Application of Machine Learning Algorithms in Nitrous Oxide (N₂O) Emission Estimation in Data-Sparse Agricultural Landscapes

Abstract

To understand if machine learning algorithms could be employed in agricultural landscapes to estimate N₂O emissions, multiple linear regression (MLR), random forest regression (RFR), support vector regression (SVR) and artificial neural network (ANN) algorithms are tested on an agricultural site in Ontario, Canada. Two scenarios, High Input (HI) and Low Input (LI), were used to check the performance of these algorithms’ using R², RMSE, VE, p-factor, r-factor and visual inspection indicators. The HI consisted of discrete measurements of N₂O, rainfall, temperature, fertilizer application dates, soil nitrate, ammonium content and pH values, whereas the LI scenario did not use the latter three. The results indicated that MLR was inapplicable as the data did not satisfy its fundamental assumptions. RFR, SVR and ANN under HI were able to capture 64% (66%), 59% (63%) and 94% (43%) of the variability of emissions within the training (testing) datasets. Subsequently, these models were able to capture 92%, 29% and 75% of high emissions (>10 gm/ha/day) within their predictive intervals of 95% confidence. RFR, SVR and ANN under the LI scenario captured 72% (68%), 61% (66%) and 81% (68%) of the variability in N₂O emissions within the training (testing) datasets. While these models were found to have comparable performance in both HI and LI scenarios, HI was found to be better at capturing high emissions. Based on the computational cost, ease in finetuning, capture of peak emissions and stable model performance, RFR and ANN are recommended to estimate N₂O emissions in the study area and similar agricultural landscapes in future studies.

1. Introduction

The Intergovernmental Panel on Climate Change (IPCC) sixth assessment report (AR6) concludes with high confidence that anthropogenic GHGs have been the dominant driver of observed global warming since the mid-20th century [1]. Of the key GHGs, N₂O is the third major gas in terms of volume, yet remains highly important due to its significant ozone depletion and global warming potential [2]. It is also reported that there is emerging N₂O–climate feedback due to the addition of nitrogen in the atmosphere and subsequent climate change, thereby highlighting the urgency to mitigate these emissions more than ever [3]. Thus, unless we understand and estimate N₂O emissions from different landscapes, we cannot mitigate relevant actions to reduce them.

Measuring N₂O emissions is possibly the only way to accurately quantify its release to the atmosphere. However, measurements can be tedious, cumbersome and resource consuming. Discrete measurements of N₂O are generally taken with manual static chambers (~1 m² coverage area) and then analyzed with gas chromatography techniques [4,5]. Despite relatively simple operating principles, minimal field requirements and low cost, chamber measurements have their own limitations. Potential leaks, the influence of disturbance of soil surface, the influence of concentrated built-up of gases and missing coverage of the spatial and temporal variability of gas fluxes are the most common issues with chamber measurements [6]. Automatic chambers can measure the fluxes continuously for longer periods but the issue of spatial coverage and other limitations still persist [7]. Also, while the static closed chambers are low cost and do not require power in the field to operate, the automatic chambers are still expensive and need power to operate continuously thus increasing the risk of power outages and instrument failures.

If the N₂O fluxes need estimation over a larger geographical domain, micrometeorological (MM) techniques like eddy covariance, eddy accumulation, relaxed eddy accumulation, flux gradient and Bowen ratio methods can be used [8]. The basic assumptions for MM techniques are that the atmospheric turbulence is stationary in nature, the terrain is flat and homogenous and there are negligible mean vertical wind velocities. While MM increases the probability to detect hot spots and hot moments of N₂O, the majority of the above stated requirements are seldom met, which leads to data quality issues along with risk of instrument failure and data acquisition. It is also computationally expensive to process the data acquired from MM and installation and operation costs are high. When field measurements are not realistic or possible over macro-scales, modeling techniques may be employed that simulate the relevant biogeochemical cycles pertaining to N₂O emissions under different environment and management conditions.

Giltrap, et al. [9] discussed that different process-based and empirical models may be employed to extrapolate the GHG field measurements to macro scales, simulate long-term emissions and run mitigation scenarios to reduce them. There have been multiple developments of N₂O emission routines in process-based models like NGAS [10], DNDC [11], APSIM [12,13], EPIC [14], DayCENT [15] and SWAT [16,17]. However, process-based models are relatively complex to understand [18,19,20], have complex algorithm and parametrization needing many and long-term data [21,22] and have their own representation on how different environmental processes affect N₂O emissions [23]. Similarly, there is still a lot of ground to cover to make these models highly efficient in capturing N₂O emissions [15,24,25,26]. Compared to the process-based models, site-specific empirical models have been found to perform better to estimate N₂O emissions [26,27,28]. Accordingly, machine learning algorithms are gaining popularity these days in N₂O estimation studies. The ability of machine learning algorithms to identify important variables, process the data and information in a shorter time than process-based models and a reduced requirement to understand the complex details of the N₂O emissions process boosts its usage in N₂O related studies [29]. Based on the input data provided to the machine learning algorithms, decisions can be made by understanding the relationship among data variables without any explicit process equations and pathways identification [23], which have often been reported in the literature with encouraging results.

Saha, Basso and Robertson [26] employed random forest regression techniques to estimate daily flux variance of N₂O in the US Midwest and found it superior to a process-based model with an explanation of 65–89% of daily fluxes on the calibrated site and around 38% of weekly and biweekly fluxes in an untested site. Once coupled with the outputs from process-based models, the revised algorithm could explain 51% of the variability of fluxes in the test sites. The random forest technique was applied by Saha, et al. [30] to estimate efficient N₂O sampling strategies. Taki, et al. [31] evaluated linear interpolation and artificial neural networks (ANNs) to fill the gaps in sub-daily N₂O emissions estimated using MM techniques in Elora research station from 2001 to 2006 and reported the outperformance of the latter. Bigaignon et al. [32] evaluated linear interpolation and ANNs to fill the gaps in daily N₂O fluxes measured using automated chambers during 2012–2016 in the southwestern part of France and recommended that the combination of both an ANN and linear interpolation yielded the best results compared to the individual ones. Goodrich, et al. [33] tested ANN and K-nearest neighbor (KNN) regression in the cattle farms of Waharoa, New Zealand, and reported the outperformance of the latter in filling the gaps of N₂O measured using the MM method. Philibert, Loyce and Makowski [29] used a random forest regression technique on the global N₂O database of Stehfest and Bouwman [34], with local environment and management variables, and reported its outperformance of several other regression models. Similarly, Maier, et al. [35] applied a marginal distribution sampling technique and RF to fill the gaps in N₂O along with other GHGs. Mehmandoost Kotlar, et al. [36] compared multiple linear regression, support vector machine and random forest regression to predict N₂O fluxes under the presence and absence of cover crops. Similarly, different machine learning algorithms are increasingly being used to estimate atmospheric [37,38] and environmental variables [39,40,41], which can support in the estimation of N₂O fluxes.

