# Impacts of Flight Operations on the Risk of Runway Excursions
## 5 Calculation and Analysis of Curve Similarity
### Curve Similarity Calculation  
In order to conduct a comprehensive study of control column operational curves, this paper adopts curve similarity as the basis, taking into account specific problems and data characteristics. It utilizes statistical indicators and measurement methods that are relevant to operational curves to effectively compare and evaluate the degree of similarity among the curves. The similarity of curves involves comparing sample curves with standard curves to analyze flight operations from multiple perspectives. This approach does not diminish the individual characteristics of operations and allows for the measurement of landing operations based on overall trends, translation changes, and scaling changes. As the lengths of the operational curves vary, interpolation is utilized to align the curve lengths. Adjusting the length of a curve using interpolation is a mathematical method used to change the length of a curve by adding or subtracting data points to a given curve. When calculating curve similarity, the code is written to set the standard curve under normal conditions as the regular curve. When wind speed exceeds 17 knots, equivalent to Beaufort scale 5, the wind-influenced curve is used as the standard curve. _Trend_: Utilizing the polyfit function in MATLAB R2018a, a least squares linear fit is applied to the standard curve and the operational curve, yielding a slope defined as trend. A trend value closer to 1 indicates that the operational curve closely aligns with the standard curve. The formula is expressed as follows:  
\[Trend=\frac{\sum X_{i}Y_{i}-\overline{X_{\text{S}}Y_{\text{S}}}}{\Sigma{X_{i} }^{2}-n_{Y_{\text{S}}}^{-2}}, \tag{1}\]  
_Vertical scaling_: This formula represents the proportional relationship between the operational curve's magnitude and the standard curve. A value greater than 1 indicates that the overall operational magnitude is less than that of the standard curve, while a value closer to 1 suggests a similarity in operational magnitude. The formula is expressed as follows:  
\[Vertical\ scaling=\frac{Y_{s}max-Y_{s}min}{Ymax-Ymin}, \tag{2}\]  
Figure 4: Selection of standard curves. _Horizon scaling_: This metric reflects the proportional relationship between the duration of the operational curve and the standard curve. The formula is expressed as follows:  
\[Horizon\ scaling=\frac{X_{s}max-X_{s}min}{Xmax-Xmin}, \tag{3}\]  
_R squared_: This metric evaluates the degree of fit between the predicted curve and the standard curve. SSR is the sum of squared residuals, and SST is the total sum of squares. The formula is expressed as follows:  
\[\text{SSR}=\sum_{i=1}^{n}(Y_{P}-Y_{m})^{2}, \tag{4}\]  
\[\text{SST}=\sum_{i=1}^{n}(Y_{s}-Y_{m})^{2}, \tag{5}\]  
\[R\ squared=\frac{SSQ}{SST}, \tag{6}\]  
Mean squared error: It calculates the average of the squared errors between the predicted curve and the actual standard curve, providing an indication of the disparity between the predicted curve and the standard curve. The formula is expressed as follows:  
\[\text{MSE}=\frac{\left(Y_{P}-Y_{S}\right)^{2}}{n}, \tag{7}\]  
_Shift_: Shift refers to the average absolute difference between the mean of the standard curve and the mean of the operational curve. The formula is as follows:  
\[Shift=\frac{\left|Y_{s}-Y_{m}\right|}{n}, \tag{8}\]  
_Curve correlation_: This metric calculates the degree of match between the operational curve and the standard curve using the Pearson correlation coefficient. The formula is as follows:  
\[Curve\ correlation=\frac{\sum_{i=1}^{n}\left(X_{i}-\overline{X_{S}}\right) \left(Y_{i}-\overline{Y_{s}}\right)}{\sqrt{\sum_{i=1}^{n}\left(X_{i}- \overline{X_{S}}\right)^{2}}}\sqrt{\sum_{i=1}^{n}\left(Y_{i}-\overline{Y_{S} }\right)^{2}}, \tag{9}\]  
where \(X\) and \(Y\) represent the abscissa and ordinate of the operational curve, _X_s and _Y_s denote the abscissa and ordinate of the standard curve, _X_m and _Y_m stand for the mean ordinate of the standard curve, and _X_p and _Y_p correspond to the abscissa and ordinate of the predicted curve.