Nabil Benzina
email: nabil.benzina@inserm.fr

Margot Morgiève

Marine Euvrard

João Flores

Alves Dos Dos Santos

Antoine Pelissolo

Luc Mallet

Personalised 360o video exposure therapy for the treatment of obsessive-compulsive disorder: A single case study

Keywords:

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Introduction 1.OCD and its treatment

With a lifetime prevalence of approximatively 2 to 3% world-wide [START_REF] Abramowitz | Obsessive-compulsive disorder[END_REF], Obsessive Compulsive Disorder (OCD) is a disabling disorder characterised by either obsessions (recurrent, persistent, intrusive and unwanted thoughts), compulsions (repetitive behaviours or mental acts performed through rigid rituals) or both symptoms. Conventional patient care relies on the one hand on selective serotonin reuptake inhibitors (SSRI) and on the other on cognitive behavioural therapies (CBT); alone or in combination [START_REF] Skapinakis | Pharmacological and psychotherapeutic interventions for management of obsessive-compulsive disorder in adults: a systematic review and network meta-analysis[END_REF]. SSRIs have shown their efficiency in reducing obsessive compulsive symptoms but only for 40 to 60% of patients [START_REF] Hirschtritt | Obsessive-Compulsive Disorder: Advances in Diagnosis and Treatment[END_REF]. CBT, and particularly the exposure and response prevention (ERP) techniques have also been demonstrated as being efficient [START_REF] Mckay | Efficacy of cognitive-behavioral therapy for obsessive-compulsive disorder[END_REF]. However, 30% of patients fail to display any significant improvement in their symptoms [START_REF] Cottraux | Combining Pharmacotherapy with Cognitive-Behavioral Interventions for Obsessive-Compulsive Disorder[END_REF] which might be explained, in the case of imaginal exposure, by the fact that 33% of the patients show poor imagery ability [START_REF] Arabian | Individual Differences in Imagery Ability and Pavlovian Heart Rate Decelerative Conditioning[END_REF] and that they attempt to set-up cognitive avoidance strategies during this kind of exposure [START_REF] Laforest | Effectiveness of In Virtuo Exposure and Response Prevention Treatment Using Cognitive-Behavioral Therapy for Obsessive-Compulsive Disorder: A Study Based on a Single-Case Study Protocol[END_REF]. Concerning in vivo exposure, this technique requires patients to put themselves in real, highly anxiety-provoking situations, resulting in a high attrition rate with 25 to 30% of patients dropping out of therapy early [START_REF] Abramowitz | The Psychological Treatment of Obsessive-Compulsive Disorder[END_REF]. Thus, 20 to 40% of patients remain unresponsive to these first-line treatments, leading to chronic disabling functional disorders [START_REF] Pallanti | Treatment-refractory obsessive-compulsive disorder: Methodological issues, operational definitions and therapeutic lines[END_REF]. We therefore need to find novel therapeutic approaches. While drug research is struggling to find new treatments, psychotherapy is most likely to offer new approaches that could improve the treatment of OCD. Indeed, the development of new technologies has given increased impetus to CBT, particularly in ERP. More precisely, the development of virtual reality (VR), made possible by the latest technological advances opens the way to a potentially increased efficiency and a broader application of these exposure techniques, especially in the most severe patients who are often reluctant to be exposed to highly anxiogenic situations. Thus, VR could allow step-by-step exposure to anxiogenic stimuli in a much more realistic and controlled manner than imaginal exposure, and in a realistic although safer situation than exposure in vivo.

VR contributions to OCD patient therapeutic care

In VR, patients are immersed interactively in real time in a two or three-dimensional computergenerated environment with the possibility to generate multimodal sensory stimuli like sounds, odours, etc... Three main categories of VR exist ranked by the degree of presence they provide: non-immersive systems where the virtual environment (VE) is viewed through a highresolution monitor; semi-immersive projection systems where the VE is generated by a graphics computer system coupled with large unidirectional projector; and fully-immersive systems consisting in either head-mounted display or three-dimensional projector (CAVE) system. VR can address some inherent limitations of the classical ERP techniques notably by its flexibility (adjustment of the environment to the patient's requirements), reproducibility (each scenario can be accurately repeated), controlled nature (adjustment of the level of the anxiety-triggering stimuli), safety (exposure to a "false" situation that can be immediately terminated with no adverse consequences), more realistic nature than in imaginal exposure and confidentiality (occurs in the consulting room thereby avoiding public embarrassment) [START_REF] North | Virtual reality therapy: an effective treatment for psychological disorders[END_REF].

VR has been extensively used for more than twenty years for anxiety disorders with rather good results, but mainly in a research framework [START_REF] Eichenberg | Virtual Realities in the Treatment of Mental Disorders: A Review of the Current State of Research[END_REF]. However, the literature on the use of VR for OCD patients remains scarce. The first VR-like study for OCD patients dates back from 1998 [START_REF] Clark | A Pilot Study of Computer-Aided Vicarious Exposure for Obsessive-Compulsive Disorder[END_REF]. It provided promising results using a vicarious learning training based on interactive pictures. OCD patients had to learn to self-expose without performing their rituals, by looking at an avatar that suffered from the same symptomatology. At the end of the sessions, these patients displayed a modest but significant reduction in symptom severity on the Yale-Brown Obsessive-Compulsive scale (Y-BOCS). Ten years later, Kim and collaborators examined VR's ability to induce anxiety for OCD participants using a head-mounted display [START_REF] Kim | Anxiety Provocation and Measurement Using Virtual Reality in Patients with Obsessive-Compulsive Disorder[END_REF]. Not only did VR trigger checking symptoms and anxiety, but this anxiety was positively correlated with the severity of the disorder and the subject's propensity to immerse in the VE. These results were reproduced in 2012 with patients suffering from symmetry/order OCD [START_REF] Kim | Provoked arrangement symptoms in obsessivecompulsive disorder using a virtual environment: A preliminary report[END_REF]. Two years later, Belloch [START_REF] Belloch | Virtual reality exposure for OCD: is it feasible?[END_REF] used a high-resolution monitor coupled with a movement tracker and demonstrated that VR is able to generate anxiety and disgust proportionally to the degree of dirtiness of the scenario administered to patients suffering from washing/contamination OCD.

Laforest described the same results using a CAVE system, for the same category of symptoms [START_REF] Laforest | Inducing an Anxiety Response Using a Contaminated Virtual Environment: Validation of a Therapeutic Tool for Obsessive-Compulsive Disorder[END_REF]. More recently, van Bennekom, using first-person videos shot in a real environment and reproducing scenarios relevant to the subjects (dirty toilet, gas stove left on, etc...) displayed on a screen, demonstrated the ability of this kind of VR to induce OCD symptoms [START_REF] Van Bennekom | A Virtual Reality Game to Assess Obsessive-Compulsive Disorder[END_REF].

Therefore, VR seems able to trigger OCD symptoms, which is a crucial step for exposure treatments, but few studies attempt to assess its efficiency to treat OCD symptoms. A decrease in OCD symptoms in three patients was reported after a VR immersive exposure (CAVE system); even though this improvement was not maintained over the long term [START_REF] Laforest | Effectiveness of In Virtuo Exposure and Response Prevention Treatment Using Cognitive-Behavioral Therapy for Obsessive-Compulsive Disorder: A Study Based on a Single-Case Study Protocol[END_REF]. Interesting results have been shown with a VR and vicarious learning procedure [START_REF] Matthews | Online computer-aided vicarious exposure for OCD symptoms: A pilot study[END_REF]. The patients had to learn to expose without performing their rituals while exploring the VE with the help of an avatar displayed on the screen. At the end of the study, the patients displayed a modest but significant reduction of the severity of OCD symptoms (a reduction of 2.4 points in the total YBOCS score on average) at one-month post-treatment.

Limitations of VR studies

These preliminary results on the use of VR as an exposure technique are limited by the small number of studies and their sample size, requiring their reproduction with larger samples and with control groups that underwent the current gold standard therapy, namely, in vivo ERP. One of the factors affecting the effectiveness of VR as an exposure technique is its ability to immerse the subject in the anxiety-provoking situation. Thus, most studies report a high degree of presence (perception of the virtual environment as real and inducing a disconnection from the real world), but we don't yet know how the behaviour in the VE can predict that in the real world. Moreover, it is not clear if the anxiety triggered during VE exposure is associated with the patient's disorder or induced by cybersickness (due to the discordance between the visual and vestibular perceptions as a result of the temporal offset between the head and the environment movements). This raises the need to evaluate anxiety by recording objective parameters such as heart rate or finger/palmar skin conductance which would be a more reliable measure of the patient's emotional state [START_REF] Laforest | Inducing an Anxiety Response Using a Contaminated Virtual Environment: Validation of a Therapeutic Tool for Obsessive-Compulsive Disorder[END_REF]; the latter not being affected by cybersickness [START_REF] Gavgani | Profiling subjective symptoms and autonomic changes associated with cybersickness[END_REF].

