Accurate Solar Radiation Forecasting Using Spectral Feature Engineering and Bayesian Optimization

Abstract

For efficient grid operation and energy management, accurate forecasting of solar radiation is essential. The unpredictable nature of weather makes this task challenging to accomplish. Existing forecasting models fail to deliver accurate results under these conditions, which results in decreased operational efficiency for renewable energy systems. We are proposing a novel methodology that combines feature engineering, machine learning, and Bayesian Optimization (BO) to obtain optimal performance. First, time frequency characteristics are extracted using a Fast Fourier Transform (FFT)-based feature engineering approach to capture dominant patterns from meteorological data. The FFT features reveal essential periodic patterns, which describe solar irradiance and its associated variables, enabling models to perform better over different time periods. The model hyperparameter tuning process, which uses Bayesian Optimization, improves prediction results. Model performance is evaluated using Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and R². The results show clear improvements across Random Forest (RF), Multilayer Perceptron (MLP), and Long Short-Term Memory (LSTM) models, with the MLP model achieving the strongest overall performance. Specifically, the MLP achieved an R² value of 0.92, with MAE and RMSE values of 1.78 and 2.75, respectively. The proposed method also demonstrates robustness under varying weather conditions and time-series cross-validation (TSCV). Overall, the combined effects of frequency-domain feature engineering and Bayesian Optimization enable robust and adaptive forecasting of solar radiation resources.

1. Introduction

The issue of energy consumption has increased the likelihood of a global energy crisis having environmental consequences. Sustainable energy is becoming a necessity, as significantly declining fossil fuel sources have raised severe economic, political, and societal challenges [1]. Continuous measurement and forecasting are quite necessary for optimizing power generation through wind, water, and solar energy sources according to energy demand. Solar energy is among the most popular renewable energy sources because it does not harm the environment, is cost-effective, and is readily available. The sun, on average, impinges Earth with about 1.5 × 10⁸ kWh of energy every year, and that is much more than what can suffice for the entire world’s energy needs [2]. The great market growth of photovoltaics (PV) has indeed been due to the decrease in the cost of PV modules, the continued support of national policies, and particularly renewable energy targets that have unlocked this potential. Accordingly, global PV capacity increased substantially, from 6.6 GW in 2006 to more than 1 TW by 2024. However, the increased benefits associated with solar energy have some disadvantages, one of them being dependence on sunlight. Other conditions that influence solar energy generation at different times include the time of day, the season, the weather, and the location on Earth. Demand forecasting effectively avoids costly errors and disruption of a day-ahead market energy network. This need arises from the global increase in energy demand and the growing environmental concerns. Quite complicated forecasting systems have been developed, particularly for renewable sources such as solar and wind power, which on their own contribute hugely to balancing grid power and optimizing resource use [3].

High forecasting accuracy is achieved by analyzing temperature, humidity, rainfall, solar radiation, cloud cover, wind speed, and wind direction. Many recent artificial intelligence (AI)-based methods have been developed to capture the non-linear aspects of renewable energy data [4]. Whereas the traditional physical model could not adequately replicate the periodicities and spectral energy distributions of renewable energy data, AI models, especially soft computing and deep learning, were quite superior in this regard. The current research shows a trend towards using sophisticated renewable energy forecasting methods, according to [5]. Energy systems now depend on AI and machine learning (ML) technologies to achieve greater operational flexibility, which enables better utilization of renewable energy resources because of their fluctuation. The system upgrades of the machine learning system improve its predictions while providing current energy distribution information. Hybrid models currently use AI together with traditional forecasting methods to create more flexible forecasting systems. The deep learning-based forecasting models demonstrate excellent accuracy and efficiency, which allows them to deliver consistent results for solar energy management and solar energy forecasting. Various machine learning models, including Random Forest (RF), Linear Regression (LR), Support Vector Regression (SVR), and Artificial Neural Network (ANN), were employed for this purpose [6]. The ANN technique primarily serves short-term solar energy forecasting because it can accurately forecast solar radiation production through its ability to model complicated nonlinear and dynamic solar radiation patterns. Researchers have found that deep learning methods perform best when they need to analyze intricate datasets that include nonlinear patterns [5]. The advanced models improve their performance, but they face challenges when dealing with multi-level data interactions, which lead to unpredictable model behavior in real-life situations that need exact energy management and grid stability forecasting [6].

The solar radiation levels exhibit non-stationary behavior because they depend primarily on three factors, which include cloud cover, atmospheric disturbances, and seasonal changes. The time-domain features fail to provide sufficient information because they obstruct seasonal and cyclical patterns that forecasters need for their predictions.

Thus, temporal models may struggle to generalize across weather conditions, reducing forecast accuracy and robustness. This shortcoming highlights the need for further representations that can better separate periodic behaviors from instantaneous fluctuations, particularly in fast-changing weather conditions.

Frequency-domain analysis is currently receiving attention in solar radiation forecasting, but most research uses Fourier transforms for signal denoising or as incorporated components of deep learning frameworks. Modern FFT-based approaches use implicit feature learning to incorporate spectrum changes directly into convolutional or lightweight neural network models without explicitly providing frequency-domain characteristics. Thus, seasonal frequencies, spectral energy distributions, and entropy-related patterns are rarely explored in a way that typical machine learning and deep learning models can use. Solar radiation forecasting has not been widely researched using explicit spectral feature engineering and rigorous hyperparameter optimization. In this study, an FFT-based spectral feature engineering approach extracts interpretable frequency-domain characteristics from meteorological time-series data and integrates them with Bayesian Optimization to increase prediction accuracy and robustness under varied weather situations. Using FFT-based features [7], models can generalize their results under a variety of conditions by transitioning from the time to the frequency. Besides feature engineering, we also employ Bayesian Optimization (BO) for optimal adjustment of hyperparameters in machine learning models. BO effectively explores the parameter space so that forecasting errors are minimal, thereby increasing predictive performance. By integrating BO, the following main advantages contribute to solar radiation forecasting.

