## 3 Shift Time Clustering on Event Logs
Here, \(Dm_{p}\) and \(X_{j,p}\) are the distance function normalized in Equation (2) and the input vector \(x_{j}\) for a specific attribute \(p\) based on the \(p\)-dimensional attribute vector of data, respectively. \(W_{i,p(s)}\) is the weight vector \(w_{i}\) for a specific attribute \(p\) based on the \(p\)-dimensional attribute vector of data in a specific iteration \(s\). \[||x_{j}-w_{i}(s)||=\sqrt{\sum_{p=1}^{dmTm}Dn_{p}(i,j)^{2}+\sum_{p=dmTm+1}^{dm }(x_{j,p}-w_{i,p}(s))^{2}}\] (3)
4. Determine the best matching unit (BMU) to find the winner neuron \(c\) by comparing the previously calculated Euclidian distances. \[c=\text{argmin}_{1\leq i\leq m}\{||x_{j}-w_{i}(s)||\}\] e. Calculate the learning rate and neighborhood function. Several learning rates and neighborhood functions have been proposed in the literature. The learning rate \(\alpha(s)\) and neighborhood function \(\alpha(s)\) are functions that decrease with the iteration \(s\). The symbol \(\alpha_{0}\) and \(\sigma(s)\) indicate the initial learning rate and initial neighborhood function, respectively. In this study, we use the learning rate \(\alpha(s)\) and neighborhood function \(\sigma(s)\) based on [20; 21] as follows: \[\alpha(s)=\alpha_{0}\cdot exp(-\frac{s}{s_{m}ax}),s=1,2,3,\cdots\] \[\sigma(s)=\sigma_{0}\cdot exp(-\frac{s}{s_{m}ax}),s=1,2,3,\cdots\] f. Calculate the neighborhood distance weights \(h_{c,i}(s)\) between the winner neuron \(c\) and every other node neuron \(i\), where the _coor_ mean coordinate of the neuron in the lattice or map is given as follows: Figure 3.0 g. Update the weight vectors of the neurons within the local neighborhood of the BMU (including the winner neuron \(c\)). We modified the updating weight mechanism as shown in Equation (4) to satisfy the time value constraint. \(wn_{i,p}(s)\) in Equation (5) is the new weight of attribute \(p\), where \(p\) is the non-time value at iteration \(s\). \(wntmp_{i}(s)\) in Equation (6) is the new temporary weight of the time value at iteration \(s\). We use the term \"temporary\" because this value is not the final value of the new weight (we need to evaluate this value with the threshold first). \(disp_{x_{i,p},w_{i,p}}\) in Equation (7) is the displacement value between \(x_{j,p}\) and \(w_{i,p}(s)\). The terms displacement and distance are not the same. The displacement value can be negative, but the distance value is always positive. \[w_{i,p}(s+1)= \begin{cases}wn_{i,p}(s),&\text{$p$ is not time-dependent}\\ wntmp_{i,p}(s)-tr,&\text{$p$ is time-dependent $\wedge wntmp_{i,p}(s)\geq tr$}\\ tr-wntmp_{i,p}(s),&\text{$p$ is time-dependent $\wedge wntmp_{i,p}(s)\leq 0$}\end{cases}  \tag{4}\]  
\[wn_{i,p}(s)+\alpha(s)\cdot h_{c,i}(s)\cdot(x_{j,p}-w_{i,p}(s)) \tag{5}\]  
\[wntmp_{i,p}(s)+w_{i,p}(s)+\alpha(t)\cdot h_{c,i}(s)\cdot disp_{x_{p},w_{i,p}(s)} \tag{6}\]  
\[disp_{x_{j,p},w_{i,p}}= \begin{cases}x_{j,p}-w_{i,p}(s),&\text{$0\leq|x_{j,p}- w_{i,p}|\leq 12$}\\ -D(x_{j,p},w_{i,p}(s)),&\text{$x_{j,p}-w_{i,p}>12$}\\ D(x_{j,p},w_{i,p}(s)),&\text{$x_{j,p}-w_{i,p}<-12$}\end{cases}  \tag{7}\]  
* Repeat steps c to g until the maximum number of iterations is reached or other stopping criteria are found. We should first determine the number of neurons in the SOM to execute it. We determine the number of neurons in the SOM using Equation (8) based on a previous study [22]. The equation calculates the number of neurons from the observations in the training data, where \(M\) is the number of neurons, and \(N\) is the number of observations in the training data. \[M\approx 5\sqrt{N} \tag{8}\]  
To compare the effect of the modified distance function of time value for different algorithms, we also implement it using \(k\)-means clustering. \(k\)-means is a centroid-based partitioning technique that uses the centroid of a cluster, \(C_{i}\), to represent that cluster [16]. Conceptually, the centroid of a cluster is its center point.