Autoregressive Neural Network Research Articles

A spatiotemporal autoregressive neural network interpolation method for discrete environmental factors

The spatiotemporal interpolation model is necessary for generating continuous distributions for spatiotemporally discrete sampling points. However, there remain challenges in spatiotemporal interpolation due to the complex spatiotemporal effect and the imprecise kernel functions. Here, we proposed a spatiotemporal autoregressive neural network interpolation model (STARNN) that incorporates adaptive spatiotemporal distance quantification and supervised learning. The 10-fold cross-validation modelling on sea surface temperature and coastal nutrients demonstrated that the STARNN model performs better than baseline models and can well depict reasonable spatiotemporal distributions for environmental factors. By proposing two stacked neural networks, the STARNN model can accurately integrate spatial and temporal distances and avoids subjective selection of the kernel function. This study developed a novel interpolation model for processing discrete spatiotemporal points by following the data-driven paradigm, which can offer decision support for simulating the spread of sea temperature anomalies and optimizing the distribution of water quality measurement stations.

Optimizing photovoltaic power plant forecasting with dynamic neural network structure refinement

Reliable prediction of photovoltaic power generation is key to the efficient management of energy systems in response to the inherent uncertainty of renewable energy sources. Despite advances in weather forecasting, photovoltaic power prediction accuracy remains a challenge. This study presents a novel approach that combines genetic algorithms and dynamic neural network structure refinement to optimize photovoltaic prediction. This methodology dynamically adjusts the neural network parameters during training, including the number of neurons, transfer functions, weights, and biases, to minimize the root mean square error. Evaluation was performed on twelve representative days using annual, monthly, and seasonal data, and a comparison was made with multiple linear regression and nonlinear autoregressive neural network models, demonstrating the approach’s effectiveness. Evaluation metrics such as mean square error, R-value, and mean percentage error reveal promising prediction accuracy. MATLAB is used for modeling, training, and testing, and a real 4.2 kW PV plant is used for validation. The results indicate significant improvements, with mean square errors as low as 20 W on cloudy days and 175 W on sunny days. The proposed methodology achieves prediction versus target regressions consistency, with R values ranging from 0.95824 to 0.99980, highlighting its efficiency in providing reliable predictions of PV power generation.

Determination of the performance of training algorithms and activation functions in meteorological drought index prediction with nonlinear autoregressive neural network

Analysis of long-term meteorological data is critical for monitoring climate trends and understanding the drought situation in a given region. In this study, monthly average precipitation data from the Niğde meteorological station in Turkey covering the period 1950–2020 were used. Within the scope of the study, seven different drought index methods were used for drought analysis, and the number and percentages of drought conditions were calculated according to these indices. For example, according to the Standardized Precipitation Index (SPI) method, the proportion of dry periods was determined as 16.2% and the proportion of humid periods as 83.8%. The Mann-Kendall trend analysis performed to determine the drought trends of the region revealed an increasing trend towards humidity in all indices (e.g., z = 1.299, p = 0.194 for SPI). In the study, 60-month drought forecasts covering the years 2020–2025 were realized using the Nonlinear Autoregressive Neural Network (NARNN) model, and the results were compared with the Autoregressive (AR) model. In the prediction performance analysis, the NARNN model showed superior prediction performance for all indices with lower RMSE values (e.g., NARNN RMSE = 0.977 for SPI; AR RMSE = 1.704). The prediction performances of different training algorithms and activation functions used in the NARNN model were analyzed. The best performance was obtained with the trainbr training algorithm and sigmoid activation function (e.g., RMSE = 0.997 for SPI). Based on these best parameters, more than 70% of the drought conditions during the 2020–2025 period were found to be normal or humid according to NARNN predictions. This study demonstrates the superiority of the NARNN model in nonlinear time series analyses and that it is a reliable tool, especially for future drought forecasts. In addition, comprehensive analyses with different index methods have significantly contributed to understanding the long-term drought trends in the Niğde region.

