Raksmey Phan

Vincent Augusto

Damien Martin

Marianne Sarazin

CLINICAL PATHWAY ANALYSIS USING PROCESS MINING AND DISCRETE-EVENT SIMULATION: AN APPLICATION TO INCISIONAL HERNIA

An incisional hernia (IH) is a ventral hernia that develops after surgical trauma to the abdominal wall, a laparotomy. IH repair is a common surgery that can generate chronic pain, decreased quality of life, and significant healthcare costs caused by hospital readmissions. The goal of this study is to analyze the clinical pathway of patients having an IH using a medico-administrative database. After a preliminary statistical analysis, a process mining approach is proposed to extract the most significant pathways from the database. The resulting causal net is converted into a statechart model that can be simulated. The model is used to understand times of occurrence of complications and associated costs. It enables the simulation of what-if scenarios to propose an improved care pathway for patients who are the most exposed, using in particular prophylactic mesh at the time of abdominal wall closure during a laparotomy on hospitalization costs and readmissions.

INTRODUCTION

The objective of process mining is to use event data to extract process-related information, e.g., to automatically discover a process model by observing events recorded by some enterprise system [START_REF] Aalst | Process Mining: Discovery, Conformance and Enhancement of Business Processes[END_REF]. Two systematic reviews related to process mining were published recently [START_REF] Maita | A Systematic Mapping Study of Process Mining[END_REF][START_REF] Gurgen Erdogan | Systematic Mapping of Process Mining Studies in Healthcare[END_REF]. 705 papers related to process mining published between 2005 and 2014 were gathered in [START_REF] Maita | A Systematic Mapping Study of Process Mining[END_REF]. For applied studies mentioning the specific application domain, "Medicine and Healthcare" is the second most studied area, and "clinical analysis pathway" is the third type of data sources used for case studies or experiments. A similar analysis was proposed in Gurgen Erdogan and Tarhan 2018 on healthcare studies. 172 papers from 2005 to 2017 have been selected. Process discovery is the most used approach when applying process mining to healthcare, meaning practitioners need more Phan, Martin, Augusto, and Sarazin knowledge about clinical pathways. It is considered as the most important activity of process mining applied on healthcare. However, there are more studies about "Healthcare process" (93) than "Clinical pathway" (59) meaning organizational problems are still more studied.

Process mining can be combined with discrete optimization [START_REF] Prodel | Discovery of Patient Pathways from a National Hospital Database using Process Mining and Integer Linear Programming[END_REF] and Monte-Carlo simulation [START_REF] Augusto | Evaluation of Discovered Clinical Pathways using Process Mining and Joint Agent-Based Discrete-Event Simulation[END_REF] to evaluate the performances of new health policies or strategies. In [START_REF] Prodel | Optimal Process Mining for Large and Complex Event Logs[END_REF], a mathematical formulation of the problem has been presented, along with a Tabu Search optimization process to mine the best process model. To reduce the computational time for large-scale data sets, a Monte Carlo sampling method is used. The ICD-10 codes (International Classification of Diseases, 10th revision) has been used for node labeling. In our paper we use a similar approach to mine clinical processes. In [START_REF] Augusto | Evaluation of Discovered Clinical Pathways using Process Mining and Joint Agent-Based Discrete-Event Simulation[END_REF] Monte-Carlo simulation has been used to estimate the impact of new policies regarding medical implants reimbursement. However evaluation and comparison of different clinical pathways with new surgical practices has not been studied.

A case study using both process mining and simulation in health-care is also proposed in Zhou et al. 2014. They used the fuzzy miner algorithm for process discovery [START_REF] Günther | Fuzzy Mining -Adaptive Process Simplification Based on Multi-perspective Metrics[END_REF]. They specifically studied the pathway of patients during a single hospital stay, starting from the admission to the release. In our paper, we focus on a macroscopic clinical pathway which can last several years from the first hospital stay to the most recent one.

CASE STUDY AND SCIENTIFIC CONTRIBUTIONS

Case study

A hernia is a protrusion of tissue or part of an organ through bone, muscular tissue or membrane. More precisely, an incisional hernia (IH) is a ventral hernia that develops after surgical trauma to the abdominal wall, including recurrences after repair of primary ventral hernias. IH repair is a common surgery that can generate chronic pain, decreased quality of life, and significant healthcare costs caused by readmissions. The laparotomy surgeries that may lead to IH are mainly colorectal surgeries, bariatric surgeries, abdominal aortic aneurysm and gynecoligic surgeries. Comorbidities like diabetes, obesity, hypertension and Chronic Obstructive Pulmonary Disease (COPD) have been identified as main risk factors.

Among patients having a laparotomy, 10% to 25% will develop an incisional hernia [START_REF] Deerenberg | Prevention and Treatment of Incisional Hernia: New Techniques and Materials[END_REF]). The ratio can be even higher, depending on the type of surgery, risk factors, suture material and surgical technique. The reported incidence is determined by several factors: (i) patient population, (ii) type of abdominal wall incision, (iii) length of follow-up, and (iv) method of IH diagnosis [START_REF] Muysoms | European Hernia Society Guidelines on the Closure of Abdominal Wall Incisions[END_REF]. For example, a midline incision due to an abdominal cancer operation is strongly associated with hernia formation, IH occurred for 41% of patients [START_REF] Baucom | Cancer Survivorship: Defining the Incidence of Incisional Hernia After Resection for Intra-Abdominal Malignancy[END_REF].

However even small incisions due to a laparoscopic surgery can generate IH. The incidence of IH after midline and off-midline incisions in patients undergoing laparoscopic colorectal surgery showed that the pooled incidence of IH was 10.6% in midline, 3.7% in transverse, and 0.9% in Pfannenstiel incisions [START_REF] Lee | Incidence of Incisional Hernia in the Specimen Extraction Site for Laparoscopic Colorectal Surgery: Systematic Review and Meta-Analysis[END_REF]. After an IH repair, the recurrence rate is above 33%, but with a re-surgery the rate is lower [START_REF] Helgstrand | Reoperation Versus Clinical Recurrence Rate After Ventral Hernia Repair[END_REF]. A patient is also likely to have a surgical site occurrence like a surgical site infection (SSI) or a seroma. The risk of SSI depends of the type of IH repair, raising from 3.4% for a laparoscopic repair to 22.1% for an open IH repair [START_REF] Kaafarani | Predictors of Surgical Site Infection in Laparoscopic and Open Ventral Incisional Herniorrhaphy[END_REF], and factors like obesity, COPD or the porosity of the mesh have a significant impact [START_REF] Sanchez | Mesh Infection in Ventral Incisional Hernia Repair: Incidence, Contributing Factors, and Treatment[END_REF].

Decrease of quality of life of patients is not the only issue. Costs of IH repair are a burden from the societal point of view. The mean total cost for an IH repair was estimated to be 6,451 euro ranging from 4,731 euro to 10,107 euro for respectively unemployed and employed patients [START_REF] Gillion | The Economic Burden of Incisional Ventral Hernia Repair: A Multicentric Cost Analysis[END_REF]. Also higher costs occur when the IH appears within 1 year versus 3 years [START_REF] Alli | Impact of Incisional Hernia Development Following Abdominal Operations on Total Healthcare Cost[END_REF].

Prophylactic mesh are promising implants that can be used during laparotomy to prevent IH [START_REF] Hernando | Prophylactic Mesh can be used Safely in the Prevention of Incisional Hernia after Bilateral Subcostal Laparotomies[END_REF]. Such medical act is safe and effectively prevents the development of IH for at least two years for abdominal aortic aneurysm [START_REF] Muysoms | Prevention of Incisional Hernias by Prophylactic Mesh-augmented Reinforcement of Midline Laparotomies for Abdominal Aortic Aneurysm Treatment: A Randomized Controlled Trial[END_REF]. It is even proposed as a standard of care for Phan, Martin, Augusto, and Sarazin patients at high risk [START_REF] Muysoms | Prophylactic Meshes in the Abdominal Wall[END_REF]. However practitioners need guidelines to find out which patients should benefit such implant.

Objectives and scientific contributions

The objective of this study is twofold:

1. Propose an analysis framework based on process mining and discrete-event simulation using medicoadministrative data to evaluate what-if scenarios of clinical pathways such as public health strategies. 2. Provide quantitative results on a case study related to IH patients clinical pathway.

The main scientific challenge of this study lies in the definition and development of the analysis framework, especially regarding the process mining causal net conversion into a discrete-event simulation model. From a medical point of view, what-if scenarios should be carefully tuned in order to get realistic results that can be used to develop new public health strategies such as new medical act reimbursement.

The remainder of this paper is organized as follows. A short literature review on clinical pathway modeling and simulation is presented in Section 1. The position of the problem is given in Section 3. A data description is provided in Section 4. The process mining approach is detailed in Section 5 along with the discrete-event simulation model generation. Numerical results on the IH case study are given in Section 6. Conclusions and perspectives are discussed in Section 7.

OVERVIEW OF THE ANALYSIS FRAMEWORK

In this paper we intend to propose an analysis framework based on process mining and discrete-event simulation using medico-administrative data to evaluate what-if scenarios of clinical pathways such as public health strategies. Such framework will be evaluated on a case study related to incisional hernia.

