# A Method for Estimating the Resultant Expanded Uncertainty Value Based on Interval Arithmetic
## 2 Reductive Interval Arithmetic Method  
The reductive interval arithmetic method, rooted in interval arithmetic originally attributed to [PERSON] [11], has been a subject of study since 1958. Concurrently, [PERSON] [12], delved into this area since 1956. It was [PERSON] who initially demonstrated the practical application of this method. Interval arithmetic facilitates the examination of interval interactions, enabling the treatment of the expanded uncertainty measure as an interval. Consequently, it can be leveraged to compute the cumulative sum of successive expanded uncertainties.  
When adhering to the classical definition of expanded uncertainty, as outlined in Equations (2) and (4), this measure can be represented as an unbiased interval with a radius equivalent to its value. Notably, in the realm of expanded uncertainty measures, the summation of two such intervals does not align with the sum derived from classical interval arithmetic, typically resulting in a smaller real value. [PERSON], proposed a modified interval arithmetic approach, termed \"reductive interval arithmetic\", denoting the capability to narrow down the resultant interval [13; 14; 15; 16].  
Considering an error signal \(e_{\Sigma}(t)=e_{0}(t)+e_{1}(t)+\ldots+e_{N-1}(t)\) comprising \(N\) partial error signals, each associated with expanded uncertainties determined at the same confidence level \(\gamma=1-\alpha\) as \(U_{0}\), \(U_{1}\), \(\ldots\), \(U_{N-1}\), respectively, the calculation of the expanded uncertainty using the reduced interval arithmetic method is expressed in the equation cited in [13]:  
\[U_{\Sigma}=\sqrt{\left[\begin{array}{c}U_{0}\\ U_{1}\\ \vdots\\ U_{N-1}\end{array}\right]^{T}\left[\begin{array}{cccc}1&h_{0,1}&\cdots&h_{0,N-1}\\ h_{1,0}&1&h_{1,N-1}\\ \vdots&&\ddots&\vdots\\ h_{N-1,0}&\cdots&\cdots&1\end{array}\right]}\left[\begin{array}{c}U_{0}\\ U_{1}\\ \vdots\\ U_{N-1}\end{array}\right]}, \tag{7}\]  
where the symbol \(h_{i,j}=h_{j,i}\) signifies the coherence coefficient value for the \(i\)-th and \(j\)-th error signal pair. These coefficients reflect the correlation between the analyzed error signals and the interdependency of their distribution shapes with the resulting error signal implementation's distribution shape. Establishing these coefficient values is a pivotal undertaking in the application of the method under discussion, detailed further in the subsequent sections of the study.  
The discussed method was proposed and used by its author to analyze the uncertainty budget of measurement chains processing series of measurement data. A feature of the indicated measurement chains is the fact that they process multiple successive implementations of error signals originating from the same source, and, therefore, have identical parameters and properties. Unfortunately, due to further research focused on other areas, subsequent works describing the indicated method have not been published, similarly to its further development. Additionally, in the English-language literature there are very few works available on the method of reductive interval arithmetic. One of its applications can be found in the work [17], however, most of the works devoted to the discussed method were published in Polish. Due to the unique features of the indicated method, the authors of this work decided to continue the research conducted by [PERSON], additionally modifying the original algorithm of its application, thus expanding the area of application of the method.