## 3 Results
### Application of the Model for Spectral Irradiance Estimation
Extending the SSI Estimation Model to Incorporate Split Conformal Prediction for Principled Uncertainty Quantification  
Our proposed approach to quantifying SSI pivots on leveraging ML. ML models inherently involve a certain level of uncertainty. As an extension of this work, to address this, we explored the incorporation of split conformal prediction into our framework to enable the generation of statistically rigorous confidence intervals for our model estimates. Split conformal prediction [57; 58; 59] constructs prediction intervals using a hold-out calibration set that is separate from the training data. After training a prediction model on the training set, a nonconformity score, in this case, the absolute error:  
\[R_{i}=|f(X_{i})-y_{i}| \tag{2}\]  
where \(f(X_{i})\) is the trained ML model applied to the \(i\)th calibration record and \(y_{i}\) is the true value of the target at the \(i\)th instance, is computed for each point in the calibration set. The interval half width \(\delta_{\alpha}\) is then calculated as the \((1-\alpha)(1+\frac{1}{n})\)th empirical quantile to ensure a coverage level of \((1-\alpha)\) on the calibration set. Prediction intervals for new data are then formed as \(f(X_{new})\pm\delta_{\alpha}\). To integrate the framework with conformal prediction, we retrained our model, this time splitting the dataset to set aside a calibration set: a total of 20% of the total dataset was isolated as an independent testing set, and the remaining 80% was result to allocate 80% as the training set and 20% as the calibration set for the conformal prediction. Then, using the trained model and the calibration set, split conformal prediction as implemented in the Python library Puncc (Predictive uncertainty calibration and conformalization) [57] was applied to evaluate the associated uncertainty of the model. We set \(\alpha=0.2\) for a coverage  
Figure 12: RAE of the estimated spectra compared to observations by the reference instrument. corresponding to 80%. Finally, the model was evaluated on the independent testing set and the performance metrics were generated. Given the substantially large size of our dataset, the additional data split to isolate a calibration set had a negligible impact on the overall model performance. We present below the performance metrics of the model integrated with split conformal prediction: Figure 13 illustrates the model's predictive performance on both the training set (green) and the independent testing set (blue) across the full spectral range of 360 nm to 780 nm. Figure 14 presents the \(R^{2}\) for the model's performance in each individual wavelength bin on both the training and testing sets. Figure 15 shows the MSE per bin, while Figure 16 illustrates the relative MSE values computed on the testing set across all 421 wavelength bins. Figure 14: \(R^{2}\) values on each of the 421 spectral bins evaluated on the training and testing sets after incorporating conformal prediction. Figure 13: Scatter diagram comparing the SSI measurements from the reference instrument on the x-axis against the SSI estimates from the ML model on the y-axis for the whole spectrum from 360 nm to 780 nm after incorporating conformal prediction. The metrics depicted in Figure 13 through Figure 16 are comparable to and closely mirror Figure 5 through Figure 8, demonstrating that the integration of uncertainty estimation can be adopted without compromising the predictive accuracy of the model. We randomly selected a subset of instances from the testing set and applied the model augmented with split conformal prediction to estimate the full spectral irradiance from 360 nm to 780 nm, along with 80% conformal prediction intervals. Figure 17 presents two examples of the estimated spectral irradiance, each accompanied by 80% conformal prediction intervals and compared against the corresponding ground-truth spectra.