# Robust Grey Relational Analysis-Based Accuracy Evaluation Method
## 3 Robust Grey Relational Analysis-Based Accuracy Evaluation Method
### Expected Valve Crossing Rate  
The expected through-the-valve rate was originally proposed by [PERSON] in 1944 as a secondary statistic for describing stochastic processes and their envelopes, and a mathematically more perfect derivation was subsequently given by [PERSON] [29] in 1960. In this paper, the expected through-the-valve rate is employed to determine the frequency at which the perturbed sequence's spatial curve intersects the reference sequence as well as the through-the-valve point neighborhood spatial position ranges, which provide positional references for the subsequent construction of interpolation points. Let \(Y(t)\) be a random sequence of consecutive states at consecutive times and at least once squarely differentiable; then, the number of times the sequence \(Y(t)\) traverses the valve \(a\) in the time \([t_{1},t_{2}]\) is said to be the number of times it penetrates the valve, and its expectation can be constructed with the help of the unit step function \(u\) as a 0-1 process function \(Z(t)\):  
\[Z(t)=u[Y(t)-a] \tag{7}\]  
here, \(u\) is the unit step function, and \(a\) is the valve value. The formal derivative of \(Z(t)\) is as follows:  
\[\ddot{Z}(t)=\dot{Y}(t)\delta[Y(t)-a] \tag{8}\]  
here, \(\delta\) is the Dirac function. Then, the number of times \(N_{a}(t_{1},t_{2})\) the random sequence \(Y(t)\) traverses the valve a in the time interval \([t_{1},t_{2}]\) is as follows:  
\[N_{a}(t_{1},t_{2})=\int_{t_{1}}^{t_{2}}\Big{|}\dot{Y}(t)\Big{|}\delta[Y(t)-a]dt \tag{9}\]  
The expected number of valve penetrations is as follows:  
\[ \begin{array}{ll}E[N_{a}(t_{1},t_{2})]&=\int_{t_{1}}^{t_{2}}E\Big{[}\dot{Y}( t)\Big{|}\delta[Y(t)-a]\Big{]}dt\\ &=\int_{t_{1}}^{t_{2}}\int_{-\infty}^{\tau_{\infty}}\int_{-\infty}^{+\infty}|j |\delta(y-1)p(y,\dot{y},t)dyd\dot{y}dt\\ &=\int_{t_{1}}^{t_{2}}\int_{-\infty}^{+\infty}|j|p(a,\dot{y},t)dydt\end{array}  \tag{10}\]  
The number of desired valve penetrations per unit time, i.e., the desired valve penetration rate, is as follows:  
\[v_{a}=\int_{-\infty}^{+\infty}|j|\ p(a,\dot{y},t)d\dot{y} \tag{11}\]  
The higher the expected through the valve rate, proving that the perturbed sequence curve traverses the reference sequence curve more times per unit of time, the smaller the distance between the relative positions of the perturbed sequence and the reference sequence within the range of a certain perturbation threshold through the valve point. The grey correlation should be given to positive gain, such as in the reference example in Figure 2, where the \(X_{1}\) and \(X_{2}\) curves are in the range of the definition, \(v_{a}(X_{1})<v_{a}(X_{2})\); however, the results of the calculation of the grey correlation of the data points 1 and 2 are \(\gamma(X_{0},X_{1})=\gamma(X_{0},X_{2})\). Therefore, it is possible to indirectly increase the weight of the influence of distance on the gray correlation by increasing the number of computed data points in the neighborhood of the penetrating valve point. In order to ensure that the spatial shape of the curve composed of the data sequence does not change, as well as to avoid the impact of the size of the local extreme difference in the neighborhood of the penetrating valve point on the distribution of the inserted data points, the creation of the data points is carried out using the unilinear interpolation, bilinear interpolation, or trilinear interpolation according to the different dimensions of the data. The arithmetic example shown in Figure 3 is validated using the Gramacy and Lee Function [30] as the basis function, as follows:  
\[f(x)=\frac{\sin(10\pi x)}{2x}+(x-1)^{4} \tag{12}\]  
here, \(f\) denote the Gramacy and Lee Function, and \(x\) is the independent variable, which is usually evaluated on the domain \([0.5,2.5]\). On the basis of this function, to add a random perturbation to construct the perturbed sequence data, sequence one is the result of a random perturbation under positively proportional weighting coefficients.