While the gap-filling technique for N₂O has not been standardized [42], several steps are currently being taken to recommend the best approach. Following the increasing usage of machine leaning algorithms in the estimation of N₂O, Dorich, et al. [43] outlined and discussed the potential strengths and challenges of five techniques of gap filling, viz., linear interpolation, random forest regression, artificial neural networks, generalized additive models and autoregressive integrated moving average models. Accordingly, this study aims to (a) evaluate the effectiveness of multiple algorithms (linear regression, random forest regression, support vector regression and artificial neural networks) in predicting N₂O emissions and (b) evaluate scenarios when readily available data is used to train the machine learning algorithms compared to the scenario when they are not. While the findings have implications for agricultural decision making, the primary focus was methodological—identifying which models perform best under different data constraints—rather than developing a directly deployable decision-support tool. The results from this study are expected to guide further studies to apply a readily available dataset for a more detailed estimate of direct N₂O emissions from agricultural farms.

2. Materials and Methods

2.1. Data

In this study, discrete measurements of N₂O (g/ha/day) were collected along with corresponding measurements of soil temperature (°C), soil volumetric moisture (%), pH, soil ammonium content (mg/kg of dry soil), soil nitrate content (mg/kg of dry soil), rainfall (mm) and average air temperature (°C) from “Baggs Farm” and the “Research North” agricultural farm in Ontario, Canada. The location of Baggs farm is 43.459° N, 80.309° W, and Research North is 43.440° N, 80.345° W. A total of 628 observations were taken during the years 2019 and 2020. In addition to the above-mentioned variables, three variables were derived, namely, days after rainfall (Daysrain), days after fertilizer application (Daysfert) and 2-day accumulation rainfall (Srain2D) from the local meteorological and management data. Similar variables are used by Saha, Kemanian, Rau, Adler and Montes [30], Saha, Basso and Robertson [26] and Taki, Wagner-Riddle, Parkin, Gordon and VanderZaag [31] in their N₂O emission estimation studies. The summary of collected data is presented in

Table 1. Summary of collected data.

Variable	Description	Units	Range in the Available Data (Min, Max)
ST	Soil temperature at 10–15 cm depth	°C	0.12, 27.75
VM	Soil volumetric moisture at 10–15 cm depth	%	5.1, 58.3
pH	pH value of the soil	-	6.08, 7.89
Daysrain	Number of days after last rainfall	-	0, 7
Rainfall	Cumulative rainfall on the day of N2O measurement	mm	0, 10.9
Airtemp	Average air temperature on the day of N2O measurement	°C	−1.8, 24.9
Daysfert	Number of days after application of nitrogenous fertilizer (inorganic)	-	7, 489
NH4	Soil ammonium content at 10–15 cm depth	mg/kg	0, 89.32
NO3	Soil nitrate concentration at 10–15 cm depth	mg/kg	0, 110.40
Srain2D	Accumulated 2-day rainfall on the day of N2O measurement	mm	0, 37.7
N2O	Average N2O flux in the given day	gm N2O-N/ha/day	−26.88, 532.34

.

For more detailed information on the collected data and field information, please refer to Ashiq, et al. [44] and Ashiq, et al. [45]. To detect whether the N₂O data had any temporal autocorrelation, the measurement intervals were calculated at each site. As these intervals were highly variable (Supplementary Table S1), a generalized least squares model with a continuous time (AR1) structure was used to check the autocorrelation. For this purpose, the corCAR1 model, which inherently accounted for irregular spacing, was used with the “nlme” package [46].

For covariates, continuous data on soil moisture, soil temperature and climate data and discrete information on crop yield and onsite mineral nitrogen are recommended for collection [43]. While the crop yield information was available [44], it was not used due to its non-differentiation at the test sites. Practically, information like Ph, NH₄ and NO₃ are not available from secondary data sources and need site specific measurements, whereas other predictor variables may be collected or estimated through external sources. For example, soil moisture and temperature may be estimated using techniques employing satellite images like SMAP [47] and Landsat [48]. Acknowledging the difficulty in estimating the data variables required to run the models, two scenarios, viz., High Input (HI), including all variables, and Low Input (LI), excluding Ph, NH₄ and NO₃ are formulated; all four algorithms were tested for both scenarios.

2.2. Training and Testing Data Sampling

Multiple studies have recommended different minimal sizes to perform statistical learning algorithms. A rule of thumb for sample size selection is based on a ratio of the number of observations to the independent predictors of 30:1 [49]. Alternatively, it can be estimated as 50 + 8 × n, where n is the number of independent variables [50]. Bujang, et al. [51] reported that for more than 10 variables, a minimum sample size of 300 should be used as a realistic approximation of estimates for the population. In other studies pertaining to N₂O emission estimation using machine learning algorithms, various sample sizes have been used. Philibert, Loyce and Makowski [29] used 594 observations from the global N₂O database [34] to train the random forest regression. Similarly, Zhuang, et al. [52] had 209 observations to apply an artificial neural network. Abbasi, et al. [53] used 26 datasets to predict N₂O emissions from paddy wetlands using artificial neural networks. As both data collection sites were in geographical proximity and the management practices were alike, the data was merged into a single dataset. It is worth noting that some researchers, like Korkiakoski, et al. [54], trained these algorithms separately for each measurement site.