Considering the technical aspects of VR, it is necessary to ensure a better ecological validity through more realistic scenarios that could be personalised and inspired by the patient's own life [START_REF] Kim | Development of a computer-based behavioral assessment of checking behavior in obsessive-compulsive disorder[END_REF]. This can be achieved by improving the objective realism of the VE on the basis of the improved graphics quality now available. One could object that previous studies have shown that these types of VE were still able to trigger OCD symptoms, but no researcher has verified that the triggered symptoms are equivalent to the ones the patients have to cope with in their daily life. Slater & Wilbur [START_REF] Slater | A Framework for Immersive Virtual Environments (FIVE): Speculations on the Role of Presence in Virtual Environments[END_REF] and Youngblut [START_REF] Youngblut | What a Decade of Experiments Reveals about Factors that Influence the Sense of Presence: Latest Findings: Fort Belvoir[END_REF] support the need of an improved objective realism of the VE in their respective review by stressing that graphics quality of the VE influences the feeling of presence of the subject. The more realistic the VE, the more immersed the subject is. Nevertheless, according to Bouchard [START_REF] Bouchard | Virtual Reality and the Training of Military Personnel to Cope with Acute Stressors[END_REF], emotion also plays an important role in the subject's immersion with a positive correlation between anxiety and the subject's immersion levels [START_REF] Bouchard | Anxiety Increases the Feeling of Presence in Virtual Reality[END_REF] and vice-versa [START_REF] Diemer | The impact of perception and presence on emotional reactions: a review of research in virtual reality[END_REF]. Besides, beyond the objective realism of the VE, the subjective realism (interpreting the VE as real) is directly involved in the immersion of the subject and is based on their expectations, beliefs and the propensity of the scenario to be meaningful to them [START_REF] Bouchard | Manipulating subjective realism and its impact on presence: Preliminary results on feasibility and neuroanatomical correlates[END_REF]. Moreover, the use of a familiar environment could improve the immersion into the VE as opposed to standardised environments [START_REF] Vecchiato | Electroencephalographic Correlates of Sensorimotor Integration and Embodiment during the Appreciation of Virtual Architectural Environments[END_REF]. It is therefore crucial to focus on ecological validity with a relevant and personalised scenario to ensure a better immersion.

Rationale and objectives of our study

In order to improve the ecological validity and the immersivity of existing VR techniques, we decided to base our therapy on first-person view 360° acquired videos. This technique allows the best ecological validity ever by completely acquiring the subject's environment in order to construct meaningful scenarios. This first-person view also allows the subject to slip into the VE as if it was reality. Some studies have shown that a subjective view is sufficient for the subject to accept the virtual body as their own despite the discrepancies in motion between the virtual moving body and the static real body; both at the subjective [START_REF] Kokkinara | First Person Perspective of Seated Participants Over a Walking Virtual Body Leads to Illusory Agency Over the Walking[END_REF] and neurophysiological level [START_REF] Pavone | Embodying Others in Immersive Virtual Reality: Electro-Cortical Signatures of Monitoring the Errors in the Actions of an Avatar Seen from a First-Person Perspective[END_REF]. The use of the first-person subjective view has also demonstrated its ability to induce cognitive changes depending on the type of the avatar and its actions [START_REF] Banakou | Virtual Embodiment of White People in a Black Virtual Body Leads to a Sustained Reduction in Their Implicit Racial Bias[END_REF]. Just as interesting , embodying an avatar that does not display the patient's symptoms induces an increased motivation to change [START_REF] Serino | A Novel Technique for Improving Bodily Experience in a Non-operable Super-Super Obesity Case[END_REF]. Lastly, the work of Clark [START_REF] Clark | A Pilot Study of Computer-Aided Vicarious Exposure for Obsessive-Compulsive Disorder[END_REF] and Matthews [START_REF] Matthews | Online computer-aided vicarious exposure for OCD symptoms: A pilot study[END_REF] suggests that embodying a subject that acts normally in anxiogenic situations sets up vicarious learning processes and increase the feelings of self-efficiency [START_REF] Bandura | Self-efficacy: toward a unifying theory of behavioral change[END_REF].

Thus, the main objective of this single case study is to evaluate the feasibility of implementing a personalised immersive 360° video exposure treatment for severe OCD refractory to conventional ERP techniques and first-line drug treatments. The secondary goal aims at evaluating the therapeutic potential of such a method by analysing the changes in a series of therapeutic response indicators (symptomatology severity, quality of life, etc…).

Methods

Participant case

Ms. V., a fifty-year old female patient, was under psychiatric care at the Albert Chenevier hospital (Créteil, France) for a severe (YBOCS score of 33) and treatment resistant contamination/washing OCD with poor insight and from which she had been suffering since her teen years. Nearly 8 hours per day were devoted to the washing compulsions. She had issues with getting dressed, had disproportionate washing protocols, could not stand being in contact with people or any "outside" objects she considered as being contaminated. She did not fear getting ill, but she was anxious about becoming "contaminated". These symptoms drove a serious disability as she was socially isolated (no one has even entered her home in eight years), had not worked for four years, had very little leisure and had hoarding issues in her house (objects fallen on the floor being considered contaminated). With regard to comorbidities, she also has a moderate dysexecutive syndrome (highlighted by a neuropsychological assessment performed before the start of the study) associated with a history of polysubstance addiction (mainly cocaine and heroin with occasional use of cannabis and MDMA; being totally abstinent for 2 years) and a moderate depressive symptomatology (BDI score of 12) without major depressive disorder (MDD).

She failed to respond to several pharmacological treatment approaches including three SSRIs at optimal dose and duration and one serotonin-norepinephrine reuptake inhibitor; each alone or in combination with lithium, atypical antipsychotics (risperidone and aripiprazole, but discontinued early by herself due to reported gonalgia) or buspirone. She failed to respond to two group CBT protocols (the first in 2010 was aborted after 4 sessions and the second in 2015 after 9 sessions) due to the influence of other participants' symptoms on her own symptomatology (generation of new obsessions) and a reluctance to perform home exercises.

She also was refractory to in vivo exposure initiated in 2016 with the patient's drop out after 3 sessions and the impossibility to perform the tasks at home due to a complete intolerance to the high level of anxiety generated. At the beginning of the study, Ms. V. had been on clomipramine treatment (75 mg/d) for 6 weeks and her treatment continued throughout this study.

Scenario filming

Three forty-five-minute scenarios were created by the patient to fit in with her everyday-life struggles such as going shopping and trying on clothes in a shop. These were rated according to a fear-scale. All the scenarios started at the patient's home, were then filmed in her neighbourhood and ended with her returning home.

We used a 360° 4K camera to film the different scenarios (KeyMission 360, Nikon). This camera allows both high-resolution 360° image acquisition and 360° sound recording in order to ensure the best quality possible (avoiding pixelization/fuzzy images) for the films for optimal immersion. During the filming sessions, the camera was carried by a female subject in our team (MM) with the same physical characteristics, and wearing the same kind of clothes as Ms. V.

Once the films were recorded, they were uploaded to a private Youtube account in order to allow the patient self-directed exposure to anxiogenic sequences at home.

Scenario administration

We used a high-end VR helmet (Homido V2) and a light and flexible audio helmet (Philips SHL3060BK). The VR helmet was combined with a smartphone (G3, LG) having the advantage of being cheap while displaying a large 5.5" screen with a ultra-high resolution (2560*1440) which is crucial for high immersivity and highly recommended in VR to avoid deleterious image pixelisation (Figure 1A). Besides, it allowed us to perform a screen mirroring on the computer to see live what the patient was looking at during the session in order to prevent avoidance (Figure 1B). During the video exposure, we recorded the palmar galvanic skin response (GSR) to monitor the patient's anxiety level (Figure 1B). When the patient was at home, she could login on a dedicated private Youtube account and use a Google Cardboard like VR helmet for self-directed exposure.

Protocol

The protocol was initiated in March 2017, at the Psychiatry Department of Albert Chenevier hospital (Créteil). This therapy was offered to the patient as part of the routine medical care provided by the unit and was approved by the ethics evaluation committee of INSERM (approval n°14-161). The patient gave her informed consent prior to the beginning of the therapy. It started with a baseline clinical assessment: obsessive/compulsive symptom level (Y-BOCS [33] and Obsessive-Compulsive Inventory Revised (OCI-R [START_REF] Zermatten | Validation of a French version of the Obsessive-Compulsive Inventory-Revised in a non-clinical sample[END_REF]); depressive symptom level (BDI [START_REF] Collet | Inventaire abrégé de la dépression de Beck (13 items): Étude de la validité concurrente avec les échelles de Hamilton et de ralentissement de Widlöcher. / The shortened Beck Depression Inventory: Study of the concurrent validity with the Hamilton Depression Rating Scale and the Widlöcher retardation Rating Scale[END_REF]); quality of life (brief version of the WHO Quality of Life scale, WHOQOL-26 [START_REF] Harper | Development of the World Health Organization WHOQOL-Bref quality of life assessment[END_REF]). 15 weekly therapeutic sessions followed. The first consisted of the validation of the patient elaborated scenarios and the explanation of the protocol and its aims.