Peanut oil price change forecasts through the neural network

Purpose For a wide range of market actors, including policymakers, forecasting changes in commodity prices is crucial. As one of essential edible oil, peanut oil’s price swings are certainly important to predict. In this paper, the weekly wholesale price index for the period of January 1, 2010 to January 10, 2020 is used to address this specific forecasting challenge for the Chinese market. Design/methodology/approach The nonlinear auto-regressive neural network (NAR-NN) model is the forecasting method used. Forecasting performance based on various settings, such as training techniques, delay counts, hidden neuron counts and data segmentation ratios, are assessed to build the final specification. Findings With training, validation and testing root mean square errors of 5.89, 4.96 and 5.57, respectively, the final model produces reliable and accurate forecasts. Here, this paper demonstrates the applicability of the NAR-NN approach for commodity price predictions. Originality/value On the one hand, the findings may be used as independent technical price movement predictions. Conversely, they may be included in forecast combinations with forecasts derived from other models to form viewpoints of commodity price patterns for policy research.

Nonlinear Autoregressive Neural Network with Exogenous Input Model Approach for Magnetic Flux Density Measured by Hall-Effect Sensor in Magnetic Spring

Nonlinear Autoregressive Neural Network with Exogenous Input Model Approach for Magnetic Flux Density Measured by Hall-Effect Sensor in Magnetic Spring

Neural network Lee–Carter model and the actuarial relevance of longevity risk assessment

Accurate forecasts of mortality and life expectancy values are crucial concerns at various decision-making levels and, in particular, in life insurance. In the context of the Lee–Carter model, we propose an approach to modeling and forecasting mortality based on the use of autoregressive neural network models with exogenous variables combined with a bootstrap scheme, to obtain both point and forecast distributions. Our proposal preserves the LC defining parameters and structure and adds flexibility at reduced costs in terms of complexity and implementability, with respect to other machine learning approaches. It allows to correct and contain the bias that the traditional linear models introduce in mortality predictions, this marking improving results from a statistical perspective and paving the way for promising future developments. The impact of the proposed novel methodology is tested in the actuarial field through case studies in the life annuities framework. In particular, it provides a much more smoothed increase in the uncertainty along the policy duration, which, in terms of risk management, implies less capital set aside against longevity risk while preserving the insurer's resilience.

Prediction of areas of Oncomelania hupensis snail spread in Anhui Province based on five machine learning models

To predict the areas of Oncomelania hupensis snail spread in Anhui Province from 1977 to 2023 using machine learning models, and to compare the effectiveness of different machine learning models for prediction of areas of O. hupensis snail spread, so as to provide insights into investigating the trends in areas of O. hupensis snail spread. Data pertaining to O. hupensis snail spread in Anhui Province from 1977 to 2023 were collected and a database was created. Five machine learning models were created using the software Matlab R2019b, including support vector regression (SVR), nonlinear autoregressive (NAR) neural network, back propagation (BP) neural network, gated recurrent unit (GRU) neural network and long short-term memory (LSTM) neural network models, and the model fitting effect was evaluated with mean absolute error (MAE), root mean squared error (RMSE) and coefficient of determination (R2). Following model training, the areas of O. hupensis snail spread were predicted in Anhui Province from 2024 to 2030. The cumulative areas of O. hupensis snail spread were 40 241.32 hm2 in Anhui Province from 1977 to 2023, and the area of O. hupensis snail spread varied greatly among years, with a periodic peak every 4 to 6 years. The fitting curves of SVR, NAR neural network, BP neural network, GRU neural network and LSTM neural network models were increasingly closer to the real curves for areas of O. hupensis snail spread in Anhui Province. The trends in areas of O. hupensis snail spread in Anhui Province from 2024 to 2030 appeared approximately "M"-shaped curves by SVR and NAR neural network models, approximately "W"-shaped curves by BP and GRU neural network models, and a unimodal conical curve by the LSTM neural network model. The LSTM neural network model had the best effect for predicting areas of O. hupensis snail spread in Anhui Province, with the RMSE of 1 277 480, MAE of 797 422 and R2 of 0.978 9, respectively. Among the five models, The LSTM neural network model has a high efficiency for predicting areas of O. hupensis snail spread in Anhui Province, which may serve as a tool to investigate the trends in areas of O. hupensis snail spread.

