# A Method for Estimating the Resultant Expanded Uncertainty Value Based on Interval Arithmetic
## 3 Coherence Factors Values Estimation
### Case of Uncorrelated Error Signals
In the work [18], it was proposed to correct the shape coefficients according to the equation:  
\[k_{i,j}=k_{j,i}=\frac{U_{i}^{2}+U_{j}^{2}}{\sum_{k=0}^{N-1}U_{k}^{2}}, \tag{12}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \(\gamma\) & **Method** & \(s_{a,b}\) & \(n\) & \(u\) & \(t\) & \(d\) \\ \hline \multirow{6}{*}{75\%} & \(U_{a}=U_{b}\) (10) & \(n\) & \(0.00000\) & \(-0.07510\) & \(-0.03966\) & \(-0.07214\) \\  & \(u\) & \(-0.07510\) & \(-0.11115\) & \(-0.11109\) & \(-0.08076\) \\  & \(t\) & \(-0.03966\) & \(-0.11109\) & \(-0.07710\) & \(-0.10742\) \\  & \(d\) & \(-0.07214\) & \(-0.08076\) & \(-0.10742\) & \(-0.02616\) \\ \cline{2-6}  & \(n\) & \(0.00000\) & \(-0.07926\) & \(-0.04082\) & \(-0.07559\) \\  & \(\sigma_{a}=\sigma_{b}\) (11) & \(u\) & \(-0.07926\) & \(-0.11098\) & \(-0.11099\) & \(-0.08057\) \\  & \(t\) & \(-0.04082\) & \(-0.11099\) & \(-0.07709\) & \(-0.10640\) \\  & \(d\) & \(-0.07559\) & \(-0.08057\) & \(-0.10640\) & \(-0.02628\) \\ \hline \multirow{6}{*}{85\%} & \(U_{a}=U_{b}\) (10) & \(n\) & \(0.00000\) & \(-0.00828\) & \(-0.02985\) & \(0.06124\) \\  & \(u\) & \(-0.00828\) & \(0.03912\) & \(-0.02600\) & \(0.14582\) \\  & \(t\) & \(-0.02985\) & \(-0.02600\) & \(-0.05733\) & \(0.05345\) \\  & \(d\) & \(0.06124\) & \(0.14582\) & \(0.05345\) & \(0.28045\) \\ \cline{2-6}  & \(n\) & \(0.00000\) & \(-0.00835\) & \(-0.03049\) & \(0.05931\) \\  & \(\sigma_{a}=\sigma_{b}\) (11) & \(u\) & \(-0.00835\) & \(0.03909\) & \(-0.02619\) & \(0.14242\) \\  & \(t\) & \(-0.03049\) & \(-0.02619\) & \(-0.05728\) & \(0.04806\) \\  &while the application of the described correction allows for obtaining results that are overestimated by approximately 5% in relation to those obtained using the Monte Carlo method (only when assuming equal expanded uncertainty values for analyzed error signals). The second significant phenomenon concerns the impact of variations in the expanded uncertainty values associated with the analyzed signal pair, for which the coherence coefficient is defined in Equation (9). To explore this effect, an experiment was conducted using the Monte Carlo method, determining the coherence coefficient for a signal pair with distinct expanded uncertainty values \(U_{a}\) and \(U_{b}\) in accordance with Equation (9), where the ratio \(\frac{U_{a}}{U_{b}}\) ranged from 1 to 10. The experimental findings for specific confidence levels are depicted in Figure 2.  
Generally, the absolute value of the coherence coefficient diminishes as the disparity between the expanded uncertainty values of the signal pair increases. However, this relationship is non-linear, as in some instances, for lower expanded uncertainty ratios, the coherence coefficient values increase before declining with larger disparities. Observing the results presented in Figure 2, it can be noticed that in the case when the summed signals are characterized by distributions close to normal, the dependence of the coherence coefficient value as a function of the ratio of the expanded uncertainty values associated with these signals becomes monotonic and begins to approach a linear function. The discussed dependence becomes strongly non-linear and devoid of monotonicity in the case of distributions that are characterized by a large width (in particular in the case of the distribution in the shape of the letter \"u\" [2]). The indicated circumstances mean that the exact value of the correction taking into account the occurring phenomena should be determined separately for each pair of signals with the given distributions; however, a universal approach is proposed in the paper, discussed later in the section.