The patient's ability to be immersed in a virtual environment (sense of presence) was also assessed using the Immersive Tendencies Questionnaire (ITQ [START_REF] Witmer | Measuring Presence in Virtual Environments: A Presence Questionnaire[END_REF]). A neutral scenario was then administered in order to get used to the system and to assess cyber-sickness with the Simulator Sickness Questionnaire (SSQ [START_REF] Kennedy | Simulator Sickness Questionnaire: An Enhanced Method for Quantifying Simulator Sickness[END_REF]). After this session, the scenarios were filmed.

Then all the sessions were organised the same way. They started with the evaluation of OCD symptom evolution since the latest session and the patient's self-exposure homework. The anticipatory anxiety level was assessed just before the VR projection with the Spielberger's State Anxiety Inventory (STAI-A [START_REF] Gauthier | A French-Canadian adaptation of the revised version of Spielberger's State-Trait Anxiety Inventory[END_REF]). The patient then visualised the scenario while palmar GSR was recorded. Immediately following the scenario projection, the level of cybersickness was assessed (SSQ). Then, the exposure session was discussed with the patient using all the data recorded during the session in order to obtain her feedback about her feelings and changes since the latest session. Between sessions, the patient was encouraged to self-expose at least once a day and to try to reproduce in real life the actions experienced in the VE.

The shift to the next most anxiogenic scenario depended on good tolerance of the previous one corresponding to: a decrease in electrodermal reactivity, the subjective evaluation of the anxiety by the patient and the behavioural manifestations of anxiety (disgust reactions, closed fist, complaining…) during exposure.

The final evaluation took place after the 15 th session and focussed on OCD and depressive symptoms and the quality of life.

Results

During the first session, Ms. V. was administered a ten-minute neutral video and showed little cyber-motion-sickness (9/48 at the SSQ). She also demonstrated ease to slip into the VE with a total ITQ score of 72/126. The following 14 sessions were dedicated to the administration of the OCD related videos (Figure 1C). The patient found the videos very immersive and also anxiogenic: she had the feeling her hands were dirty and wanted to wash them. She was living the scene as if she was really in it and had obvious reactions of disgust. Visual avoidance of the anxiogenic parts of the environment was often present at the beginning. She was thus verbally encouraged to visualize these parts when visual avoidance was detected (while minimizing these interventions to limit the impact on immersion) by the monitoring of her gaze through the screen mirroring. Progressively, visual avoidance disappeared and she displayed good habituation with a decrease in the electrodermal reactivity, the STAI-A score (46 to 37/80 for the first scenario and 67 to 37/80 for the second) and the behavioural markers of anxiety (more relaxed attitude).

Although she did not self-expose regularly to videos (due to the use of her own smartphone, less powerful than the one used in session, with a "Google like" cardboard that we provided her, resulting in cybersickness) as assessed by the low number of views on YouTube, Ms. V.

showed a moderate clinical improvement (Figure 2) with a 5-point drop in the Y-BOCS score.

At OCI-R, there was a decrease of one point on the washing dimension, 3 points in the accumulation dimension but also a decrease of 4 points in the order dimension. The decline in the latter two dimensions reflects her clinical improvement, allowing Ms. V. to deal with hoarded objects and thus be less concerned with order. In addition, depressive symptomatology improved with the BDI score dropping 8 points to 4. This clinical improvement was reflected in real life by an expansion of Ms. V's repertoire of actions, with the possibility, for example, of taking her phone in her bag, reproducing a shopping scene in a store, hugging a friend (with less disgust according to her). More generally, she reported more voluntary exposure to anxiogenic stimuli ("this is the first time I challenge myself a little bit every day"). Among the actions that Ms. V. manages to carry out again, some were not included in any of the scenarios presented, indicating a generalisation process. In addition, Ms. V. reported that she sometimes surprised herself by not paying attention to "dirt", indicating a possible reduction in attention bias. In addition, Ms. V. accepted a temporary job when she had not worked for 4 years. All these elements have contributed to an improvement in her quality of life, as evidenced by the increase in WHOQOL-26 perceived physical and mental health scores (Figure 2).

Discussion

This proof of concept study aimed at first highlighting the feasibility and the good tolerance of personalised immersive 360° video exposure to treat OCD. The acceptance and compliance of the patient towards this therapy despite the severity of her disorder is an encouraging evidence that personalised immersive 360° video exposure can be easily implemented to treat patients with severe symptoms and a major reluctance to accept more classical ERP techniques.

The technology underlying this type of virtual exposure is also a guarantee of its facilitated implementation due to the use of inexpensive equipment (approximately $400 for a complete video recording and viewing system). Concerning in situ video recording of exposure scenarios, it is quite easy to imagine involving the paramedical staff in a hospital environment or alternatively the patient's family in a logic of empowerment of the patient and his relatives.

Considering self-exposures, the experience of our patient using her own smartphone which proved to be unsuitable for 360° 4K video streaming, demonstrates that it is necessary for future studies to provide patients with equipment that is compatible with the latency-free streaming of this video type.

Considering the therapeutic potential of immersive 360° video exposure, these preliminary results are promising with, in addition to clinical improvement (5-point drop at the Y-BOCS), a substantial improvement in her quality of life (resulting in a 25-point improvement in the physical health dimension and a 19-point improvement in the psychological health dimension of WHOQOL-26).

Nevertheless, the patient was on clomipramine treatment initiated 6 weeks before the start of the protocol. The clinical improvement could thus be related to this drug and not to video viewing, or at least, the drug could have potentiated the exposure therapy. Although the oral dose of clomipramine was well below the recommended dose in OCD (75 mg/d for an effective dose in OCD around 250 mg/d [START_REF] Hirschtritt | Obsessive-Compulsive Disorder: Advances in Diagnosis and Treatment[END_REF]), the high interindividual variability in the clomipramine pharmacokinetics and the lack of plasma measurements prevent this hypothesis from being ruled out. Another confounding factor is the presence of depressive symptoms at the start of the study which is an influencing factor of OCD severity, even without comorbid MDD [START_REF] Rickelt | The relation between depressive and obsessivecompulsive symptoms in obsessive-compulsive disorder: Results from a large, naturalistic follow-up study[END_REF]. Thus, our results could be explained by the depressive symptom improvement. However, it has been shown that depressive symptomatology does not influence treatment response but rather it is OCD symptom improvement that leads to changes in depressive symptoms [START_REF] Anholt | The impact of depression on the treatment of obsessive-compulsive disorder: Results from a 5-year follow-up[END_REF], allowing us to rule out this hypothesis. It is also necessary to mention the particular clinical profile of this patient. Indeed, her multiple comorbidities, first and foremost her history of polysubstance addiction complicated by a dysexecutive syndrome, make her case not generalizable to the general clinical population, which makes it necessary to continue this work on a more representative sample. Finally, it is necessary to stress that a decrease in the YBOCS score of only 15% does not allow the patient to be classified as responsive to the treatment according to commonly accepted criteria (reduction of the YBOCS score by at least 25% [START_REF] Hirschtritt | Obsessive-Compulsive Disorder: Advances in Diagnosis and Treatment[END_REF]). However, it is also important to consider other indicators of therapeutic response that are equally important in routine clinical practice, foremost among which is the subject's quality of life. It is thus clear that the quality of life has improved significantly in a patient who was initially confined to her home because of her disorder. However, due to the uncontrolled nature of this pilot study, it is not possible to reliably link this improvement to a specific effect of the therapy. Indeed, as in any psychotherapy, there is a common factor (the "dodo bird effect" [START_REF] Marcus | Is the Dodo bird endangered in the 21st century? A meta-analysis of treatment comparison studies[END_REF][START_REF] Wampold | A meta-analysis of outcome studies comparing bona fide psychotherapies: Empirically, "all must have prizes[END_REF] or more commonly the contextual or placebo effect) which alone can explain a more or less important proportion of the therapeutic response. This technique is in line with an increasingly promoted approach of patient empowerment, enabling patients to become partners and actors in their psychotherapeutic care. Moreover, due to its high level of personalisation, the technique is consistent with the development of personalised psychiatry by allowing the specificity of the patient's symptoms to be taken into account, as well as the patient's own environment [START_REF] Briffault | From e-Health to i-Health: Prospective Reflexions on the Use of Intelligent Systems in Mental Health Care[END_REF].

