Yosra Chaher

Jean-Pierre Belaud

Hervé Pingaud

Managing Open Innovation in Connected Health through a Living Lab

Keywords: open innovation, connected health, living lab, healthcare services, innovation engineering

The digital footprint marks all parts of society and impacts our lifestyles. Beyond perceived changes in individual way of life, digital revolution also affects sociotechnical systems, collective structures or communities that delivers primary services for citizens. Healthcare is deeply impacted by such a digital mutation at many levels (patient expectations, caregivers practices and organization). We are probably witnessing the emergence of smart and personalized healthcare with a very large scope. To be included in this dynamics, many healthcare organizations now feature collaborative approaches and open innovation management to improve their performance in a sustainable perspective. But the complexity of systems is often a limit to their progress. This paper discuss the place and role that could be played by new utilities that are naturally based on open innovation approach: the living labs. In connected health where many stakeholders should interact, they are naturally expanding. Using an example, we show the gains in value chain that this new envision of open innovation could provide. In parallel, the paper argues for a development of open innovation engineering in such labs. It leads to the conclusion that a reference framework is helpful in such a perspective.

I. INTRODUCTION

In today's fast-changing world, competitive organizations are those that quickly transform new ideas into new products, services and organizations. In fact, innovation is the cornerstone for a successful and sustainable society since it stimulates new needs of consumers while searching solutions that meets their expectations. Innovation practices promote creativity in many kind of organizations. It often leads to competitive advantages in terms of cost, brand image and value for companies. But innovation also open new perspectives to imagine new ways of life in modern societies into which citizen empowerment has become a political reality. So, it leads to the emergence of new problems which must be addressed at the right level, and finding this level is still an open question.

Intrinsically, an innovation process is difficult to understand and always depicted as very risky to execute. That is why, managers seek to better control their innovation process, making them more efficient on all the functions of the corresponding value chain: from the sourcing of ideas to the breakthrough in the markets. Beyond this challenge, the rising pressure of time and limited capacity of investment have been at the root of an ongoing trend to shift from a closed model of innovation to more open innovation strategies. The difficulty to rationalize innovation management rises when many independent actors have to share common objectives, skills, resources, projects and savings. The efficiency of collaboration adds a new dimension to the challenge of innovation. If it is a greater challenge to tackle, open innovation is potentially an answer to the question of conciliation between company's vision of market needs and society yearnings. One difficulty in this perspective is to capture knowledge from potential future consumers/users and to streamline the information into well-designed projects.

Although they are widely studied in a multidisciplinary context, innovation and open innovation remain in their infancy in terms of engineering and a lot of research has still to be done to recognize this relatively new engineering paradigm beside more improved ones like design engineering or production engineering, inside the discipline of industrial engineering. We are searching for concepts, approaches, methods and tools for a rationalization of open innovation engineering. In this context, our research aims to imagine a methodological reference framework adapted to a collaborative space into which stakeholder representatives are working together whatever the process phase in progress, trying to reduce many sources of uncertainties. Our framework will try to fill this gap, provide indications on who and why people shall work, how to proceed and on what subject they will have to focus on. Intensive use of digitalization in product design as well as evidence-based involvement of people in IT services foster an integrated approach of both product and service design in R&D activities. As a matter of fact, new areas are investigated with this idea and the health sector is one of them.

Open innovation in connected health area is a high level challenge for our society and has been identified as so by the WHO at a global scale. Health is integral part of public policies, subject to strong regulations at the national level, operated by care facilities, but also more and more by self-managed persons. Indeed, connected health promotes use of IT to help patients manage their own diseases (chronic diseases). A new concept of expert patient is born from this observation. In fact, each initiative in this particular sector involves at least four categories of stakeholders (patients, technology suppliers, health practitioners and public policy institutions). So, the relevant open innovation process is complex due to human factors and needs to be managed with the diversity of stakeholder requirements, among them, we could say the criticality of service quality for patients, the respect of caregivers skills and associated responsibilities, the adequacy of technology and risk management for providers and finally the cost control for public institutions.