• We introduced an FFT spectral feature engineering procedure so that solar radiation model forecasts could become more accurate.
• Bayesian Optimization (BO) of the hyperparameters of machine learning models ensured that their performance and predictive accuracy were vastly improved.
• Even while the time-related structure of solar radiation data is preserved, assessments of models are strictly carried out by the Time-Series Cross-Validation (TSCV) methodology.
• Based on SHAP, distribution plots, and performance curves, the model’s performance is compared under seasonal variations and complex data patterns.
• Performance evaluation of machine-learning models is performed under multiple weather conditions, namely, Clear Sky, Cloudy, High Humidity, and Mixed Conditions.

The remaining sections of this paper are structured as follows. Following a review of prior work and the research methods, we present an overview of the dataset, the preprocessing stages, and the FFT-based framework. This paper analyzes environmental elements that affect solar irradiance and discusses how the framework captures complicated relationships. We explain the TSCV approach, model applications, and performance evaluation, followed by a discussion analyzing the experimental data and situating the suggested forecasting paradigm within existing research.

2. Related Work

The global transition to sustainable energy has raised interest in renewable energy, particularly solar power. Future energy scenarios will rely heavily on solar energy due to significant expansion and technological advances [5]. The process of managing electricity distribution becomes more difficult because solar power generation depends on unpredictable weather conditions. As a result, addressing these issues requires accurate forecasting of solar radiation [8].

Li et al. [9] utilized numerical weather prediction (NWP) models to forecast solar irradiance on the basis of atmospheric conditions. Although NWP models are useful, results indicate that they can be further improved. Of utmost importance is the quality and accuracy of the meteorological data that serve as the inputs. This is a major drawback in any place where there is not sufficient weather data. Mittal [10] examined time-series methods, including ARIMA and exponential smoothing, for the forecasting of solar energy. These methods have been effective in using historical data for short-term forecasting, assuming that future trends will become similar to those of historical data, which is not always the case, especially with the continuous changes in weather conditions.

Regression models using linear or non-linear analysis of solar radiation data were studied by Antoñanzas [11]. The effectiveness of such models is highly dependent on the complexity of the selected model. Simple models may lead to underfitting, while complex models may lead to overfitting. Ramirez-Vergara [12] investigated the probabilistic forecasting of solar radiation using Markov Chain and Bayesian probabilistic approaches, where prediction uncertainty is quantified, which is an essential factor in the risk management of power systems. However, in effecting its application, relatively very large datasets and processing resources not commonly available have to be used. Voyant [12,13,14] incorporated supervised machine learning techniques in their research, such as support vector regression. These models exhibit quite believable assumptions but do not promise very high accuracy in predictions. This is because efficacy in historical data can be maximized only when appropriate input factors and models are employed. They involve a very significant amount of computing resources, especially when using a large number of databases. The prediction accuracy of the above forecasts is increased due to their incorporation of nonlinear data interactions.

Ahmad and Chenn [15] assert that ensemble approaches like bagging and boosting are effective. With many models combined, increased forecasting precision will be achieved. Such processes are very complex and tend to be resource-intensive despite inherent advantages. Their research pointed out that ensemble approaches like bagging, boosting, and random forests are effective in enhancing accuracy in prediction through the combination of many models. The strategies do enhance predictive accuracy, but their implementation becomes difficult because they need powerful computing systems, which demand extensive computer resources.

Kumari and Toshniwa [14] used deep learning to forecast solar irradiance via CNNs. More importantly, CNNs are great at recognizing spatial patterns, which makes them really effective in the analysis of satellite pictures. Still, they might be constrained by huge datasets and resource-poor processing usage. Rajagukguk [16] examined sequential data LSTM networks for long-term dependency. Due to their substantial processing requirements and extensive datasets, Long Short-Term Memory networks are highly effective in forecasting solar radiation and power output. While an LSTM model may have yielded superior predictive accuracy through its extended memory capacity, the model did not exhibit learning behaviors.

Alameen et al. [17] produced accurate solar irradiance data by Generative Adversarial Networks (GANs) to enhance the quality of training data, which in turn increases prediction accuracy. GANs are beneficial in many aspects; however, for their utility, they require high computation and model skill. The framework of flexible hybrid ensemble FHE in Song 18 establishes a system that selects base models according to their performance in prediction errors to achieve better forecasting results through hybrid model feature combination. Integrated models provide superior accuracy compared to their standalone counterparts, but they demand additional processing power to operate.

Significant advancements have been made in the field of solar forecasting; however, numerous challenges persist due to the complexities and variability inherent in weather patterns and solar radiation. The development of models that deliver both precise results and fast processing times presents a major obstacle [18]. The methodology we present here provides a solution to the existing problem. We applied FFT-based spectral feature engineering to identify essential frequency patterns that additional analysis methods fail to detect. Traditional time-based modeling methods fail to detect yearly cycles and recurring trends that remain hidden in their time-based modeling approaches. The model improves its ability to recognize fundamental patterns in solar and meteorological data changes, which leads to better and more precise predictions. The results demonstrate that further feature engineering work is needed to improve solar radiation prediction results through spectral analysis, which will enhance accuracy, stability, and operational value.