Our methodology is illustrated in Figure 1. Data is collected from medico-administrative database, as presented in Section 4. Data Extraction (step 1) consists of querying the database. At this step, we also validate the cohort with health practitioners using statistical analysis. We then convert the original data into event logs in order to be able to apply process mining (step 2). In this second step, we use existing mining algorithms such as extensions of Fuzzy Miner [START_REF] Günther | Fuzzy Mining -Adaptive Process Simplification Based on Multi-perspective Metrics[END_REF]) on which the Disco software is based. Finally, the model is converted again into a discrete-event simulation model in order to perform what-if scenarios simulations of new clinical pathway strategies (step 3). The following sections of this paper present the medico-administrative data used for this study (Section 4). Then the process mining approach and the simulation model generation are described in Section 5. The performance of the approach is evaluated on a real case study related to clinical pathway of patients having incisional hernia in Section 6.

Medico-administrative database

Data is extracted from the French national hospital discharge database (PMSI: "Programme de Médicalisation des Systèmes d'Information") using the International Classification of Diseases (ICD-10) and the French classification called "Common Classification of Medical Acts" (Classification Commune des Actes Médicaux in French, CCAM 2010). Surgeries done by laparotomy represent 431,719 patients in 2010. There are 71% of females and the average age is 42, due to the high number of caesarean section into this population. Among these patients, 22,088 (5%) have had a first incisional hernia (IH) in the following years. Patients having at least one IH are around 62 years old and are in majority women (55%). Figure 2 summarizes this first statistical analysis. According to the ICD-10, there are about 556 codes related to the laparotomy surgical operations. In this paper we focus only on laparotomies generating the highest volumes of IH according to two criteria.

1. The first criterion is the number of patients: we selected laparotopy surgeries performed on at least 100 patients. Such selection allows to avoid irrelevant surgeries for this study. 2. The second criterion is the rate of IH: we decided to select only laparotomy surgeries that lead to at least 10% of occurrence rate of IH; such decision allows to avoid surgeries representing a high number of patients but with a low incidence rate of IH. For example, surgeries like "Scheduled cesarean section by laparotomy" having 59,000 patients with an IH occurrence rate of only 0.9% are put aside.

As such, all the following statistics and analysis will be more accurate and not diluted. With this selection the number of studied laparotomy codes has been reduced to 65 (top-65). This top-65 represents 71,863 patients with an average rate of incisional hernia in the next 5 years of 17%.

Patients having an IH are twice more likely to have at least one comorbidity factor in comparison to someone without an IH. On the top-65, a study of the impact of the 4 selected comorbidity factors has been done: Chronic Obstructive Pulmonary Diseases (COPD), Diabetes, Hypertension and Obesity. Results are displayed in Table 1. It shows that patients with at least one of those comorbidities are more likely to have IH in the next 5 years (obesity seems to be the most impactful factor).

Phan, Martin, Augusto, and Sarazin

CLINICAL PATHWAY PROCESS MINING AND SIMULATION

This section introduces the process mining approach used in this paper. The process model and the preprocessing data step will be presented. The readers are referred to [START_REF] Aalst | Process Mining: Discovery, Conformance and Enhancement of Business Processes[END_REF][START_REF] Günther | Process Mining in Flexible Environments[END_REF][START_REF] Augusto | Evaluation of Discovered Clinical Pathways using Process Mining and Joint Agent-Based Discrete-Event Simulation[END_REF] for more details on the basics of process mining including formal definitions of concepts of events, traces, and logs.

Data conversion for process mining

A data conversion procedure is necessary before applying process mining to medico-administrative data. The objective of this procedure is to obtain an event log, the basic material to apply any process mining algorithm. Contrary to [START_REF] Augusto | Evaluation of Discovered Clinical Pathways using Process Mining and Joint Agent-Based Discrete-Event Simulation[END_REF] where event labels are Diagnosis Related Groups (DRG), important information to define nodes of our process model are contained in medical acts performed during a patient stay in the hospital. Thus we need to extract both Diagnosis Related Groups (DRG) that characterize a hospital stay, as well as medical acts performed during a hospital stay. To extract such information, we need to duplicate a hospital stay into several log entries. Figure 3 shows an example of this conversion step: for Patient A who had 3 medical acts during a hospital stay labeled "IH", we generate 3 events in the log. Then it is necessary to focus on one medical act that is incisional hernia repair in the clinical pathway of patients. It is the inclusion point of our patients and events before and after this central event should be distinguished. An example is proposed in Figure 4 where a patient has a blood infusion before and after an incisional hernia that should be discriminated.

Process mining and event refinement

Using the converted data, we are able to apply any process mining algorithm. In our case we use Disco1 , a process mining tool to generate graph in which each node contains the CCAM code of the surgery whereas the frequencies are displayed for each node and arc. The next step is to define medical acts that are relevant regarding incisional hernia occurrence analysis. For example it is not relevant to know that 2 months before a laparotomy a patient had an ossiculoplasty (an operation to repair, reconstruct and improve the movement of the tiny bones of hearing in the middle ear). Based on practitioners expertise, we focus only on the CCAM codes given by Table 2.

Table 2: Incisional hernia repair codes in the CCAM.

CCAM-Code Description

LMMA004 Cure of postoperative IH of the anterior abdominal wall w/ mesh, by direct approach LMMA010 Cure of postoperative IH of the anterior abdominal wall w/o mesh, by direct approach LMMC015 Cure of postoperative IH of the anterior abdominal wall w/ mesh, by laparoscopy Due to space limitation, we focus in this paper on the analysis of three clinical pathways regarding IH recurrence and occurrence: (i) for general patients (Figure 5), and (ii) for non-obese patients and for obese patients (Figure 6).

Discrete-event simulation model generation

To generate the simulation model, we use a state chart conversion procedure as defined in [START_REF] Augusto | Evaluation of Discovered Clinical Pathways using Process Mining and Joint Agent-Based Discrete-Event Simulation[END_REF]. This is a step by step conversion procedure in order to produce a simulation model using as input a causal net generated using a process mining approach such as Fuzzy Miner. The input material of our procedure is a process model. Such model is denoted Clinical Pathway State Chart and is defined as follows. Definition 1 (CP State Chart) A Clinical Pathway State Chart (CPSC) is a state chart M which underlying graph is a bigraph (bipartite graph) having the following properties:

• The two types of states of the bipartite graph are Care-states and Wait-states. By definition, two states of the same type cannot be linked by a transition. • Each Wait-state has exactly one input transition and one output transition: the probability of activating the output transition of a Wait-state is always equal to 1. • The sum of all probabilities of output transitions of a Care-state is equal to 1.

According to Definition 1, a Care-state may be followed by {0; n} Wait-state(s) whereas a Wait-state is always followed by exactly one Care-state: the probability of the transition between a Wait-state and a Care-state is equal to 1.

The resulting simulation model has been implemented using Anylogic and is illustrated in Figure 7. Parameters and scenarios are detailed in the following Section.

Phan, Martin, Augusto, and Sarazin

NUMERICAL RESULTS

The goal of the simulation is to evaluate the economic impact of using a prophylactic mesh after a laparotomy to prevent incisional hernia. Clinical studies are currently done and seem to be encouraging. In this paper we focus on the following questions:

• Does using a prophylactic mesh after a laparotomy shorten the clinical pathway of patients?

• Are hospitals and healthcare systems going to save money thanks to the use of a prophylactic mesh at the time of initial laparotomy closure?

Phan, Martin, Augusto, and Sarazin

Design of experiments

The simulation has been done with Anylogic simulation software. Below is shown a scheme of the simulation. It takes into account only patients with a laparotomy in the top-65 of the previous data analysis. Possible events for those patients are:

• Use of a prophylactic mesh.

• Incisional hernia occurrence.

• Incisional hernia recurrence.

The simulation also takes into account the difference between obese and non-obese patients. The most critical post-operative complication, the risk of having a mesh ablation after the use of a prophylactic mesh and after an incisional hernia occurrence is also considered; note that other complications can occur after the use of mesh but the lack of information about these complications the associated costs in particular have limited our capacities to take them into consideration in this evaluation.

The two main key performance indicators are: (1) The average direct cost for one patient after the laparotomy until a potential incisional hernia recurrence; and (2) the average number of hospital stays for one patient after a laparotomy until a potential incisional hernia recurrence.

Figure 8 summarizes the simulation scheme. We simulated 72,000 patients (from 2010) having a laparotomy among the top-65, with Mesh ablation rate of 3% and obesity rate of 13% (obtained from data analysis). We compare population with prophylactic mesh and without.

Data

Information used for these simulations come from previous data analysis and hospital reimbursement data. Only occurrence rates for population with prophylactic mesh are extracted from recent articles [START_REF] Hernando | Prophylactic Mesh can be used Safely in the Prevention of Incisional Hernia after Bilateral Subcostal Laparotomies[END_REF][START_REF] Muysoms | Prevention of Incisional Hernias by Prophylactic Mesh-augmented Reinforcement of Midline Laparotomies for Abdominal Aortic Aneurysm Treatment: A Randomized Controlled Trial[END_REF][START_REF] Muysoms | Prophylactic Meshes in the Abdominal Wall[END_REF]. Table 3 gives the incisional hernia occurence and recurrence rates for obese and non-obese patients dependent on whether they have (or not) a prophylactic Phan, Martin, Augusto, and Sarazin mesh. Of course, all patients that have incisional hernia recurrence also have a prophylactic mesh. We can notice that the occurrence rate for patients with prophylactic mesh is very small. Table 4 gives the cost value for each type of surgical operation (incisional hernia repair, incisional recurrent hernia repair, prophylactic mesh and mesh ablation) dependent on patients with or without obesity. Note that the cost value for prophylactic mesh is given by Medtronic.