Enhanced forecasting of emergency department patient arrivals using feature engineering approach and machine learning

BackgroundEmergency department (ED) overcrowding is an important problem in many countries. Accurate predictions of ED patient arrivals can help management to better allocate staff and medical resources. In this study, we investigate the use of calendar and meteorological predictors, as well as feature-engineered variables, to predict daily patient arrivals using datasets from eleven different EDs across three countries.MethodsSix machine learning (ML) algorithms were tested on forecasting horizons of 7 and 45 days. Three of them – Light Gradient Boosting Machine (LightGBM), Support Vector Machine with Radial Basis Function (SVM-RBF), and Neural Network Autoregression (NNAR) – were never before reported for predicting ED patient arrivals. Algorithms’ hyperparameters were tuned through a grid-search with cross-validation. Prediction performance was assessed using fivefold cross-validation and four performance metrics.ResultsThe eXtreme Gradient Boosting (XGBoost) was the best-performing model on both prediction horizons, also outperforming results reported in past studies on ED arrival prediction. XGBoost and NNAR achieved the best performance in nine out of the eleven analyzed datasets, with MAPE values ranging from 5.03% to 14.1%. Feature engineering (FE) improved the performance of the ML algorithms.ConclusionAccuracy in predicting ED arrivals, achieved through the FE approach, is key for managing human and material resources, as well as reducing patient waiting times and lengths of stay.

Непараметрическая оценка прогностической точности моделей на примере временны́х рядов концентраций метана в атмосферном воздухе арк­тического острова Белый

Over the past few decades, the number, variety, and complexity of time series forecasting models have grown. There has also been an increased interest among researchers in comparing and assessing the predictive accuracy and performance of models, and in determining which models are more accurate. The researchers propose using the hypothesis testing approach to assess the performance of a time series forecast model. They obtained data for the study while monitoring the dynamics of ground-level concentrations of the main greenhouse gases on the Arctic Island of Bely, Yamalo-Nenets Autonomous Area, Russia. A total of three models based on autoregressive neural networks with exogenous input (NARX) were considered to predict changes in methane concentration in the surface layer of atmospheric air. The performance of the models was assessed using 8 “traditional” indices (correlation and determination coefficients, Wilmott goodness-of-fit indices, mean absolute error, relative mean square error, etc.) and the proposed permutational approach using the hypothesis testing method. In general, based on the data under study, the estimates of the permutation approach and other indicators of accuracy and error coincide, but there are discrepancies. Effect sizes do not always determine the statistical significance of differences.

Time Series Decomposition of Land Surface Temperature for Long-Term Trend Forecasting and Impact on Nesting Sea Turtle Habitats in the Arabian Gulf

Abstract Improving land surface temperature (LST) modeling is vital for mitigating climate change effects on various ecosystems and marine habitats such as important sea turtle habitats. Over the past decade, extreme temperatures have likely significantly affected nesting sea turtle habitats in the Arabian Gulf, with predominantly female hatchlings creating an imbalance in the sex ratio. Such shifts have profound implications for these habitats’ long-term survival and conservation management. This study leverages statistical machine learning models to measure ongoing temporal variations in LST. We break down the LST time series into trend, seasonal, and noise components using classical decomposition methods like X11, Seasonal Extraction in autogressive integrated moving average (ARIMA) Time Series (SEATS), and the seasonal and trend decomposition using local regression (Loess) (STL) approach. The long-term trends in LST data are driven by climate change rather than seasonal fluctuations. We employed neural network autoregression (NNAR), bagged error, trend, and seasonal (BaggedETS), exponential smoothing models, and the STL method to project future LST values. We also explored advanced forecasting models like dynamic harmonic regression; trigonometric seasonality, Box–Cox transformation, autoregressive moving average (ARMA) errors, and trend and seasonal components (TBATS); and seasonal autoregressive integrated moving average (SARIMA) for comparative performance analysis. Extended warm periods were identified for Abu Ali Island between 2017 and 2018 through several decomposition methods, likely linked to the 2015/16 El Niño event. We also conducted a marine heatwave (MHW) analysis from 2010 to 2020, establishing a pronounced impact of the 2015/16 El Niño on the Arabian Gulf. In nesting beach environments with high LST, marine heatwaves could have a significant impact on sea turtle populations without human intervention such as artificially cooling the nest temperature. The SARIMA model showed higher forecasting precision for in situ weather data, while the NNAR model demonstrated superior performance with remotely sensed data.