To ensure that all algorithms selected in this study were using the same data instances for training and testing, we fixed the seed for a random number generator (set.seed variable) in R and identified the unique and non-duplicated random rows of the dataset to split the data for training and testing purposes in a 70:30 ratio [53]. By doing this, it is ensured that the testing data is excluded before the commencement of model training and when the validation is performed, it is performed on completely independent data. Some studies split the data into an additional bin of training, testing and validation sets to avoid duplication of data during training and testing and to avoid possible overfitting. Many other studies also use K-fold cross-validation to generalize the model and reduce the risk of overfitting. However, for this study, we have kept the simple approach of a single split into two independent datasets, but for individual algorithms like RFR, we have employed bootstrapping techniques to avoid overfitting.

The seed number that was able to split the entire data into training and testing datasets of similar data statistics was finalized to ensure that the characteristics (values) of N₂O during the training period were not too different than those in the testing period. A similar approach of fixing the random number generator to reproduce the results is followed in other N₂O studies [35].

2.3. Multiple Linear Regression (MLR)

Regression analysis is a statistical technique that estimates the relationship among variables with a reason and result relation [55]. In MLR, a linear relationship is estimated between a dependent variable (predictand: Y in Equation (1)) and multiple independent variables (predictors: X₁, X₂, … X_n in Equation (1)).

$$Y={\beta }_0+{\beta }_1\ast X_1+ {\beta }_2\ast X_2+\cdots +{\beta }_n\ast X_n+\epsilon ,$$

(1)

where ${\beta }_0$ is the intercept, ${\beta }_1$… ${\beta }_n$ are the parametric coefficients and $\epsilon$ is the residue error.

This best fit linear relationship is used to predict the observed instances and the error of MLR ($\epsilon$) is expected to follow a Gaussian distribution [56]. There are some pre-requisites of applying MLR to predict a variable of interest, of which, no multi-collinearity among predictors, no autocorrelation among the observed values, no random predictors (each predictor follows a distribution) and homoscedasticity of residuals are the key ones. If there is no linear relationship among the predictors and predictand, the predictor variables may need transformation. In this study, the assumptions needed to apply MLR on the study dataset were tested and only when they were deemed satisfactory was the appropriate MLR technique applied. If the majority of the assumptions were unmet, MLR was still applied to check if there was any merit to its application in N₂O estimation in the study sites.

To test multicollinearity among predictor variables, the R package “car” [57] was used. The package estimates the Variance Inflation Factor (VIF) to detect the multi-collinearity of model parameters by estimating the correlation and its strength between the independent predictor variables. VIF values of 1 or less imply no correlation, values of 1-5 imply moderate correlation and values above 5 imply strong multicollinearity, meaning the variables are recommended to be removed from the model as they are expected to bring in redundancy in model performance [58]. As the N₂O variables were measured in discrete intervals and not continuous ones in our study, the autocorrelation criterion was not considered. To check the linear relationship between the predictors, a diagnostic plot of residual errors and fitted values was inspected. If the line is horizontal in nature and without a distinct pattern, it is an indication of linear relationship. Similarly, the quantile plots of residuals can be inspected to check if the residuals are normally distributed and nominally follow a 1:1 line. For the homoscedasticity (homogeneity of variance of residuals), residuals are inspected in a scale-location plot. If a relatively horizontal line is seen in this plot with equally spread plots, it indicates that there is no problem of homoscedasticity and MLR can be applied. All these analyses were performed in R software, version 4.2. For MLR, four scenarios of model building are assumed:

(1)

HILMo: N₂O is influenced by all independent predictors of Table 1. There is no interaction among the independent predictors.

(2)

HILM: N₂O is influenced by all independent predictors of Table 1. There is interaction among ST, VM, Ph and NO₃.

(3)

LILMo: N₂O is influenced by independent variables under LI. No Ph, NO₃ or NH₄ data values are used to train the model. There is no interaction among the independent predictors.

(4)

LILM: N₂O is influenced by independent variables under LI. Interaction among the ST, VM and Srain2D was permitted.

The ordinary least squares method is used to estimate the parameters of the MLR, which used mean square error as the model objective function.

2.4. Random Forest Regression (RFR)

RFR is a supervised machine learning algorithm that uses the results from multiple random decision trees predicting a continous variable [59]. Accordingly, it can be expected that the output of an RFR is more accurate than a single decision tree. For a better understanding of RFR, refer to Supplementary Figure S1.

RFR creates a set of decision rules based on the predictor input, then these trees run parallely and independently to provide their individual predictions. The random seed selected at the start of study was used to generate training and testing datasets at a 70:30 ratio. Then, this former data is used to create a bootstrap sample of observations (“in-bag”) and some data is left out as “out-of-bag”. Secondly, the explanatory variables and their number (mtry) are randomly selected for each tree. A tree is built using the “in-bag” data and “mtry” variables by recursively partitioning the initial data into smaller groups called nodes via binary splits based on a single predictor. Each split is finalized by comparing the splits with all predictor variables. Tree growth generally ends when the number of observations in the terminal node reaches five or fewer for regression. This process is repeated for a number of trees (ntree) to bootstrap samples and trees.

To finalize the number of trees in the forest and the variables in each tree to best estimate the N₂O, we ran a permutation of 1–9 variables and 1–400 trees. The R package “randomForest” [59] was used to fit random forest regression on our dataset. The mean square error is used to identify the variables of importance in the RFR algorithm and the number of variables in each tree and numer of trees are finalized for the best R² value between different simulated and observed N₂O values during both the training and testing of the datatsets. Unlike MLR, which has multiple pre-requisites on predictor-predictand relationships, RFR does not need any prior relationship between them, thus making it one of the most powerful and easy to apply machine learning algorithms for both classification and regression problems.

