# Short-Term Wind Speed Forecasting Using Statistical and Machine Learning Methods
## 3 Analysis of the Forecasting Results
### Prediction Intervals Evaluation  
Forecasting using models is known to be filled with much uncertainties given good point forecasts performance evaluation. In this section, the methods for quantifying the uncertainties in point forecasting models as well as in the combined point forecasting model are described. To measure uncertainties, the needed feature to be used is the prediction interval. Along with making use of AQRA for forecast combination, it was also used to construct prediction intervals through the lower and upper bound estimates for the models and at different quantiles [21]. The quantiles considered are the 90 th, 95 th and 99 th quantiles, respectively. To quantify uncertainties, five prediction interval evaluation metrics were used in this study and the results are summarised in Table 7. The five metrics are the prediction interval nominal coverage (PINC), the prediction interval coverage percentage (PICP), the prediction interval normalised average width (PINAW), the prediction interval normalised average deviation (PINAD) and the prediction interval coverage average normalised width (PICAW). The theoretical formulations of these metrics are given in Section 2 (Equations (19) and (21)-(26)). These uncertainty metrics relating to quantifying the uncertainty in wind speed forecasting, using the individual point forecasting models and the AQRA forecasting combination model through the use of AQRA to construct prediction intervals whose values according to these metrics, are presented in Table 7.  
Figure 7: Density plots for PIW of BNN through AQRA. In Table 7, five metrics are used to evaluate PIs from individual and the combined models. Four models in total are evaluated using these metrics. PICP is used to ascertain the reliability of the constructed PIs. Therefore, the more the actual values are covered by the PI, the higher are the PICP values [21]. Furthermore, PICP value is expected to be greater or equal to the PINC or confidence value else deemed as invalid [20]. Accurate and satisfactory PI performance is indicative of higher PICP values and lower PINAW values [20; 21]. Both PICP and PICAW values indicate the quality of the constructed PI. Hence, a high PICP value with small PICAW value is required for a quality PI construction [21]. In addition, given a high PICP, the deviation value of the PIs from the actual value is expressed by PINAD. Hence, a higher PICP should give a lower PINAD value, showing less deviation from the actual value [31]. The discussion for Table 7 follows from the foregoing theoretical and literature findings. Using PICP at 90% PINC, all forecasts are valid and AQRA has the highest value and presents more coverage than the rest of the models. SGB is next and the last is BNN showing that the PI from BNN covers less actual values than the rest. Taking PICP and PINAW, AQRA presents both the highest PICP value and the lowest PINAW, thus the most reliable model at this confidence level. This is followed by BNN and the least is GAM. Although the PI from BNN covers fewer actual values, the covered values present a more closely fit weight than SGB and GAM. Taking PICP and PICAW, AQRA is the model with the best quality PI since it has the lowest PICAW value and the highest PICP value followed by BNN and the least is SGB. The degree of deviation from the actual value is shown by PINAD and BNN has the least degree of deviation followed by AQRA; GAM has the highest degree of deviation from the actual value. The ABLL and AAUL columns represent the number of actual values that are not within the range of PIW. They represent the number of actual values that are below the lower limit and the actual values that are above the upper limit, respectively. For the 95% confidence value, SGB recorded the highest PICP value, GAM is next and then AQRA, while BNN has the lowest PICP value. At the 99% confidence values, SGB and BNN give higher PICP values than GAM and AQRA.