Non-linear Dynamic Behavior Identification of a Quadcopter F450 Using an Artificial Neural Network-Based NARX Model

A quadcopter drone is an extremely complex, multi-variable, highly nonlinear, and underactuated system characterized by its six degrees of freedom controlled by only four actuators as inputs. This highlights the importance of employing advanced algorithms for its identification. Therefore, this research aimed to use a nonlinear neural network model to identify the dynamic behavior of a quadcopter based on the commercially available F450 frame. Data acquisition involved four experiments in a controlled environment for both roll and pitch angles, recording the signal duty cycles and the quadcopter’s attitude. Then, the selected non-linear autoregressive neural network model with exogenous inputs (N-NARX) model was trained using the acquired data along with the Levenberg-Marquardt algorithm. Afterwards, the response of the quadcopter’s actual attitude angles from the validation dataset was analyzed against the predicted values generated by the neural model, obtaining an 89.44% fit with an RMSE of 2.25% for the roll angle and an 89.29% fit with an RMSE of 2.20% for the pitch angle. Both attitude angles were subjected to a statistical cross-correlation validation to assess their relationship at different time lags, observing a solid settling within the confidence bands at a 95% level. It was concluded the proposed neural network model can effectively capture the quadcopter’s nonlinear dynamics.

Comparing classic time series models and state-of-the-art time series neural networks for forecasting as-fired liquor properties

Abstract The properties of as-fired black liquor dictate kraft recovery boiler operation. If these properties could be forecasted, operations could be adjusted to optimize boiler performance. Here, we compare the performances of classic time series models and two state-of-the-art time series neural networks for forecasting as-fired liquor heating value, viscosity, and boiling point rise at a Canadian mill. Additionally, we show that, like classic time series models, autoregressive neural networks can be regarded as functions of unknown disturbances, which is useful in comparing model complexities. Our results show that classic time series models can accurately forecast as-fired liquor properties and that classic time series models perform comparably to state-of-the-art time series neural networks. We suspect this is due to the high autocorrelation of mill data that results from frequent measurements relative to long residence times. This autocorrelation is suspected to attenuate the cross-correlations between upstream disturbances and as-fired liquor properties. As a result, neural networks, which are useful for accommodating non-linear cross-correlations and dynamics, struggle to outperform classic time series models and may not always be appropriate for forecasting chemical process parameters.

Aging trajectory prediction of lithium-ion batteries based on mechanical-electrical features via nonlinear autoregressive and regression neural networks

The prediction of the lifespan of lithium-ion batteries (LIBs) is crucial for gaining early insights into battery degradation, enabling the optimization of battery usage and management strategies. This work assesses the expansion and aging characteristics of LIBs under various operating conditions and develops a combined machine-learning method to predict the aging trajectory of LIBs based on mechanical and electrical features. To avoid errors caused by secondary data processing, five mechanical-electrical features are directly extracted from the stress and electrochemical characteristic curves to reflect the battery aging state. A nonlinear autoregressive (NAR) neural network is adopted to predict the change trend of aging features (AFs) using historical data. Subsequently, the nonlinear regression (NR) neural network is employed for aging trajectory prediction using the aging feature dataset, which consists of both historical and predicted data. The experimental results demonstrate that the proposed method can ensure that the mean absolute error (MAE) in aging trajectory prediction is within 0.25 %. The method comprehensively analyzes and utilizes the mechanical-electrical properties of LIBs to enhance the predictive accuracy, expanding the scope of the predictive model in practical applications.

Quantifying uncertainty in neural network predictions of forced vibrations

AbstractThe prediction of forced vibrations in nonlinear systems is a common task in science and engineering, which can be tackled using various methodologies. A classical approach is based on solving differential (algebraic) equations derived from physical laws ('first principles'). Alternatively, Artificial Neural Networks (ANNs) may be applied, which rely on learning the dynamics of a system from given data. However, a fundamental limitation of ANNs is their lack of transparency, making it difficult to understand and trust the model's predictions. In this contribution, we follow a hybrid modelling approach combining a data‐based prediction using a stabilised Autoregressive Neural Network (s‐ARNN) with a priori knowledge from first principles. Moreover, aleatoric and epistemic uncertainty is quantified by a combination of mean‐variance estimation (MVE) and deep ensembles. Validating this approach for a classical Duffing oscillator suggests that the MVE ensemble is the most accurate and reliable method for prediction accuracy and uncertainty quantification. These findings underscore the significance of understanding uncertainties in deep ANNs and the potential of our method in improving the reliability of predictive nonlinear system modelling. We also demonstrate that including partially known dynamics can further increase accuracy, highlighting the importance of combining ANNs and physical laws.