2.5. Support Vector Regression (SVR)

Initially proposed by Vapnik [60], support vector machine (SVM) is an effective multi-purpose analytical tool that is mostly employed for classification problems. Recently, it has been advanced to use in regression problems (hence support vector regression (SVR)). It is an algorithm that can handle linear and non-linear data and also has an outlier detection ability [61]. In this supervised algorithm, each data point is plotted in a multi-dimensional space, and the correspinding value of each predictor variable represents a coordinate. A best fitting hyperplane is then estimated, i.e., the one that has maximum number of data points. Contrary to the MLR, where the objective was to minimize the error between the predicted and the observed values, SVR tries to fit an optimal plane within a pre-defined threshold (epsilon). This threshold is the distance between the hyperplane and the boundary line. This concept of maximal margin thus allows SVR to be viewed as a convex optimization problem that is penalized with a cost parameter to avoid overfitting. Additionally, there is no pre-requisite in SVR that the explanatory variables follow a particular distribution.

To construct a model in SVR, we need to specify an appropriate kernel function, which could be either linear, polynomial, sigmoid or radial. A radial basis kernel function was selected in this study, which transformed our data from the non-linear relationship to a linear plane where SVR was fitted and then the data was back-mapped to the original non-linear space [62]. SVR with radial basis function is reported to perform better for N₂O emission estimation by Mehmandoost Kotlar, Singh and Kumar [36]. The SVR is fine-tuned by varying the error and penalty cost parameters. We defined the range of error from 0 to 1 (increments of 0.1) and cost parameter from 1 to 250 (increments of 0.5). The advantage of SVR is that it is robust to outliers and there is no pre-requirement for scaling or physcially transforming the variables. Also, the hyperplane that is close to the training data points is ignored and optimally placed to prevent it from overfitting on training data [63]. In this study, the R package “e1071” [64] was used to perform SVR.

A non-linear SVR function F(x) is presented in Equation (2).

$$F\left(x\right)=\sum _{i=1}^M({\propto }_i-{\propto }_i^{\ast }) K (X_i,X)+\delta$$

(2)

where M = number of observed data points, α is the lagrange multiplier, K is the transformation kernel and δ is the bias [65]. For a detailed explanation of SVR, please refer to Shaahmadi, et al. [66].

2.6. Artifical Neural Networks (ANNs)

An ANN is a model mimicking the workflow process of neurons in a human brain by continuously learning with each data instance of the training data. It is a multi-layer perceptron (MLP) with at least three layers: input, hidden and output [67]. Represented as nodes and links, each layer receives input from the previous layer’s output. Generally, the number of neurons in the first layer is the same as the number of predictors available in our dataset. Thus, each node in the first layer receives data values of corresponding variables for an instance, then passes it to the second round with corresponding weights in each link and a bias value. In the second layer (or subsequent layers), a hidden layer exists, which receives a summation of weights and corresponding data values with an individual activation function. Similarly, these values are then passed to the next layer, if any, and this procedure is followed. By the end of hidden layer, the output is then generated at the output node. Based on the prediction made and actual value at the output node for this instance, the corresponding biases are adjusted and another data instance is passed through the model. In this study, we used R package “neuralnet” [68] to fit ANNs to the data. When the ANN was used on original data, the models did not converge. So, the data was scaled before using the ANN.

Since it was likely that we were dealing with the outliers in the predictor and predictand variables in our study, an appropriate scaling technique was required. Thus, we used a robust scaling technique that used the median and interquartile range of each variable to maintain the relative distances between non-outlier datapoints. This is a technique frequently used by researchers in N₂O emissions studies [69]. The equation used to scale the data is presented in Equation (3).

$$F{X}_n={\displaystyle \frac{X_i-Q_1}{Q_3-Q_1}}$$

(3)

where FXn = normalized data value, X_i is the data value, Q₁ is the first quartile and Q₃ is the third quartile.

2.7. Comparison Metrics

While each algorithm discussed above uses mean square error as the objective function in our study, we have identified Coefficient of Determination (R²), root mean square error (RMSE) and volumetric efficiency (VE) metrices to evaluate the performance of the algorithms to predict the best estimate of the N₂O.

R², ranging from 0 (poor model) to 1 (perfect fit), explains how well the input features explain the variability in emissions. Technically, it reflects how well the model fits but not the accuracy of the individual predictions. From a decision-making perspective, a high value (>0.7) would indicate a model capturing the overall emission patterns, which would be useful for exploring key drivers and informing high-level policies regarding those drivers. RMSE is expressed in the same units as the variable and ranges from 0 (perfect prediction) to infinity. It quantifies the average magnitude of prediction errors and helps to assess how closely the model predicts actual emission values, supporting applications like emission forecasting, compliance monitoring or planning emission abatement strategies. For multiple models with a comparable capture in the variance of observed values (similar R² values), RMSE provides information regarding which model to select based on their lowest RMSE estimates. However, RMSE is sensitive to outliers and deviation from bigger observed values can distinctly increase it, thus warranting a careful scrutinization before finalizing the selection of best model. However, since N₂O values are desired to be captured for their peak emission periods, RMSE remains a relevant comparison metric to evaluate the performance of models estimating N₂O values. Similarly, VE, ranging from 0 (no efficiency) to 1 (perfect volumetric match), evaluates model’s ability to replicate cumulative emissions and thus emphasizes aggregate accuracy over individual predictions. Thus, for applications like total emission accounting or long-term nutrient management, a high VE would indicate dependable cumulative estimates. As well as these indices revealing the performance of each model to estimate the best possible estimate of N₂O, we also identified the associated uncertainty in each prediction from the models using prediction intervals (PIs) [70].

For each algorithm, the best estimates of predicted values are associated with the prediction interval (PI), which is a quantification of the possible observed values for a given instance of input variables. For data with low variability, these PIs are relatively narrow, whereas for variables like N₂O, which is known to have lot of variation, we can expect a wider prediction band. For MLR, we used the inherent prediction intervals using the standard Gaussian technique. Each band is generated at 95% probability, which indicates that for any given large database of measured N₂O values and their respective co-variates, there is a 95% probability that any measurement of N₂O will fall within the generated band. For RFR, SVR and ANN, the bootstrapping technique is used to generate the predicted values of N₂O for a given data instance and then later summarized with the standard Gaussian technique. The bootstrapping technique has been tested and verified on RFR [71,72], SVR [73,74] and ANNs [75,76]. Two additional statistical factors, viz., the p-factor and r-factor, are also calculated to quantify these uncertainties. The p-factor, ranging from 0 to 1, indicates the fraction of observed values captured within model’s 95% prediction uncertainty bounds. Thus, a p-factor closer to 1 would mean that the model is favorable for uncertainty-aware decision making, especially in N₂O modeling where capturing variability is critical. Similarly, the r-factor can range from 0 (tight band) to values >2 (very wide band) and is recommended when the values are less than 1 [77] for precise predictions. It is computed as average width of the uncertainty band divided by the standard deviation of measured variable [78]. This is important for high-resolution decisions such as site-specific management and needs to be balanced with the p-factor to avoid overconfidence. All these metrics form a robust evaluation framework, as R² and VE assess model adequacy, RMSE ensures quantitative reliability and the p and r factors support decisions under uncertainty. Thus, it is recommended to analyze all metrices before interpreting the models for environmental and agricultural decision making. All metrics were calculated using the “hydroGOF” package in R [79].

