Gaussian Process Regression for Predictive But Interpretable Machine Learning Models: An Example of Predicting Mental Workload across Tasks

Simulating root length density dynamics of sunflower in saline soils based on machine learning

The increasing effects of climate change require the accurate measurement and prediction of carbon emissions from building construction operations. This study aims to assess the performance of Gaussian Process Regression (GPR) and Support Vector Machine (SVM) in predicting emissions and determine which model is best suited for improving sustainable construction practices. The GPR and SVM models were trained and tested using empirical datasets with varying parameter values. Model performance is evaluated using correlation coefficient (R), root mean square error (RMSE), and mean absolute error (MAE). The analysis reveals that GPR consistently performs better than SVM across diverse datasets, especially in intricate and varied data situations. The superior performance of GPR highlights its robustness and versatility in handling the complexities of construction-related emissions data. The results indicate that GPR has the potential to assist significantly many stakeholders in the construction sector, including planners and environmental managers, in making better-informed decisions that are in line with carbon reduction objectives. Further studies could examine the incorporation of further predictive variables into the GPR model, such as economic indicators and the effects of material innovation.

Gaussian process regression with skewed errors

Forecasting short-term solar radiation is crucial for many solar energy applications. Additionally, solar energy has a lower environmental impact than conventional sources like fossil fuels and can be used for investment purposes through the construction of large solar farm facilities. To test, evaluate, and compare various solar radiation models, short-term observations of meteorological, astronomical, computational, and geographical data were collected at two distinct locations from 2012 to 2015. In this study, seven machine learning models were employed: multi-layer perceptron (MLP), feedforward backpropagation algorithm (FFBP), autoregressive integrated moving average (ARIMA), linear regression (LR), radial basis function neural network (RBFNN), random forest (RF), and Gaussian process regression (GPR) models. These models were used to forecast hourly global solar radiation (GSR) using the aforementioned data as model input. The performance of the selected models' forecast accuracy was thoroughly examined by assessing it for a typical day, for four seasons, and under three sky conditions. The RF model can forecast GSR with satisfactory accuracy, and MLP and GPR models provide better accuracy than LR, FFBP, RBF, and ARIMA models. For example, the R2 value range of RF are 0.9621 for Tetuan site and 0.9534 for Tangier site, respectively. Meanwhile, RF, MLP, and GPR models under-forecast few high radiation values on clear days, which may due to the differences in training and testing data ranges and distributions of the sky conditions. Finally, the obtained result of this study indicate that the proposed RF model is a reliable alternative for short-term global solar radiation forecasting due to its high forecast accuracy.

Gaussian process regression to determine water content of methane: Application to methane transport modeling

The growth of rice is a sequence of three different phenological phases. This sequence of change in rice phenology implies that the condition of the plant during the vegetative phase relates directly to the health of leaves functioning during the reproductive and ripening phases. As such, accurate monitoring is important towards understanding rice growth dynamics. Leaf Area Index (LAI) is an important indicator of rice yields and the availability of this information during key phenological phases can support more informed farming decisions. Satellite remote sensing has been adopted as a proxy to field measurements of LAI and with the launch of freely available high resolution Satellite images such as Sentinel-2, it is imperative that accurate retrieval methods are adopted towards monitoring LAI at irrigated rice fields. Here, we evaluate the potential of a hybrid radiative transfer model (i.e., PROSAIL - Gaussian Process Regression (GPR), for estimating the phenological dynamics of irrigated rice LAI using imager derived from the Sentinel-2 multispectral instrument. LAI field measurements were obtained from an experimental rice field in Nasarawa state, Nigeria during the dry season. We used the PROSAIL radiative transfer model to create a look up table (LUT) that was subsequently used to train a GPR model. Afterwards, we evaluated the potential of the hybrid modelling approach by assessing the overall model accuracy and the extent to which LAI was able to accurately predict LAI during key rice phenological phases. We compared the predicted hybrid GPR LAI values with LAI values generated from the SNAP toolbox, based on a hybrid Artificial Neural Network (ANN) modelling approach. Our results show that the overall predictive accuracy of the hybrid GPR model (R2 = 0.82, RMSE = 1.65) was more accurate than that of the hybrid ANN model (R2 = 0.66, RMSE = 3.89) for retrieving LAI values from Sentinel-2 imagery. Both models underestimated LAI values during the reproductive and ripening phases . However, the accuracy during the phenological phases were more significant when using the hybrid GPR model (P < 0.05). During the different phenological phases, the hybrid GPR model predicted LAI more accurately during the reproductive (R2 = 0.7) and ripening (R2 = 0.59) phases compared to the hybrid ANN reproductive and ripening phases. When monitoring LAI phenological profiles of both hybrid models, the hybrid GPR and ANN models underestimated LAI during the reproductive and ripening phases. However, the ANN model underestimations were statistically significantly greater than those for the hybrid GPR model (P < 0.05). Our results highlight the potential of hybrid GPR models for estimating the phenological dynamics of irrigated rice LAI from Sentinel-2 data. They provided more accurate estimation of LAI patterns from varying nitrogen and water applications than hybrid ANN models.

