1.2. Survey of Related Literature
Some authors have applied various methods incorporating spatial analysis. Andre et al. [
2] used a spatial–temporal model for short-term solar irradiation forecasting. They used a spatial–temporal vector autoregressive model, which was designed in such a way that it could handle sparse spatial–temporal data. An iterative strategy in the model process selected related stations and eliminated insignificant predictor variables.
In a study by Liu et al. [
3], an ensemble-temporal deep learning approach incorporating multi-sites in a spatial power grid was used to predict solar power. Variational modeling was used to predict uncertainty, and the proposed methods estimated uncertainty very well. Yang [
4] used the ultra-fast pre-selection method to solve the lasso problem, that of having an insufficient degree of freedom and curse of dimensionality. The variables selected via ultra-fast were then used to develop a lasso-temporal model. Their results showed that their proposed algorithm did not need meteorological priors and provided the best forecasts.
Another study by Eschenbach et al. [
5] was based on forecasting solar irradiation based on various machine-learning methods. The methods used were ARX (autoregressive with exogenous inputs), NN (Neural Network), RRF (random regression forest), and RT (regression trees). Their results showed that for a short lead time, NN produced more accurate results as well as for dense-temporal input data.
Kim et al. [
6] applied a temporal method that combined public and satellite data. Based on a satellite from South Korea, SVR (Support Vector Regression), ANN (Artificial Neural Network), ARIMAX (Autoregressive Moving Average), and DNN (Deep Neural Network) were used based on short interval forecasts of GHI data. Their results showed that the models based on temporal and spatial characteristics gave better forecasts than the other models that were based on numerical data using weather variables.
Zhang et al. [
7] applied a Spatial–Temporal Gaussian process state space model with the Kronecker structure on weather data from Colorado and Global Historical Climatology Network. The other objective was to come up with a kernel that will be used for the Colorado and GHCN (Global Historical Climatology Network) data. To estimate the hyper parameters of the Spatial–Temporal Gaussian model, the Kalman Filter and smoother was used. The results showed that the forecasting performance improved from the results by [
8] based on the desirable properties of the Matern kernel used for both datasets.
Hamelinjck et al. [
9] applied variational spatio-temporal regression coupled with Gaussian processes. A non-conjugate GP technique produced a sparse state model using separable Markov kernels. The results of the proposed methodology proved to be more accurate and efficient due to filtering parallelization of spatial locations, sparsity, and application of variational Gaussian process analysis. The proposed method outperformed the baseline methods.
Agoua et al. [
10] forecasted photovoltaics using a probabilistic spatial–temporal model approach that used datasets from plants that were close to each other. This technique incorporates short dry forecasting of periods of 0–6 h. Quantile regression was combined with Lasso, which was used for variable selection. The proposed methodology showed superior performance in comparison to KDE (Kernel Density Estimation), which was used as the benchmark model.
Banerjee et al. [
11] developed a temporal–spatial model using predictive process modeling. The method reduced computational burden by reducing the modeling space to a lower dimensional subspace. The results of the proposed methodology addressed the problem of misspecification of the model using a larger dataset, which was achieved by applying the induced specification predictive process.
Luttinen et al. [
12] did a Bayesian analysis using sea surface temperature data. They used a probabilistic factor analysis using spatiotemporal data. Gaussian process priors were used for the factors and loading matrix. According to their results, the Gaussian Process Factor Analysis outperformed the Bayesian Principal Component analysis.
Tomizawa and Yoshida [
13] applied Gaussian Process regression with various Gaussian random fields to data with problems of spatial variability. The maximum likelihood method was used to estimate the random and fluctuating fields’ scale. Their results showed that the model with the Whittle Matern kernel was the best for the random component.
Comber et al. [
14] used Gaussian Process splines regression taking into account the variational of geographical areas. The technique is a smoothing parameter used in splines regression combined with Gaussian Processes, optimizing the GP splines regression. The model showed predictions that were more accurate because of accommodating heterogeneity.
Another research was done by Najimbi et al. [
15], they used probabilistic GPR to forecast solar power using meteorological data. Short-term forecasting was used based on
k-means clustering. A Matern
covariance function was used. A 5-fold validity test set, holding out 30 random days, was used to validate the applied method. Root mean square error was reduced due to the application of the proposed methodology.