3. Results

3.1. Testing Autocorrelation

While the data used in this study is treated as discrete measurements, it should be checked whether temporal autocorrelation exists among the measurement sites, which would affect the applicability of the models. This can easily cause overfitting and biased evaluation and uncertainties may be underestimated. Since the dates between measurements were highly variable across the measurement sites, a continuous time model (AR1) was used in conjunction with a model that inherently used irregular intervals [80]. The results suggested that within each site and across the dates of measurement the estimated AR1 autocorrelation parameter (Phi) was 0.024, which is very close to zero, indicating very weak or negligible autocorrelation. Conversely, if it were close to 1, it would have indicated a strong temporal autocorrelation. With this test, it was concluded that the N₂O levels at the measurement sites (Figure 1) were almost uncorrelated with the next date and autocorrelation will not bring any bias to the further testing of the models.

3.2. Training and Testing Data Sampling

To demonstrate that a bias starts from as early as selecting a random seed to split the training and testing data for similar analysis, we performed a bootstrap of the simple linear regression of N₂O with all variables used in this study. The seed number to split the entire dataset into random subsets of training and testing data (in 70:30 ratio) increased by 1, from 1 to 1000. It was observed that there could be a significant bias arising from the selection of a seed number in the algorithm applied later (Supplementary Figure S2). To ensure that all algorithms selected in this study are applied on same dataset, which is reflected properly in both training and testing samples, a seed number of 16 was selected to ensure similar data characteristics (Figure 2). It is to be noted that selecting the best performing seed here could induce an additional bias, so the seed yielding close to median performance yet capturing the range of N₂O data and its statistical characteristics was selected.

With seed number finalized as 16, it was observed that N₂O values are distributed proportionately during the training and testing periods. The first quartile of N₂O values during the training (testing) periods were 0.7 (0.8), the medians were 1.5 (1.6), the means were 11.1 (9.5) and the third quartiles were 4.3 (4.6) gm/ha/day. This makes our technique similar to a stratified sampling technique where the continuous target variable is binned into quartiles and the training/testing split is performed proportionately. Similarly, it was ensured that both training and testing data had their peak emissions and sinks of N₂O values.

3.3. MLR

When the training data was subjected to MLR without any scaling or transformation of its predictor variables, it was observed that multiple predictors were heavily influenced by multicollinearity (

Table 2. Variance Inflation Factor of different linear regression models selected in the study.

Variable	VIF-HILMo	VIF-HILM	VIF-LILMo	VIF-LILM
ST	7.644	14,585	7.028	29.359
VM	2.732	10,799	2.578	17.846
Ph	1.236	98.59
Daysrain	1.79	2.19	1.784	51.775
Rainfall	1.422	1.64	1.414	1.441
Airtemp	5.236	6.07	5.099	5.816
Daysfert	1.721	1.94	1.153	1.215
NH4	1.066	1.29
NO3	1.65	166,292
Srain2D	1.65	1.84	1.603	1.832
ST: VM		8233		10.519
ST: Daysrain				35.761
ST: Ph		14,886
VM: Ph		10,517
VM: Daysrain				39.08
ST: NO3		166,779
VM: NO3		133,735
Ph: NO3		167,053
ST: VM: Ph		8094
ST: VM: Daysrain				12.059
ST: VM: NO3		139,166
ST: Ph: NO3		168,306
VM: Ph: NO3		131,980
ST: VM: Ph: NO3		137,431

).

Generally, it is recommended to remove the variables with high multicollinearity (>5) from the analysis, as they are expected to bring in redundancy to the model and their omission would not significantly alter the model performance. However, when we omitted the variables with multicollinearity, the performance of the model dropped significantly (Supplementary Table S1).

Similarly, when the plots of residuals were inspected for homoscedasticity, it was found that the residuals were not spread equally along the range of predictors that indicated heteroscedasticity and thus the violation of the assumption for MLR (Supplementary Figure S3). The assumption of linearity among variables and the output was also inspected (Supplementary Figure S4), which showed that there was a pattern in the residuals indicating a non-linear relationship between the predictors and predictand. Thus, it was summarized that MLR cannot be applied to the study site to predict N₂O emissions without further intervention of data. Since the N₂O emission values had negative numbers as well (indicating sinking of N₂O into the soil), log-transformation could not be performed, which could have been an alternative in scenarios where data variables are not linearly related [81]. The results are quite expected, as it is well known that N₂O generation is a highly non-linear process and depends on a multitude of environmental variables [16,82].

Even if all these assumptions are ignored and MLR is applied on the given data, it is observed that the technique does not explain more than 30% of the variability of the N₂O fluxes during the training and 33% of the fluxes during the testing (Supplementary Table S2). A relatively old study by Ciarlo, et al. [83], which fitted MLR on soil moisture to explain the variations in N₂O, also reported that none of the MLR models were able to satisfactorily explain the N₂O emissions properly. Similarly, an even older study by Clayton, et al. [84] reported that MLR was explaining about 28% of the variability in N₂O fluxes from a fertilized grassland. With site-specific variables, MLR may be able to explain more variability in the N₂O fluxes than for a mixed dataset [85]; however, this explanation is still much lower. Mehmandoost Kotlar, Singh and Kumar [36] also reported that MLR was not able to estimate N₂O emissions with satisfactory accuracy. For a coarse time-scale (e.g., yearly), MLR may still explain more than 70% of the variability in N₂O fluxes in a similar study to ours [86].