Results

Simulations have been done to analyse results about general population, to compare obese and non-obese patients and to evaluate the total cost for each case. In the following, column "Prophylactic Mesh" presents the two scenarios: with and without mesh; column "#stay" gives the average number of times that patients come to the hospital for mesh ablation, incisional hernia occurrence or incisional hernia recurrence and thus should be as small as possible; and finally column "Cost by patient" gives the average cost value for each patient.

Table 5 shows that for the general population, the number of stays is divided by 2.7 when comparing population without mesh to population with mesh. The average direct cost for each patient is also decreased by 45% by using a prophylactic mesh for every patient. These results show the importance to consider the use of prophylactic mesh on the general population.

Since obesity increases the risk of incisional hernia, Table 6 gives results regarding the same analysis on obese and non-obese population. In this case, we consider that a non-obese patient is always associated Phan, Martin, Augusto, and Sarazin with the lowest cost at each stage of his medical pathway, whereas an obese patient is always associated with the highest cost at each stage of his medical pathway (note that the highest cost in the medical pathway is usually associated to morbidly obese patients). As it is shown in Table 6, for non-obese patients the use of a prophylactic mesh still decreases largely the average number of times that patients come to the hospital (#stay), even if the average cost value for each patient (Cost/patient) is similar. Regarding the obese population, the amount of stays (#stay) is divided by 4 when a prophylactic mesh is used and the average cost value is decreased by 65%. In addition to the previous analysis about general population, these results highlight the benefit of using prophylactic mesh for obese patients. A cost analysis is finally provided. Table 7 presents the total cost for general population (column "total"), for non-obese (column "Non-obese") and obese (column "Obese") population. Results show that the main savings come from the prevention of incisional hernia for obese patients. Indeed, the cost savings for non-obese patients represent only 7% of the savings whereas there make up 87% of the population.

CONCLUSIONS AND PERSPECTIVES

Process Mining is an observed behaviour approach that uses event-log data to describe "what really happened"; so it leads to realistic simulation models. With this approach, it is easy to have a model that generates events for statistical analysis. Such analysis (combining process mining and simulation) shows that the use of prophylaxis mesh significantly decreases the cost of medical care and enhances the clinical pathway (by reducing the average number of times that patients come to the hospital) after a laparotomy and particularly for obese patients. However these simulations have multiple limitations. Indeed, (1) it does not consider post operative complications like seroma or surgical site infection; (2) in case of multi operations by laparotomy, it is supposed that the surgery site is always the same; and (3) the previous laparotomy before the IH is selected however it could come from an anterior surgery. Also we could not take into account the impact of the experience of the surgeon and the type of closure after the laparotomy.

The proposed methodology has proven effective and brings new insight for practitioners. However, the method is difficult to generalize to other data sources or case studies because of the special features related to each medical case. For future works, we intend to develop a generic data extraction for process mining and simulation that could be embedded in a software. Phan, Martin, Augusto, and Sarazin

Figure 1 :

 1 Figure 1: Global methodology.

Figure 2 :

 2 Figure 2: Extraction of 22,088 patients with IH from 431,719 patients with laparotomy.

Figure 3 :

 3 Figure 3: Conversion of a stay into 3 surgeries.

Figure 4 :

 4 Figure 4: Disctinction between events before and after an IH.

Figure 5 :

 5 Figure 5: Process mining on general patients population.

Figure 6 :

 6 Figure 6: Process mining on non obese (left) and obese (right) patients.

Figure 7 :

 7 Figure 7: Resulting simulation model.

Figure 8 :

 8 Figure 8: Simulation scheme.

Table 1 :

 1 Rate of incisional hernia dependent on comorbidities.

	Factor	COPD	Diabetes	Hypertension	Obesity
		no	yes	no	yes	no	yes	no	yes
	#patients 67,131 4,732 61,386 10,477 41,287 30,576 63,131 8,732
	IH rate	16%	20%	16%	19%	15%	19%	15%	31%

Table 3 :

 3 Incisional hernia rate depending on prophylactic mesh.

		Occurrence		Recurrence
	Prophylactic mesh No obesity Obesity No obesity Obesity
	no	15%	31%	-	-
	yes	3%	5%	13%	19%

Table 4 :

 4 Surgical operation cost value depending on patients obesity.

		No obesity Obesity
	Incisional hernia repair	1,972e 10,894e
	Incisional hernia recurrence repair	3,333e 18,411e
	Prophylactic mesh	200e	200e
	Mesh ablation	2,320e 18,088e

Table 5 :

 5 Table of simulation for the General Population.

	Prophylactic Mesh #stay Cost/patient
	no 0.2	926e
	yes 0.07 512e

Table 6 :

 6 Number of stays and cost per patient.

	Non-obese	Obese
	Prophylactic Mesh #stay Cost/patient #stay Cost/patient
	no 0.17 369e	0.38 4,621e
	yes 0.07 348e	0.10 1,595e

Table 7 :

 7 Costs evaluation.

	Prophylactic Mesh Total	Non-obese	Obese
	no 66 million e 23 million e 43 million e
	yes 36 million e 21 million e 15 million e
	save 30 million e 2 million e	28 million e

Fluxicon -DISCOver your processes: https://fluxicon.com/disco/

ACKNOWLEDGMENTS

Authors would like to thank Mrs. Linda Spinnler and Mr. Philippe Gravagna from the Medtronic group (Sofradim Production) who supported this study.

AUTHOR BIOGRAPHIES

RAKSMEY PHAN received his Ph.D. degree from the Institut Suprieur d'Informatique, de Modlisation et de leurs Applications (ISIMA), France, in 2013 and his engineering degree, in 2010. Currently, he is a researcher-engineer in the Center for Health Engineering and in the IEOR team of CNRS UMR 6158 LIMOS, EMSE. His fields of interest include Operational Research, Databases and Internet of Things. His e-mail and web addresses are raksmey.phan@mines-stetienne.fr and http://emse.raksmey.fr/, respectively.

DAMIEN MARTIN received his engineer degree (equivalent to a Master of Science) from IMT Mines Albi-Carmaux, France in 2019. His email address is damien.martin@mines-albi.fr.

VINCENT AUGUSTO is a Professor of industrial engineering in the Center for Health Care Engineering at Mines Saint-Etienne. He received his Ph.D. in Industrial Engineering from Mines Saint-Étienne, France, in 2008 and the Habilitation à Diriger des Recherches degree from the University of Saint-Étienne, France, in 2016. He was a visiting scholar at CIRRELT (Centre Interuniversitaire de Recherche sur les Réseaux d'Entreprise, la Logistique et le Transport), University of Laval, Quebec, Canada in 2009 and 2015. His research interests include modeling, simulation, optimization of healthcare systems and their supply chains, process mining and machine learning applied to healthcare. His email address is augusto@emse.fr.

MARIANNE SARAZIN is a public health doctor with a doctorate in Life Sciences from Mines Saint-Étienne, France. She is the head of the Medical Information Department of the Mutualiste Sanitary Group of Saint-Étienne and a collaborator in the Center for Health Care Engineering at Mines Saint-Etienne. She is also with the UMRS 1136 Inserm lab, specialized in the modeling of epidemics. Her email address is marianne.sarazin@iplesp.upmc.fr.


Hugo De Oliveira
email: hdeoliveira@hevaweb.com

Vincent Augusto

Baptiste Jouaneton

Ludovic Lamarsalle

Martin Prodel

Xiaolan Xie

Optimal Process Mining of Timed Event Logs

Keywords: process mining, event log, time modeling, tabu search, healthcare data, patient pathways

The problem of determining the optimal process model of an event log of traces of events with temporal information is presented. A formal description of the event log and relevant complexity measures are detailed. Then the process model and its replayability score that measures model fitness with respect to the event log are defined. Two process models are formulated, taking into account temporal information. The first, called grid process model, is reminiscent of Petri net unfolding and is a graph with multiple layers of labeled nodes and arcs connecting lower to upper layer nodes. Our second model is an extension of the first. Denoted the time grid process model, it associates a time interval to each arc. Subsequently, a Tabu search algorithm is constructed to determine the optimal process model that maximizes the replayability score subject to the constraints of the maximal number of nodes and arcs. Numerical experiments are conducted to assess the performance of the proposed Tabu search algorithm. Lastly, a healthcare case study was conducted to demonstrate the applicability of our approach for clinical pathway modeling. Special attention was paid on readability, so that final users could beneficially use the process mining results.

Introduction

The digital revolution affects all industries and businesses and produces a huge amount of data. Numerous decision aid analytical methods and tools are available to take advantage on relevant data. Machine Learning methods have been widely used. Supervised and unsupervised learning for knowledge discovery, when applied to matrix data where each row is an observation characterized by features in columns, have spread over.

Knowledge discovery from healthcare data, such as patient lifetime hospitalization history is presently not optimal yet crucial. Therefore, we want to find common patterns, process models, or care pathways of hospitalization histories for cohorts honed to a specific time periods. Such studies are important to detect relevant "causal" relationships or transitions, to check the conformance of practice to guidelines, etc. An example of a causal relation could be "most patients of a given surgery had the same prior underlining condition or past medical event".

Traditional machine learning techniques are not well suited to generate process models from data. Process mining well suited for this purpose, and has been first formalized in 2004 [START_REF] Van Der Aalst | Workflow mining: discovering process models from event logs[END_REF], followed by developments in various fields [START_REF] Maita | A systematic mapping study of process mining[END_REF] including healthcare [START_REF] Erdogan | Systematic mapping of process mining studies in healthcare[END_REF]. Temporal information such as the time between two events and the number of repetitions of a given event in the past are particularly important in process modeling and prediction. In healthcare, a second hospital visit shortly after the first, unfortunately is likely to be an undesirable complication or result of the earlier admission. A patient having been hospitalized several times for a given disease or had a much longer prior hospital stay before recovery is more likely to need a non-standard surgery than a patient having no or just one past hospitalization. Unfortunately, such features are rarely taken into account by process mining approaches. For example, repeated events are prohibited for the sake of visibility and time is not considered during model construction. A previous study on the understanding of process models found that the average connector degree and density are two identified factors which induce negative effects on comprehension (at a fixed size) [START_REF] Reijers | A study into the factors that influence the understandability of business process models[END_REF].