Conclusion

In conclusion, this study has produced promising results but needing to be replicated on a larger and more representative sample before the recognition of personalised exposure therapy by immersive 360 o video as the technique of choice in the therapeutic arsenal of OCD. Indeed, it would enable a low financial cost solution that can be easily replicated to bring the patient's environment within the therapist's consulting room for a more easily accepted exposure to anxiogenic stimuli that are relevant to the patient and therefore a lower drop-out rate.

FiguresFigure 1 .

 1 Figures

Figure 2 .

 2 Figure 2. Clinical and life quality improvement.

Acknowledgements

We thank Ms. V. for her helpful and dedicated involvement in this study.

Disclosure Statement

The authors declare no conflicts of interest.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.


Karim Kilani

The Gini coefficient and negative income: A probabilistic approach

Keywords: Adjusted Gini index, Lorenz function, Negative income, Quantile function JEL classification. C43, D63

published or not. The documents may come

The Gini coefficient and negative income: A probabilistic approach

Introduction

The treatment of inequality in the case of wealth distributions with a strictly negative component is rather sparsely addressed in the literature on this research question. However, the issue is quite relevant from an economic point of view, for example in the framework of a cost-benefit analysis aiming at evaluating policies where some individuals would suffer significant losses while others would make profits.

If we look at this problem by analyzing the usual Lorenz curve, the latter immediately decreases, and dramatically so if the negative component of the wealth distribution is significant, before rising again and ending up in the usual area where we expect it to be, just beneath the diagonal of perfect equality. As a result, the Gini index, which measures the area between that diagonal and the Lorenz curve, can exceed unity, which is quite detrimental to the applicability of this index. This happens when the integral of the negative part of the Lorenz function exceeds that corresponding to the positive part.

In this paper, we follow the proposal of [START_REF] Chen | The gini coefficient and negative income[END_REF], who published a rather intuitive and simple way to adjust the Gini index in order to keep it within the proper limits, i.e. between zero and one. We generalize their approach, which was limited to the case of statistical data, to the probabilistic framework including any distribution with finite and strictly positive mean. Among other things, we propose a new interpretation of their adjusted index in terms of a ratio involving mathematical expectations of random variables of wealth rather than integrals of portions of the Lorenz function. We will also examine the case of some usual distributions in order to show the operational character of this approach, beyond the reservations that could be formulated about this adjusted index.

The rest of the paper is organized as follows: after having specified in section 2 the framework and the very general assumptions on the wealth distributions we study, we recall in section 3 the definition of the Lorenz function as an integral of the quantile function, as well as the essential properties of the latter two functions that will be useful for the sequel. In the section 4, we remind the definition of the Gini index and we show that for distributions which have a fairly large negative part, this index can exceed unity, as has already been pointed out since the papers of [START_REF] Schutz | On the measurement of income inequality[END_REF] or [START_REF] Chen | The gini coefficient and negative income[END_REF]. We then explain, following the method advocated by [START_REF] Chen | The gini coefficient and negative income[END_REF], how to adjust the Gini index by adding twice the area of the negative part of the Lorenz curve to the denominator of the Gini index. In section 5, we treat the case of discrete distributions and show that we do indeed recover the adjusted index of the statistical case as accurately derived by [START_REF] Berrebi | The Gini coefficient and negative income: A comment[END_REF] in a very short note correcting the paper of [START_REF] Chen | The gini coefficient and negative income[END_REF]. Finally, in the section 6, we treat two cases of continuous distributions: the uniform distribution where the values of the Gini index and the adjusted index are explicit and a generalization of the Pareto distribution where the Gini index is explicit while the adjusted index requires some basic numerical computations.

Assumptions

We consider a wealth distributed among the individuals of a population. It can be a discrete distribution, a continuous one, or even any mixture of these two types of distributions.

The so-called distribution is modeled by means of a real random variable W defined on a probability space where P denotes its probability function.

In this paper, we allow that a part of the individuals in the population receives a nega-tive income. However, we make the assumption that the mathematical expectation of W , the wealth, is finite and strictly positive 0 < µ ≡ E(W ) < ∞. This means that we analyze the inequalities generated by the distribution of wealth in a population where gains more than compensate for losses.

The Lorenz Function

Let us recall the definition of the quantile function, widely used in the literature on inequalities, which is the generalized inverse of the cumulative distribution function F(x) ≡ P(W ≤ x). More formally, it is defined by [START_REF] Parzen | Nonparametric statistical data modeling[END_REF]

w(p) ≡ sup x : F(x) ≤ p , 0 ≤ p < 1.
This is the maximum wealth held by 100 × p% of the least wealthy individuals. We recall that the quantile function is non-decreasing and continuous to the left [START_REF] Resnick | Extreme values, regular variation and point processes[END_REF].

The Lorenz function for its part is defined as follows [Gastwirth, 1972[START_REF] Arnold | Pareto distributions[END_REF]]

L(p) = 1 µ p 0 w(z)dz, 0 ≤ p ≤ 1. (1)
This is the share of cumulative wealth held by 100 × p% of the least wealthy individuals.

It is clear that L(0) = 0 and that, since the integral of the quantile function is the mathematical expectation [see, e.g. [START_REF] Parzen | Quantile probability and statistical data modeling[END_REF], we have

L(1) = (1/µ) 1 0 w(p)d p = 1.
Thanks to the fundamental theorem of analysis, L is a derivable function whose derivative

is L ′ (p) = w(p)/µ, 0 ≤ p ≤ 1.
It is thus a convex function, decreasing in the interval [0, F(0)] and thus non-positive. On the other hand, it will be strictly increasing in the interval (F(0), 1]. There exists thus a unique proportion p ⋆ in the last mentioned interval such that

L(p) ≤ 0 ⇐⇒ 0 ≤ p ≤ p ⋆ . (2)
By continuity, we therefore have

L(p ⋆ ) = 0. ( 3
)
The share of cumulative wealth held by individuals with a level of wealth below w(p ⋆ ) is zero. Moreover, if the variable is almost surely non-negative, we will have p ⋆ = F(0).

Otherwise, if the wealth distribution exhibits a non-zero mass negative part, we will have

p ⋆ > F(0), with L(p) < 0 if 0 < p < p ⋆ and L(p) > 0 if p > p ⋆ .
We can deduce that the positive and negative parts of the function of L which respectively verify

L + (p) = max(L(p), 0); L -(p) = -min(L(p), 0). ( 4
)
In the rest of the paper, we will compare the current distribution whose Lorenz function is L to two extreme distributions of wealth. The first is the one where wealth is distributed in a perfectly equitable way, i.e. all individuals receive the same wealth, and whose Lorenz function is given by

L(p) ≡ p, 0 ≤ p ≤ 1. ( 5
)
Recall that the Lorenz function satisfies the following inequality: L ≤ L, formally proved in [START_REF] Thompson | Fisherman's luck[END_REF] Theorem 1] which deals with the general case we are considering here, i.e. a possibly negative part for the wealth distribution but with a strictly pos-itive mean. In general, this inequality is strict, except when the distribution concentrates all its mass in a single value, in which case there is equality over the whole probability interval.

The second limiting Lorenz function is the one where the entire wealth is held by an infinitesimal fraction of the individuals of the population, which is given by

L(p) ≡        0, 0 ≤ p < 1; 1, 0 ≤ p < 1. ( 6
)
4 The Gini index and its adjustment

One way of thinking of the Gini index, which is far from being unique, is the following:

We seek to measure the distance that separates the Lorenz function L from the two extreme functions L and L given respectively by ( 5) and ( 6), and representing respectively a completely egalitarian and a totally unequal situation .

More formally, recall the definition of the L 1 -norm which states

|| f || ≡ 1 0 f (p)d p, (7)
where f is any non-negative integrable function. The Gini index can be expressed with respect to this norm in the following way:

G ≡ || L -L|| || L -L|| . ( 8
)
It is thus the ratio between the distance which separates L from L and the distance separat-ing L and L. Note that G ≥ 0, with equality only in the case of an almost-surely constant wealth distribution.

Using the expressions for the extreme Lorenz functions seen in the above section, we can rewrite the Gini index in a more usual form

G = 2 1 0 [p -L(p)]d p = 2 µ 1 0 w(p)pd p -1. ( 9
)
The second form is obtained through an integration by parts which will prove useful in the rest of the paper.

Another way to write the Gini index is to make use of the positive and negative parts We conclude that a necessary and sufficient condition that positions the value of G with respect to unity is given by:

G ⋚ 1 ⇐⇒ ||L -||| ⋚ ||L + ||. (10)
Thus, it may happen that the wealth distribution admits a negative part that has excess mass, generating a very high level of inequality, so that the Gini index even goes beyond unity.

A reasonable idea would be to modify the denominator of the Gini index given in its form (8) to obtain an adjusted index that remains within the usual range, i.e. between 0 and 1. The revised index could be designed as follows

G ⋆ ≡ || L -L||| || L|| + ||L -|| . ( 11
)
Since the difference between the denominator and numerator of the expression on the right-hand side of the above equation is ||L + ||, it is clear that we do have G ⋆ ≤ 1.

