# A Method for Estimating the Resultant Expanded Uncertainty Value Based on Interval Arithmetic
## 3 Coherence Factors Values Estimation
### Case of Correlated Error Signals  
Dealing with correlated error signals presents a significantly more intricate scenario. In such instances, the coherence coefficient value is a product of the convolution of the distributions' shapes and their correlation coefficient. The existing literature [13; 15] delves into analytical techniques for determining coherence coefficient values in correlated scenarios. Nonetheless, these methods pertain to specific signal parameters, necessitating a strategy for estimating these coefficients akin to the approach taken for uncorrelated signals. However, before delving into estimation methods, it is imperative to establish a clear understanding of how a pair of correlated error signals is defined. In experimental settings, a correlated signal pair \(e_{a}(t)\) and \(e_{b}(t)\) is characterized by:  
\[e_{b}(t)= \begin{cases}r_{a,b}e_{a}(t)+\sqrt{1-r_{a,b}^{2}}e_{c}(t)&\text{where $r_{a,b}>0$}\\ r_{a,b}e_{a}(t)-\sqrt{1-r_{a,b}^{2}}e_{c}(t)&\text{where $r_{a,b}<0$}\end{cases} , \tag{16}\]  
where the signal \(e_{c}(t)\) is a signal with a given distribution, uncorrelated with the signal \(e_{a}(t)\), and \(r_{a,b}=r_{b,a}\) is the Pearson correlation coefficient of signals \(e_{a}(t)\) and \(e_{b}(t)\), determined according to the Equation (6). It is crucial to acknowledge that the definition outlined in Equation (16) may not precisely mirror the relationships within the analyzed measurement setup. In such instances, the definition's formulation should be adjusted to align with the specific case under investigation. Subsequently, the dependencies discussed in the subsequent subsection should be determined to adhere to the established methodology. For a signal pair governed by the relationship in Equation (16), the coherence coefficient value, contingent on the expanded uncertainty values and correlation coefficient, can be computed using Equation (9). Figure 3 illustrates the coherence coefficient values' dependency for a signal pair with defined probability density function shapes, assuming equivalent standard uncertainties across the signals. Figure 4 showcases the coherence coefficient values' correlation with varying expanded uncertainty values and correlation coefficients for signal pairs sharing a similar distribution type. In this analysis, the distribution of \(e_{a}(t)\) aligns with the first type, while the distribution of \(e_{c}(t)\), as previously outlined in Equation (16), corresponds to the second type. Upon scrutinizing the outcomes of the conducted experiments, it is evident that the coherence coefficient value for a correlated signal pair is contingent on both the standard uncertainty ratio of these signals and the Pearson correlation coefficient. Establishing a relationship akin to that described in Equation (14), allowing for a reasonably accurate estimation of the coherence coefficient utilizing the shape coefficients determined from Equation (11), proves to be exceedingly challenging in this scenario. The discussed relationships are solely explored for signal pairs; complexity escalates with an increasing number of error signals. It can be noticed that in the case of the results presented in Figure 3, one can draw conclusions similar to those discussed earlier for the results presented in Figure 2. Regardless of the adopted confidence level, the value of the coherence coefficient is directly proportional to the value of the Pearson correlation coefficient of the analyzed signals. The main reason for the non-linearity of the discussed characteristics is the fact that in the case of no correlation (for \(r_{a,b}=0\)), the value of the coherence coefficient is nonzero for signal pairs whose shape of the resultant distribution of implementation values deviates from the shape of the normal distribution. As depicted in Figure 3, the relationships become notably more non-linear with higher confidence levels.