3. Proposed Methodology

The methodology research includes three stages, which start with data preprocessing and proceed to spectral feature extraction before ending with model optimization as a means to discover time-based patterns that enhance prediction accuracy. The raw data is first preprocessed to obtain clean and usable datasets for further exploration. This followed the exploratory analysis to reveal insights and trends in the data. In the next step, a spectral feature engineering technique from FFT will be used, thereby forming the core of the methodology to improve forecasts by extracting the best periodic and dominant frequency components from meteorological time-series data. BO is used in parallel, involving initialization, surrogate model development, acquisition function maximization, and an objective function evaluation. As shown in Figure 1, the integrated process involves both FFT-based feature extraction and BO to attain a very adaptable and precise solar radiation forecast. All experiments were conducted in a controlled computational environment to ensure reproducibility. The simulations were performed on a system equipped with an Intel Core i7 processor and 16 GB RAM, running Windows 10 (64-bit). The proposed forecasting framework was implemented in Python (version 3.9). Data preprocessing and analysis were carried out using NumPy (v1.23) and Pandas (v1.5), while Scikit-learn (v1.2) was used for conventional machine learning models. Deep learning models were implemented using TensorFlow (v2.11), and result visualization was performed with Matplotlib (v3.7). Bayesian Optimization was applied using standard optimization routines available within the Python environment.

3.1. Data Overview and Preprocessing

The dataset used for this study is entirely based on real-world meteorological measurements, instead of synthetic, simulated, or numerically generated data. Data were obtained from a publicly accessible IEEE repository and consist of multi-year observational recordings of solar radiation, as well as associated meteorological variables. All variables correspond to actual measured environmental conditions, contain no missing values, and therefore provide a robust and reliable foundation for long-term statistical analysis, seasonal evaluation, and time-series. The features chosen for this study comprise 12 constituents that are suitable for developing effective solar radiation forecasting models, as demonstrated in

Table 1. Dataset in the solar radiation forecasting model.

Feature	Datatype	Description
index	int64	Index of the record
site	int64	Site number where the data was collected
hum	int64	Relative Humidity (%)
julian	int64	Julian day of the year
dayl	int64	Day length in seconds
prcp	int64	Precipitation (mm)
srad	float64	Solar radiation (W/m2)
swe	int64	Snow water equivalent (mm)
tmax	float64	Maximum temperature (°C)
tmin	float64	Minimum temperature (°C)
vp	float64	Vapor pressure (Pa)
wp	float64	Wind speed (m/s)

. These include day length (dayl), precipitation (prcp), snow water equivalent (swe), maximum temperature (tmax), minimum temperature (tmin), vapor pressure (vp), and solar radiation (srad). The dataset is suitable for time-series analysis and modeling because it maintains complete data over multiple years and provides accurate data with continuous coverage. The dataset used in this research study can be accessed through the IEEE portal at this link: https://dx.doi.org/10.21227/ssj7-9p64 (accessed on 12 January 2026). The complete data processing pipeline is depicted in Figure 1. In the very first step, the data are cleaned to eliminate all invalid data points and outlier records. Min–Max and Z-score normalization methods are used to transform the data into standard formats. Feature selection is applied to reduce the number of input variables, resulting in improved model performance and efficiency. The outlier identification methods are used to detect and eliminate data points that would negatively affect prediction accuracy.

Table 2. The statistical analysis of the variables.

Parameter	Count	Mean	Std. Dev.	Min	Q1 (25%)	Median	Q3 (75%)	Max
Julian day	2555	183.00	105.39	1.00	92.00	183.00	274.00	365.00
Day length (s)	2555	43,200	6130.80	34,214	37,325	43,200	49,075	52,186
Precipitation (mm)	2555	3.18	7.31	0.00	0.00	0.00	3.00	86.00
Solar radiation (MJ/m2)	2555	330.81	101.21	57.60	257.60	332.80	416.00	566.40
Snow water equivalent (mm)	2555	0.00	0.00	0.00	0.00	0.00	0.00	0.00
Max temperature (°C)	2555	8.00	0.00	−8.50	−7.00	−1.50	3.00	8.50
Min temperature (°C)	2555	−1.50	6.00	−19.98	−5.50	−1.50	0.00	5.00
Vapor pressure (Pa)	2555	560.00	109.00	320.00	400.00	560.00	722.00	1029.50
Max forecasted temp (°C)	2555	46.40	9.45	37.40	34.70	34.70	37.50	65.70
Min forecasted temp (°C)	2555	29.30	8.65	22.10	19.40	19.40	24.00	39.90
Average temperature (°C)	2555	3.25	3.25	−4.25	−2.75	3.25	6.40	13.80
Temperature range (°C)	2555	9.50	9.50	8.50	8.50	10.50	10.50	12.50

presents statistical data that shows how the solar radiation forecasting dataset distributes its data points and their distribution across different ranges. The average meteorological conditions show solar radiation at 330.81 MJ/m² and precipitation at 3.18 mm. Solar radiation shows high variability because the standard deviation reaches 101.21 MJ/m², while precipitation maintains constant levels. Solar radiation measurements range between 57.6 MJ/m² and 566.4 MJ/m². The data distribution shows a median solar radiation value of 332.8 MJ/m², which results in an uneven distribution pattern. Accurate forecasting algorithms that account for changeable weather conditions affecting solar energy production would require this information. The statistical parameters presented in Table 2 encompass both raw input features and derived variables employed for data characterization, whereas the feature set in Table 1 includes solely the inputs utilized for model training.

3.2. Exploratory Data Analysis

The exploratory data analysis (EDA) corresponding to the study dataset showed statistical significance. After analyzing dataset balance, it seemed to have enough range to appropriately model the solar radiation levels that should be present. A correlation matrix was used to refer to cross-variable correlation for dataset structural characteristics, which are measured in terms of balance and solar radiation (srad) distribution of the dataset. From the histogram of srad values, it could be deduced that srad values were quite normal, slightly skewed upwards. Most srad values tended to be around 300, which indicates that the data collection was balanced from 200 to 450 MJ/m². The little skewness towards the high end requires caution against model bias. Figure 2 presents an analysis of the distribution and variability of solar radiation (srad) labels within the dataset, focusing on balancing aspects. The correlation matrix of the dataset features was also analyzed to understand the strength and direction of linear relationships.