Equivalent and arguably more operational ways of expressing this adjusted Gini index are as follows

G ⋆ = G 1 + 2||L -|| = 1 - 2||L + || 1 + 2||L -|| . ( 12
)
This adjustment is in every respect equivalent to the one suggested by [START_REF] Schutz | On the measurement of income inequality[END_REF] and formalized, but only in the statistical case, by [START_REF] Chen | The gini coefficient and negative income[END_REF], which we shall verify in the next section.

In the same vein as the expression for the Gini index obtained in (9), both these alternative expressions for the integral for the additional area used in the index adjustment will be useful in the following

||L -|| = - p ⋆ 0 L(p)d p = 1 µ p ⋆ 0 w(p)pd p. ( 13
)
Let us consider again the expression (9) which gives the Gini index, in the second integral form. By performing a change of variable, we obtain

G = 1 µ 1 0 w( √ p)d p -1.
Since w( √ p) is nothing else than the quantile function of the maximum of two variables W 1 ,W 2 , independent and of the same law as W ,1 so that

G = E(max(W 1 ,W 2 )) -µ µ By noticing that |W 1 -W 2 |/2 = max(W 1 ,W 2 ) -(W 1 +W 2 )/2
, we deduce this other expression of the Gini index

G = E(|W 1 -W 2 |/2) µ . ( 14
)
In this form, the index is interpreted as the average of the mean absolute half-difference between the wealth of two individuals taken at random from the population, divided by the mean wealth. If we now consider the expression for the adjusted index given by ( 12), it can also be written, by combining it with the latest equation highlighted, as

G ⋆ = E(|W 1 -W 2 |/2) µ + 2 p ⋆ 0 w(p)pd p
.

By applying a change of variable to the integral that appears in the denominator of the above equation, we can write this expression in the following way

G ⋆ = E(|W 1 -W 2 |/2) µ + 2(p ⋆ ) 2 1 0 w(p ⋆ × p)pd p .
We notice that w(p ⋆ × p) corresponds to the quantile of the truncation of the variable W to the right of w(p ⋆ ), i.e. the variable where we retain only the realizations which satisfy the condition W ≤ w(p ⋆ ). By denoting W ⋆ this truncated variable, we end up with

G ⋆ = E(|W 1 -W 2 |/2) µ + (p ⋆ ) 2 E(max(W ⋆ 1 ,W ⋆ 2 )) , (15)
where W ⋆ 1 ,W ⋆ 2 , are two independent random variables having the same distribution as W ⋆ .

Thus, the adjusted index adds to the mathematical expectation of wealth in the denominator of the Gini index the weighted mathematical expectation of the maximum wealth of two individuals taken at random independently from the subpopulation of less wealthy individuals whose gains and losses just about even out.

In Expression ( 15), the weighting of the mathematical expectation by the square of p ⋆ may seem surprising at first sight. It is due to the fact that we draw two truncated random variables which must be less than w(p ⋆ ). The probability of obtaining them corresponds precisely to this weighting.

The case of discrete distributions

Let us consider an ordered list of n ≥ 2 wealth data w 1 ≤ • • • ≤ w n and their associated (nonzero) probabilities p 1 , • • • , p n , this list could be finite or infinite (in which case n = ∞).

Throughout this section, we will assume that the mean wealth is strictly positive µ =

∑ n 1 w i p i > 0, in accordance with our assumptions in the previous sections.

If W is the random variable that takes the values w 1 , • • • , w n , with associated probabilities p 1 , • • • , p n , the Lorenz function corresponding to that discrete variable is

L(p) = i ∑ 1 y j + p -π i p i y i , π i-1 ≤ p < π i , i = 1, • • • , n, (16)
value of w i while π i ≡ ∑ i 1 p j are cumulative probabilities. It goes without saying that we have ∑ n i y i = 1. In order for our notations to be completely defined, we assume that a sum whose lower bound is greater than its upper bound is zero.

To obtain the Gini index, one may use the expression of the index G in its second integral form which appears in ( 9), which gives us

G = 2 n ∑ 1 π i π i-1 (y i /p i )pd p = n ∑ 1 γ i -1, (17)
where

γ i ≡ (2π i -p i )y i , i = 1, • • • , n.
Let us then define the unique index k ≥ 0 that verifies the following condition

2 k ∑ 1 y i ≤ 0 ∧ k+1 ∑ 1 y i > 0 . (18)
Necessarily, p ⋆ lies within the interval π k , π k+1 . Solving (3) with L given by ( 16), we find an explicit value for this critical probability given by

p ⋆ = π k - ∑ k 1 y i y k+1 p k+1 . (19)
We can now calculate an expression for the term that adjusts the Gini index using (13)

p i π i y i ∑ i 1 y j γ i ∑ i 1 γ j 1 -

Some continuous distributions

We will now provide some illustrations for continuous distributions with negative support.

Uniform distribution

We analyze the uniform law on the interval [-a, 2a], where 0 ≤ a < 1, whose Lorenz function is

L(p) = p(p -a) 1 -a , 0 ≤ p ≤ 1 (23)
It has a negative part on the interval (0, a), and we have p ⋆ = a. We obtain the following Gini index

G = 1 3(1 -a) . ( 24
)
It decreases with a, from infinity near -1 to 1/3 in the neighborhood of a = 0, passing by 1 when a = 1/3. Noticing that

||L -|| = 1 1 -a a 0 (2p -a)pd p = a 3 6(1 -a)
, the adjustment of the Gini index is as follows

G ⋆ = 1 3(1 -a) + a 3 . ( 25
)
The following table provides values for the Gini index and the adjusted index.

Table 2: Standard and adjusted Gini index for the uniform distribution on [-a, 2a] a 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 G 0.33 0.41 0.50 0.62 0.78 1.00 1.33 1.89 3.00 6.33 G ⋆ 0.33 0.37 0.42 0.47 0.54 0.62 0.71 0.80 0.90 0.97 Figure 2 shows the Lorenz curve of a uniform distribution of wealth taken in the interval (-0.8, 1) and allocated, this time, among a continuum of families. The curve is this time a piece of parabola with a rather pronounced negative part, as for the Schutz data, and becomes positive from p ⋆ = 0.8.

  of the Lorenz function. Using also the fact that || L -L|| = || L|| -||L + || + |L -||, we deduce the following alternative writing G ≡ || L|| -||L + || + |L -|| || L|| .

Figure 1 :

 1 Figure 1: The Lorenz curve corresponding to the Schutz data

Figure 2 :

 2 Figure 2: Lorenz curve of the uniform distribution on (-0.8, 1)

Table 1 :

 1 Computation of standard and adjusted Gini indices for Schutz data

	i	w i

Since the cumulative distribution function of max(W 1 ,W

) is given by F 2 (x), the inequality F 2 (x) ≤ p becomes equivalent to the following: F(x) ≤ √ p.

In the case where the distribution is strictly positive, i.e. when w 1 > 0, we will arbitrarily set, preserving the hypothesis that a sum where the upper and lower bounds are reversed, k = 0. In this case, the value of p ⋆ defined in (19) will be zero.

with the value of p ⋆ calculated just above, to obtain

We conclude that the adjusted Gini index given by ( 12) for discrete distributions is, by combining the expressions ( 17) and ( 20), equal to the following ratio

In the statistical case treated by [START_REF] Chen | The gini coefficient and negative income[END_REF] and which corresponds to the particular case where p i = 1/n and

with n finite, we obtain, by replacing these values in (21), the expression below

This expression corresponds precisely to the one found by [START_REF] Berrebi | The Gini coefficient and negative income: A comment[END_REF], who corrects an error that crept into the formula of a footnote of the paper of [START_REF] Chen | The gini coefficient and negative income[END_REF].

As a very simple illustration, we use the data provided in [START_REF] Schutz | On the measurement of income inequality[END_REF], taken up by [START_REF] Chen | The gini coefficient and negative income[END_REF], on the wealth of ten families listed as (-500, -300, -300, -100, 200, 300, 300, 400, 500, 500), with an average value of 100.

The Gini index, by applying formula (17), is thus: G = 2.94 -1 = 1.94 and exceeds unity in this case. It is clear, from the sixth row of the table where ∑ 6 1 y j = 0, that we have, according to (18), k = 6; by applying (19), it follows: p ⋆ = π 6 = 0.8. Therefore,

14, so that G ⋆ = 1.94/2.14 = 0.9065.

Generalized Pareto distribution

We now consider a generalization of the Pareto distribution whose Lorenz function is of the following form:

where 0 ≤ a < 1 is a scaling parameter whose interpetation is provided below, and where ξ < 1 as a shape parameter.