Explainable and Reduced-Feature Machine learning models for shape and drag prediction of a freely moving drop in the sub-critical Weber number regime

Accurately predicting the shape and drag of a moving drop is crucial in many spray applications. However, due to the complex interaction between the drag force and drop shape deformation, accurate prediction of drop shape and drag from Computational Fluid Dynamics (CFD) simulation requires large computational resources and time. A novel data-driven approach using NARXNN (Non-linear Auto Regressive eXogenous input Neural Network) is proposed in this study, which recurrently predicts the drop shape and drag for a given Weber number (We) and Reynolds number (Re), in the sub-critical We regime where there is no drop breakup. The average error in radius and drag coefficient for the test cases, as low as 0.36% and 0.49%, respectively, is achieved. Most importantly, SHAP (SHapley Additive exPlanations) analysis is performed to identify important features for the predictions, hence enabling the explainability of the physics involved in the predictions of machine learning (ML) models for drop deformation and the interaction between the drop and gas flows. Global interpretation of SHAP values identifies We and Re as the most important features to predict the drop shape and drag. Based on the SHAP analysis, several reduced-feature models are developed. All the reduced-feature models can predict drop shape and drag coefficient within 4% average error in radius and drag coefficient for any We and Re combinations within the prescribed parameter space. The ML models developed in this study demonstrate a tremendous computational advantage over the traditional CFD simulations. The drop shape and drag coefficient evolution can be predicted within seconds for a single case, whereas the CFD simulation takes several days to a week on a single processor.

Individual tree mortality: Risks of climate change in the eastern Brazilian Amazon region

The mortality of trees in humid tropical forests plays a fundamental role in understanding forest development, particularly after disturbances such as those caused by logging and extreme weather events. The aim of this study was to evaluate estimates of individual tree mortality following Reduced Impact Logging (RIL) in the Eastern Brazilian Amazon at biennial intervals from 2005 to 2012. RIL is based on operations planning, personnel training, and investments in forest management, and harvesting through RIL must: (a) minimize environmental damage, (b) diminish operation cost by increasing work efficiency, and (c) reduce operational waste. A mortality model was constructed based on the estimation of three distance-independent competition-indices (DII) and five models for predicting the probability of individual tree mortality. The Kolmogorov-Smirnov statistical test was used to determine the most representative model, from which a Neural Network Autoregressive (NNAR) model was constructed to forecast mortality after RIL. Mortality data was correlated with the El Niño–Southern Oscillation (ENSO) and climate (Rainfall, Maximum, Minimum, and Average air temperature). The tested models showed similar and accurate estimates with R2 exceeding 0.90, although underestimation and overestimation trends were observed. The NNAR satisfactorily represented species mortality over the simulated years. The period from 2012 to 2014 was characterized by a Neutral and Weak El Niño event, and exhibited the highest mortality value for a 25 cm DBH (diameter at breast height), the smallest DBH class measured in this study. In the correlation matrix analysis, maximum air temperature showed the highest positive correlation with trees mortality. Despite the challenges in estimating individual tree mortality in tropical forests after selective logging, accurate estimates were achieved using traditional regression techniques and NNAR. These results can support technical and silvicultural decisions regarding forest management in the Eastern Amazon region of Brazil.

Forecasting mortality and DALYs from air pollution in SAARC nations

This study investigates the projected impact of air pollution on mortality and Disability-Adjusted Life Years (DALYs) across SAARC countries. Utilizing Time Series and Machine Learning methodologies such as Autoregressive Integrated Moving Average, Exponential Smoothing, and Neural Network, the research aims to accurately forecast the mortality and DALYs attributed to air pollution from 2020 to 2030. Statistical analyses reveal a consistent upward trend in deaths and DALYs during the forecasting period, primarily driven by Ambient Particulate Matter Pollution (APM) and Ambient Ozone Pollution (AOP). Comparing the predictive accuracy of the models, Neural Network outperformed other methods, as indicated by Root Mean Square Error (RMSE) values. Specifically, the study finds that deaths and DALYs due to Ambient Particulate Matter pollution are least prevalent in the Maldives, while India and Pakistan exhibit the highest rates, and deaths and DALYs due to Ambient Ozone pollution are lowest in the Maldives and highest in Bangladesh and Pakistan. Moreover, deaths and DALYs attributed to Household Air Pollution (HAP) are lowest in Pakistan and highest in Nepal. These findings underscore the urgent need for air pollution control measures and informed policymaking in SAARC countries to mitigate the escalating health burden associated with air pollution.

Analyzing and forecasting air pollution concentration in the capital and Southern Thailand using a lag-dependent Gaussian process model.