It is also observed that none of the MLR techniques selected in this study could capture the hot moments of N₂O, as indicated by the predictive intervals shown as red error bars in Figure 3 for the HILM and LILM scenarios.

While in MLR models the predictive intervals were able to capture the emissions within 100 gm/ha/day and thereby have a p-factor of 0.84, the bigger values were left out, which is not desired from a N₂O emission simulation model. Accordingly, it was concluded that MLR is not the right algorithm to estimate N₂O emissions in our study area.

3.4. RFR

Compared to the MLR models, RFR was observed to have a distinct outperformance in the simulation of the N₂O emissions in the study area. It is also observed that RFR can easily overfit (Figure 4). The combination of “mtry” and “ntree” variables in RFR for HI revealed very high R² values (~0.9–0.97) during the training period and yielded very poor performance during the testing period (~0.1–0.4). For HI, it was observed that if the number of variables to include in a tree was kept as a lower number (2–4) and the number of trees was increased, a more favorable performance could be seen during both the training and testing periods.

While inclusion of multiple trees in an RFR increases its computational cost, it was not found to be significant in our study (training the data with multiple trees took a few extra minutes). For the LI scenario, the performance of RFR for different numbers of trees and variables is presented in Supplementary Figure S5. It is observed that for LI, in contrast to HI, the inclusion of more variables was found to have better performing models. However, for an optimal number of four variables in a tree and with a higher number of trees, the performance of the model for both the training and testing datasets was better. It is also observed that as the number of trees exceeded 40, the decrease in RMSE was not significant (Supplementary Figure S6), thereby prompting the number of trees to be limited to 50 for both models. Thus, an RFR algorithm with the architecture of variable/number of trees as 2:48 (4:40) was selected for the HI (LI) scenario in this study.

The HI RFR model was able to explain 65% of the variability in the measured N₂O fluxes with an RMSE of 27.9 (20.0) gm/ha/day of the training (testing) datasets. Similarly, it was found to have 43% (35%) volumetric matches to the observed measurements for the training (testing) datasets. Interestingly, the LI RFR model was found to have better performance in estimating N₂O emissions during both the training and testing periods. LI RFR was able to explain 72% (68%) of the variability of measured fluxes, with lower errors of 23.9 (19.2) g/ha/day and better volumetric matches of 52% (46%) during the training (testing) periods.

The model performance statistics for these two scenarios are quite interesting, as the LI RFR model is found to perform better than the HI RFR model, despite using fewer predictors, and can be described as a classic example of the “less is more” principle in machine learning. The reason for this could be the collinear variables (air temperature and soil temperature) or possible noise from NH₄ and NO₃ measurements. The higher number of covariates means more competition for split importance (evident from variable importance plots prepared in Figure 5), as opposed to the LI model where fewer features allowed the model to focus more clearly on highly relevant predictors. These performance metrices, along with a visual plot of HI and LI RFR, are presented in Supplementary Figure S7.

In the HI RFR, the five most important variables in terms of their influence on increasing the mean square error (MSE) are Srain2D, Daysfert, NH₄, Rainfall and NO₃. The gradual decline in importance indicates that the model drew from many variables. Similarly, under the LI RFR scenario, Daysfert was the most important variable, followed by air temperature and Srain2D. The overall shape also has a steep drop-off, indicating a heavy reliance on a few key features. Since Daysfert, Srain2D and Airtemp were consistently among the important predictors in both scenarios, this suggests that the timing of fertilization and recent rainfall events are the key drivers of N₂O emissions. This aligns well with the established fertilizer-driven N₂O emission hypothesis, especially when favorable environmental conditions persist (seen through the importance of rainfall and temperature in the plots).

The predictive intervals of both the HI and LI RFR models are presented for higher N₂O emissions (for instance where the observed N₂O values were > 10 gm/ha/day) in Figure 6. It is observed that both HI and LI RFR have the capability to capture 90% of the emissions within their 95% predictive intervals. Similarly, the average width of the uncertainty band was found to be 1.5 for HI and 1.3 for LI, indicating a slightly better performance for the LI scenario. However, it is observed that the peak emissions (for, e.g., 300–400 gm/ha/day) were not encapsulated within the predictive intervals of these models. The reasons for this could be that the number of samples with relatively lower N₂O values is significantly higher than the samples with higher values and have been used to train the models. This also means that more peak emissions need to be included in the training dataset.

While these results were for the best estimate of N₂O values, it is equally important to understand the uncertainty arising from the model itself. These uncertainties are quantified through the band of predicted data by different trees of each scenario. However, there could be an additional uncertainty stemming from the model itself, which is not represented through such analysis. For this purpose, techniques like Bayesian models [87] may be employed.

RFR was reported to explain nearly 90% of the variability of yearly N₂O fluxes over the LUCAS points in Europe [88]. Similarly, Villa-Vialaneix, et al. [89] reported that as number of data instances in the training data increases, RFR performs better in N₂O estimation. For a training data of 500 instances, it was reported to explain 70% of the variability of fluxes. A different application of RFR for N₂O emissions due to biochar was reported to explain 24% of the variability [90]. Thus, it could be summarized that RFR had a much better overall performance compared to MLR in the study area; moreover, the data indicate that even the LI scenario could perform as well as or even better than HI in regions with limited data. One of the key disadvantages of techniques like RF could be that, with respect to a MLR, they cannot precisely analyze the impacts of changes in different input variables in N₂O emissions [29]. For example, even if the soil moisture is changed to 100%, with the RFR model we selected, most of the trees might not use the soil moisture as a variable, so the aggregated results might not show the anticipated change in N₂O values.

3.5. SVR

It is observed that the SVR algorithms trained in this study were not sensitive to the epsilon parameter. Only the changes in the cost value of model changed the RMSE computed by comparing the measured and model-simulated values. Villa-Vialaneix, Follador, Ratto and Leip [89] fixed the value of epsilon to 1 and only fine-tuned the SVR based on the kernel and cost parameters in a similar study. Figure 7 presents the calibration summary of the HI scenario; similar performance was observed for the LI scenario, as shown in Supplementary Figure S8.