The correlation matrix is depicted through a heatmap visualization (Figure 3). Each cell displays the correlation coefficient between two features. The hue gradient in the heatmap indicates the magnitude and orientation of correlations. Dayl, tmax, tmin, and vp were positively connected with srad, showing that greater values of these variables on a given day increase solar radiation. Increased rainfall decreases solar radiation (srad). Another finding was that tmax, tmin, and vp correlated strongly, suggesting multicollinearity. This analysis was particularly essential for feature selection and for clarifying the fundamental relationships within the data, thereby informing the modeling process.

3.3. Spectral Feature Engineering Using FFT

The FFT-based feature engineering framework is structured as a pipeline for obtaining rich spectral information from meteorological time-series data, shown in Figure 4. It entails beginning with applying the FFT, which covers the conversion from the temporal signal state to the frequency domain. Spectral analyses are carried out to identify dominant frequencies, evaluate power distributions of the dominant frequencies, and estimate the spectral entropy. The meteorological time-series data were analyzed using the FFT, which operated on daily recorded data. The researchers used a 365-day rolling window to study how solar radiation and its related meteorological variables behaved throughout different seasons and annual cycles. The researchers performed discrete Fourier transform calculations on each window to analyze the resulting magnitude spectrum, which showed the major frequency components. The five most prominent frequencies that had the highest spectral magnitudes functioned as the selected representative features. Spectral energy and spectral entropy values were extracted to define how frequency-domain information distributed itself across different complexity levels. The researchers used rolling aggregation to smooth these spectral features, which helped them achieve better stable temporal patterns before they used the features in machine learning and deep learning models. The quality check evaluates spectral features according to their clarity and their ability to show different types of spectral content. We use the raw time-domain features when the features show weak performance or excessive background noise. Through rolling mechanisms, the acquired spectral features undergo aggregation, which helps to remove fluctuations while the system checks for stable temporal patterns. The smoothed temporal features undergo clustering or binning based on frequency similarity, which allows them to become possible features for machine learning or deep learning development. The process of performance assessment leads to spectral parameter refinement, which results in a stronger framework that better adapts to forecasting requirements.

3.4. Machine Learning Models and Hyperparameter Configuration

The research investigated multiple machine-learning models, which included Random Forest (RF), Linear Regression (LR), Decision Tree (DT), K-Nearest Neighbors (KNN), Multilayer Perceptron (MLP), Long Short-Term Memory (LSTM), and Gated Recurrent Unit (GRU) to determine their ability to detect hidden patterns in large datasets while studying solar energy generation data. In the present study, we used Linear Regression and Decision Trees as baseline standard models, which enabled a clear assessment of the performance between basic models and advanced machine learning and deep learning systems. The models provide benefits because they enable both performance evaluation and computational power assessment, which helps researchers understand how much spectral feature engineering boosts performance versus model complexity. The research framework evaluates its effectiveness through both basic and advanced models, which test its performance across multiple modeling approaches. Hyperparameters that are known to influence model performance were selected and fine-tuned for each model. For the RF and DT models, the following parameters were tuned to maximize accuracy and generalization: number of trees (n_estimators), tree depth (max_depth), and splitting criteria (min_samples_split and min_samples_leaf). Other hyperparameters were also tuned to enhance the ability of the MLP, LSTM, and GRU neural network models to capture temporal dynamics and nonlinear behavior of the data in the respective architecture of each model, including computational structure, number of hidden units (hidden_layer_sizes), activation functions, dropout rates, and learning rate initialization (learning_rate_init). The optimized hyperparameters for model accuracy are shown in

Table 3. Models and their corresponding hyperparameters.

Model	Hyperparameters	Optimal Values
RF	n_estimators, max_depth, min_samples_split, min_samples_leaf, max_features	200, 20, 2, 5, ‘sqrt’
LR	fit_intercept, normalize	True, False
DT	max_depth, min_samples_split, min_samples_leaf, max_features	15, 2, 3, ‘auto’
KNN	n_neighbors, weights, algorithm, leaf_size	21, ‘distance’, ‘auto’, 30
MLP	hidden_layer_sizes, activation, solver, learning_rate_init, dropout	(100, 50), ‘relu’, ‘adam’, 0.005, 0.2
LSTM	units, activation, dropout, recurrent_dropout	128, ‘tanh’, 0.3, 0.2
GRU	units, activation, dropout, recurrent_dropout	100, ‘relu’, 0.2, 0.2

.

3.5. Model Optimization and Performance Evaluation

Bayesian Optimization is one of the most promising methods for optimizing high-cost evaluating complex functions, mostly used in hyperparameter tuning of machine learning models. In this scheme, one tries to make a probabilistic model for the objective function and then uses that model to select the most promising hyperparameters for further evaluations. The core of BO is the Gaussian Process (GP), which serves as a surrogate model to predict the objective function $f\left(x\right)$

$$f\left(x\right)\sim GP\left(m\left(x\right), k\left(x, x^{\prime }\right)\right)$$

(1)

where

• $m\left(x\right)$ is the mean function (often assumed to be zero if no prior knowledge is available).
• $k\left(x, x^{\prime }\right)$ is the kernel (covariance) function, which defines the covariance between any two points $x^{\prime }$and $x$.

A commonly used kernel is the squared exponential (radial basis function, RBF) kernel, expressed as:

$$k\left(x, x^{\prime }\right)=\exp \left(-{\displaystyle \frac{1}{2l^2}}\ast {\left|\left|x - x^{\prime }\right|\right|}^2\right)$$

(2)

where

• l is the length scale, a hyperparameter controlling the smoothness of the function.