Starting from these limitations regarding time consideration and comprehension, an extension of existing optimal process mining theory is presented. Thus, the main scientific contribution of this paper is the mathematical formalization of a new ascending and time-dependent process mining approach, structured on a grid for a better representation and understanding. Reminiscent of Petri net unfolding [START_REF] Giua | Control of Safe Ordinary Petri Nets Using Unfolding[END_REF], ascending representation forbids loops and confusing backward transitions, whereas the time-dependent feature considers time patterns of event logs during optimization. Descriptors are introduced, characterizing event logs complexity and process models structure. The replayability indicator [START_REF] Prodel | Optimal process mining for large and complex event logs[END_REF] is slightly modified and acts as a key performance indicator to evaluate the resulting process models. Finally, a new Tabu search procedure for process mining optimization is presented, tested and validated through a series of designed experiments as well as a real-life healthcare case study.

The rest of this paper is organized as follows. A literature review focused on recent work in process mining with a particular focus on healthcare is presented in Section 2. General definitions of event log representation are given in Section 3. A mathematical formulation of the new process models and optimization problems is presented in Section 4. Section 5 describes the process discovery methods involved and Section 6 details in depth computational experiments designed to test methods on different simulated event logs. A case study based on real data is presented in Section 7. Finally, conclusion and perspectives are given in Section 8.

Literature Review

The raison d'être of Process mining is to discover, monitor and ameliorate actual processes as they occur by extracting knowledge from event logs readily available in I.T. systems. Process mining is situated between Big Data and Data Mining on one side, and Business Process Modeling and Analysis on the other. The field of process mining can be divided in 3 main areas: process discovery, conformance checking and extension of a model [START_REF] Van Der Aalst | Introduction. In Process Mining: Discovery, Conformance and Enhancement of Business Processes[END_REF].

Recently, two systematic studies mapped out Process Mining [START_REF] Maita | A systematic mapping study of process mining[END_REF][START_REF] Erdogan | Systematic mapping of process mining studies in healthcare[END_REF], which are valuable to describe the scope and the dynamics of this field. Maita et al. regrouped in their study 705 papers about process mining from 2005 to 2014 [START_REF] Maita | A systematic mapping study of process mining[END_REF]. The number of publications addressing process mining has significantly increased from 2005 to 2014. "Discovery" is the main purpose for the use of process mining (71% of papers), with "graph structure-based techniques" being the most common intersection. For studies mentioning a specific application domain, "Medicine and Healthcare" is the second most frequent domain just after the overall sector "Enterprise". Moreover, "clinical analysis pathway" ranks third as data sources used for case studies or experiments. Importantly, the majority of these studies does not mention any process mining tool being used. For the rest, ProM is the most often mentioned with in-house frameworks a distant second.

A similar work [START_REF] Erdogan | Systematic mapping of process mining studies in healthcare[END_REF] on healthcare studies considers 172 papers from 2005 to 2017. Observations detail a rapid expansion of the field, giving new opportunities for further research and practice. Process discovery appears as the most important activity of process mining when applied to healthcare. Studies on "Healthcare process" (93) are more common than "Clinical pathway" (59). Furthermore, studies corresponding to "Multiple hospital" data are less frequent (14) compared to "Single hospital" data (130). The same observation is true for studies regarding "Multiple department" data [START_REF] Van Der Aalst | Introduction. In Process Mining: Discovery, Conformance and Enhancement of Business Processes[END_REF] and "Single department" data (122). Finally, the number of studies which propose a new tool, model or metric is low (17%). This limited body of relevant papers in our field seems to imply the originality of our contribution.

Since the later systematic review on healthcare, new process mining contributions applied to healthcare have been proposed. Kusuma et al. compiled a literature review of process mining in cardiology [START_REF] Kusuma | Process mining in cardiology: A literature review[END_REF]. Promising opportunities to assist medical experts in care analysis were shown, although few formal process mining methodologies were included. Litchfield et al. [START_REF] Litchfield | Can process mining automatically describe care pathways of patients with long-term conditions in uk primary care? a study protocol[END_REF] introduced a study protocol to apply process mining to primary care in the UK for the first time. The use of orthodox process mapping in addition to data-driven process mining is presented as useful to identify differences and similarities. Similar works on patient pathways mining were published using discrete optimization [START_REF] Prodel | Discovery of patient pathways from a national hospital database using process mining and integer linear programming[END_REF] and simulation [START_REF] Prodel | Evaluation of discovered clinical pathways using process mining and joint agent-based discrete-event simulation[END_REF]. The most related paper to our work is of [START_REF] Prodel | Optimal process mining for large and complex event logs[END_REF], where a mathematical formulation of the problem was presented, along with a Tabu search optimization process to search for best process model. In addition, to reduce the computation time for large-scale data sets, they used a Monte Carlo sampling method. As healthcare data often contains hierarchical structure (ICD-10 codes for example), this data characteristic for labels was considered during the optimization process [START_REF] Prodel | Evaluation of discovered clinical pathways using process mining and joint agent-based discrete-event simulation[END_REF]. The large scale problematic and the hierarchical structure of labels were also addressed.

There are two limitations. First, qualitative representation of pathways with repeated events is not readily understood by decision makers and clinicians. Indeed, a repeated event pattern in the event log will be represented by a loop, which does not take into account the linear characteristic of patient pathways. More importantly, the time is not considered during the optimization process. The including of time as a parameter could be beneficial to highlight hidden time patterns contained in the event log.

Therefore, two extended process mining approaches are presented in this paper. New descriptors to characterize data sets and newly formulated process models are also presented. The Tabu search used by Prodel et al. [START_REF] Prodel | Optimal process mining for large and complex event logs[END_REF] is tested and compared to other methods. Moreover, an improved version of the Tabu Search algorithm adapted to new process model is suggested.

Event, Trace and Log

This section provides a formal description of data involved including events, traces and event logs. Definition 1. (Event). Each event denoted e is a couple (a, t) where a ∈ A is an element of a finite set A of labels corresponding to the event class of e, and t ∈ T with T = N or R corresponding to the event time also called time-stamp. An event e is then equivalently defined by the two following functions:

• label(e) = a called labeling function;

• time(e) = t called timing function.

Definition 2. (Trace).

A trace is a sequence of events denoted as σ = e 1 , e 2 , . . . , e m with m ∈ N * such that e k ∈ A × T and time(e k ) < time(e k+1 ). A trace leads to the following functions:

• trace(e k ) denoting the trace ID of each event;

• position(e k ) = k denoting the order of the event in the trace;

• |σ| = m denoting the length of the trace. Definition 3. (Event log). An event log is a set of traces denoted as L = {σ 1 , σ 2 , . . . , σ n } with n ∈ N * . An event log contains all input data of this paper. Without loss of generality, we assume that each label appears at least once in the event log L, i.e. ∀a ∈ A : ∃σ ∈ L, e ∈ σ | e = (a, t). The set of possible combinations of labels and positions in L is: The diversity measures characterize the log's complexity: a high diversity means an increased number of different elements in terms of labels and causal relations. Also, the distribution of trace lengths is a valuable predictor to assess event logs' complexity (let |σ| mean , |σ| std , |σ| max and |σ| min be its mean, standard deviation, maximum and minimum values, respectively). Notations of event logs are summarized in Table 1.

A lab,pos = σ=e1,...,
Example 1. Table 2 shows a short event log related to patient hospitalization pathways. Each row is a hospitalization event. Events with the same ID are ordered by increasing time stamp and represent a trace. Each trace is a patient's hospitalization history. Each first event of a patient has a timestamp set to 0. By clustering hospitalization by main diagnosis, the set of labels is as follows :

• A lab = {I500, I44.2, I621, G935, E149, I272}.
Alternative event clustering is possible by taking into account more detailed information such as the duration and secondary treatments during the hospitalization.

Thus, event-position set is as follows:

• A lab,pos = {(I500, 1), (I44.2, 2), (I621, 3), (G935, 4), (E149, 2), (I272, 3), (I500, 4), (I44. Basic and timed transitions are:

• T C = {(I500, I44 .
2), (I500, I621), (I500, G935), (I44.2, I621), (I44.2, G935), (I621, G935), (I500, E149), (I500, I272), (I500, I500), (E149, I272), (E149, I500), (E149, I44.2), (I272, I500), (I272, I44.2))}

• T DC = {(I500, I44.2), (I44.2, I621), (I621, G935), (I500, E149), (E149, I272), (I272, I500))};

• T tC = {(I500, I44.2, 30), (I500, I621, 72), (I500, G935, 103), (I44.2, I621, 42), (I44.2, G935, 73), (I621, G935, 31), (I500, E149, 5), (I500, I272, 93), (I500, I500, 145), (I500, I44.2, 180), (E149, I272, 88), (E149, I500, 140), (E149, I44.2, 175), (I272, I500, 52), (I272, I44.2, 87), (I500, I44.2, 35))};

• T tDC = {(I500, I44.2, 30), (I44.2, I621, 42), (I621, G935, 31), (I500, E149, 5), (E149, I272, 88), (I272, I500, 52), (I500, I44.2, 35))}.

