# Robust Grey Relational Analysis-Based Accuracy Evaluation Method
## 2 Accuracy Assessment Based on GRA
### The Mean Square Deviation Distance-Accuracy Model  
Further, we need to construct a mean-variance distance-accuracy model to further map the gray correlation to accuracy via the mean-variance distance. The function for converting mean square deviation distance to accuracy can be selected based on specific circumstances. Generally, the mean square deviation distance \(d(\mathbf{X},\mathbf{Y})\) is transformed into a distance under a standard model \(d_{0}(\mathbf{X},\mathbf{Y})\) for calculation. The transformation relationship is as follows:  
\[d_{0}(\mathbf{X},\mathbf{Y})=(d_{\text{max0}}-d_{\text{min0}})\times\frac{d(\mathbf{X},\bm {Y})-d_{\text{min}}}{d_{\text{max}}-d_{\text{min}}}+d_{\text{min0}}\;if\;d(\bm {X},\mathbf{Y})\in[d_{\text{min}},d_{\text{max}}] \tag{4}\]  
In the equation, \(d_{\text{max0}}\) represents the maximum mean square deviation distance under the standard model. \(d_{\text{min0}}\) represents the minimum mean square deviation distance under the standard model. At the same time, the median of the mean square deviation distances under the standard model \(d_{mean0}\) is defined as follows:  
\[d_{mean0}=\frac{d_{\text{max0}}+d_{\text{min0}}}{2} \tag{5}\]  
The accuracy corresponding to this situation is \(C_{mean}=0.5\); meanwhile, the accuracy corresponding to the upper limit of the mean square deviation distance \(d_{\text{max}}\) is referred to as the lower limit of accuracy, denoted as \(C_{\text{min}}\); the accuracy corresponding to the lower limit of the mean square deviation distance \(d_{\text{min}}\) is referred to as the upper limit of accuracy, denoted as \(C_{\text{max}}\). The accuracy corresponding to the reference mean square deviation distance \(d_{ref}\) is referred to as the reference accuracy \(C_{ref}\), which serves as the passing line for simulation accuracy. In engineering, the reference accuracy is commonly set to \(C_{ref}=0.6\)[27; 28]. Furthermore, the mean square deviation distance-accuracy standard model is as follows:  
\[C(\mathbf{X},\mathbf{Y})=\frac{1}{2}\left[\frac{e^{-\alpha(d_{0}(\mathbf{X},\mathbf{Y})-d_{ \text{min0}})}-e^{\alpha(d_{0}(\mathbf{X},\mathbf{Y})-d_{\text{mean0}})}}{e^{-\alpha (d_{0}(\mathbf{X},\mathbf{Y})-d_{\text{mean0}})}+e^{\alpha(d_{0}(\mathbf{X},\mathbf{Y})-d_{ \text{mean0}})}+1}\right]\;if\;d_{0}(\mathbf{X},\mathbf{Y})\in[d_{\text{min0}},d_{ \text{max0}}] \tag{6}\]  
here, \(C(\mathbf{X},\mathbf{Y})\) is the accuracy between \(\mathbf{X}\) and \(\mathbf{Y}\); \(d_{0}(\mathbf{X},\mathbf{Y})\) is the distance under a standard model; \(d_{mean0}\) is the mean square deviation distances under the standard model; \(\alpha\) is the shape factor of the curve, and the shape of the curve can be altered by changing \(\alpha\). According to the study in reference [11], the shape factor generally exhibits better versatility when set to 0.5. The standard model curve is illustrated in Figure 1, which shows the mean square deviation distance-accuracy model. By utilizing the aforementioned two models, the grey relational degree can be converted into mean square deviation distance, which is then further converted into accuracy, thereby completing the accuracy evaluation of dynamic time series in simulation models based on time-domain analysis [13]. In the study presented in reference [6], it was proposed that one reason why the grey relational degree cannot determine the geometric similarity of time series data is due to its inability to ensure order preservation during calculations. This means that it cannot distinguish between positive and negative deviations between simulation and experimental data, thus making it difficult to assess the consistency of geometric shapes across continuous time-varying data. When applying the grey relational analysis method to evaluate model accuracy, potential disturbance factors must be considered.