We assume that the mean of the underlying wealth distribution is 1a, without loss of generality, to ensure a simple understanding of the parameters of the distribution. Indeed, since L ′ (0) = -a/(1a) < 0, the value -a represents the lower bound on the support of the distribution. The distribution contains a positive and a negative part. However, we have to consider three different patterns depending on the scale parameter of the distribution:

2. If ξ = 0, by applying l'Hospital's rule, it is an exponential distribution with as Lorenz function:

The support is in this case bounded and given by:

For all these cases, the Gini index amounts to

The value of p ⋆ as a nonzero solution of L(p ⋆ ) = 0 cannot be obtained explicitly, as in the case of the uniform distribution discussed above. It can only be evaluated numerically.

The Lorenz function given by ( 26) satisfies the following differential equation

Recalling that the adjustment area can be seen in two alternative forms in (13), we obtain the following equation

Solving the latter equation, we obtain

Thus, summarizing the previous results, we find the following adjusted Gini index

Figure 3 corresponds to the Lorenz curve of a generalized Pareto distribution with a = 0.5 and ξ = 0.5, with a Gini index equal to G = 4/3. The value where the curve crosses the x-axis, computed numerically, is p ⋆ = 0.8889. This allows us to find the adjusted Gini index of G ⋆ = 0.9557.


Lcsm

Patrice Berthod

Yassin Hamini

Lionel Aranda

Influence of tantalum on the rates of high temperature oxidation and chromia volatilization for cast (Fe and/or Ni)-base low carbon alloys

The high temperature oxidation behaviour of chromia-forming superalloys can be influenced by the presence of tantalum, an element which is often added to the alloy in order to reinforce it by solid solution strengthening or by forming refractory carbides in grain boundaries.

The aim of this work is to study the oxidation kinetics of an iron-base alloy and an iron-nickel base alloy containing 0.2%C and 3%Ta at several high temperatures, with comparison with previous results obtained for more simple alloys with the same carbon content but without tantalum, and for a nickel base alloy with 0.2C and 3%Ta.

Two alloys, Fe30Cr0.2C3Ta and FeNi30Cr0.2C3Ta, were synthesized by high frequency induction melting under 300mbar of pure Ar. Thermogravimetry tests were performed at 1000°C, 1100°C and 1200°C for 50 hours under a dry industrial air flow at 1.5 L/h. The oxidation constants were determined by plotting the mass gain measures according to equation {m x dm/dt = Kp -Kv x m} and extracting the parabolic constant Kp and the volatilization constant of chromia Kv.

The presence of tantalum, that can lead in Fe-base and FeNi-base carbon-containing alloys to an interdendritic carbides network exclusively constituted of TaC carbides, greatly influences the high temperature oxidation behaviour. More precisely the parabolic constant Kp and the volatilization constant Kv increases for the first one at the highest temperatures, but decreases for the second one since tantalum seems to induce a relative protection of chromia against volatilization of chromia. But such an effect only exists for low carbon alloys since tantalum should on the contrary enhance the rates of the oxidation phenomena when the carbide density is double, as observed in another study.

y = 1,09E-08x + 1,59E-10 0 2E-11 4E-11 6E-11 8E-11 1E-10 1,2E-1 0 1,4E-1 0 1,6E-1 0 1,8E-1 0 2E-10 -0,007
-0,0 06 -0,005 -0,004 -0,0 03 -0,002 -0,001 0 -m (g/cm2) m*(dm/dt) (g2/cm4/s)

  = Kp -Kv . m}-type treatment of mass gain results and validation of the obtained Kp and Kv by comparison between real curve and the mathematical curve plotted with these values (here: Fe30Cr0.2C3Ta at 1100°C)

  Surfaces of the two alloys Fe30Cr0.2C3Ta (left) and (Fe=Ni)30Cr0.2C3Ta (right) oxidized at 1200°C (top) and 1000°C (bottom) SEM examinations were done on the external oxides and on the sub-surface metallurgical state of samples. Internal oxidation was characterized by measuring the surface fraction of the internal (Ta, Cr) oxides of tantalum, and by calculating the average value of an equivalent thickness (equal to micrograph length x oxide surface fraction).

  ± 2 (4.0*) 1.3 ± 0.2 ( 0.9*) Values of the average thicknesses of external chromia and of the internal (Cr,Ta) oxide *: values deduced from the former multiplied by 0.707 since durations were 100h instead of 50h n. o. r. : no external oxide remained on the sample after cooling) Effect of the base element M on the mass gain curves of the M-30Cr-0.2C-3Ta alloys when oxidized at 1200°C (top) and 1000°C (bottom) presence of 3%Ta on the mass gain curves of Fe30Cr0.2C and Ni30Cr0.2C based alloys when oxidized at 1200°C (top) and 1000°C (bottom) : excellent f it between t he experimental m ass gain curve and t he m athematical o ne Normal char acter s: good fit between the exper imental mass gain curve and the mathematical one Italic character s: not good fit between the experimental mass gain curve and the mathematical one Values of Kp and Kv issued from the {dm/dt = Kp -Kv . m} treatment of the thermogravimetry results for all alloys at the three temperatures Kv (x 10

?) Kp (x 10 -12 g 2 .cm -4 .s -1 )

	-10 -2 .s -1 g.cm ) Fe30Cr0.2C FeNi30Cr0.2C Ni30Cr0.2C Fe30Cr0.2C FeNi30Cr0.2C Ni30Cr0.2C	temperature 1200°C 1100°C 1000°C 1200°C 1100°C 1000°C 1200°C 1100°C 1000°C 1200°C 1100°C 1000°C 1200°C 1100°C 1000°C 1200°C 1100°C -10.6 (temperature 0%Ta 3%Ta 434 (catastrophic oxidation) 204 109 31 -21 (?) / 112 / 52 / -2.3 (?) 143 117 52 56 16 0%Ta 3%Ta 160 (catastrophic oxidation) 98 159 19 8.6 / 119 / 23 / 1.4 82 137 28 42 1000°C 5.7 2.0


Ramona Elali
email: ramona.elali@univ-paris1.fr

Elena Kornyshova
email: elena.kornyshova@cnam.fr

Rébecca Deneckère
email: rebecca.deneckere@univ-paris1.fr

Camille Salinesi
email: camille.salinesi@univ-paris1.fr

Domain Ontology Construction with Activity Logs and Sensors Data -Case Study of Smart Home Activities

Keywords: Process Mining, Ontology, Sensors, Events Logs, Smart Home

Process mining relies on activity logs to discover process models, check their conformance, enhance processes, and recommend the next activity. On another side, many environmental factors such as time, location, weather, and profile are obtained from many sources, such as sensors, external systems, outside actors, or domain knowledge bases, and could also enhance recommendations. The existing research mainly focuses on single activity log datasets; only a few consider combining various sources. Our main goal is to provide better inputs to process discovery and better recommendations. In this paper, we focus on the combination of activity logs and sensors data with domain ontology as an intermediate step to attaining our goal. We use a case study of smart home activities to test this combination.

Introduction

An enormous amount of events data and sensor data is being recorded daily whilst the propagation of Information and Communication Technologies (ICT) is taking its rise worldwide across all domains. Data has become available everywhere, hence, the challenge becomes to turn out this data into value to get insight, increase productivity, produce better performance, and save costs [START_REF] Aalst | Process Mining: Data Science in Action[END_REF]. Process Mining (PM) focuses on the generated activities from business processes that can be used as a recommendation technique to direct the user on which next activity to follow according to his current activity [START_REF] Khodabandelou | Process Mining Versus Intention Mining[END_REF][START_REF] Compagno | Exploring Digital Conversation Corpora with Process Mining Corpus Pragmatics[END_REF]. PM 's main objective is to discover, check the conformance, and enhance the process models that are based on event logs [START_REF] Aalst | Process Mining: Data Science in Action[END_REF].

Ontology is an important subject in the domain of IS [START_REF] Woznowski | A Human Activity Recognition Framework for Healthcare Applications: Ontology, Labelling Strategies, and Best Practice[END_REF]. Many researchers are using ontology to classify domain knowledge [START_REF] Mansingh | Using ontologies to facilitate postprocessing of association rules by domain experts[END_REF] such as concepts, all types of entities, and the existing relationships between them [START_REF] Viinikkala | Ontology in Information Systems[END_REF]. Ontology organizes information in a natural, logical, and systematic way. Therefore, contextual information can be structured into a domain ontology by describing the relationships between the entities of the domain ontology and the contextual information.

In this paper, we propose an approach that combines events logs with contextual data obtained from sensors by using domain ontologies. Nowadays, contextual data can be obtained through many techniques and from various sources. We believe that contextual data can enrich the event logs by providing insights about users' behavior. We test and illustrate this proposal by applying it to a case study of Smart Home daily living activities. We choose this specific case study because this domain is rich in contextual information due to the presence of many sensors. Hence, we use the dataset provided by BP-Meets-IoT 2021 Challenge [START_REF] Koschmider | BP-Meets-IoT 2021 Challenge Dataset[END_REF]. This dataset contains an event log of the activities of daily living in addition to the sensor's log data. We are focusing on this case study to construct a specific domain ontology that identifies the links between the different person's activities and the sensor's data.

