## 3 Methodology
### Predictive Process Monitoring  
Process Data DefinitionPP formalizes the task of forecasting process outcomes from partial execution traces. In the context of sepsis care, this translates to predicting whether a patient will be admitted to the ICU based on their evolving hospital trajectory. Mathematically, a process event \(e\in E\) represents a timestamped action in a patient's care pathway, structured as a tuple \((a,c,t_{\text{start}},t_{\text{complete}},\upsilon_{1},\ldots,\upsilon_{n})\), where \(a\) denotes the activity label (e.g., ER Registration, CRP Measurement), \(c\) identifies the unique patient case, and \(t_{\text{start}}\) and \(t_{\text{complete}}\) correspond to the start and completion timestamps. The attributes \(v_{i}\) encapsulate event-specific clinical markers such as lactic acid levels or compliance with SIRS criteria. The event universe \(E=A\times C\times\mathbb{T}\times\mathbb{T}\times V_{1}\times\cdots\times V_{n}\) captures all possible interactions in the care pathway. Projection functions allow the extraction of individual event components, facilitating a granular analysis of activities, timestamps, and clinical parameters. A patient's care trajectory is represented as a trace \(\sigma_{c}=\left\langle e_{1},\ldots,e_{\left\langle e_{i}\right\rangle}\right\rangle\), where events are ordered by start time. For predictive monitoring, the prefixes \(\text{hd}_{i}(\sigma_{c})=\left\langle e_{1},\ldots,e_{i}\right\rangle\) denote partial execution sequences, and suffixes \(\text{tl}_{i}(\sigma_{c})\) represent the remaining events. The event \(\log E_{C}=\left\{\sigma_{c}\mid c\in C\right\}\) aggregates all patient trajectories, serving as the foundation for predictive modeling. Data PreprocessingThe dataset \(\mathcal{D}=\left\{(\mathbf{x}_{i},y_{i})\right\}\) is constructed via a feature function \(\text{feat}:E^{*}\rightarrow\mathbb{R}^{d}\), where \(\mathbf{x}_{i}=\text{feat}(\text{hd}_{i}(\sigma_{c}))\) encodes key attributes relevant to ICU admission risks. Temporal features capture elapsed time since triage and intervals between critical actions such as antibiotic administration. Clinical attributes include blood test results, leukocyte counts, and indicators of sepsis severity based on SIRS criteria. Sequential patterns track the frequency of ICU transfers and variations in discharge protocols, revealing deviations from standard care pathways. To enhance predictive accuracy, the original event log was transformed into a structured dataset incorporating 30 additional process-specific features. Temporal dynamics were represented through metrics such as elapsed time between consecutive events, average hour of care activities, and total duration from registration to discharge. Activity patterns captured workflow inefficiencies by analyzing the maximum frequency of repeated activities. Sequential transitions were encoded through binary indicators for critical event pairs, such as IV Antibiotics + ICU Transfer, highlighting deviations from expected clinical progressions. A structured preprocessing pipeline ensured statistical rigor and interpretability. Outlier removal was applied to filter extreme values in event durations and laboratory results, refining the dataset to 995 high-confidence cases. Robust scaling standardized numerical features, such as leukocyte counts and time intervals, using median and inter-quartile range adjustmentsto mitigate the influence of extreme values. Class imbalance, a critical challenge given the ICU admission rate of approximately 10%, was addressed through stratified sampling, preserving the natural distribution in training, calibration and test sets without resorting to synthetic data generation, thereby maintaining the temporal integrity of patient trajectories. _Supervised Learning._ The predictive task is formulated as a binary classification problem that maps partial traces to ICU-admission outcomes.