## 3 Proposed Metrics of Protocol Compliance
### Compliance per Patient  
In [13], all the memberships of all the activities of all patients are averaged, ignoring the information about which activities were performed for which patient. Moreover, averaging may not necessarily be the best aggregating operator. In some cases, one may suggest that the protocol is compliant with the degree to which the least compliant activity is compliant or most of the activities are compliant. Therefore, there can be a need to first estimate the compliance level per patient and, in the second step, aggregate those compliance levels. We suggest using the ordered weighted averaging (OWA) operator [66; 67] for this purpose. For completeness, let us first repeat the definition of an OWA operator. An ordered weighted averaging (OWA) operator of dimension \(n\) is a mapping  
\[F:[0,1]^{n}\longrightarrow[0,1] \tag{2}\]  
if associated with \(F\) is a weighting vector  
\[W=[w_{1},\ldots,w_{n}]^{T} \tag{3}\]  
such that:  
\( \begin{array}{ll}1.&w_{i}\in[0,1]\), for all \(i=1,\ldots,n\), \\ 2.&\sum_{i=1}^{n}w_{i}=1\), and \\ \end{array} \)  
\[F(a_{1},\ldots,a_{n})=W^{T}B=\sum_{j=1}^{n}w_{j}b_{j} \tag{4}\]  
where \(b_{j}\) is the \(j\)-th largest element in the set \(\{a_{1},\ldots,a_{n}\}\), and \(B=[b_{1},\ldots,b_{n}]\). \(B\) is called an ordered argument vector if for each \(b_{i}\in[0,1]\), \(j>i\) implies \(b_{i}\geq b_{j}\), \(i=1,\ldots,n\). An equivalent notation is one with increasing sorting. The minimum can correspond to the universal quantifier _for all_, while the maximum can correspond to the existential quantifier _for at least one_. Moreover, by an appropriate choice of the weighting vector \(W\), between \(W=[1,0,\ldots,0]\), as in the maximum type aggregation, and \(W=[0,\ldots,0,1]\), as in the minimum type aggregation, we can obtain an aggregation operator corresponding to \"intermediate\" _linguistic quantifiers_, e.g., _at least a half, most_, _almost all_, etc. For regular non-decreasing monotone quantifiers, [PERSON] [66] generates the weighting vector \(W=[w_{1},\ldots,w_{n}]^{T}\) as:  
\[w_{i}=\mu_{Q}\Big{(}\frac{i}{n}\Big{)}-\mu_{Q}\Big{(}\frac{(i-1)}{n}\Big{)}, \hskip 28.452756 pti=1,\ldots,n \tag{5}\]and since, by definition, \(\mu_{Q}(0)=0\) and \(\mu_{Q}(1)=1\), then \(w_{1}+\cdots+w_{n}=1\). Therefore, an OWA operator may be viewed to provide, in general, a linguistic quantifier driven aggregation [67]. Please note that an averaging can also be performed with an OWA operator with the weighting vector \(W=[\frac{1}{n},\ldots,\frac{1}{n}]\). In our context, first we calculate the compliance level for each trace (patient) \(t_{j},j=1,\ldots,m\) as  
\[c_{OWA}(t_{j})=F(\mu_{p}(a_{j1}),\ldots,\mu_{p}(a_{i_{j^{\prime}}}))=\sum_{k=1 }^{r_{j}}w_{k}b_{k} \tag{6}\]  
where \(r_{j}\) is the number of activities in the trace \(t_{j}\), and \(b_{k}\) is the \(k\)-th largest element in the set \(\{\mu_{p}(a_{j1}),\ldots,\mu_{p}(a_{i_{j^{\prime}}})\}\), i.e., set of compliance degrees of individual activities in trace \(t_{j}\). \(w_{k}\) are the weights calculated according to (5). The quantifier can be defined linguistically, such as _for all, most_, _almost all_, depending on the user's preference. This quantifier enables setting a fuzzy threshold to what percentage of patient population is expected to follow the protocol and to what extent. For instance, in cancer treatment protocols, around 80% of patients follow the protocols. Next, we aggregate those compliance levels of the patients. For this purpose, we can also use an OWA operator. \[C_{OWA}(P)=F(c_{OWA}(t_{1}),\ldots,c_{OWA}(t_{m})) \tag{7}\]  
Again, the quantifier can be defined linguistically, such as _for all, most_, _almost all_, depending on the user's preference, or one can use an average. In such case, we can interpret the obtained value. For instance, if we use a linguistic quantifier _most_ to calculate the weighting vectors, the obtained value can be interpreted as the degree to which most patients have most of the activities compliant with the protocol.