Finally, diversity descriptors are:

• div e = 6; div e,p = 8; div causal = 6; div t-causal = 7.

Grid Process Model Optimization Problem

This section is dedicated to discover process models that best fit the event logs given in Section 3. For this purpose, first the concepts of grid process models and time grid process models are introduced. Subsequently, the fitness measure is introduced to measure how well a process model captures the causal relations of an event log. We terminate by a formal definition of the process model optimization problem.

Grid Process Models Definition 7. (Grid process model). A grid process model of a given log

L is a triplet G-P sM = (N, E, L)
where:

• N is a set of nodes partitioned into K disjoint subsets called layers, i.e. N = N 1 ∪• • •∪N k , N k ∩N l = ∅; • E ⊂ N × N is a set of edges such that (x, y) ⊂ E with x ∈ N k , y ∈ N l implies k < l, i.e. the process
model is acyclic with edges going from lower layers to higher layers;

• L : N → A is the labeling function of the nodes.

From the above definition, one can also define the position function

P(x) = k, if x ∈ N k . As a result, (x, y) ⊂ E implies P(x) < P(y).
The main difference from the previous process model definition in [START_REF] Prodel | Optimal process mining for large and complex event logs[END_REF] is the possibility for an event to appear at various positions in a trace. For example, an event happening both at the beginning and at the end of a patient pathway could now be described in the process model by two nodes, one with low and the other with a high position. In this case, loops on the same node and backward edges are not allowed anymore. An example of a G-P sM and its process model equivalent are given in Figure 1. Example 2. The Figure 1a shows a grid process model G-P sM = (N, E, L) with:

N = (n 1 , n 2 , n 3 , n 4 , n 5 ) E = (e 1 , e 2 , e 3 , e 4 , e 5 ) L(n 1 ) = A P(n 1 ) = 1 e 1 = (n 1 , n 3 ) L(n 2 ) = B P(n 2 ) = 1 e 2 = (n 1 , n 4 ) L(n 3 ) = C P(n 3 ) = 2 e 3 = (n 2 , n 3 ) L(n 4 ) = A P(n 4 ) = 3 e 4 = (n 3 , n 4 ) L(n 5 ) = C P(n 5 ) = 3 e 5 = (n 3 , n 5 )
An equivalent classic process model is presented in Figure 1b, without duplicated label nodes, allowing loops and backward transitions.

Time Grid Process Models

To the best of our knowledge, in existing process mining approaches, time-related information is added after model discovery and remains descriptive. Significantly, we provide a new approach to include timerelated information within the optimization process of building a process model.

Definition 8. (Time grid process model).

A time grid process model of a given log L is a four-uplet T G-P sM = (N, E, L, T ) where:

• (N, E, L) is a grid process model with eventually multiple edges between nodes;

• T : E → T × T associates a time interval [a (x,y) , b (x,y) ] to each edge (x, y) ∈ E.
As shown in Figure 2, using this definition, previous unique edges between two nodes in G-P sM are replaced by multiple possible edges in T G-P sM , each of them having its own time interval. Definition 8 ensures that a given causal relation, with a given time value between two events, will be characterized by a unique possible edge with same starting and ending node in the process model. The uniqueness of characterization will be useful for graph construction and replayability defined thereafter.

Example 3. The Figure 2 shows a time grid process model T G-P sM = (N, E, L, T ) with:

N = (n 1 , n 2 , n 3 , n 4 ) E = (e 1 ,
e 2 , e 3 , e 4 , e 5 , e 6 , e 7 )

L(n 1 ) = A P(n 1 ) = 1 e 1 = (n 1 , n 2 ) T (e 1 ) = [10, 24] L(n 2 ) = B P(n 2 ) = 2 e 2 = (n 1 , n 2 ) T (e 2 ) = [30, 35] L(n 3 ) = C P(n 3 ) = 2 e 3 = (n 1 , n 2 ) T (e 3 ) = [45, 62] L(n 4 ) = A P(n 4 ) = 3 e 4 = (n 2 , n 4 )
T (e 4 ) = [0, 5] e 5 = (n 2 , n 4 ) T (e 5 ) = [START_REF] Kusuma | Process mining in cardiology: A literature review[END_REF]25]

G-PsM TG-PsM

G-P sM = (N, E, L) T G-P sM = (N, E, L, T ) N = N 1 ∪ • • • ∪ N k N = N 1 ∪ • • • ∪ N k L(x) ∈ A, x ∈ N L(x) ∈ A, x ∈ N P(x) = k, x ∈ N k P(x) = l, x ∈ N l T ((x, y)) = [a (x,y) , b (x,y) ], (x, y) ⊂ E Div N = |{L(n)} n∈N | Div N = |{L(n)} n∈N | Div E = |{(L(x), P(x), L(y), P(y))} (x,y)⊂E |

Process Model Complexity Characterization

The larger a process model is, the better it represents traces from an event log, but the drawback is that the more convoluted it becomes for someone to understand. Building a process model while controlling its complexity is crucial. A process model's complexity is described by its number of nodes |N | and edges |E|. Process model diversities are defined similarly to event log diversities (Definition 6). Definition 9. (Node and edge diversities). For a process model G-P sM or T G-P sM , the node diversity is Div N = |{L(n)} n∈N |. For a time grid process model T G-P sM , the edge diversity is Div E = |{(L(x), P(x), L(y), P(y))} (x,y)⊂E |.

Diversity descriptors characterize the variety of nodes and edges of a process model. A high diversity means that only few nodes (or edges) have the same label, whereas a low diversity indicates many similarly labeled nodes (or edges). For instance, in the G-P sM of Figure 1a, |N | = 5, Div N = 3 and |E| = 5, and in the T G-P sM presented in Figure 2, |N | = 4, Div N = 3, |E| = 7 and Div E = 3. This simple example highlights the increase of |E| for T G-P sM compared with that of G-P sM . Notations of the process models are summarized in Table 3.

Replayability

To evaluate the capacity of a process model to represent a trace, a new replayability score has been devised to match the newly defined grid process models. Initially, preliminary definitions are required. Algorithm 1 Grid and Time grid replayability games.

1: Initialization 2: z ← 0, δ ← 0, φ ← 0, m ← 0 3:
Find cm the first replayed event of σ 4:

If no event of σ replayed: 5:

return z, δ, φ 6:

Else: 7:

Set N actual as the node which replayed cm 8: Remark 1. For any transition in a trace we have, by definition, {time replayability} ⇒ {basic replayability} but {basic replayability} {time replayability}. To calculate the replayability of a trace, an algorithmic procedure is presented, named Grid replayability game (or Time grid replayability game) (Algorithm 1). The grid replayability game starts from c m , with m being the index of the first event of σ replayed in G-P sM (line 3). If c m is replayed by several nodes of G-P sM , the node with the lowest position is chosen (line 7). The next event c m+1 of σ possibly replayed by G-P sM is sought, with a strictly superior node position than the previous node which replayed c m (lines [START_REF] Reijers | A study into the factors that influence the understandability of business process models[END_REF][START_REF] Silverman | Density Estimation for Statistics and Data Analysis[END_REF]. If the transition (c m , c m+1 ) is not basic-replayed by G-P sM , the transition is said strongly-forced. If the transition ((c m , c m+1 ), t m,m+1 ) is not time-replayed by T G-P sM , the transition is said time-forced (lines [START_REF] Van Der Aalst | Introduction. In Process Mining: Discovery, Conformance and Enhancement of Business Processes[END_REF][START_REF] Van Der Aalst | Workflow mining: discovering process models from event logs[END_REF]. This process is repeated until the last replayable event is reached (line 10). If at least one event of σ has not been replayed while being in between two replayed events, it is said skipped (lines 17-18).

z ← z +
The strictly ascending condition of Definition 7 ensures that transitions are achieved by increasing positions during the replayability game. Thus, some events of σ might not be replayed during the game, even if they would have been according to Definition 10. Based on these new replayability games, adapted replayability score functions are introduced. Definition 11. (Replayability score). Considering the Grid replayability game, the replayability score of a sequence σ in a G-P sM or T G-P sM is defined as follows:

R(G-PsM, σ) or R(TG-PsM, σ) = ( z |σ| -α * δ -β * φ |σ| ) +
where:

• z is the number of events of σ replayed by G-P sM ;

• δ a binary variable equal to 0 if no event of σ is skipped;

• φ is the number of (timed) strongly-forced transitions;

• α, β are weighting factors.

Proposition 2. For a given trace σ, a process model T G-P sM and a G-P sM obtained with nodes and simple edges of T G-P sM : R(TG-PsM, σ) ≤ R(G-PsM, σ)

Proof. Proposition 2 translates the strictest character of the time grid replayability compared to grid replayability, for fixed coefficients.

Problem Formulation for Process Model Discovery

Let L be an event log, the process model optimization problem consists in determining an optimal grid process model G-P sM defined on L, maximizing the replayability and under some process model complexity constraints.