The next section will focus on the related works. Section 3 will present our proposal which we illustrate in Section 4. We conclude in Section 5.

Related Works

PM purposes to enhance, improve, discover, or check the conformance of activityoriented process models from event logs [START_REF] Aalst | Process Mining: Data Science in Action[END_REF][START_REF] Khodabandelou | Process Mining Versus Intention Mining[END_REF]. Event logs are the main input of information for the mining techniques [START_REF] Diaz-Rodriguez | Literature Review about Intention Mining in Information Systems[END_REF]. However, PM approaches do not usually take into account the context aspect behind the recorded user activities. By using various sources of logs, the discovered process models can offer better guidance through processes. In [START_REF] Van Eck | Enabling process mining on sensor data from smart products[END_REF], a process mining approach is suggested in the context of sensor data where sensor data were transformed into event logs to be used as input in the PM techniques. A comprehensive review on event logs preparation techniques from raw of data was introduced in [START_REF] Diba | Extraction, correlation, and abstraction of event data for process mining[END_REF]. In this review, the authors describe three steps of data preparation: event data extraction from heterogeneous data sources, event correlation, and event abstraction.

Ontology is now widely used in the Information Systems and Artificial intelligence world [START_REF] Viinikkala | Ontology in Information Systems[END_REF], where many researchers classify concepts, entities, and the relationship between different concepts in domain knowledge [START_REF] Mansingh | Using ontologies to facilitate postprocessing of association rules by domain experts[END_REF][START_REF] Viinikkala | Ontology in Information Systems[END_REF]. An ontology-based personalized recommender approach to provide a more precise and personalized recommendation was presented in [START_REF] Ge | An ontology-based method for personalized recommendation[END_REF]. In [START_REF] Obeid | Ontology-based recommender system in higher education[END_REF], a combination between the ontology-based recommender system and machine learning techniques is used to provide a personalized recommendation for students. An ontology-based recommendation approach relying on collaborative filtering that achieves a better recommendation in e-Learning systems was presented in [START_REF] Zhang | Ontology-Based Collaborative Filtering Recommendation Algorithm[END_REF]. In [START_REF] Middleton | Ontology-Based Recommender Systems[END_REF], a classification for research papers was done using ontology then they used collaborative techniques to recommend to the users interesting papers. In addition, activity recognition in Smart Home is a crucial topic to study while working on process mining. In [START_REF] Woznowski | A Human Activity Recognition Framework for Healthcare Applications: Ontology, Labelling Strategies, and Best Practice[END_REF], the authors proposed a hierarchical ontology to annotate the human activity in a home environment for healthcare taking into consideration the social context of the activity, the room, and the physical state. [START_REF] Chen | A Knowledge-Driven Approach to Activity Recognition in Smart Homes. Knowledge and Data Engineering[END_REF] developed a knowledge-driven approach by creating ontologies for the sensors and the activities for the purpose of activity recognition. In addition, in [START_REF] Salguero | Using Ontologies for the Online Recognition of Activities of Daily Living[END_REF], they have developed an ontology for the activities of daily living in a smart environment to use as input for a supervised learning technique for online activity recognition. Also, an Activity of a Daily Living ontology-based approach was developed in [START_REF] Chen | Ontology-based activity recognition in intelligent pervasive environments[END_REF][START_REF] Bae | An ontology-based approach to ADL recognition in smart homes[END_REF] for activity recognition in a context-aware smart home environment.

Approach to Combine Sensors Data and Event Logs Using Domain Ontology

In our view, we believe that ontologies can help to combine sensors data and activity logs. Firstly, all the data acquired from the sensors and all the activities of the events log can be semantically interpreted in an ontology specific to the domain. Secondly, all the links between the ontology concepts (especially the links between the sensors elements and the activities elements) must be defined. For instance, a specific value of sensor data can trigger the start, or the end, of an activity.

To combine sensors data and event logs using domain ontology we have to proceed in three steps (see Fig. 1): (1) create the sensor ontology based on the sensors data, (2) create the activity ontology based on the events log and (3) refine the domain ontology by creating the links between the two preceding ontologies (white arrow on the second part of Fig. 1).

Fig. 1. Domain Ontology Construction Approach.

If the events log and the sensor data are related to existing ontologies, we can proceed directly to step 3 to identify the links between them.

In this work, we focus on step 3 with two main hypotheses: H1: It is possible to identify the ontological links between sensors data and activities using the timestamp on data and events. H2: It is possible to identify the ontological links between sensors data and activities by association rules.

Timestamps. The timestamp is considered the only attribute that can let us find the relationship between the activity event and the sensor. We developed a Python algorithm that takes as input the events log and sensor log. The two logs need to be sorted according to the timestamp field. This algorithm is composed of two main steps. In the first step, the start and completion of each activity should be identified. Then, the timestamp field is used to pivot the events log according to the start and completion of each activity. Then in the second step, a join was done between the two logs using the timestamp field to create the link between the sensor data and the event data.

Domain Ontology Construction

Association rules. Association rule mining is the technique of finding relationships between different items. It is commonly used in Market

Basket Analysis, where it identifies which items are frequently purchased together for marketing purposes. It is the process of finding frequent patterns, associations, causal structures, and correlations within different datasets existing in different types of databases like transactional databases, relational databases, or any other form of databases [START_REF] Zaki | Data Mining and Analysis: Fundamental Concepts and Algorithms[END_REF][START_REF] Sunita | A Comparative Study of Association Rule Algorithms for Course Recommender System in E-learning[END_REF]. The purpose of association rule mining is to find the rules that allow us to guess the appearance of a specific item after the occurrences of other items in the transaction. It is one of the top concepts of machine learning rule-based techniques that discover interesting relations between items in big databases by indicating the frequent associations' rules that are composed of an antecedent and consequent [START_REF] Zaki | Data Mining and Analysis: Fundamental Concepts and Algorithms[END_REF]. Association rules allow uncovering the hidden relationships between the items that are frequently used together. It is represented by if-then statements (an antecedent itemset and a consequent itemset). Apriori [START_REF] Agrawal | Fast algorithms for mining association rules[END_REF] is one of the top standard algorithms for frequent itemset mining and association rule learning. In this work, we used Association Rule mining techniques since we needed to find rules between the sensors and the activities. To apply the Apriori algorithm, first of all, we needed to transform the dataset into a transaction log that is supported by the algorithm. We labeled each log by day date and then we combined both logs and sorted them according to the day and the timestamp. Then we transformed the combined log into a transaction log which is supported by Apriori. In addition to the transaction log, Apriori needs as input multiple parameters such as the minimum support and minimum confidence. Support indicates how often an itemset is frequent in the transactions. Confidence indicates how often the rule (if → then) statement is found true.

We illustrate our proposal through a Smart Home case study in the following section.

Case Study

The rapid evolution of artificial intelligence endorses the existence of smart homes and make it crucial to study this domain. Numerous studies are done on smart home [START_REF] Tiwari | Changing World: Smart Homes Review and Future[END_REF][START_REF] Alam | A review of smart homes -Past, present, and future[END_REF][START_REF] De Silva | State of the art of smart homes[END_REF][START_REF] Alaa | A review of smart home applications based on Internet of Things[END_REF]. Hence, our case study will be focused on smart home since this field is rich in different types of sensors. We need to find the link between the person's activities and the sensors to find later how the context can affect the user's intention. Data preprocessing includes standard activities aiming at enhancing data quality: cleaning, integration, and transformation [START_REF] Tamilselvi | An efficient preprocessing and postprocessing techniques in data mining[END_REF].

In our experiment, we are using the dataset provided by BP-Meets-IoT 2021 Challenge [START_REF] Koschmider | BP-Meets-IoT 2021 Challenge Dataset[END_REF]. This dataset consists of two people's activities logs in addition to the sensor logs recorded during 21 consecutive days between 16 March 2020 and 6 April 2020. While studying the dataset, we found 53 activities and 16 sensors. Both logs contained noise data; some duplicated records, in addition to some records that don't represent any type of activities or sensors. We cleaned the dataset from the noise by removing those tuples, and we obtained 23166 records in the events log and 39304 records in the sensors log. A sample of each of the datasets is shown in Fig. 2. Each record in the event logs is identified by the Activity Name while each record in the sensor logs is identified by the Sensor Name and both logs are categorized by the Timestamp and the Resource Id.

Fig. 2. Event Logs and Sensor Logs Samples.