The performance summary of SVR in predicting N₂O values under the HI and LI scenarios is presented in Supplementary Figure S9. The R² value of the SVR model under the HI scenario was found to be 0.59 (0.63) for the training (testing) data. Similarly, the average deviations in terms of RMSE values were found to be 28.6 (26.0) g/ha/day for the training (testing) datasets. The volumetric match in terms of VE explained that SVR under the HI scenario was unable to capture the volume of N₂O during the testing period, despite it capturing 38% of the volumetric efficiency during training period. Like RFR, the LI scenario was found to have a comparable performance to that of the HI scenario. It is observed that under LI, the R² values estimated from the SVR are 0.61 (0.66), the RMSE values are 28.3 (23.4) gm/ha/day and the VE values are 0.34 (0.19) during the training (testing) periods. While the deterministic best estimates from the SVR are found to be comparable for both scenarios, the uncertainty in N₂O emissions are better reflected in the HI scenario (Figure 8). However, it was observed that the prediction interval of SVR had poor coverage and underestimated the uncertainty. The narrow predictive intervals relative to the variability of observations also mean that the model was overconfident and not a good fit to estimate N₂O emissions and the uncertainty in them in the study area.

Villa-Vialaneix, Follador, Ratto and Leip [89] reported that SVM (or SVR) could explain nearly 85% of the variation in the N₂O fluxes for a sample size of 500. Mehmandoost Kotlar, Singh and Kumar [36] also reported that SVR had better performance in estimating N₂O emissions compared to other models. However, in our study, we found that SVR was comparable to RFR in predicting a deterministic value for N₂O emissions. Also, compared to other algorithms, it was found that SVR is the most time consuming, as it essentially converts the entire data into a kernel matrix and takes a long time to fine-tune [89]. SVR had a better performance than RFR in estimating soil organic carbon stocks, explaining 60% of the variability in the study by Were, et al. [91]. However, owing to the higher computational expense, a similar accuracy to that of RFR to estimate the N₂O emissions and the relatively low predictive intervals in this study, it was inferred that SVR was inferior to RFR in the study area for N₂O emissions estimation.

3.6. ANN

Since there is no structured methodology to construct an ANN, it can have multiple hidden layers and nodes [92]. Thus, a thumb rule is followed in this study to construct a feed-forward ANN. Since the architecture was allowed to expand to two hidden layers, multiple ANN models were constructed with the number of nodes in each hidden layer ranging from 1to 10 for HI and from 1 to 7 for LI. Thus, a total of 100 ANNs were evaluated for the HI scenario and 49 ANNs were evaluated for the LI scenario. Accordingly, it was observed that for a HI ANN, the model was prone to overfitting (observed from high R² values during training and low R² values during testing). Accordingly, a few pockets of optimal performance were identified that had relatively similar performance during both the training and testing periods. It was observed that during testing under the HI scenario, three pockets (5,8; 5,4; 8,8) had good performance; we selected the latter one (Figure 9).

Similar calibration statistics for the LI scenario are presented in Supplementary Figure S10, where we finalized an ANN with 5:4 nodes in the two hidden layers. While it is generally recommended to have fewer nodes in hidden layer to prevent the overfitting of ANNs, other studies pertaining to N₂O estimation have used higher numbers of nodes in their hidden layers [23]. In general, it was observed that the ANN had high accuracy in capturing the measured N₂O emissions in the study area (Supplementary Figure S11). The HI ANN could explain 94% (44%) of variability of measured N₂O, the RMSE values were 10.8 (35.2) gm/ha/day and the volumetric efficiencies were 0.59 (−0.1) for training (testing) periods. It was again observed that for the LI scenario, the results are more favorable, explaining 81% (68%) of the variability and having RMSE values of 18.9 (21.7) gm/ha/day and volumetric efficiencies of 0.41 (0.26) during training (testing) periods. Villa-Vialaneix, Follador, Ratto and Leip [89] reported similar performances for an ANN and RFR for a sample size of 500.

Accordingly, the uncertainty was estimated for values of more than 10 gm/ha/day and showed that the HI ANN was able to capture the N₂O emissions with good accuracy (Figure 10). The model captures 75% of all high emissions within its prediction intervals of 1.25 times the observed variability, reflecting a balanced uncertainty range. The LI ANN had a slightly lower coverage (p = 0.64) but narrower uncertainty levels (1.07), showing slightly higher confidence with narrow spread. It exhibited a good fit for the majority of peak emissions, but a few extreme errors persisted, giving a slight edge to HI ANN.

Taki, Wagner-Riddle, Parkin, Gordon and VanderZaag [31] used ANNs and a linear interpolation technique to fill the gaps in sub-daily N₂O emissions and reported that the ANNs constructed for each year were able to explain 41% of the variability in fluxes. Abbasi, Luithui and Abbasi [53] used an ANN to estimate the N₂O from paddy fields using predictors like soil characteristics, fertilizer inputs and rice cultivar yields and reported excellent performance (R²~0.99) for the testing period. It can be argued that the number of data instances used to train and test the ANN in the study of Abbasi, Luithui and Abbasi [53] was very low compared to ours (only 26 in total), which could be the reason for better performance. However, it can still be argued that, for a relatively smaller data size, having more layers and hidden nodes in each ANN can easily lead to overfitting.

The ANN frequently encountered problems while training the algorithms with the training dataset. Without scaling the data, none of the algorithms converged to a final value and thus the model could not be trained. Even when scaled, one had to be quite careful about the local minima values and the model thus needed multiple initializations of weights and repetition of runs. It was also highly sensitive to the number of predictor variables selected in the first input layer, as all the weights had to be initialized and trained accordingly [89]. Also, it was very prone to overfitting, as observed in Figure 9 and Figure S10.