Over the last decade, innovations in satellite remote sensing (RS) and data science have widened the scope and relevance of agricultural monitoring and management applications at farm and territorial scales. Recently launched and upcoming hyperspectral satellite missions (e.g., ASI-PRISMA, DLR-ENMAP, Planet-Tanager, ESA-CHIME, Kuva-Hyperfield) provide high spectral resolution (&lt; 10 nm) across the 400-2500 nm range, opening new frontiers for assessing biophysical and biochemical functional traits of agroecosystems, while advancements in machine learning (ML) and artificial intelligence (AI) allow the efficient exploitation of the information content of high-dimensionality spectral datasets. Here we summarize the results and lessons learned from three experiments in different European agricultural systems (croplands and grasslands), analysing how the synergy between hyperspectral imaging spectroscopy (field and satellite), ML, and foundation models could support adaptive agroecosystem management through the retrieval of vegetation and soil properties related to the nutrient cycle. The three case studies are:(1) Assessment of biomass and nutritional quality of Alpine pastures: Gaussian Process Regression (GPR) models were calibrated on 250 vegetation samples and field spectra collected in 2024 and 2025 from semi-natural pastures in Valtournanche (Aosta, Italy) and Val Camonica (Brescia, Italy). PRISMA-derived maps for biomass and leaf-level protein, and fiber content showed good accuracy against in-situ data (LAI [-] R2 = 0.71, RMSE = 0.89; Biomass [g · m-2] R2 = 0.67, RMSE = 178.71; DM [%] R2 = 0.65, RMSE = 2.70; CP [%] R2 = 0.58, RMSE = 0.52; ADF [%] R2 = 0.45, RMSE = 2.42; NDF [%] R2 = 0.50, RMSE = 0.61), allowing mapping of pasture metabolizable energy in support of grazing management.(2) Monitoring of nutritional status of paddy fields: GPR models were developed on 200 vegetation samples and field spectra collected in 2024 and 2025 in several fields located in the Ferrara region (Italy). These models, demonstrated on PRISMA and EnMAP time-series, effectively monitored crop development across a temporally and spatially independent test set (LAI [-] R2 = 0.83, RMSE = 0.30; Fresh Biomass [g · m-2] R2 = 0.72, RMSE = 627.67; LCC [μg · cm-2] R2 = 0.58, RMSE = 3.40; LNC [μg · cm-2] R2 = 0.34, RMSE = 22.51; CNC [g · m-2] R2 = 0.56, RMSE = 0.77).(3) Retrieval of Soil Organic Carbon (SOC) and soil Nitrogen (N) on arable lands: A transformer-based, sensor-agnostic deep learning architecture was fine-tuned on open global spectral libraries. When applied to EMIT and Tanager-1 imagery over the Po Plain (Italy) and Northern Netherlands regions, the model yielded high accuracy (SOC [%] R2 = 0.61, MAE = 0.37; N [%] R2 = 0.68, MAE = 0.12) against 289 independent field observations.Our findings demonstrate that satellite hyperspectral spectroscopy can complement operational multi-spectral missions, adding key information about agroecosystems nutritional status. Furthermore, we show that the use of pre-trained ML and AI models on global spectral libraries and field reflectance data allows accurate retrieval even in the absence of ground truth acquisition synchronous to the satellite overpass, offering a potential scalable solution for agroecosystems management and monitoring at landscape scale.

Enhancing estimation of water quality index using stacking machine learning techniques: The case of Southern Bug River.