The acquisition function is designed to decide where to sample next. Two commonly used acquisition functions are Expected Improvement (EI) and Upper Confidence Bound (UCB) [19].

$$EI\left(x\right)=E\left[\max \left(f\left(x\right)-f\left(x^{+}\right), 0\right)\right]$$

(3)

where

• $f\left(x\right)$ is the current best observed value.
• $f\left(x^{+}\right)$is the predicted value of the objective function at point $x^{+}$.

The Upper Confidence Bound $UCB$ is another commonly used acquisition function, which provides a trade-off between exploration and exploitation by considering both the mean $\mu \left(x\right)$and the uncertainty standard deviation $\sigma \left(x\right)$of the surrogate model:

$$UCB\left(x\right)= \mu \left(x\right)+ \kappa \ast \sigma \left(x\right)$$

(4)

where

• $\mu \left(x\right)$ is the predicted mean at point $x$ from the Gaussian Process.
• $\sigma \left(x\right)$ is the predicted standard deviation (uncertainty) at point $x$.
• $\kappa$ a parameter that controls the balance between exploration and exploitation.

Once the acquisition function selects a new point $x$, the objective function $f\left(x\right)$ is evaluated. This evaluation provides feedback to update the model. The objective function $f\left(x\right)$can be a measure of model performance, such as Mean Squared Error (MSE)

$$MSE={\displaystyle \frac{1}{n}}\ast \Sigma {\left(y_i-{\widehat{y}}_i\right)}^2$$

(5)

where

• $y_i$ represents the actual values.
• ${\widehat{y}}_i$ represents the predicted values from the model.
• $n$ is the number of data points.

The new data point from the evaluation step is added to the dataset, and the surrogate model (GP) is updated accordingly. This involves recalculating the posterior mean and covariance:

$${\mu }_{post\left(x_{\ast }\right)}= \mu \left(x_{\ast }\right)+ K\left(x_{\ast }, X\right)\ast K{\left(X, X\right)}^{-1}\ast \left(y - \mu \left(X\right)\right)$$

(6)

where

• ${\mu }_{post\left(x_{\ast }\right)}$ is the updated mean for the new point $x_{\ast }$.
• $K\left(x_{\ast }, X\right)$ is the covariance between the new point $x_{\ast }$ and the observed data points $X$.
• $K{\left(X, X\right)}^{-1}$ is the covariance matrix of the observed data points.
• $y$ is the observed value of the objective function at the points in $X$.

3.6. Evaluation Metrics

To assess model performance, we employed MAE, RMSE, and R² metrics. MAE and RMSE point out the errors that are considerably large and show some insights regarding those errors that indicate the discrepancies in predicted values from true values. R² depicts fitment; however, it does not explain solar data well under extreme conditions. This demonstrates that the balanced combination encompasses both criteria of accuracy and robustness.

$$MAE=\left({\displaystyle \frac{1}{n}}\right)\ast \Sigma \left|y_i-{\hat{y}}_i\right|$$

(7)

$$RMSE=\left({\displaystyle \frac{1}{n}}\right)\ast \Sigma {\left(y_i-{\hat{y}}_i\right)}^2$$

(8)

$$R^2=1-((\Sigma {\left(y_i-{\hat{y}}_i\right)}^2 /\Sigma {\left(y_i-\overline{y}\right)}^2))$$

(9)

4. Results

In this section, we present and explain the results of our experiments to validate the efficiency of our proposed forecasting models. All machine learning (ML) models were developed using Python with the Scikit-learn library. The performance of the model is assessed in terms of three standard metrics: MAE, RMSE, and R². These measures examine predictive accuracy, magnitude of error, and model fit, respectively, providing a robust framework for judging the effectiveness of the DTIFS framework. The improvement seen across the various models due to Bayesian optimization has been greatest with the MLP model among others.

Table 4. Model performance comparison using baseline, random search, grid search, and Bayesian optimization.

Model	Baseline			Random Search			Grid Search			Bayesian Optimization
	MAE	RMSE	R2	MAE	RMSE	R2	MAE	RMSE	R2	MAE	RMSE	R2
LR	3.18	3.98	0.80	3.15	3.95	0.81	3.12	3.90	0.81	3.10	3.85	0.82
RF	2.52	3.12	0.88	2.49	3.08	0.89	2.48	3.07	0.89	2.47	3.06	0.90
DT	2.77	3.47	0.82	2.74	3.44	0.83	2.72	3.42	0.83	2.70	3.40	0.84
KNN	3.48	4.29	0.75	3.45	4.27	0.76	3.44	4.26	0.76	3.42	4.25	0.77
MLP	1.83	2.82	0.90	1.80	2.78	0.91	1.79	2.76	0.91	1.78	2.75	0.92
LSTM	2.47	3.27	0.84	2.44	3.24	0.85	2.42	3.22	0.85	2.40	3.20	0.86
GRU	2.58	3.36	0.83	2.55	3.33	0.84	2.54	3.32	0.84	2.53	3.30	0.85

indicates the comparison between the basic model setups and three hyperparameter tuning methods, which include random search, grid search, and Bayesian Optimization, for all evaluated models. The tuning strategy shows a consistent improvement pattern that starts from baseline settings and reaches its highest point with Bayesian Optimization. The Bayesian Optimization method produces the best MAE and RMSE results and the highest R² values for all models, resulting in improved prediction accuracy compared to both baseline configurations and simpler tuning methods. The absolute magnitude of improvement introduced by Bayesian Optimization shows clear progress, and its impact becomes evident when compared with random and grid search methods. The Random Forest model experiences an MAE reduction from 2.52 to 2.47, while R² increases from 0.88 to 0.90 after applying Bayesian Optimization. The Decision Tree, Linear Regression, and KNN models demonstrate the same pattern of continuous improvement, as Bayesian Optimization achieves better results than random and grid search for all evaluation metrics, although the models’ limited hyperparameter spaces result in smaller performance gains. Neural network models demonstrate the most pronounced effects from Bayesian Optimization. The MLP model shows the largest improvement, with MAE decreasing from 1.83 to 1.78, RMSE decreasing from 2.82 to 2.75, and R² increasing from 0.90 to 0.92. The LSTM and GRU models exhibit similar trends, with consistent reductions in MAE and corresponding increases in R². Overall, the results indicate that Bayesian Optimization performs best for models operating in high-dimensional hyperparameter spaces that require complex searches, as structured optimization enables more effective exploration than random or exhaustive tuning approaches.