Last, but not least, medical practices are changing and science delivers really new artefacts that will deeply modify our relationships to healthcare practices in the next decades. There is a move away from the medical curative model to a social interconnected health perspective into which prevention and self-management of patients will be paramount [START_REF] Van Geenhuizen | Living labs in health innovation: Critical factors in their application[END_REF]. Let's give some illustrations about this. Following regulatory obligations, telemedicine services are an example of connected health expansion in France. Services are widespread at the regional level and stimulate eHealth policy development [START_REF] Islam | The internet of things for health care: a comprehensive survey[END_REF]. The ''Hôpitaux 2012'' plan and the Law on Hospitals, Patients, Health, and Territory (la loi HPST) are worth mentioning in this regard [START_REF] Artmann | Dumortier`e-health strategies -country brief: France[END_REF]. It is a reality that health care systems are under pressure as a consequence of the ageing population and fast demographic growth with chronical diseases increase, in a situation of decreasing financial budgets. Therefore, the connected health sector has been declared as a component of the national economic strategy in July of 2016. In the text that describes the strategy, open innovation is enhanced and the inclusion of living labs in the innovation process is recommended.

This paper is organized as follows: we first provide a literature review on recent developments and our positioning in relation to the existing works before we explain our research approach. Finally, we present our findings before concluding.

II. LITERATURE REVIEW

A. Research context and problematic

As we aforementioned, innovation has become a strategic issue for organizations as well as for the society of the future. It has even attracted the attention of researchers who focus on innovation engineering. Innovation is a complex paradigm that requires concepts, methods and tools in order to develop new products, processes or services. Upstream in the innovation process, to solve inventive problems during preliminary design, researchers and engineers use traditional methods such as: the C-K approach, the brainstorming theory and the trial and error approach. Their major drawbacks are known: a random process, a lack of systematization and a huge part let to talent of people. To overcome these drawbacks, we work on the emergence of more structured innovation management, based on an engineering framework with an emphasis put on systematization of the upstream phase of the innovation process, in the stages of formulation/problem resolution and the generation of innovative ideas [START_REF] Penide | Innovative process engineering: a generic model of the innovation process[END_REF]. Research works have been carried out by Penide [START_REF] Penide | Innovative process engineering: a generic model of the innovation process[END_REF] and Lopez Flores [START_REF] Flores | Collective intelligence to solve creative problems in conceptual design phase[END_REF] to develop models in order to manage the innovation process. For reference [START_REF] Penide | Innovative process engineering: a generic model of the innovation process[END_REF], innovation is a process of problem resolution and knowledge transformation, e.g., the first activity aims to define or redefine a problem. The corresponding model proposed attempts to align operational, supporting, and management activities, which inspire us for constructing the innovation life cycle thereafter. Nevertheless, reference [START_REF] Penide | Innovative process engineering: a generic model of the innovation process[END_REF] didn't take into account the collaborative aspect of open innovation as has been done in reference [START_REF] Flores | Collective intelligence to solve creative problems in conceptual design phase[END_REF]. In fact, [START_REF] Flores | Collective intelligence to solve creative problems in conceptual design phase[END_REF] proposed to define a conceptual framework for the development of a Computer Aided Innovation tool in open innovation mode supported by the Web 2.0 technologies. Finally, [START_REF] Flores | Collective intelligence to solve creative problems in conceptual design phase[END_REF] used the notion of collective intelligence to improve the innovation process which has been considered as a linear one, and this doesn't match with the reality. Consequently, our research is in line with these previous works.

Let's precise that the corresponding approaches have in common to understand innovation as a process of information management and knowledge transformation. Optimally, we will mix the business process model of [START_REF] Penide | Innovative process engineering: a generic model of the innovation process[END_REF] to optimize the use of the Computer Aided Innovation tool of [START_REF] Flores | Collective intelligence to solve creative problems in conceptual design phase[END_REF].

This concept has been growing since the term was coined by Chesbrough in 2003. This paradigm considers that organizations can expand their innovation capacities beyond their own boundaries. Indeed, it implies inbound and outbound knowledge flows across the organizational boundary. [START_REF] Chesbrough | Open Innovation: The New Imperative for Creating and Profiting from Technology[END_REF] describes this concept as a phenomenon of companies making greater use of external ideas and technologies in their own business, and letting unused internal ideas and technologies go outside for others to use in their business. Many organizations now feature collaborative practices in their activities, even in R&D activities, such as IBM and Procter & Gamble, who have shrunk their internal R&D departments and successfully tapped into external sources of innovation [START_REF] Galbraith | Open innovation in connected health: an empirical study and research agenda[END_REF]. Hence, the concept of open innovation [START_REF] Huizingh | Open Innovation: State of the Art and Future Perspectives[END_REF] has been introduced and defined as a paradigm that assumes that organizations can and should use external, and internal and external paths to market, as the organizations look to advance their technology. Therefore, our innovation framework is adapted to a collaborative space between organizations working together. It promotes greater a social dimension with controlled information and knowledge exchanges. Consequently, we will focus on potential interactions among diverse actors of the innovation ecosystem.

