances the model’s generalization ability and prediction accuracy. Furthermore, the implementation of RFE methods based on multiple base models diminishes the risk of overfitting [69]. The gradual elimination of unimportant features substantially decreases the model’s sensitivity to noise and redundant features. This process results in improved model robustness and stability.

4.3. Advantages of Individual Machine Learning Models

The utilization of seven different machine learning algorithms enables a comprehensive approach to data modeling and analysis. Each algorithm operates uniquely, capturing diverse data patterns and associations [70]. For example, tree-based algorithms excel at handling non-linear relationships [71,72], while support vector machines are effective at managing high-dimensional data [71,73]. Combining these algorithms, we can thoroughly explore the dataset’s features and patterns, enhancing the flexibility and adaptability of the prediction models. Moreover, employing multiple machine learning algorithms helps mitigate the limitations and biases associated with relying on a single algorithm. It is observed that no single model consistently outperforms others when constructed under varying conditions and using different data characteristics. This highlights that each machine learning algorithm operates under its own set of assumptions and limitations [74]. Using multiple algorithms enables the construction of models across diverse conditions, facilitating model comparison and selection. Assessing how different algorithms perform on the same dataset helps identify their strengths, weaknesses, and suitability for yield prediction. This aids in choosing the most effective algorithm or a combination of algorithms. In this study, the Cubist model demonstrated promising predictive performance across different scenarios, achieving a maximum R² of 0.715. Additionally, the utilization of multiple machine learning algorithms enables the exploration of the importance and influence of feature factors. The assessment of feature importance may vary among algorithms. Comparing the feature selection outcomes of multiple algorithms, we gain insights into the extent to which different features contribute to yield prediction. This optimization of feature selection and model interpretation capabilities enhances our understanding of the significance of various features in the context of yield prediction [75].

4.4. Advantages of the BMA Model

The BMA model utilized in this study achieved reliable yield predictions with improved accuracy compared to the seven independent machine learning models. The highest R² increased to 0.725, and the lowest RMSE decreased to 0.258. The BMA model’s success can be attributed to two main factors. Firstly, it efficiently extracted information from seven existing machine learning models, avoiding uncertainties associated with individual model parameters and structures [33]. Secondly, the BMA model derives prior probabilities based on the performance of each individual model, enhancing prediction accuracy by aggregating models with strong performance [76]. Figure 6 illustrates the normalized weights of each individual machine learning model in constructing the BMA model. The SVM model holds the highest weight in the 2020 dataset, while the Cubist model carries the greatest weight in both the 2022 and, 2020 and 2022 datasets. Notably, these individual machine learning models are the top performers, as confirmed in validation approaches A and B. This demonstrates that the BMA model leverages the best-performing models to enhance accuracy and reliability, consistent with the findings in a previous study [36]. Figure 7 displays the observed and predicted values of the best individual machine learning models and BMA models constructed using data from different years. Examining the optimal separate prediction models for the three datasets reveals that the difference between the predicted and observed values fluctuates within a specific range. The smallest fluctuation range is observed in the 2022 dataset, ranging from −1.99 to 2.20 t·ha⁻¹. For the BMA model, the fluctuation range of the difference between predicted and observed values is even smaller, fluctuating between −1.54 and 2.24 t·ha⁻¹. Figure 8 presents the observed and predicted values of the yield prediction models, based on validation methods A and B. Both the optimal individual machine learning model and the BMA model are included for comparison. It is evident that the fluctuation range of the difference between the predicted and observed values is smaller for the BMA model compared to the individual machine learning model. Furthermore, the fluctuation range for validation approach A is significantly reduced compared to validation approach B. This demonstrates that the BMA model effectively reduces model uncertainties, enhancing its overall reliability. Additionally, the results obtained from validation methods A and B indicate that the BMA model consistently outperforms the standalone machine learning model on unseen datasets. This highlights the superior generalization ability of the BMA model. Furthermore, the novel approaches used in this study show promise, the data collected from only one sampling time (grain filling stage) may not be sufficient to provide highly accurate yield predictions. Therefore, the sampling times selected for this study were 6 May 2020 and 10 May 2022, both at similar nodes during the grain filling period of winter wheat. This enabled us to explore the performance of yield prediction models at the same fertility nodes in different environments. In the future, studies could consider multiple sampling sessions throughout the growing season to enhance the robustness of the data and improve prediction accuracy. We observed significant variation and dispersion between the observed and predicted yields. The observed discrepancies may be attributed to the inherent variability in plant growth conditions.

4.5. Analysis of Model Generalization Capabilities

Utilizing a one-year training set and a one-year validation set represents a common approach for evaluating the generalizability and generalization capacity of machine learning models. This method effectively assesses model performance on unseen data, ensuring the model’s reliability for real-world applications. Initially, constructing the model using two years of data allows for the incorporation of historical insights and trends, leading to enhanced accuracy and stability [77]. Training the model on a two-year dataset allows it to consider factors like seasonality and environmental conditions. This approach results in a more comprehensive and representative model [31]. Additionally, employing one year’s data as the training set in this study enables the model to learn historical patterns and correlation patterns, ensuring a proper fit to the data. Using another year’s data as the validation set tests the model’s performance on new data, assessing its ability to generalize and remain stable. Among the models created using validation methods A and B, the Cubist and GLM models show the best predictive abilities. In contrast, the RF model, despite having the highest accuracy, exhibits poor predictive performance and struggles to provide reliable predictions. This outcome may be attributed to the model’s one-year training data not aligning well with the unseen data or a lack of features. Subsequent studies will consider incorporating more than three years of data for in-depth exploration.

5. Conclusions

In this study, we developed yield prediction models using RFE methods, seven individual machine learning algorithms, and the BMA ensemble learning algorithm. The models, developed from UAV-collected RGB and MS features in 2020 and 2022, consistently demonstrated high accuracy. Whether using RGB features, MS features, or RGB + MS features, all seven models showed reliable performance. Among the models constructed with recursive feature selection, the Cubist model demonstrated the best performance (R² = 0.715, RMSE = 0.831 t·ha⁻¹, MSE = 0.691 t·ha⁻¹). Furthermore, the BMA model consistently outperformed the individual machine learning models in all cases, achieving a maximum R² of 0.725. Training our yield prediction models with data from one year and validating them with data from another year proved effective. Notably, the Cubist and GLM models displayed robust predictive performance. The BMA model continued to surpass the performance of the individual machine learning models, achieving the highest accuracy in validation approach A (R² = 0.673, RMSE = 0.560 t·ha⁻¹, MSE = 0.313 t·ha⁻¹). This demonstrates the superior generalization ability of the BMA model, making it a promising choice for future research involving multi-year data prediction.