(GridOpt) max

G-PsM=(N,E,L) R(G-PsM, L) (2)
with

R(G-PsM, L) = 1 |L| σ∈L R(G-PsM, σ)
subject to

E ⊆ N × N (3)
max(P(x) x∈N ) ≤ p max (4) |N | ≤ U N ( 5
)
|E| ≤ U E (6)
where R(G-PsM, σ), R(G-PsM, L) ∈ [0, 1], p max ∈ N * is the maximum position for the process model construction, U N ∈ N * and U E ∈ N are the process model node and edge complexity bounds, respectively. The problem of determining an optimal time grid process model, denoted as TimeGridOpt is similar, with the following supplementary constraint:

∀e ∈ E, T (e) ∈ T E (e) (7)
where

T E (e) = ([a j , b j ]) j∈N * is a pre-defined set of disjointed intervals, with ∀e ∈ E, ∀I = [a, b] ∈ T E (e), a, b ∈ N and a < b.
The purpose of this constraint is to have accurate but specific values of time intervals for each edge. Thus, we first define judicious intervals for each edge, before optimization of the time grid replayability score. For example, let x and y be two nodes of a time grid process model T G-P sM . After looking at all possible causal transitions (L(x), L(y)) in event log L, we obtain D the distribution of corresponding time values. Let m -and m + be the maximum and minimum values of D, respectively. An optimal solution in terms of complexity and replayability is to take the single edge with the full interval [m -, m + ]. Inconveniently, this does not highlight time particularities and specific intervals. This is why dividing the interval [m -, m + ] in relevant sub-intervals {[x i , y i ]} i where min i x i ≥ m -, max i y i ≤ m + and i {[x i , y i ]} = ∅ would prove more advantageous.

For both problems, an optimal process model has to be found in terms of nodes (with their labels and positions) and (time-) edges (basic or with a specific time interval). Complexity being a hard constraint, it can be useful to first define an optimal solution without complexity constraints, to illustrate the complexity of a potential optimal model. Proposition 3. For any log L with p max = max σ∈L |σ|, U N = div e,p and U E = σ∈L (|σ| -1), there exists a process model G-P sM such that R(G-P sM, L) = 1.

Proof. All traces are perfectly replayed in a process model with A lab,pos as the set of nodes and arcs connecting nodes of position p to position p + 1 corresponding to direct causal relations from events of position p in some trace.

The Figure 3 shows an example of the process models described previously, with A lab = {A, B, C, D, E, F } and |σ max | = 6. Proof. Trivial as, at each step of the replayability game, the closest event of σ with label equal to the label of a node of higher position gives the next node. This is independent of arcs.

Optimization and Process Discovery

This section presents five algorithms : Random Search (RS), Frequent Model (FM), Spring Search (SS), Tabu Search (TS) and Tabu Search with Optimal Edges (TSOE). These algorithms are used to solve the optimization problems GridOpt and TimeGridOpt (defined through Equation 2). Among them, one heuristic (SS) and two meta-heuristics (TS and TSOE) are tested, modifying a current graph solution (or creating a neighborhood of new solutions). Transformation of the current solution is done by achieving some moves of nodes and edges. Before presenting these methods, the data preparation process is detailed in the following.

Data Preparation

To select nodes or edges to add (or delete) during the optimization process, a function f n : A× 1, p max → N is defined. For each tuple (l, p) with l ∈ A and p ∈ 1, p max , a value corresponding to its number of appearances within the event log is assigned. An event's position is either its real position in its trace if |σ| max ≤ p max , or a rescaled value to ensure all positions to be between 1 and p max if |σ| max > p max . Similarly, a function f e : T C → N is defined regarding appearance of transitions in event log. An extension for T G-P sM is also proposed, considering transitions and time intervals to compute f e values. In the following, "promising" is employed to characterize a node n = (l, p) or an edge e = (x, y) (with T (e) for T G-P sM ) with a high value for f n (l, p) or f e (L(x), L(y)). These functions are useful to select promising nodes or edges to add during the optimization process.

Algorithm 2 Tabu search with Optimized Edges (TSOE) for GridOpt.

Spring Search (SS) for GridOpt

A simple greedy heuristic, called Spring search (iterative jumps) is initially utilized to solve the GridOpt problem. During the search, new solutions are proposed by iteratively increasing and decreasing the size of a solution, which brings diversity Each iteration consists of four steps: (1) delete all edges of the current solution, (2) delete K non-promising nodes, (3) add K new promising nodes, and (4) add U E edges (the edge complexity bound), selected from the subset of possible edges (depending on nodes present in the G-P sM ). At the end of an iteration, the obtained G-P sM becomes the current solution.

Tabu search (TS) for GridOpt

Similarly to [START_REF] Prodel | Optimal process mining for large and complex event logs[END_REF], a Tabu search is implemented to solve GridOpt. Because the strictly ascending condition increases the dependence of edges towards nodes, only one move is used to generate neighborhoods. At each iteration, a neighborhood is made of non-tabu neighbors which are obtained from the current solution in 4 steps : (1) delete a non-promising node and its surrounding edges, (2) consider the number of deleted edges as a budget X to reassign, (3) add a promising new node, and (4) add X new promising edges respecting the strictly ascending condition. Each neighbor is evaluated by computing replayability, and the best neighbor is kept as current solution.

Tabu Search with Optimized Edges (TSOE) for GridOpt

According to Proposition 4, edges do not intervene in the choice of the next node reached during the replayability game. Considering a solution without edges G-P sM * , the replayability R(G-P sM * , σ) of a trace σ ∈ L will have all possible transitions from a node x to a node y forced during the replayability game. If an edge from x to y is added to the solution, the replayability score R(G-P sM

* , L) = 1 |L| σ∈L R(G- P sM * , σ) will increase, by a coefficient c (x,y) = β |L| σ∈L f φ (σ,(x,y)) |σ|
, with f φ (σ, (x, y)) = 1 if transition (x, y) has been forced during the replayability game of σ, 0 otherwise. Thus, by computing c (x,y) for every possible transition (x, y), keeping top-U E transitions produces the best-edge configuration from G-P sM * , respecting constraint (6) of Equation 2.

From these observations, an adapted version of Tabu Search is defined, consisting in searching only for solutions without edges, and then for every solution in evaluating the best edges to add for an optimized replayability score. This method's advantage drastically reduces the search space to graphs without edges. Tabu Search with Optimized Edges is further detailed in Algorithm 2.

Kernel Density Estimation

TSOE for TimeGridOpt

Before solving TimeGridOpt, accurate but specific values of time intervals T E (e) for possible edges e ∈ E need to be defined. First, causal transitions of event logs are analyzed to determine corresponding time distributions. Thus, a 1-D clustering method based on Kernel Density Estimation (KDE) is used to construct the set of possible edges to add during the process model optimization. KDE is the construction of an estimate of the density function from observed data, using a kernel function [START_REF] Silverman | Density Estimation for Statistics and Data Analysis[END_REF]. Applied to each time distribution, the resulting function will be used to define clusters by considering local maxima of the function as centers of clusters. For each maximum, an interval [x max -, x max + ] is defined, as shown in Figure 4. A low value of gives small and precise clusters; a high value decreases precision, replays more transitions and increases the global replayability score.

Finally, the approach is the same as for TSOE GridOpt, except that edges to add come with the most suited time interval as found by 1-D clustering. For an unconnected solution, the algorithm will choose the top-U E timed-edges to get the final process model solution.

Computational Experiments

Log Generation

The following design of experiments, to test and compare the performances of the previously defined algorithms on various event logs, is presented here. Event logs of various sizes were generated to match real-life variability in data sets. All experiments were performed on an Intel Core i7 processor (2.8 GHz), 16 GB RAM, and Windows 10 OS. The algorithms were coded in Python 3.6.

Log Generation for GridOpt (without time)

Event logs are generated from process models, which must be created first. Three parameters are required to create a G-P sM : (i) a maximum position p max , (ii) an event diversity div e , and (iii) an event-position diversity div e,p . For a given combination of these three, a fully connected G-P sM is randomly created with all the edges respecting the strictly ascending condition. Then, traces are generated by selecting a graph's node (with higher probability for nodes at lower positions) and following a path in the model until a terminal node is reached (a node without any outgoing edge). A number N noise = Z * σ |σ| of noisy random elements (not in the graph) is added to traces, at random positions. We arbitrarily set Z = 0.1 (10% of noise in event logs). Resulting traces make a log.

Log Generation for TimeGridOpt (with time)

Logs are generated in the same way as for GridOpt, except that edges of generated models are now split in two categories:

• Without time pattern (each transition resulting from this type of edge will have a time between events respecting uniform law U(a, b));

• With time patterns (a set {N (µ i , s 2 i )} i∈N * of Gaussian distributions is defined, each transition following one of these laws, randomly selected).

The goal of the optimization is now to highlight temporal patterns in the discovered model with multiple timed edges. The following distributions are used for the log generation: with s the standard deviation. The allocation of edges to patterns (1-4) is uniform. For double or triple time patterns, the choice among distributions is equiprobable. The value of s is set to 25, to test the robustness of the method for time with variability while keeping patterns identifiable. The edge constraint U E is set to 80 (4 × U N ) to allow the model to add multiple edges. Other parameters are set as for previous design of experiments. For KDE clustering, = 0.05 × max(time-stamp) to have a precise time interval for edges.

Design of Experiments

For each of 15 combinations of (div e,p , div e , p max ) used to create logs, 10 random G-P sM are created. From each G-P sM , a log of 1,000 traces is generated. 5 methods are then applied to solve the problem: Random search (RS), simple Frequency model (FM) obtained by only taking most frequent nodes and edges, Spring search (SS), Tabu search (TS) and Tabu Search with Optimized Edges (TSOE). Figure 5 summarizes the design of experiments at hand. For the search algorithms, stopping criteria are the maximum number of iterations (x = 250) or the number of iterations without improvement (x = 25). The neighborhood's size for TS and TSOE is empirically set to 15, based on previous tests showing small variability in the best obtained solution's replayability. Similarly, the Tabu list's size is set to 15. The size constraints (number of nodes, number of edges and maximal position) are constant throughout the entire experimental design to always get an interpretative and comprehensive model. Constant parameters and constraint values are summarized in Table 4. Configurations of the design of experiments are listed in the left part of Table 5. For TimeGridOpt, configurations 2, 4 and 15 are tested.