Once the implemented spectral feature engineering and optimization were applied, the second test proved quite significant, which can be seen from

Table 5. Effect of feature engineering on model performance.

Model	Before Feature Engineering			After Spectral Feature Engineering
	MAE	RMSE	R2	MAE	RMSE	R2
RF	2.51	3.13	0.87	2.36	3.05	0.90
LR	3.15	3.95	0.78	3.08	3.85	0.81
DT	2.72	3.40	0.81	2.61	3.29	0.83
KNN	3.62	4.28	0.74	3.48	4.15	0.77
MLP	1.82	2.80	0.89	1.68	2.55	0.91
LSTM	2.47	3.25	0.83	2.33	3.15	0.85
GRU	2.59	3.38	0.80	2.44	3.21	0.82

. The largest improvement in this respect was observed for the MLP, wherein MAE reduced from 1.82 to 1.68, RMSE decreased from 2.80 to 2.55, and R² rose from 0.89 to 0.91, which translates into better accuracy and stability. However, RF also improved by decreasing MAE from 2.51 to 2.36 and increasing R² from 0.87 to 0.90. There was a modest improvement for LR and KNN, while DT showed a mixed bag of benefits, showing MAE improvement from 2.72 to 2.61 and R² improvement from 0.81 to 0.83. The LSTM and GRU were also found to benefit from feature engineering, as measured by MAE and R². The feature engineering enhancement differentially improved all models, but particularly MLP and RF, to produce increased accuracy and reliability in the prediction of solar radiation.

MLP showed superior prediction over all other models across different weather scenarios: Clear Sky (CS), Cloudy (CL), High Humidity (HH), and Mixed Conditions (MC). Under Clear Sky, MLP achieved its best performance with an R² of 0.92. On the other hand, RF was fairly accurate and stable across scenarios. KNN and LR, however, showed poor performance under conditions of High Humidity and Mixed Conditions, indicating their lack of robustness. In contrast, although LSTM and GRU gave fairly uniformly good results, their accuracy could not stand up to that of MLP and RF. This proved that weather variability has an impact on forecasting accuracy, whereas the more sophisticated models, such as MLP, tend to be more robust against environmental fluctuations.

Table 6. Model performance under different weather conditions.

Weather Condition	Model	Testing MAE	Testing RMSE	Testing R2
Clear Sky	RF	2.47	3.06	0.90
	LR	3.10	3.85	0.82
	DT	2.70	3.40	0.84
	KNN	3.42	4.25	0.77
	MLP	1.78	2.75	0.92
	LSTM	2.40	3.20	0.86
	GRU	2.53	3.30	0.85
Cloudy	RF	2.52	3.10	0.89
	LR	3.15	3.95	0.80
	DT	2.75	3.45	0.82
	KNN	3.45	4.30	0.76
	MLP	1.80	2.80	0.91
	LSTM	2.45	3.25	0.85
	GRU	2.55	3.35	0.84
High Humidity	RF	2.55	3.20	0.88
	LR	3.20	4.05	0.79
	DT	2.80	3.50	0.81
	KNN	3.50	4.35	0.75
	MLP	1.85	2.85	0.90
	LSTM	2.50	3.30	0.84
	GRU	2.60	3.40	0.83
Mixed Conditions	RF	2.50	3.15	0.89
	LR	3.12	3.95	0.81
	DT	2.72	3.45	0.83
	KNN	3.40	4.25	0.76
	MLP	1.75	2.70	0.93
	LSTM	2.38	3.18	0.87
	GRU	2.50	3.28	0.86

summarizes model performances over different weather scenarios.

Forecasting models use time-series cross-validation to maintain temporal ordering and prevent data leakage [20]. Model stability evaluation in the temporal domain was conducted using Time-Series Cross-Validation (TSCV), which involves segmenting the dataset into four chronological folds, the results of which are presented in

Table 7. Time-series cross-validation results.

Fold	Model	Training Data	Testing Data	MAE	RMSE	R2
Fold 1	RF	January–March	April	2.48	3.08	0.88
	LR	January–March	April	3.15	3.90	0.80
	DT	January–March	April	2.72	3.42	0.82
	MLP	January–March	April	1.80	2.72	0.92
	KNN	January–March	April	3.55	4.30	0.75
	LSTM	January–March	April	2.50	3.18	0.86
	GRU	January–March	April	2.58	3.26	0.85
Fold 2	RF	February–April	May	2.42	3.05	0.89
	LR	February–April	May	3.10	3.85	0.81
	DT	February–April	May	2.68	3.38	0.83
	MLP	February–April	May	1.75	2.60	0.93
	KNN	February–April	May	3.47	4.15	0.77
	LSTM	February–April	May	2.45	3.15	0.87
	GRU	February–April	May	2.55	3.25	0.86
Fold 3	RF	March–May	June	2.35	3.00	0.90
	LR	March–May	June	3.05	3.80	0.82
	DT	March–May	June	2.61	3.32	0.84
	MLP	March–May	June	1.70	2.58	0.93
	KNN	March–May	June	3.43	4.18	0.78
	LSTM	March–May	June	2.43	3.10	0.86
	GRU	March–May	June	2.50	3.22	0.85
Fold 4	RF	April–June	July	2.38	3.02	0.89
	LR	April–June	July	3.08	3.80	0.81
	DT	April–June	July	2.65	3.35	0.83
	MLP	April–June	July	1.72	2.62	0.93
	KNN	April–June	July	3.52	4.25	0.76
	LSTM	April–June	July	2.40	3.05	0.87
	GRU	April–June	July	2.53	3.18	0.84

