# A Method for Estimating the Resultant Expanded Uncertainty Value Based on Interval Arithmetic
## 3 Coherence Factors Values Estimation
### Case of Uncorrelated Error Signals  
For uncorrelated error signals, the methodology presented in [18] offers a viable solution. This approach involves initially determining the shape factor value \(s_{a,b}\) for a signal pair \(e_{a}(t)\) and \(e_{b}(t)\) with specified parameters. Subsequently, this factor is adjusted based on the actual parameters of the studied signals. This technique found practical application in the study by [19], with the correction initially proposed in [18]. The correction primarily accounted for the assumptions of the central limit theorem in computing the resultant expanded uncertainty value. Further refinements, including adjustments to address discrepancies in the expanded uncertainty values of the examined error signals, are elaborated upon in detail. Assuming two uncorrelated error signals \(e_{a}(t)\) and \(e_{b}(t)\), whose expanded uncertainty values are \(U=U_{a}=U_{b}\), the value of the shape factor \(s_{a,b}\) can be determined for the pairs of signals discussed in the form:  
\[s_{a,b}=s_{b,a}=\frac{U_{a,b}^{2}-U_{a}^{2}-U_{b}^{2}}{2U_{a}U_{b}}=\frac{U_{ a,b}^{2}}{2U^{2}}-1, \tag{10}\]  
where the expanded uncertainty values for both signals need to be determined at the same confidence level \(\gamma\). It is worth mentioning that with a sufficiently large dataset (approximately 100,000 implementation values) for the signals in question, the coefficient's value can be estimated using the Monte Carlo method. In cases where equal expanded uncertainty values are assumed (\(U_{a}=U_{b}\)), the coefficient's value is influenced solely by the probability density function shape of the signals under analysis. Alternatively, an approach to determining the shape factor values could involve assuming identical standard uncertainties (\(\sigma=\sigma_{a}=\sigma_{b}\)). Under this assumption, Equation (10) is expressed as:  
\[s_{a,b}=s_{b,a}=\frac{U_{a,b}^{2}-c_{a}^{2}\sigma_{a}^{2}-c_{b}^{2}\sigma_{b} ^{2}}{2c_{a}\sigma_{a}c_{b}\sigma_{b}}=\frac{U_{a,b}^{2}}{2c_{a}c_{b}\sigma^{2 }}-1, \tag{11}\]  
However, the values obtained in this way may differ from those obtained by adopting the assumption described for Equation (10). The values of shape factors should be deter mined for pairs of all unique shapes of error signal distributions occurring in the analyzed measurement chain. Table 1 presents the acquired shape factor values for typical probability density distribution functions, considering the specified confidence level and the method employed for determining these values. Upon examining the data within the table, it becomes evident that the shape factor values escalate with the rise in confidence level. Additionally, there is a trend where values calculated using Equation (10) are generally higher than those derived from Equation (11). The impact of the method used to ascertain shape factor values on the efficacy of the proposed approach for determining the resultant expanded uncertainty value is explored in subsequent sections of the study. Upon analyzing Equations (9)-(11), it is observed that the shape coefficients determined through the chosen method align with the coherence coefficients for signals sharing identical parameters to those used in the shape factor calculations. Any alterations in signal parameters or their quantity lead to variations in coherence coefficient values. Hence, a methodology for estimating coherence coefficients based on shape coefficient values and the current signal parameters needs to be proposed. The initial aspect to address pertains to the central limit theorem. As the number of summed independent random variables increases, the probability density distribution of their sum tends towards a normal distribution [1]. Referencing the outcomes in Table 1, it is apparent that the closer the distributions of a pair of summed signals resemble the normal distribution, the closer the shape coefficient value approaches zero. Consequently, as the quantity of summed signals rises, the subsequent coherence coefficients utilized in Equation (7) tend towards zero.