GridOpt Results

The Figure 6 shows the evolution of replayability during the optimization process for each method, specifically applied to configurations 2 and 15. Graphs used to generate event logs are also tested on noised data to compare results of different methods with the initial model used for trace generation, without size limitations ("ROOT" in the figure). Median replayability among 10 event logs for each configuration is presented versus the number of iterations (maximum value for the number of iterations of each method is set to the minimum stopping criterion among 10 replications). ROOT and FM models are obtained by noniterative methods, their means are displayed by horizontal lines on the same figure for comparison purposes. For RS, an improving solution is rarely obtained, and the stopping criterion is more quickly reached compared to other iterative methods. The margin for improvement during search is small for complex data sets as visible by comparison of Figure 6b and Figure 6a. Furthermore, the gap between ROOT model and search solution strongly increases from Figure 6a to Figure 6b.

Event log description for GridOpt, design of experiments and resulting replayability of the best mined models are given in Table 5. Computation times are presented in Table 6. Neighborhood searches (TS and TSOE) systematically outperform other methods (including the heuristic SS and the frequency model FM). TSOE outperforms TS on 9 out of 15 data configurations, especially when div e = 5. Otherwise, TSOE and TS perform equally. TSOE scores ranges from 0.26 to 0.90. Lower values (< 0.30) are obtained for complex data configurations (div e,p = 300 and div e = 100), due to the model size constraints. The unconstrained model used for event log generation (ROOT, where |N | = div e,p ) systematically scores at 0.90 ± 0.01. Visual representations of the best models mined by TSOE are presented in Figure 7. Visualization of a process model is possible via a tablet application developed by the company HEVA for that purpose. Each graph is read from left to right, increasing positions. Circles represent nodes of the model, and flux from circles represents edges. The size of nodes and edges are proportional to the number of traces replayed by them during the replayability game. The first qualitative observation is the repetition of events with the same label (Figure 7a), with Label 1 or Label 4. The strong decrease in replayability score from Figure 7a to Figure 7b is visible in the decrease in node and edge size, as fewer traces are well represented. If we focus on edges, Figure 8 highlights this strong decrease. Within the optimization for edges, the leeway in replayability is reduced because of the decrease in the number of patients going through edge pathways (172 vs 38 patients in the example of Figure 8). For this reason, the effect of edge optimization in TSOE is less visible in more complex data sets as Figure 7b compared to Figure 7a.

TimeGridOpt Results

Results are presented in time values 100, 200 or 300) is also given. The replayability of the best mined models with TSOE for TimeGridOpt is slightly inferior to TSOE for GridOpt on the same event log. The number of incoherent edges, i.e. the edges not respecting time patterns, is low (5.6%, 7.0% and 6.0%) and thus is encouraging for the methodology presented. These incoherent edges characterize traces with transitions generated from no time pattern edges (25% of transitions following {U(0, 400)}) or noisy transitions obtained after adding noise to the event log. Visual representations of the best models are shown in Figure 9 for configuration 2. The general shape of the process model obtained by solving TimeGridOpt is similar to previous G-P sM graphs (Figure 9a). The time-focused representation of T G-P sM highlights the type of edges obtained after the optimization, corresponding to the amount of timed edges it contains. According to the simulated event log, the time pattern edges could be of 3 types: simple (one interval centered in 200), double (2 intervals centered in 100 and 300) or triple (centered in 100, 200 and 300), as shown in Figure 9b.

Real-life Case Study

Diabetes Mellitus

Diabetes Mellitus (DM) is a group of metabolic disorders, resulting in chronic hyperglycemia due to unregulated insulin secretion and/or action. Common forms of DM are type-1 and type-2. Type-2 diabetes is the most common form (90-95% of patients). It is mainly characterized by insulin resistance and relates to the lifestyle, physical activity, dietary habits and heredity. Type-1 diabetes is less frequent (5-10%) and is due to destruction of β cells of the pancreas [START_REF] Kharroubi | Diabetes mellitus: The epidemic of the century[END_REF]. Data Mining methods have been widely applied to DM data and supervised learning prevails (85% of studies). Moreover, clinical data sets were the most used [START_REF] Kavakiotis | Machine learning and data mining methods in diabetes research[END_REF]. Our method, unsupervised Process Mining, adds a new angle and diversity to existing approaches in DM research. This real-life case study shows how the newly developed approach helps to analyze patient pathways before the appearance of four identified complications.

Data and Methodology

Data originates from the French National Health Insurance (CNAM), where a group of 50,000 patients suffering type-1 or type-2 diabetes in 2008 was constructed. Within this population, 5,714 patients developed at least one of the following complications until 2016: stroke, amputation, infarctus or TCKD (Terminal Chronic Kidney Disease). For each complication and for each patient, a 2-year period of medical history was analyzed. A time grid process model was built for each complication over these 2-year periods. TSOE algorithm was used with the following parameters : models' size is |N | = 20, |E| = 4 × |N |, and otherwise as in Table 4. Events of different categories were available:

• Hospitalizations (diabetes, cardiovascular, surgery...);

• Complications (stroke, amputation, infarctus, TCKD);

• Other medical events (dialysis , insulin, emergency without hospitalization).

Other follow-up exams, much more frequent in patient pathways (around 70% of the events), were also available: general practitioner visits, glycated hemoglobin tests (HBA1C), glycemia tests, creatinine tests, etc. Discussions with medical experts led to the non-consideration of these exams as key nodes for the process model. Instead of studying their sequence and successions in the pathway, they were simply and usefully quantified within the period between two nodes (i.e. on an edge). The quantification of such events was performed on the final mined model: for each patient crossing an edge during the replayability game, a list of frequent exams is computed, and median values for each frequent exam are printed on the edges. An unconnected grid process model with best time grid process model's nodes was created first. Then, a number of edges |E|, equal to the Div E , are used to connect the grid process model using optimized edges.

Results

Descriptors and replayability performances are presented in Table 8. The evolution of the optimization for each event log is presented in Figure 10. Event log analyses using the descriptors show pathway differences between stroke, infarctus, amputation and TKCD. Indeed, the first two complications are characterized by shorter traces (|σ| q1,m,q3 : (1, 3, 4) and [START_REF] Giua | Control of Safe Ordinary Petri Nets Using Unfolding[END_REF][START_REF] Kavakiotis | Machine learning and data mining methods in diabetes research[END_REF][START_REF] Kusuma | Process mining in cardiology: A literature review[END_REF] vs. [START_REF] Kavakiotis | Machine learning and data mining methods in diabetes research[END_REF][START_REF] Kusuma | Process mining in cardiology: A literature review[END_REF][START_REF] Prodel | Discovery of patient pathways from a national hospital database using process mining and integer linear programming[END_REF] and [START_REF] Kharroubi | Diabetes mellitus: The epidemic of the century[END_REF][START_REF] Litchfield | Can process mining automatically describe care pathways of patients with long-term conditions in uk primary care? a study protocol[END_REF][START_REF] Prodel | Evaluation of discovered clinical pathways using process mining and joint agent-based discrete-event simulation[END_REF]), that is to say short and unstructured pathways compared to the other two. This difference between complications is also highlighted by the best final replayability scores: amputation and TCKD have lower scores (R T G : 0.75 and 0.70) compared with those of stroke and infarctus (0.87 and 0.86) because longer and more complex pathways are less easily replayed in a graph than shorter ones. These observations are illustrated by Figure 11. An example of frequent events' information can be seen in Figure 11b where a pattern of diabetes hospitalization before stroke is highlighted. For 206 patients concerned, time between events was 242 days on average. As an example of frequent exams, the median number of general practitioner visits is displayed ("MG : 5"). The grid structure, which allows duplicate labels in a process model, is particularly suitable in this case study. As shown in Figure 11a, a high number of "Other hospitalizations excluding surgery" and "Cardiology hospitalizations" are interesting patterns revealed by the grid process model. Time pathway analysis gives further opportunities for understanding patient pathways. As an example, the process model relating to the complication "amputation" (Figure 11a) shows globally unique short time pathways. On the opposite, the process model relating to the complication "stroke" (Figure 11b) presents diverse time pathways, with not only short duration transition.

Conclusion and Future Research

An extended methodology to create suitable process models for healthcare applications has been presented. Its Scientific contributions are multiple. New process models considering a grid structure and time patterns were mathematically defined. We formulated a set of descriptors to characterize the structure of such a process model and event log complexity. The establishment of a new property for grid process models leads to a novel search algorithm to mine optimized process models. The search incorporates the grid structure and includes time patterns upon construction of the process model. Computational experiments validate the overall performance of this approach. The interest of neighborhood-based searches to solve the problem was quantitatively shown: Tabu Search with Optimized Edges is more efficient for a small event diversity. A qualitative observation was made regarding the grid structure, representing with more fidelity the linearity of patient pathways over time. This improves the visualization of repeated events. The advantage of considering time within optimization was also spotlighted. In addition, the applicability of the method and the interest in patient pathways analysis is demonstrated by a case study. The grid structure, the time patterns and the display of certain frequent events on edges provide interpretative highlights for medical staff and decision makers.