Su kaynaklarının planlanması ve tarımsal su yönetimi için referans bitki su tüketiminin (ET0) doğru tahmin edilmesi oldukça önemlidir. Bu çalışmada üç farklı makine öğrenimi yönteminin, yapay sinir ağları (YSA), rassal ağaç (RT) ve gauss süreç regresyonu (GPR), yarı nemli iklim koşullarına sahip bir bölgeden alınan iklim verileri kullanılarak günlük ET0’ı tahmin etme performansları araştırılmıştır. Modelleri eğitmek ve test etmek için Bafra’da 2018–2020 döneminde toplanan günlük meteorolojik verilerin beş girdi kombinasyonu [günlük ortalama (Tort), maksimum (Tmax) ve minimum sıcaklık (Tmin), rüzgar hızı (u2), bağıl nem (RHort) ve güneşlenme şiddeti (Rs) hem tam hem de eksik kombinasyonları dahil] kullanılmıştır. Makine öğrenimi modellerinin performansı ise FAO-56 Penman Monteith (FAO-56 PM) standart denklemi ile karşılaştırılmıştır. Modellerin doğruluğunu değerlendirmek için dört farklı istatistiksel performans indeksi kullanılmıştır [belirleme katsayısı (R2), ortalama mutlak hata (MAE), hata kareleri toplamının karekökü (RMSE) ve Nash–Sutcliffe etkinlik katsayısı (NSE)]. Sonuçlar, her beş senaryoda da YSA modellerinin RT ve GPR modellerinden daha iyi performans gösterdiğini ortaya koymuştur. Tüm verilerin girdi olarak kullanıldığı beşinci senaryo için YSA modelleri tarafından tahmin edilen ET0 değerlerinin FAO-56 PM değerleri ile iyi bir uyum içinde olduğu belirlenmiştir (R2=0.988, MAE=0.166, RMSE=0.205, NSE=0.984). Bununla birlikte meteorolojik veriler tam olmadığında bile sadece Tort, RHort ve Rs ile oldukça yüksek günlük ET0 tahminleri elde edilmiştir. Genel olarak, radyasyona dayalı makine öğrenimi modelleri, sıcaklığa dayalı makine öğrenimi modellerinden daha iyi performans göstermiştir. Sonuçlar, yarı nemli iklim özelliklerine sahip bir bölgede YSA modelinin ET0'ı oldukça verimli ve doğru bir şekilde modellemek için etkili bir şekilde kullanılabileceğini göstermektedir.

The Gaussian process regression (GPR) model, which is a powerful machine learning tool for probabilistic prediction, is introduced into slope displacement prediction. Using this model, the displacements of the slope of the permanent ship lock of the Three Gorges Project, the Wolongsi slope, and the high slope of Longtan hydropower station were predicted. In addition, the predictive uncertainty index (PUI) for describing the uncertainty of the predicted results was proposed, and the corresponding classification of the PUI was established. The study results demonstrate that the GPR model can self-adaptively acquire model parameter values and has satisfactory adaptability for predicting nonlinear time series of slope displacement. The proposed PUI and its classification based on the GPR model enable quantitative uncertainty analysis and, in turn, reliability evaluation of the predicted results. The GPR model provides a new approach to displacement prediction and safety management in slope engineering.

Due to the recent advances in digitisation of the manufacturing industry and the generation of manufacturing data, there is increasing interest to integrate machine learning on the shop floor to improve efficiency and quality control. Ultrasonic welding is an emerging joining process used in various manufacturing industries, and is an energy efficient, cost-effective method of joining similar or dissimilar materials. However, the quality of the joint achievable is heavily dependent on process input parameters. In this study, a Gaussian Process Regression (GPR) model is developed to map the relationship between process parameters and joint performance for ultrasonically welded aluminium joints, with a view to improving quality control in a manufacturing setting. Initially, a 33 full factorial design of experiments is conducted to investigate the influential parameters, then a GPR model is trained on the experimental data. In-process sensor data is also used to infer process performance. To assess the prediction performance of the model, ten unseen parameter combinations are predicted and compared to their respective experimental result. The model demonstrates a high level of accuracy producing a Pearson’s correlation coefficient of 0.982 between the predicted and actual results for all data. The mean relative predictive error for unseen data is 7.35%.KeywordsUltrasonic weldingMachine learningGaussian process regressionQuality control

PurposeEnsuring the early detection of structural issues in aircraft is crucial for preserving human lives. One effective approach involves identifying cracks in composite structures. This paper employs experimental modal analysis and a multi-variable Gaussian process regression method to detect and locate cracks in glass fiber composite beams.Design/methodology/approachThe present study proposes Gaussian process regression model trained by the first three natural frequencies determined experimentally using a roving impact hammer method with crystal four-channel analyzer, uniaxial accelerometer and experimental modal analysis software. The first three natural frequencies of the cracked composite beams obtained from experimental modal analysis are used to train a multi-variable Gaussian process regression model for crack localization. Radial basis function is used as a kernel function, and hyperparameters are optimized using the negative log marginal likelihood function. Bayesian conditional probability likelihood function is used to estimate the mean and variance for crack localization in composite structures.FindingsThe efficiency of Gaussian process regression is improved in the present work with the normalization of input data. The fitted Gaussian process regression model validates with experimental modal analysis for crack localization in composite structures. The discrepancy between predicted and measured values is 1.8%, indicating strong agreement between the experimental modal analysis and Gaussian process regression methods. Compared to other recent methods in the literature, this approach significantly improves efficiency and reduces error from 18.4% to 1.8%. Gaussian process regression is an efficient machine learning algorithm for crack localization in composite structures.Originality/valueThe experimental modal analysis results are first utilized for crack localization in cracked composite structures. Additionally, the input data are normalized and employed in a machine learning algorithm, such as the multi-variable Gaussian process regression method, to efficiently determine the crack location in these structures.

