## 2 Accuracy Metrics
### Scale-Dependent Metrics  
Metrics based on absolute errors or on squared errors are called scale-dependent metrics [34]. The scale-dependent metrics have the same scale as the original data [34] and provide errors in the same units [35]. However, the scale-dependent metrics can be difficult  
Figure 2: Degree of correlations. Five degrees of correlations: very weak (from 0.0 to 0.2 or from 0.0 to \(-\)0.2), weak (from 0.2 to 0.4 or from \(-\)0.2 to \(-\)0.4), moderate (from 0.4 to 0.6 or from \(-\)0.4 to \(-\)0.6), strong (from 0.6 to 0.8 or from \(-\)0.6 to \(-\)0.8), very strong (from 0.8 to 1.0 or from \(-\)1.8 to \(-\)1.0).  
to compare for series that are on different scales or that have different units. For example, if a prediction error is 10 units, the gravity of the error cannot be evaluated unless the level of gravity is also provided [36].  
Although the scale-dependent metrics are not unit-free, they are favored in machine learning evaluation. The commonly used scale-dependent metrics are MSE, RMSE and MAE. The calculation methods of the three metrics are defined as [37]  
\[\text{MSE}=\frac{1}{n}\overset{n}{\underset{1}{\sum}}\left(\text{D}_{\text{ pre}}-\text{D}_{\text{act}}\right)^{2} \tag{3}\]  
\[\text{RMSE}=\sqrt{\frac{1}{n}\overset{n}{\underset{1}{\sum}}\left(\text{D}_{ \text{pre}}-\text{D}_{\text{act}}\right)^{2}} \tag{4}\]  
\[\text{MAE}=\frac{1}{n}\overset{n}{\underset{1}{\sum}}|\text{D}_{\text{pre}}- \text{D}_{\text{act}}| \tag{5}\]  
MSE measures the mean squared error between the predicted value and actual value. For every data point, the distance is measured vertically from the actual value to the corresponding predicted value on the fit line, and the value is squared. Subsequently, the sum of all the squared values is calculated and divided by the number of points. Therefore, the unit of MSE is the square of the original unit. Due to the squaring of errors, the negative values and positive values do not cancel each other out.  
The range of MSE is \((0,+\infty)\); the smaller the MSE value is, the higher the accuracy of the prediction model. The perfect value of MSE is 0, indicating that the prediction model is perfect. MSE is defaulted as the loss function of linear regression in machine learning.  
RMSE measures the average magnitude of error between the predicted value and actual value. Thus, RMSE is the average distance measured vertically from the actual value to the corresponding predicted value on the fit line. Simply, it is the square root of MSE.  
In the same manner as MSE, the range of RMSE is \((0,+\infty)\); the smaller the RMSE value is, the higher the accuracy of the prediction model. In contrast with MSE, the units of RMSE are the same as original units, making the RMSE more interpretable than MSE.  
MAE is a metric used to measure the average magnitude of the absolute errors between the predicted value and actual value. The MAE is often called the mean absolute deviation (MAD) [35; 36; 38]. The range of MAE is (0,+\(\infty\)); the smaller the MAE value is, the higher the accuracy of the prediction model. The advantage of MAE is that the unit of MAE is the same as original data, and it is easy to calculate and understand. The MAE is often used as a symmetrical loss function [36].  
Both MAE and RMSE express the average magnitude of prediction error with the units of the original data. In comparison with MAE, the RMSE has a relatively high weight for large errors, because the errors are squared before averaging. If prediction errors are normally distributed, the MAE and RMSE can be switched with each other with Equation (6), which is defined as [35; 36]  
\[\text{RMSE}=0.8\text{MAE} \tag{6}\]  
In summary, the scale-dependent metrics MSE, RMS, and MAE penalize errors according to their magnitude [35]. The disadvantage of these three metrics is that they are not unit-free, and it is difficult to compare predictions with different units. Moreover, MSE, RMSE and MAE interpret only the magnitude of error, but do not indicate the direction of error [38].