There was not any defined domain ontology given with the dataset. Moreover, despite previous works already having defined ontologies for the daily activity recognition in smart homes [START_REF] Woznowski | A Human Activity Recognition Framework for Healthcare Applications: Ontology, Labelling Strategies, and Best Practice[END_REF][START_REF] Chen | A Knowledge-Driven Approach to Activity Recognition in Smart Homes. Knowledge and Data Engineering[END_REF][START_REF] Salguero | Using Ontologies for the Online Recognition of Activities of Daily Living[END_REF][START_REF] Chen | Ontology-based activity recognition in intelligent pervasive environments[END_REF][START_REF] Bae | An ontology-based approach to ADL recognition in smart homes[END_REF], none of them were matching the data of our dataset. Thus, we had to define this domain ontology ourselves. After a deep analysis of the data of the sensors and events log, we were able to identify the different parts of the domain ontology. To construct our domain ontology -which is specific to our dataset and case study -we have defined two main ontologies (Smart home activity ontologysub-section 4.1 and smart home sensor ontologysub-section 4.2). Their links are detailed in sub-section 4.3. To ease the understanding of our work, we propose here only a simplified view of the two ontologies where all the links between the concepts are "is-a" relationships. We define the more complex associations between the concepts in a specific meta-model (cf. Fig. 10).

Smart Home Activity Ontology

The Activity ontology (Fig. 3) contains the different types of activities and actions that can happen inside a home. Each activity should be completed by a human inside a fixed location. The Activity ontology is divided into groups of activities (see Fig. 3): Bathroom Activity, Clean Activity, Work Activity, etc... Each activity group inside the Activity ontology has its own instances. As an example, the Kitchen Activity has a set of real-life activities that happen inside the kitchen (see Fig. 4). Drink Water, Go Kitchen Shelf, Put Plate to Sink, Wash Dishes, etc... are instances of Kitchen Activity group.

Smart Home Sensors Ontology

Smart Home Sensors Ontology includes three parts: Sensor, Position, and Human ontologies.

The sensor ontology contains all the information related to the sensors such as sensor type, value, value range, and location. According to our dataset, we were able to identify different types of sensors such as Environment sensors, Kitchen Sensors, etc. In Fig. 5, we have presented the high level of the ontology sensor, where the sensors are divided into different types and each type contains its related sensors.

For each location inside the home, there are a set of sensors related to the location such as Kitchen Sensor, Bedroom Sensor, and Living Room Sensor. And there are some sensors that are common to all the house as the Environment Sensor, Power Sensor, and Water Sensor. Fig. 6 shows an example of instances of Kitchen Sensor. Fridge Door Contact Sensor, Cooked Food Sensor, Food Sensor, and Unwashed Dishes Sensor are all sensors located inside the kitchen. Each sensor works differently according to its type, and it has its own range of values [START_REF] Koschmider | BP-Meets-IoT 2021 Challenge Dataset[END_REF]. For instance, the Unwashed Dishes sensor is used to count the number of unwashed dishes inside the kitchen sink, its value is greater than 0 when there are dishes inside the sink to wash, otherwise, it will be 0. While the Fridge Door Contact Sensor indicates when the fridge door is opened or closed, so its value is 0 when it is closed and 1 once it is opened. The Food Sensor is used as a food counter inside the home, so its value is equal to or greater than 0. As for the Cooked Food Sensor, it works the same as the Food Sensor, but it counts only the available cooked food inside the home. Each position category has its own instances that represent the different fixed locations inside the home. As an illustration, the Kitchen Position contains the different positions inside the conditions such as the dining table, the fridge, the kitchen shelf, and the kitchen sink (see Fig. 8). Then, Human ontology is a needed domain to have to characterize the human profile (see Fig. 9). It categorizes the different person types that perform the different activities. It can be divided into two main subdomains: the initial home residents (inhabitants) and the guests (see Fig. 9).

Links between Sensors and Activity Ontologies

We can identify different relationships between the defined ontologies (see Fig. 10). Each human does activities. Sensors affect activities and can be positioned at a specific position. Each activity can be followed by one or many activities (that can be done sequentially or not). Each activity can happen at a particular position at home and be affected by sensors. Each position can be related to sensors and activities. Fig. 11 illustrates an example of the links between the ontologies. When the human 'goal is Washing Dishes then the set of activities that are related to this goal will take place inside the kitchen; the inhabitant will go to the kitchen sink, put the plate inside the sink and start to wash dishes. In the meantime, the sensor of Unwashed Dishes will increase its value according to the plates that are being washed. The Water Use sensor will increase its value and the Position sensor will indicate that the human position is next to the Kitchen Sink.

The established relationships were used to test the established hypotheses. As presented in Section 3, we had two main hypotheses: it is possible to identify the ontological links between sensors data and activities using the timestamp on data and events (H1), and it is possible to identify the links by association rules (H2).

To validate H1, an automatic test was done using Python as a programming language. 100836 records were obtained corresponding to 53 different activities and 16 different sensors. The obtained data was cleaned because it was containing some noise data thus the remaining was 99975 records. We obtained 112 correspondences between

Conclusion

This research paper describes an approach to constructing domain ontologies using sensors data and events logs. Sensor, Position, Human, and Activity are the ontologies that were defined. Our main objective in this work was to find relations between sensors and activities. We were able to identify and validate the link between sensor data and activities, based on a Smart Home case study. However, the limited amount of data prevented us to find more interesting links between activities and sensors.

A future experiment on a larger dataset will provide us with a more complete set of links Then, we will continue our main research proposal by using these ontologies as input to build process models by precising the links between sensors and activities in order to identify which data can trigger an activity. Our main goal is to be able to make recommendations to users on the fly, following these contextual data and the event logs.

Fig. 3 .

 3 Fig. 3. Smart Home Activity Ontology.

Fig. 4 .

 4 Fig. 4. Smart Home Activity Ontology with Details about the Kitchen Activity.

Fig. 5 .

 5 Fig. 5. Sensor Ontology High Level.

Fig. 6 .

 6 Fig. 6. Kitchen Sensor Instances Example.Position ontology (Fig.7) represents the different parts of the home where the activities can take place and the sensors are attached. Position ontology is divided into 6 main positions categories (see Fig.8): Entrance Position, Exercise Place Position, Bathroom Position, Kitchen Position, and Living Room Position.Each position category has its own instances that represent the different fixed locations inside the home. As an illustration, the Kitchen Position contains the different positions inside the conditions such as the dining table, the fridge, the kitchen shelf, and the kitchen sink (see Fig.8).

Fig. 7 .

 7 Fig. 7. Position Ontology High Level.

Fig. 8 .

 8 Fig. 8. Kitchen Position Instances Example.

Fig. 9 .

 9 Fig. 9. Human Ontology High Level.

Fig. 10 .

 10 Fig. 10. Links between the Ontologies.

Fig. 11 .

 11 Fig. 11. Example on the Link between the Different Ontologies' Domains.

Table 2 .

 2 Sample of the obtained rules.

	Rule

activities and sensors. We compared the mapping results with the results that we obtained manually while analyzing the data and deduced that for each occurred activity there is one or multiple sensors triggered. An extraction is given in Table 1 to illustrate the obtained links between activities and sensors using the timestamp data (the complete list can be found in [START_REF] Elali | Data Exploration of BP-Meets-IoT 2021 Challenge Dataset: Correspondences & Associations Rules Result[END_REF]). In this manner, we were able to validate H1 and identify the link between the activities and the sensors while relying only on the timestamp. To validate H2, a second test was done using Association Rules by applying the Apriori algorithm [START_REF] Agrawal | Fast algorithms for mining association rules[END_REF] on the dataset selecting only the logs that are relevant to one person. After combining the logs, we obtained 410 transaction records. We did three experiments. We set the confidence value to 80% in all the experiments to get strong rules. The minimum support value is set through the process of trial and error. In the first experiment, we set the value of minimum support to 0.15 and we obtained 56 rules. In the second experiment, we set the minimum support value to 0.05 and we obtained 8565 rules. And in the third experiment, we set a small value of minimum support equal to 0.03 and we obtained 34623 rules. We analyzed the generated rules manually and found that the result of the second experiment is more realistic due to the value of the minimum support and the confidence. In Table 2, we present a small extraction of the obtained rules [START_REF] Elali | Data Exploration of BP-Meets-IoT 2021 Challenge Dataset: Correspondences & Associations Rules Result[END_REF] with the corresponding confidence degrees for each rule.

Thus, despite the limitation of Apriori to perform well when the itemset is large, we were able to generate rules that associate activities with sensors and validate H2.

On the whole, this experiment has validated the existing relationship between the sensors and the activities. We were able to automatically find the rules that we have found before manually while analyzing the dataset. For an instance, when a person goes to the kitchen next to the shelf; the position sensor will locate his position accordingly, so the next activity will be to get a glass then move to the sink where the 'water use' sensor will directly reflect the water consumption while filling the glass with water. And after drinking water, the 'unwashed dishes' sensor will increment its value.