In another paper, Najimbi et al. [
16] predicted solar power using weather variables. Eight different partitions were used to cluster the data using k-means clustering, and GPR was used based on the Matern
covariance function. The Elbow and Gap techniques were used to develop optimal clusters, and the results showed that the forecasting error was reduced.
Paiva et al. [
17] performed a comparative analysis on multilayer perceptron (MLP) artificial neural networks and multigene genetic programming (MGGP). The assessments indicated that MGGP gave more accurate and fast results in single predictions, and the ANN performed better for ensemble forecasts. Wang et al. [
18] used a cluster-based analysis on ultra-short-term wind power using a hierarchical directed graph method and dynamic–temporal correlation. They first defined three nodes based on wind power, wind speed, and target nodes, and defined input sample and correlation matrices that are temporal-based to evaluate the correlation of neighboring wind farms. They also used directed edges to connect various nodes to obtain a hierarchical-based graph form, which was later applied to train the prediction model. The proposed model outperformed the other benchmark models used.
The world meteorological organization produced a guide highlighting the major sources of uncertainty in forecasted values related to the weather as the process of producing a forecast, its interpretation, and atmospheric unpredictability (Gill et al. [
19]). Most of the studies reviewed in this research on forecasting global horizontal irradiance (GHI) using spatial regression considered only the spatial feature, leaving out weather variables and the variability in GHI. Additionally, most classical approaches for predicting GHI rely mainly on a single power plant. The current study focuses on a multi-site approach to modeling and forecasting GHI using Gaussian process models and including a nonlinear trend covariate. A summary of previous studies on modeling solar radiation based on spatial analysis is given in
Table 1.
1.3. Research Highlights and Contributions
The current study presents an in-depth analysis and spatial–temporal predictive modeling of GHI at radiometric stations in sparse geographical regions. To the best of our knowledge, this is the first study to be carried out using South African data. The study proposes the standardization of predictions, which improves forecast accuracy. Another highlight is in the selection of the validation sets. Eight radiometric stations were used in the study, with three locations in each validation set. This resulted in fifty six possible validation sets. From the randomly selected five sets, an in-depth analysis was carried out to investigate how the proposed models would perform, especially those stations that are far apart in Euclidean space.
Based on the literature review discussed in
Section 1.2, the highlights and contributions made in this study are summarized as follows. The research focused on moving from a single site to a multi-site forecasting approach, that is, using geographically sparse data. Multi-site research is expected to give a diverse or large sample that is good enough to come up with significant associations between meteorological stations; hence, it enhances the statistical power of the model. Previously, solar forecasting has been based on single-site analysis. Exploring the temporal forecasting approach is expected to improve modeling accuracy.
We predicted solar irradiation using spatial regression coupled with Gaussian process modeling using Bayesian inference. GP regression has proved to be a very powerful tool for modeling the variability and uncertainty of solar irradiation [
20], though it is very computationally expensive. The spatial analysis then improves the predictive accuracy since it reduces the computational burden in regression analysis. The combination of these two methods thus produces a hybrid model that produces favorable results.
A comparative analysis was done between the proposed models, GP Spatial and GP-AR Spatial, and the linear spatial model, which was used as a benchmark model. The evaluation metrics used to assess the accuracy of the models are MAE (Mean Absolute Error), RMSE (Root Mean Square Error), CP (Coverage Probability) and CRPS (Continuous Rank Probability Score).
Standardization of the forecasts made from the selected model was also done in this research. The standardized forecasts were then added to the original forecasts, improving the forecast accuracy. During forecasting, multicollinearity can be introduced if independent variables are correlated. This problem compromises the model’s statistical significance and produces imprecise coefficients. To overcome this problem, we used ElaticNet, one of the shrinkage methods used in variable selection. ElaticNet is a supervised algorithm that identifies the variables strongly associated with the response variable. The nonlinear rend variable is another important variable that resulted in a significant improvement in the forecast accuracy. Finally, we combined forecasts using QGAM (Quantile Generalized Additive Model) and MCQRNN (Monotone Composite Quantile Regression Neural Networks). Combining forecasts is a very effective tool for reducing errors.
The following sections are organized as follows.
Section 2 briefly discusses the temporal GP and GP-temporal autoregressive models, including the benchmark model Linear-temporal Regression. The section also presents a discussion of Bayesian inference, which is used in the computation of the parameters. The results of the exploratory analysis of GHI proposed methods and the benchmark model are given in
Section 3. The conclusion is given in
Section 4.