# A Method for Estimating the Resultant Expanded Uncertainty Value Based on Interval Arithmetic
## 1 Introduction
Unlike the standard uncertainty metric, it enables the assessment of the likelihood of obtaining an error signal value equal to or less than the assumed expanded uncertainty value, with the flexibility to freely set the confidence level within the range \(\gamma\in(0;100)\%\). Nevertheless, the challenge with this metric lies in the intricate calculations involved in determining its final value, especially when multiple error signals perturb the output quantity. If the uncertainty budget encompasses \(N\) error signals \(e_{0}(t)\), \(e_{1}(t)\), \(\ldots\), \(e_{N-1}(t)\), with variances \(\sigma_{0}^{2}\), \(\sigma_{1}^{2}\), \(\ldots\), \(\sigma_{N-1}^{2}\) correspondingly, the standard uncertainty of the signal \(e_{\Sigma}(t)=e_{0}(t)+e_{1}(t)+\ldots+e_{N-1}(t)\) is defined by the equation cited in [1]:  
\[\sigma_{\Sigma}=\sqrt{\left[ \begin{array}{c}\sigma_{0}\\ \sigma_{1}\\ \vdots\\ \sigma_{N-1}\end{array} \right]^{T}\left[ \begin{array}{cccc}1&r_{0,1}& \cdots&r_{0,N-1}\\ r_{1,0}&1&r_{1,N-1}\\ \vdots&&\ddots&\vdots\\ r_{N-1,0}&\cdots&\cdots&1\end{array} \right]\left[ \begin{array}{c}\sigma_{0} \\ \sigma_{1}\\ \vdots\\ \sigma_{N-1}\end{array} \right]}, \tag{5}\]  
where symbol \(r_{i,j}\) denotes subsequent Pearson correlation coefficients, determined on the basis of the equation:  
\[r_{i,j}=r_{j,i}=\frac{\sigma_{i,j}^{2}-\sigma_{l}^{2}-\sigma_{l}^{2}}{2\sigma _{l}\sigma_{j}}, \tag{6}\]  
wherein the mentioned signals may represent both sources of errors related to type A and type B uncertainties. When dealing with the expanded uncertainty metric, utilizing Equation (5) is unfeasible. Various methods in the literature offer ways to determine the resulting expanded uncertainty value, such as the distributions propagation method [3; 4], the expanded uncertainty combination rule method [5; 6], and the fuzzy sets-based method [7]. However, these approaches often rely on analytical techniques, necessitating intricate calculations involving convolution operations. In practical measurements, the Monte Carlo method is frequently employed [8; 9], or calculations are simplified by assuming compliance with the central limit theorem conditions, as suggested in the guide [1]. Nonetheless, each of these methods has its specific limitations, as elaborated upon later in the text. Considering the existing knowledge and challenges identified, the objective is to propose a method for determining the expanded uncertainty value with reduced computational complexity (in comparison to the Monte Carlo method) while maintaining an acceptable level of accuracy. The Monte Carlo method, renowned for its precision, serves as the benchmark method in this study. It is anticipated that the values obtained through the proposed method should deviate from the reference values by no more than \(\pm 5\%\). Additionally, the research will present results obtained through the classical approach [1], which assumes adherence to the central limit theorem conditions. Owing to the complexity associated with the other aforementioned methods, their results are not included in the analysis. The article is structured into six sections. Section 1 serves as an introduction, outlining the work's motivation, current knowledge, and key assumptions. Section 2 elaborates on the fundamental assumptions of the proposed method in a general context. Section 3 details the process of determining coherence coefficient values for both uncorrelated and correlated error signals. In Section 4, a simulation experiment is conducted to validate the method's efficacy, presenting the obtained results. Section 5 showcases practical examples applying the discussed method, including one that addresses an existing measurement setup detailed earlier in the work [10]. Section 6 encapsulates the key findings from the study, highlighting the most important advantages and disadvantages of the method. A crucial component of the project is the literature review, essential for consolidating information related to the analysis method under discussion.