4. Discussion

Measurement of N₂O emissions from fields in their natural settings using standard techniques may be the best way to really understand their spatio-temporal characteristics. Despite this knowledge, for several reasons, it may not be feasible to employ such techniques over a large area or a long period. With emerging research and the accessibility of data, data readily available from secondary sources can be used in estimating the N₂O emissions from an agricultural landscape. Some ML techniques have used readily available data like soil moisture, temperature and electrical conductivity to explain up to 64% of the variability in methane emissions from paddy fields [88]. This study also shows that ML algorithms like ANN, RFR and SVR could potentially capture trends in N₂O emissions using data like rainfall, temperature, number of days after application of nitrogenous fertilizer, soil moisture and soil temperature. Information on in situ rainfall and temperature can be easily accessed from local weather stations and fertilizer application days can be approximated by talking with local farmers. Similarly, during the past decade, ML techniques have used satellite images to approximate soil moisture [93,94,95], pH [93], temperature [96,97] and, if needed, other atmospheric variables [37,38]. Thus, the novelty of this study is the demonstration that ML techniques can make use of such readily available secondary information to estimate N₂O emissions, which are a product of a very complex biogeochemical process. These techniques can be also used to fill the gaps in the existing time-series of measured N₂O emissions [31,43].

While evaluating the applicability of models, a single performance metric cannot explain the model’s ability for capturing the trends, magnitude and uncertainty of highly variable N₂O fluxes. Thus, in this study, multiple performance indicators were used to robustly evaluate model performance; R² and VE assess model adequacy, RMSE ensures the quantitative reliability and applicability for total emissions accounting and the p-factor and r-factor quantify the uncertainty. These metrices also have relevancy from a decision-making perspective, as models with higher uncertainty (e.g., SVR in our study) could not provide sufficient confidence for fine-tuning decisions, while models with more reliable performance (like RFR and ANN) enable risk-aware decision making. Thus, different models provide value in making decisions pertaining to nitrogen application timing, emissions accounting, climate smart farming initiatives, etc., where both accuracy and uncertainty are important.

It should also be highlighted that several improvements can be made to future studies on similar applications. The common issue of potential overfitting, which was identified across all algorithms, can be addressed through several procedures. If the data size is not large, it is better to reduce the number of variables (avoiding redundancy), which means simpler models (like fewer hidden layers in an ANN) can be built with fewer parameters. Similarly, penalty parameters can be specified in the loss function to discourage the complexity. Future studies are recommended to adopt a K-fold cross-validation technique to generalize the models. Similarly, the bootstrapping techniques employed to quantify the uncertainty in the N₂O emissions estimations are likely to capture the aleatoric uncertainty but might not be enough to handle the epistemic uncertainty arising from the models themselves. Thus, for future studies, it is recommended to further explore the use of Bayesian techniques. Many studies have used the hierarchical Bayesian model for N₂O, assuming lognormal probability distribution [87], or used these techniques to calibrate the existing model parameters, with great improvement [98,99]. It should also be acknowledged that the peak emissions were not captured with great accuracy in our case, the reason for which is likely the small number of peak emissions instances available to train the models. Additionally, covariates known to influence the peak N₂O emissions should be identified from the literature and included in such modeling experiments.

5. Conclusions

Four generic machine learning algorithms, viz., multiple linear regression (MLR), random forest regression (RFR), support vector regression (SVR) and artificial neural networks (ANNs), were tested for their applicability to predicting N₂O emissions from agricultural lands in Kitchener, Ontario, Canada. A set of readily available weather and management variables like rainfall, temperature, number of days after rainfall, 2-day accumulated rainfall, soil moisture, soil temperature and number of days after nitrogenous fertilizer application were collected, along with soil measurements of ammonium, nitrate, and pH on the days when discrete measurements of N₂O were taken from the study site. Two scenarios, viz., High Input (HI) and Low Input (LI), were devised with the inclusion and exclusion of the last three variables (soil nitrate, soil ammonium and pH values) from the dataset. A total of 628 observations were then split into training and testing datasets in a 70:30 ratio in such a way that both datasets included similar statistical characteristics and non-duplicate data. Temporal autocorrelation was tested before any of the models were fitted with the data. To eliminate biases from the selection of training data, it was ensured that all algorithms used consistent training and testing datasets that were independent and representative in nature. The mean square error was used as an objective function to fine-tune each algorithm. Additionally, R², RMSE, VE, p-factor and r-factor metrices were estimated for each algorithm to compare their ability to predict N₂O emissions in the study area. Based on the analysis, the following conclusion were drawn.

• There is a need to appropriately split the data into training and testing datasets in such a way that the basic characteristics of the dataset are preserved. There is a considerable band of uncertainty in the results predicted by algorithms trained on different sets of training data.
• The dataset violated the basic assumptions of the MLR application, such as high multicollinearity, heteroscedasticity and no linear relationship between the predictors and predictand N₂O variable, thereby rendering the application of MLR invalid. Despite the violation of these assumptions, MLR could not explain more than 30% of the variability of N₂O emissions.
• RFR, SVR and ANN could subsequently explain more than 66%, 63% and 43% of the variability of an unseen test dataset when the models were trained for the HI scenario.
• The RFR, SVR and ANN models trained under the LI scenario were found to have comparable performance with models trained under the HI scenario, with subsequent explanations of 68%, 66% and 68% of the variability of the unseen test dataset.
• The peak emissions were better captured by an ANN under the HI scenario, followed by an ANN under the LI scenario, RFR under the HI scenario and RFR under the LI scenario. However, none of the models were able to capture all of the peak emissions within their uncertainty bands.
• Prediction uncertainties were best captured by the RFR and ANN algorithms, with SVM performing very poorly.
• All algorithms were prone to overfitting, thus demanding the careful selection of model parameters and representative training and testing datasets.
• Considering the computational cost, ease in fine-tuning the model, stability of the model results for bootstrapped datasets and a relatively easier interpretation, RFR followed by ANN is recommended for N₂O estimation in future studies in the study area.
• There is a merit in using machine learning algorithms trained with local data to estimate N₂O emissions in conjunction with readily measurable weather and management variables. However, additional variables known to influence peak emissions need to be included, along with more instances of peak emissions.
• Future studies are recommended to use K-fold cross-validation to generalize the model and avoid biased performance metrices. Similarly, techniques capable of handling epistemic uncertainty in addition to the aleatoric uncertainties stemming from data are recommended.