## 3 Proposed Metrics of Protocol Compliance
### Sugeno Integral-Based Compliance Measure  
The difficulty with the OWA-based method is the weighting vector. Even though the weights can be calculated from a linguistic quantifier, several possibilities may exist. This can be considered a drawback of those methods. In addition, the obtained numbers might be relatively difficult to interpret (e.g., low values in these cases can be considered less informative.) Therefore, aggregation using a Sugeno integral offers an interesting solution. First, let us reiterate the definitions of fuzzy measure and Sugeno integral. Let \(X=\{x_{1},x_{2},x_{3},\ldots,x_{n}\}\) be a finite set. A fuzzy measure [(69; 70)] on \(X\) is a function \(g:2^{X}\rightarrow[0,1]\) such that:  
* \(g(\varnothing)=0\)
* \(g(X)=1\)
* If \(A\subseteq B\) then \(g(A)\leq g(B)\)\(\forall A,B\in 2^{X}\)  
Moreover, there is a special case of fuzzy measure called a \(\lambda\)-fuzzy measure or Sugeno \(\lambda\)-fuzzy measure [(71)]. The \(\lambda\)-fuzzy measure is a fuzzy measure \(g\) that satisfies the \(\lambda\)-rule, i.e., there exists \(\lambda\in(-1,\infty)\) such that \(g(E\cup F)=g(E)+g(F)+\lambda g(E)\cdot g(F)\) whenever \(E\cap F\
eq\varnothing\). When \(g(X)=\sum_{i=1}^{n}g(\{x_{i}\})\), the parameter \(\lambda=0\); we then obtain probability measure. The values \(\{g(\{x_{i}\})|i=1,2,\ldots,n\}\) are called the fuzzy densities. Knowing the densities, parameter \(\lambda\) can be uniquely determined by the equation [(71)]:\[g(X)=\frac{1}{\lambda}\left[\prod_{i=1}^{n}(1+\lambda\cdot g(\{x_{i}\}))-1\right] \tag{11}\]  
This equation comes directly from the \(\lambda\)-rule and the fact that \(X=\bigcup_{i=1}^{n}\{x_{i}\}\). In our work, we use \(\lambda\)-fuzzy measures. Let \(g\) be a fuzzy measure and \(h\) be a function \(h:X\rightarrow[0,1]\). Moreover, assume that \(\{x_{i}\}\) are ordered so that \(h(x_{1})\geq h(x_{2})\geq\ldots\geq h(x_{n})\). Then, the discrete Sugeno integral [71] of a function \(h\) with respect to \(g\) is a function \(S_{g}:[0,1]^{n}\rightarrow[0,1]\) such that  
\[S_{g}(h)=\max_{i=1,\ldots,n}\left[\min(h(x_{i}),g(A_{i}))\right] \tag{12}\]  
where \(A_{i}=\{x_{1},x_{2},...,x_{i}\}\). In our context of protocol compliance, the Sugeno integral can have a very interesting interpretation. Let us consider a single protocol execution. The value of k% for the Sugeno integral would mean that at least k% of activities are compliant with at least k% degree. On the protocol level, we could say that at least k% traces are compliant with the k% degree. As discussed previously, we first calculate the compliance level for a trace \(t_{j}\), \(j=1,\ldots,m\). Trace \(t_{j}\) had \(r_{j}\) activities \(a_{kj}\), \(k=1,\ldots,r_{j}\). In our context, set \(X\) is the set of activities performed in the trace \(t_{j}\), so \(X=\{a_{1j},\ldots a_{r_{j}}\}\). We consider two possibilities for the densities. In the first case, all activities are equally important and a density of an activity \(a_{kj}\) is  
\[g_{1}(a_{kj})=\frac{1}{r_{j}} \tag{13}\]  
In this case, the densities add up to one, hence \(\lambda=0\), and we use a probability measure. In the other case, we incorporate the importance \(u_{kj}\) of the activity \(a_{kj}\). We assume that the importance is a number from interval \([0,1]\), where \(1\) denotes the highest and zero denotes the lowest level of importance/criticality. Here, the density of an activity \(a_{kj}\) is  
\[g_{1}(a_{kj})=\frac{u_{kj}}{r_{j}} \tag{14}\]  
In this case, several densities will be less than \(\frac{1}{r_{j}}\), and therefore, the sum will be less than \(1\), producing a \(\lambda\) that is bigger than \(0\). Intuitively, activities that are more important have a larger measure. They also have more influence on the compliance level. Certainly, other choices of densities can be followed. We define the compliance for a trace as the Sugeno integral of the function \(f(x)=x\) with respect to measure \(g_{1}\).