The air pollution problem has now amassed worldwide attention due to its multifaceted harm to human health. Exploring the concentration of air pollution and improving forecast have important consideration worldwide. In this research, we analyze the air pollution concentration of Southern Thailand and compare it with the central region. Also, we proposed a methodology based on the lag-dependent Gaussian process (LDGP), a Bayesian non-parametric machine learning model, with a stable optimization approach, which is a cluster-based multi-starter technique based on the Nelder-Mead optimizer. This model also provides the confidence band for forecasted values. We also used autoregressive deep neural network (AR-DNN), autoregressive random forest (AR-RF), gradient boosting (GB), and K-nearest neighbors (KNN) models. A comparison of the proposed methodology was performed on the daily air pollution data collected from the southern provinces and also from the capital of Thailand from 1 January 2018 to 31 December 2022. We used well-established performance evaluation measures to compare the performance of the models. To evaluate the bias due to overfit, we performed a tenfold cross-validation for all the pollutants in each region and compared the models to choose the best one. Moreover, we explored the concentration of air pollution in these regions. Results of descriptive analysis revealed that Bangkok had a much higher concentration of air pollution as compared to the southern region. However, the southern region had higher exposure to PM air pollutants as per WHO recommendations and also had higher exposure to O3 and CO levels. The proposed LDGP model outperformed the other machine learning models for forecasting all air pollutants. Hence, it is recommended to be used by experts for further research and studies with different kernel functions. This research is also expected to contribute to local government planning and prevention and worldwide use of the same methodology for the sustainability of public health.

Use of Deep-Learning-Accelerated Gradient Approximation for Reservoir Geological Parameter Estimation

The estimation of space-varying geological parameters is often not computationally affordable for high-dimensional subsurface reservoir modeling systems. The adjoint method is generally regarded as an efficient approach for obtaining analytical gradient and, thus, proceeding with the gradient-based iteration algorithm; however, the infeasible memory requirement and computational demands strictly prohibit its generic implementation, especially for high-dimensional problems. The autoregressive neural network (aNN) model, as a nonlinear surrogate approximation, has gradually received increasing popularity due to significant reduction of computational cost, but one prominent limitation is that the generic application of aNN to large-scale reservoir models inevitably poses challenges in the training procedure, which remains unresolved. To address this issue, model-order reduction could be a promising strategy, which enables us to train the neural network in a very efficient manner. A very popular projection-based linear reduction method, i.e., propel orthogonal decomposition (POD), is adopted to achieve dimensionality reduction. This paper presents an architecture of a projection-based autoregressive neural network that efficiently derives an easy-to-use adjoint model by the use of an auto-differentiation module inside the popular deep learning frameworks. This hybrid neural network proxy, referred to as POD-aNN, is capable of speeding up derivation of reduced-order adjoint models. The performance of POD-aNN is validated through a synthetic 2D subsurface transport model. The use of POD-aNN significantly reduces the computation cost while the accuracy remains. In addition, our proposed POD-aNN can easily obtain multiple posterior realizations for uncertainty evaluation. The developed POD-aNN emulator is a data-driven approach for reduced-order modeling of nonlinear dynamic systems and, thus, should be a very efficient modeling tool to address many engineering applications related to intensive simulation-based optimization.

Wind Speed Modeling under Quasi‐Linear Autoregressive Neural Network Model for Prediction of Production Power

Accurate wind speed modeling is beneficial for the design of wind energy conversion systems. Models of wind speed are used to assess the adequacy and dependability of a power supply. However, precise wind speed modeling is challenging due to the sporadic availability of wind speed. In this note, we propose a wind speed model with an autoregressive (AR) structure. A hybrid model is developed under linear and nonlinear parts based on a quasi‐linear autoregressive exogenous neural network (Q‐ARX‐NN). The model's structure is composed of a regression vector and its coefficients. The coefficients are divided into linear and nonlinear coefficients. A set of linear coefficients is identified under the algorithm of least square error (LSE), and a set of nonlinear coefficients is modeled by using a neural network to refine the residual error of the nonlinear part. Some artificial neural network (ANN) models can be set as nonlinear part sub‐models to sharpen the model's accuracy. The proposed model is tested for wind speed modeling to estimate wind energy production. Various nonlinear parts of the sub‐model are tested, such as neural networks, radial basis function networks, and ANN networks. Moreover, we evaluate the effects of the order of the model by varying hidden and output nodes, which can be summarized as the number of coefficients of the regression vector. Using specific wind turbine performance data, prediction models estimate production power. © 2024 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.