. The method maintains temporal integrity for subdivided data while conducting model tests throughout seasonal changes. MLP delivered the best performance, which showed continuous MAE and RMSE reduction together with R² values between 0.92 and 0.93. RF achieved second place, with an R² range between 0.88 and 0.90. KNN ranked as the weakest performer among the three because it experienced significant drops in performance during early folds, which demonstrated its high sensitivity to time-based changes in solar data. LSTM and GRU maintained stable performance across all folds, but they achieved lower accuracy rates than MLP and RF. We applied a four-fold rolling-window TSCV to maintain temporal continuity while ensuring that all training windows contained sufficient seasonal information.

The experimental design enables observation of model performance differences resulting from different model architectures and feature selection methods, rather than from variations in datasets or evaluation procedures. The proposed FFT-based framework demonstrates superior performance compared to CNN and LSTM methods because it allows identification of specific frequency-domain features of the system. The proposed method enables direct encoding of dominant periodicities, seasonal cycles, and spectral energy patterns, whereas CNN and LSTM models rely on automatic feature extraction from time-domain sequences. The explicit spectral representation of data reduces learning difficulty, improves model stability, and enhances generalization ability under severe weather variability.

The scatter plot shows the relationship between predicted and actual values for solar radiation output, with the data points clustering around the ideal diagonal line, indicating strong predictive accuracy. Figure 5 shows the performance of various machine learning models, comparing the performance of different machine learning models. With the lowest MAE of 1.78 and RMSE of 2.75, and the highest R² (0.92), MLP is the best model and performs better than all others. This shows how reliable and precise MLP is at predicting solar radiation. The next closest is RF, which is not far behind but has a lower R² value. Conversely, models with lower R² and higher error, like KNN and LR, show poorer predictive abilities. All findings support that MLP is the most accurate model for forecasting solar energy generation in this study.

As shown in Figure 6, the smoothed cumulative error distribution (CED) plot clearly indicates that the MLP model achieves the most reliable prediction, with more than 90% of the predictions being less than a 3 MJ/m² error threshold. Random Forest and LSTM models roughly follow the same trends, showing a great consistency with respect to their predictability. KNN and LR fall far behind in slow cumulative growth in predictive accuracy. Again, the steepness of the MLP curve proves how well this model generalizes on varying conditions. Such outcome results confirm the success of deep learning, further improved by engineered features and BO. The same pattern results have also authenticated the robustness of the simulation model.

A thorough study of the 3D surface that represents day length and temperature against solar irradiance clearly exhibits evidence of nonlinear interactions among these variables, as illustrated in Figure 7. With an increase in day length, solar irradiance initially increases, following a seasonal pattern aligned with that of the solar cycle. Temperature positively affects irradiance, but beyond about 30 °C, the effect appears to plateau, probably due to thermal saturation or atmospheric attenuation. The joint effect maximizes irradiance under conditions of long day length and moderate-to-high temperatures. These findings endorse the rationale behind including temporal variables in the forecasting model and thus justify the use of spectral feature engineering and MLP-based architectures for adequately modeling seasonal and nonlinear behaviors in solar radiation prediction.

Figure 8 depicts the residual distributions of seven models, which provide insight into how each of those models performed both in prediction accuracy and consistency. The distributions of LSTM and GRU are compact, symmetric, and zero-centered, which means that their error variance is low, along with bias, both being characteristics of an equally structured and generalized model. The performance of MLP is more or less comparable but relatively breached at the end. In turn, there are wider and skewed distributions in Linear Regression and KNN with obvious bias, thus representing under- or over-predictions systematically. Random Forest sits between the two extremes, having medium dispersion and a few extreme residuals. These residual characteristics increase the credibility for the understanding that deep learning models (LSTM and GRU) indeed outperform other, more straightforward or “shallow” models for capturing nonlinearities and temporal dependencies that are embedded in solar irradiance forecasting.

5. Discussion

This section presents a comprehensive analysis of the experimental results, situates the proposed forecasting framework within the framework of existing research, critically evaluates its robustness and constraints, and delineates potential avenues for future research and practical application.

5.1. Interpretation of Key Results

The proposed methodology, which utilizes FFT-based spectral feature engineering together with Bayesian Optimization, shows better solar radiation forecasting results than the existing methods, as demonstrated in Table 4 and Table 5. The evaluated models show increased R² values together with decreased MAE and RMSE values, which proves the framework works for machine learning and deep learning systems because it operates across all model types. The MLP model demonstrates a prediction error decrease with increasing explanatory power through Bayesian Optimization, while structured tuning methods provide better results for models working in higher-dimensional hyperparameter ranges (Table 4). The hyperparameter landscape exploration allows these models to use spectral feature engineering better because they can identify more complex feature representations. The results shown in Table 5 demonstrate that when FFT-based features are added to the system, the performance increases slightly but not significantly. The frequency-domain representations allow models to directly identify existing dominant periodicities and seasonal cycles found in solar radiation data, while raw time-domain inputs only contain this information as hidden patterns. The feature engineering method develops the model’s ability to detect periodic patterns that stay unchanged while it adapts to fast-changing situations.

5.2. Contribution with Respect to the State of the Art

The evaluation of the proposed method was conducted through a comparative analysis with existing solar radiation forecasting research studies that have been documented in the scientific literature.

Table 8. Performance comparison of the proposed approach with existing solar radiation forecasting studies.

