2.14 Combination of SVM, WT, ANN, and GA methods

Wavelet transform (WT) is a time-series analysis tool that decomposes a signal into a representation demonstrating time-series details and trends as a function of time [22]. Mohammadi et al. [23] uses a hybrid of support vector machine and wavelet transform algorithm (SVM-WT) for extracting features of solar radiation data. In addition, the artificial neural network (ANN) and genetic algorithm (GA) are combined to forecast daily global solar radiation. The results show that the SVM-WT method improves the forecasting accuracy by recognizing the patterns and providing better training for the neural network.

#### 2.15 Clustering-based multi-model method

Wu and Chan [24] propose a novel multi-model framework for solar radiation forecasting. The framework is based on the assumption that there are several patterns in the stochastic component of solar radiation series. The time series data are first classified into multiple subsequences. The k-means algorithm is then applied to group the subsequence into different clusters. Finally, the time-delay neural network (TDNN) is trained to model a specific pattern in each cluster. The pattern corresponding to the current time is then determined for the forecasting purpose. This process is followed by selecting the appropriate trained TDNN model. The comparison analysis of the proposed forecasting method with autoregressivemoving-average (ARMA) model shows that the proposed model provides superior performance.

#### 2.16 Data-driven method

Aerosol Optical Depth (AOD) and the Angstrom Exponent data are used in [25] and are included in several data-driven models for hourly solar radiation forecasting. Several machine learning methods including multilayer perceptron (MLP), SVR, k-nearest neighbors (kNN), and decision tree regression are used as the datadriven models and evaluated for their forecasting accuracies. The evaluation results show that the MLP method outperforms other data-driven forecasting models.

#### 3. Performance metrics

Several indexes are presented for evaluating the performance of time-series prediction methods. These indexes are used to represent the forecast error. Lower error values correspond to more accurate forecasting. This section provides the definitions and mathematical representations for most commonly used error indexes in forecasting.

Note that SActual is the observed value, S ^ is the predicted value (forecast), and N is the total number of observations.

#### 3.1 MSE

The mean square error (MSE) is used as the accuracy performance indicator given by:

$$\text{MSE} = \frac{1}{N} \sum\_{n=1}^{N} \left( \hat{\mathbf{S}}(n) - \mathbf{S}\_{\text{Actual}}(n) \right)^2 \tag{1}$$

#### 3.2 nMSE

Normalized mean square error (nMSE) is an estimator of overall deviations between the forecasted and measured samples. It is defined as follows:

$$\text{nMSE} = \frac{N \sum\_{n=1}^{N} \left(\hat{S}(n) - S\_{Actual}(n)\right)^2}{\sum\_{n=1}^{N} \hat{S}(n) \cdot \sum\_{n=1}^{N} S\_{Actual}(n)} \tag{2}$$

#### 3.3 RMSE

The square root of MSE is given by:

$$\text{RMSE} = \sqrt{\frac{1}{N} \sum\_{n=1}^{N} \left( \hat{S}(n) - S\_{Actual}(n) \right)^2} \tag{3}$$

#### 3.4 nRMSE

Normalized root mean square error (NRMSE) is calculated as follows:

$$\text{nRMSE} = \frac{\sqrt{\frac{1}{N} \sum\_{n=1}^{N} \left(\hat{\mathbf{S}}(n) - \mathbf{S}\_{\text{Actual}}(n)\right)^{2}}}{\frac{1}{N} \sum\_{n=1}^{N} \mathbf{S}\_{\text{Actual}}(n)}\tag{4}$$

#### 3.5 MAE

Mean absolute error is calculated as follows:

$$\text{MAE} = \frac{1}{N} \sum\_{n=1}^{N} \left| \hat{\mathbf{S}}(n) - \mathbf{S}\_{Actual}(n) \right| \tag{5}$$

Pattern Recognition and Its Application in Solar Radiation Forecasting DOI: http://dx.doi.org/10.5772/intechopen.83503

#### 3.6 nMAE

Normalized mean absolute error (nMAE) is given by:

$$\text{nMAE} = \frac{1}{N} \sum\_{n=1}^{N} \frac{\left| \hat{\mathbf{S}}(n) - \mathbf{S}\_{Actual}(n) \right|}{\max(\mathbf{S}\_{Actual}(n))} \tag{6}$$

#### 3.7 MAPE

Mean absolute percentage error is calculated as follows:

$$\text{MAPE} = \frac{1}{N} \sum\_{n=1}^{N} \left| \frac{\hat{\mathbf{S}}(n) - \mathbf{S}\_{Actual}(n)}{\mathbf{S}\_{Actual}(n)} \right| \tag{7}$$

#### 3.8 MBE

Mean bias error (MBE) is given by:

$$\text{MBE} = \frac{1}{N} \sum\_{n=1}^{N} \hat{\mathbf{S}}(n) - \mathbf{S}\_{\text{Actual}}(n) \tag{8}$$

#### 3.9 nMBE

Normalized mean bias error (nMBE) is calculated by normalizing the MBE as follows:

$$\text{InMBE} = \frac{\sum\_{n=1}^{N} \hat{S}(n) - S\_{Actual}(n)}{\sum\_{n=1}^{N} S\_{Actual}(n)} \tag{9}$$

#### 3.10 MABE

The MABE represents the mean absolute bias error as follows:

$$\mathbf{MABE} = \frac{\mathbf{1}}{N} \sum\_{n=1}^{N} \left| \hat{\mathbf{S}}(n) - \mathbf{S}\_{\text{Actual}}(n) \right| \tag{10}$$

#### 3.11 SMAPE

The SMAPE represents symmetric mean absolute percentage error as follows:

$$\text{SMAPE} = \frac{1}{N} \sum\_{n=1}^{N} \frac{\left| \hat{\mathcal{S}}(n) - \mathcal{S}\_{\text{Actual}}(n) \right|}{\left( \hat{\mathcal{S}}(n) + \mathcal{S}\_{\text{Actual}}(n) \right)^{2}} \times 100 \tag{11}$$

#### 3.12 Forecast skill

The forecast skill is an error index which provides accuracy comparison between any given forecasting model and the persistence forecasting method. The forecast skill is calculated by:

$$\text{Forecast Skill} = \left( 1 - \frac{\text{RMSE}}{\text{RMSE} \left( \text{Persistance} \right)} \right) \tag{12}$$

A forecast skill of 1.0 implies an unattainable perfect forecasting, and a forecast skill of 0.0 indicates a performance similar to the persistence method. Besides, negative values show lower forecasting accuracies compared to the persistence method. Detailed explanations and applications of the forecast skill can be found in [26].

Table 1 provides a summary of the solar radiation forecasting methods explained in subsections 2.1–2.16, along with the error metrics used for each method.



Pattern Recognition and Its Application in Solar Radiation Forecasting DOI: http://dx.doi.org/10.5772/intechopen.83503

#### Table 1.

Summary of the pattern recognition-based solar forecasting methods.