Long-term average solar radiation prediction via seasonal Global Horizontal Irradiance (GHI) forecasting is increasingly leveraging Machine Learning (ML) to uncover complex relationships in historical GHI and climatic parameters surpassing traditional methods. This study investigates seasonal GHI forecasting across four climatically distinct Indian locations namely Chennai, Jaisalmer, Leh and Mawsynram using data from the National Solar Radiation Database. Following rigorous data quality assessment and feature selection based on Spearman's correlation and Mutual Information, eight key features are identified for training three ML models: Efficient Linear Regression (ELR), Regression Trees (RT) and Gaussian Process Regression (GPR). The GPR model demonstrates superior accuracy in seasonal GHI prediction across diverse Indian climatic conditions making it ideal for reliable and cost-efficient Stand-Alone Photovoltaic (SAPV) systems. It is observed that the GPR model achieves the lowest Root Mean Square Error (RMSE) of 0.0030, Mean Absolute Error (MAE) of 0.0022 and Coefficient of Determination (R²) of 0.9999 achieving a reduction of 189.1% in RMSE, 190.09% in MAE and 20.56% improvement in R² compared to ELR. Furthermore, the GPR model surpasses the RT model with a reduction of 124.05% in RMSE and 111.1% in MAE, together with an improvement of 0.2604% in R². While RT provides reasonably accurate forecasts, ELR shows the least reliability. These results confirm GPR's exceptional precision and ability to capture nearly all dataset variability.

This paper describes a real-time data collection framework and an adaptive machining learning method for constructing a real-time energy prediction model for a machine tool. To effectively establish the energy consumption pattern of a machine tool over time, the energy prediction model is continuously updated with new measurement data to account for time-varying effects of the machine tool, such as tool wear and machine tool deterioration. In this work, a real-time data collection and processing framework is developed to retrieve raw data from a milling machine tool and its sensors and convert them into relevant input features. The extracted input features are then used to construct the energy prediction model using Gaussian Process (GP) regression. To update the GP regression model with real-time streaming data, we investigate the use of sparse representation of the covariance matrix to reduce the computational and storage demands of the GP regression. We compare computational efficiency of sparse GP to that of full GP regression model and show the effectiveness of the sparse GP regression model for tracking the variation in the energy consumption pattern of the target machine.

In this work, four mainstream machine learning (ML) techniques are used to evaluate the bulk porosity and static elastic modulus of weathered Yungang Grottoes sandstone. Datasets are gathered from the experiments, which includes 432 groups effective experimental data including 8 inputs features. bulk porosity and static elastic modulus were considered as outputs to determine the weathering degrees of Yungang Grottoes sandstone. The 4 performance criteria were used to evaluate the ML models. Results demonstrate that the Artificial Neural Network (ANN) is the best-fitted models for estimating the bulk porosity and static elastic modulus compared to Multiple Linear Regression (MLR), Support Vector Regression (SVR), Gaussian Process Regression (GPR). The accuracy of the trained model for static elastic modulus is slightly higher than that of bulk porosity. The GPR and ANN model can accurately predict the bulk porosity and static elastic modulus in training stages. The ANN with multi-hidden layers developed is competent with high degree of precision and generalization ability for bulk porosity and static elastic modulus compared to other selected regression-based ML models (MLR, SVR, and GPR). The coefficient of determinations of ANN in the range of (0.9537–0.9641) during the testing stages is more stable and higher than that of (0.8883–0.9453) other built ML models. The prediction efficiency of pretrained ANN model was well adjusted for the actual and forecast datasets at the training and testing stages, and the error range was no more than 0.7% and 0.15 GPa at both stages of prediction for bulk porosity and static elastic modulus respectively. And the ANN based static elastic modulus prediction model’s error proportions significantly decreased and were confined to a modest range between + 10% and − 10%. The proposed surrogate models are valid for the bulk porosity ranging from 7 to 14% and the static elastic modulus ranging from 0.7 to 1.4 Gpa, which can be utilized for the accurate and fast prediction of the weathering degrees of Yungang Grottoes sandstone.