Study	Method	Feature Domain	MAE	RMSE	R2
[11]	NWP-based	Physical	3.95	5.82	0.78
[13]	ARIMA	Time-domain	4.21	6.10	0.74
[16]	SVR	Meteorological	3.12	4.45	0.81
[9]	CNN	Spatial	2.64	3.98	0.86
[21]	LSTM	Time-domain	2.31	3.42	0.88
This work	MLP + FFT + BO	Spectral + Time	1.78	2.75	0.92

presents a detailed comparison between the proposed FFT-based framework and representative state-of-the-art methods, including time-domain statistical models, machine learning approaches, and deep learning models. The proposed MLP + FFT + BO approach shows better performance than all previously reported methods through its lower MAE and RMSE values and higher R² measurement, as shown in Table 8. The proposed framework uses spectral feature engineering to capture seasonal patterns and frequency-based patterns in solar irradiance data because it combines traditional time-domain approaches with deep learning-based models. The combination of these elements results in better forecasting performance because it enables the model to remain stable across various operational scenarios. The results demonstrate that the proposed method contributes new knowledge to the field by showing how frequency-domain data and Bayesian hyperparameter optimization work together to improve solar radiation forecasting accuracy.

5.3. Robustness and Generalization of the Proposed Approach

The proposed forecasting framework is tested for its ability to withstand different weather conditions and different time periods. Table 6 presents the model performance results for Clear Sky, Cloudy, High Humidity, and Mixed Conditions, which show that the MLP model achieves better MAE, RMSE, and R² results than all other models. Random Forest also shows stable performance across scenarios, whereas KNN and Linear Regression perform poorly under high variability conditions. Temporal generalization is further validated using Time-Series Cross-Validation, with results reported in Table 7. The relatively stable performance across all folds indicates that the proposed approach maintains predictive accuracy across seasonal transitions while preserving the temporal structure of the data. The robustness of the framework is further supported by the cumulative error trends shown in Figure 6 and the SHAP-based feature importance analysis demonstrated in Figure 9. The SHAP-based analysis further supports robustness by showing that the same physically meaningful predictors remain dominant across models and conditions, indicating consistent generalization rather than dataset-specific fitting.

5.4. Limitations of the Study

Despite the promising results, certain limitations of the proposed approach should be acknowledged. The first factor is that the effectiveness of FFT-based spectral feature engineering requires assessment of meteorological data input quality, along with its continuous stream. The residual distribution analysis, shown in Figure 8, demonstrates that some models produce wider and skewed error distributions, which makes them more vulnerable to both noise and data variations. The combination of FFT parameter selection and window size determination, together with frequency resolution choice, affects both spectral feature extraction and model performance. The current implementation achieved outstanding results in the present study, yet researchers must examine parameter sensitivity across multiple datasets. The Bayesian Optimization method incurs extra computational costs because it requires multiple model assessments. The resource expenses, presented as proof of enhanced accuracy in Table 4 and Table 5, do not permit real-time operation in situations where resources are limited. The study dataset contains data from only one specific geographical area. The results from Table 6 and Table 7 demonstrate strong internal robustness and temporal generalization, yet further validation across multiple locations and climatic regions is necessary to confirm their broader applicability.

5.5. Future Research Directions and Deployment Potential

Research needs to apply the proposed framework on multiple-site datasets that contain data from different climate zones to assess its ability to predict spatial patterns. Nonstationary solar radiation patterns will receive better representation through multi-resolution time-frequency techniques, which include wavelet and hybrid FFT-wavelet methods. The research will use probabilistic forecasting extensions to measure their ability to predict upcoming uncertain events, which serves as a fundamental requirement for effective energy management. The use of adaptive and automated methods for spectral parameter selection will improve system reliability for the analysis of historical data. The method uses feature importance analysis to explain results, which helps build systems that operate reliably under all weather conditions. The framework provides solutions for forecasting solar radiation on both day-ahead and intra-day bases, which helps with energy storage planning and grid stability improvement for smart grid and energy management systems. The method achieves optimal performance for renewable energy forecasting in real-world situations through the effective optimization of its computational workflows.

6. Conclusions

In this study, we propose a novel algorithm combining FFT with BO for solar radiation forecasting. While the algorithm is developed to find meaningful patterns in solar data by designing specific features, BO is used to tune the model parameters. The proposed approach was applied to MLP and successfully achieved an R² of 0.91, MAE of 1.68, and RMSE of 2.55, outperforming traditional models. The results show that domain-specific knowledge, feature engineering, and hyperparameter optimization can help improve prediction accuracy. Time-series cross-validation was used to test the model, which showed that the proposed approach outperforms and is more stable than other recent approaches across different time horizons. In summary, the combination of the FFT framework and BO provides more reliable solar energy predictions, which are required for the integration of renewable energy into the electrical grid and for improved power system management. From a theoretical perspective, this study demonstrates the value of integrating explicit spectral feature engineering with machine learning and deep learning models for non-stationary solar radiation forecasting. By transforming meteorological time-series data into the frequency domain, the proposed framework enables more effective learning of dominant periodicities and seasonal patterns that are often obscured in purely time-domain representations, thereby improving model generalization under variable weather conditions. From a practical standpoint, the proposed FFT-based framework enhances forecasting accuracy and robustness, supporting more reliable grid operation, photovoltaic generation scheduling, and energy management. The combination of spectral feature engineering with Bayesian Optimization also improves adaptability across different data characteristics and computational settings, making the approach suitable for operational use. Future research may explore multi-resolution spectral techniques, such as wavelet or hybrid FFT–wavelet methods, as well as probabilistic forecasting to quantify uncertainty. Further studies could also examine the generalization of the proposed framework across different geographic regions, climatic conditions, and higher temporal resolutions.