Three opportunities for future work come to mind. Firstly, a focus on optimization performances for complex data sets will be made, by considering less strict constraints (nodes and edges). During the experiments presented in this work, constraints were specifically set to obtain an overall comprehensible model capable of being simply visually interpreted. However, increasing the complexity of the process model can be achieved if interactive tools permit the exploration of the final model wherein key elements are able to be clearly discerned. Secondly, studying the relation between event log descriptors, graph constraints and replayability of the best models minded is of important interest as well. Any results rendered will be useful for the calibration of constraints, particularly for the third research axis. Eventually, future research should focus on creating a methodology to perform supervised learning with traces as input data, whereas current state-of-the-art classification methods only take "flattened data" as input (vectors of features). A process model optimized for classification purposes will produce an explainable predictive model.

  Equivalent classic process model.

Figure 1 :

 1 Figure 1: Example of a grid process model and its classic process model equivalent.

  e 6 = (n 3 , n 4 ) T (e 6 ) = [2, 50] e 7 = (n 3 , n 4 ) T (e 7 ) = [75, 100]

Figure 2 :

 2 Figure 2: Example of a time grid process model.

Definition 10 .

 10 (Replayability). An event e is said replayed by a process model G-P sM if label(e) = L(x) for some node x of G-P sM . A causal relation (e k , e l ) is said basic replayed by G-P sM if label(e k ) = L(x) and label(e l ) = L(y) for some edge (x, y) of G-P sM . A timed causal relation (e k , e l , time(e l ) -time(e k )) is said time-replayed by an T G-P sM if label(e k ) = L(x), label(e l ) = L(y) and time(e l ) -time(e k ) ∈ [a (x,y) , b (x,y) ] for some edge (x, y) of the T G-P sM .

Figure 3 :

 3 Figure 3: Example of a process model described in Proposition 3.

Figure 4 :

 4 Figure 4: Illustration of KDE clustering.

1 .

 1 No time pattern U(0, 400); 2. Simple time pattern ({N (200, s 2 )}; 3. Double time pattern ({N (100, s 2 ), N (300, s 2 )}; 4. Triple time pattern ({N (100, s 2 ), N (200, s 2 ), N (300, s 2 )});

Figure 5 :

 5 Figure 5: Schematic representation of the design of experiments. Replayability parameters β 0.1 γ 0.1 Search parameters K (SS) 2 Neighborhood's size (TS and TSOE) 15 Size of Tabu list 15 Max. number of iterations 250 Max. number of iterations without improvement 25 Constraints Number of nodes UN 20 Number of edges UE 40 (2×UN ) Maximal position pmax min(10, |σ|max)

Figure 6 :Figure 7 :

 67 Figure6: Replayability versus the number of iterations: 6 different methods applied to three logs for the GridOpt problem; log of configuration 2 (Fig.6a) and configuration 15 (Fig.6b).

  Focus on the edges for T G-P sM models.

Figure 9 :

 9 Figure 9: Examples of the best T G-P sM models mined by the TSOE algorithm

Figure 10 :

 10 Figure 10: Replayability versus iterations for the four complication event logs.

Table 1 :

 1 Definition 4. (Causal relations). For each trace σ = e 1 , . . . , e m in L, all pairs of labels (label(e k ), label(e l )) such that 1 ≤ k < l ≤ m are called causal relations and pairs (label(e k ), label(e k+1 )) are called direct causal relations. Let T C be the set of all causal relations and T DC be the set of direct causal relations. Timed causal relations set: T tC = {(label(e k ), label(e l ), time(e l ) -time(e k ))} 1≤k<l≤m σ=e1,...,em∈L Timed direct causal relations set: T tDC = {(label(e k ), label(e k+1 ), time(e k+1 ) -time(e k ))} 1≤k<m σ=e1,...,em∈L Descriptors Traces length description: |σ| mean , |σ| std , |σ| max , |σ| min Event diversity: div e = |A| Event-position diversity: div e,p = |A lab,pos | Causal relation diversity: div causal = |T DC | Timed causal relation diversity: div t-causal = |T tDC | Event log notations. • causal relation diversity div causal = |T DC |; • timed causal relation diversity div t-causal = |T tDC |. Proposition 1. If A = ∅ and |σ| > 1 for at least one trace σ, then 1 ≤ div e ≤ div e,p ≤ σ∈L |σ| and 1 ≤ div causal ≤ div t-causal ≤ σ∈L (|σ| -1)

	{(label(e 1 ), 1), . . . , (label(e m ), m)}
	em∈L
	Definition 5. (Timed causal relations). All triplets (label(e k ), label(e l ), time(e l ) -time(e k )) such that
	(label(e k ), label(e l )) ∈ T C (or T DC ) are called timed causal relations (or direct timed causal relations). Let
	T tC and T tDC be the set of all timed causal relations and the set of direct timed causal relations.
	Definition 6. (Diversity measures). The event log L has the following diversity measures:
	• event diversity div e = |A|;
	• event-position diversity div e,p = |A lab,pos |;

Table 2 :

 2 An example of an event log of patient pathways.

	2, 5)}.

Table 3 :

 3 Process model related notations.

  If c m+1 can be replayed in G-P sM by a node Nnext with P(N actual ) < P(Nnext): 12: z ← z + 1, N actual ← Nnext (with the lowest position) 13:If no edge exists between N actual and Nnext (or if the transition between (cm, c m+1 ) cannot be time replayed (Time grid replayability game)):

		1
	9: Trace crossing
	10:	While m < |σ|:
	11:
	14:	φ ← φ + 1
	15:	m ← m + 1
	16: Skipped elements analysis
	17:	If at least one element of σ has been skipped (not been replayed between replayed elements):
	18:	δ = 1
	19: Conclusion
	20:	return z, δ, φ

Table 4 :

 4 Search parameters and constraints used for design of experiments.

Table 7 .

 7 For each data configuration, the mean number of incoherent edges (edges which do not correspond to any defined pattern through design of experiments, by not containing div e p max AVG STD AVG STD AVG STD AVG STD AVG STD AVG STD

		Data		RS	FM	SS	TS	TSOE	ROOT
	div e,p 1 25	5	5	0.33 0.08 0.87 0.00 0.86 0.00 0.87 0.00 0.90 0.00 0.91 0.00
	2	25	5	25	0.15 0.09 0.77 0.02 0.74 0.02 0.81 0.02 0.84 0.02 0.90 0.01
	3	100	5	25	0.24 0.04 0.70 0.01 0.70 0.02 0.75 0.01 0.77 0.01 0.90 0.01
	4	100	5	50	0.14 0.08 0.66 0.01 0.65 0.02 0.72 0.01 0.74 0.01 0.89 0.01
	5	100	50	5	0.19 0.04 0.34 0.01 0.35 0.02 0.44 0.01 0.44 0.02 0.90 0.01
	6	100	50	25	0.15 0.04 0.34 0.03 0.33 0.03 0.42 0.02 0.42 0.02 0.90 0.01
	7	100	50	50	0.11 0.02 0.34 0.04 0.33 0.03 0.41 0.02 0.42 0.02 0.89 0.01
	8	100 100	5	0.13 0.02 0.23 0.04 0.25 0.03 0.36 0.03 0.36 0.02 0.90 0.01
	9	100 100	25	0.13 0.04 0.29 0.03 0.27 0.04 0.37 0.02 0.38 0.02 0.90 0.01
	10 100 100	50	0.10 0.03 0.30 0.02 0.29 0.03 0.38 0.03 0.38 0.02 0.89 0.01
	11 300	50	25	0.15 0.02 0.26 0.02 0.26 0.01 0.31 0.01 0.31 0.02 0.89 0.01
	12 300	50	50	0.13 0.02 0.24 0.01 0.24 0.02 0.30 0.01 0.31 0.01 0.89 0.01
	13 300 100	5	0.13 0.02 0.20 0.01 0.21 0.01 0.26 0.01 0.26 0.01 0.89 0.01
	14 300 100	25	0.11 0.01 0.21 0.02 0.21 0.02 0.27 0.02 0.28 0.02 0.89 0.01
	15 300 100	50	0.10 0.02 0.21 0.02 0.21 0.02 0.27 0.02 0.29 0.02 0.89 0.01

Table 5 :

 5 The replayability score of the best models mined by different methods: average and standard deviation.

	Data	RS		SS		TS		TSOE
		AVG	STD	AVG	STD	AVG	STD	AVG	STD
	1	21	9	39	9	622	151	1479	392
	2	15	1	33	6	731	155	1825	419
	3	22	7	40	11	665	134	1591	619
	4	19	5	38	6	873	234	1930	598
	5	34	12	47	16	878	363	1828	362
	6	18	5	39	12	711	191	1463	321
	7	18	5	33	7	726	162	1541	407
	8	15	1	40	12	679	184	1182	335
	9	15	<1	33	10	678	162	1513	398
	10	15	<1	35	8	689	150	1416	454
	11	19	5	38	8	639	99	1353	411
	12	18	2	38	8	745	182	1404	334
	13	15	<1	37	6	785	213	1354	347
	14	16	1	43	8	656	132	1406	253
	15	16	<1	36	7	674	173	1497	257

Table 6 :

 6 Computation time (in seconds) of each method: average and standard deviation.

	Data	Replayability	Time		Incoherent edges
	Config.	AVG	STD	AVG	STD	AVG
	2	0.81	0.02	9820	3048	5.6%
	4	0.72	0.01	7965	2547	7.0%
	15	0.28	0.02	10999	3688	6.0%

Table 7 :

 7 Best models mined by TSOE for TimeGridOpt: replayability, time (in seconds) and percentage of incoherent edges.

Table 8 :

 8 Diabetes event log descriptors and replayability score.

(a) Configuration 2 (b) Configuration 15Figure 8: Focus on the edges for G-P sM models.

Acknowledgement

The authors wish to thank Chris Yukna for his help in proofreading.