# CvAMoS--Event Abstraction Using Contextual Information
## 3 Background  
In this section, we provide the basic notions that will be used for the rest of the paper. Please note that all definitions presented in this section are already utilized in the process mining community and we use them according to standard literature.  
**Definition 1** (Event).: _Let us call the set of case identifiers \(\mathcal{C}\), let \(\mathcal{A}\) be the set of possible activity names, and let \(\mathcal{X}=\mathcal{N}\times\mathcal{V}\) be the set of extra attributes, each of which comprises a name (\(\mathcal{N}\) is the set of all possible names) and a value (\(\mathcal{V}\) is the set of all possible values). The set of all possible events, also known as event universe, is \(\mathcal{E}=\mathcal{C}\times\mathcal{A}\times\mathcal{P}(\mathcal{X})\) (Here, \(\mathcal{P}\) indicates the powerset, i.e., the set of all sub-sets). An event\(e=(c,a,X)\in\mathcal{E}\) is an element from the event universe._  
In the context of this paper, the set \(\mathcal{A}\) contains activity names at the finest granularity level, e.g., the sensors being triggered. The set of all possible sequences of events is \(\mathcal{E}*\). When multiple events share the same case identifier, we group them together into a sequence: \(s=\langle(c_{1},a_{1},X_{1}),(c_{2},a_{2},X_{2}),\dots\rangle\in\mathcal{E}*\) (with \(c_{1}=c_{2}=\dots\)).  
**Definition 2** (Trace).: _For each sequence of events sharing the same case ID, we consider the activities these events refer to and we call this trace. For example, given \(s=\langle(c_{1},a_{1},X_{1}),(c_{2},a_{2},X_{2}),\dots\rangle\in\mathcal{E}*\), such that all events in \(s\) have the same case ID (i.e., \(c_{1}=c_{2}=\dots\)), the trace is \(\langle(a_{1},X_{1}),(a_{2},X_{2}),\dots\rangle\)._  
We refer to \(|t|\) as the length of trace \(t\). To access the \(i\)-th element of trace \(t\), we use the notation \(t(i)\in\mathcal{A}\times\mathcal{P}(\mathcal{X})\). Given two traces \(t_{1}=\langle(a_{1}^{1},X_{1}^{1}),(a_{2}^{1},X_{2}^{1}),\dots,(a_{n}^{1},X_{ n}^{1})\rangle\) and \(t_{2}=\langle(a_{1}^{2},X_{1}^{2}),(a_{2}^{2},X_{2}^{2}),\dots,(a_{m}^{2},X_{ n}^{2})\rangle\), it is possible to concatenate them such that \(t_{1}\cdot t_{2}=\langle(a_{1}^{1},X_{1}^{1}),\dots,(a_{n}^{1},X_{n}^{1}),(a_{ 1}^{2},X_{1}^{2}),\dots,(a_{m}^{2},X_{m}^{2})\rangle\).  
**Definition 3** (Sub-trace operator).: _Given a trace \(t=\langle(a_{1},X_{1}),\dots,(a_{n},X_{n})\rangle\), we can use the sub-trace operator \(t[i,k]\) to extract the sequence of events in the trace such that \(t[i,k]=\langle(a_{i},X_{1}),(a_{i+1},X_{i+1}),\dots,(a_{i+k},X_{i+k})\rangle\) where \(1\leq i<n\) and \(0<k\leq n-i\). Please note that the result of the application of the sub-trace operator is a trace._  
**Definition 4** (Event-log).: _An event log\(E_{L}:\mathcal{A}*\rightarrow\mathbb{N}\) is a multi-set of traces where, for each trace, we can know how often it is actually observed in the data._To know the multiplicity of a trace \(t\) in the event log \(E_{L}\), we use the notation \(E_{L}(t)\). The notation \(t\in E_{L}\) indicates that trace \(t\) appears at least once in \(E_{L}\) (i.e., \(E_{L}(t)>0\)). With \(|E_{L}|\), we indicate the number of traces in the log, including their multiplicity.