However, the large number of definitions, with so much different points of view, that we could found in the scientific literature about innovation shows that it is a complex paradigm, and that, for engineering purposes, it must be studied as a set of inter-related concepts. As it is a dynamic in an organization supported principally by motivated people that drives innovation, business process are also useful to explain how to do and capture the different ways of making progress over time without being too much unstructured. Today, there are several forms of innovation that have been introduced as a form of specialization of the global concept. But there is a lack of a global view of this different views and the tentative of unification of them could be of added value to encapsulate knowledge about them, as well as to minimize the features that have to be analyzed when talking about a special application. That is why, we decided to develop a methodological reference framework which raises a problem of knowledge, life cycle analysis and information system as a support.

B. Living lab: new innovation tool

In this very intense collaborative context of knowledge sharing, new entities have appeared to enhance the open innovation paradigm. In fact, new spaces of co-creativity have been recently created, like the Fab Labs [START_REF] Troxler | Bending the Rules. The Fab Lab Innovation Ecology[END_REF] and the Living Labs [START_REF] Salminen | Facilitating user driven innovation trough a Living Lab[END_REF]. The Living Lab concept originates from Prof William Mitchell, MediaLab and School of Architecture and City Planning, MIT, Boston. Living Labs represent a user-driven research methodology for sensing, prototyping, validating and refining complex solutions in multiple and evolving real life contexts. The first Living Labs created from the initial ideas were in the area of smart/future homes. In those settings real people (visitors) were observed in their usage of emerging technologies in the setting of a real home. In many of the implementations people stayed in these homes for several days or weeks [START_REF] Eriksson | Living labs as a multi-contextual R&D methodology[END_REF]. A major contribution to the living lab concept came from research on lead users as an important origin of innovations causing a greater emphasis on involving users more actively and early in the new product, process and service development [START_REF] Van Geenhuizen | Living labs in health innovation: Critical factors in their application[END_REF]. So, the living labs promote the open innovation approach [START_REF] Schuurman | The impact of living lab methodology on open innovation contributions and outcomes[END_REF]. The European Network of Living Labs (ENoLL) counts today almost 400 historically recognized Living Labs and more than 170 active members. They defined the living labs as user-centered, open innovation ecosystems based on systematic user co-creation approach, integrating research and innovation processes in real life communities and settings. They added that living labs operate as intermediaries among citizens, research organizations, companies, cities and regions for joint value co-creation, rapid prototyping or validation to scale up innovation and businesses.

Health is not an area insensitive to this new innovation tool since it is currently subject to important changes. E-health or connected health are new areas that have emerged, not only as a new specialty in medicine but also a new paradigm of health consideration for people [START_REF] Pingaud | System thinking in healthcare systems: The Connected Health Lab experience[END_REF]. As a consequence, several living labs work for the health care sector. For example, the Medical Field Lab (MFL) in Maastricht (Netherlands) and the Health Innovation Lab (HIL) in Copenhagen (Denmark). In France, the General Council for the Economy, Industry, Energy and Technologies (CGEIET) launched a working group in 2009, in consultation with the Ministry of Health. All stakeholders, public and private, were represented. About fifteen French Living Labs have been identified in the Health and Autonomy sector at a national scale.

According to [START_REF] Pallot | The living lab approach: A user centred open innovation ecosystem[END_REF], a Living Lab is currently considered as a user-centered open innovation ecosystem integrating research and innovation within a Public-Private-People partnership through an iterative Experiential Design process [START_REF] Pallot | A holistic model of user experience for living lab experiential design[END_REF]. It means that within a Living Lab, the focus on the user covers the entire innovation process as shown in Fig. 1, where all the stakeholders are collectively engaged in creating and validating innovative solutions in a kind of "knowledge refinery". First, they begin by an analysis of user's needs in which there is a proposal for several investigation tracks and representation to collect and organize knowledge on the ecosystem and to classify identified needs. The second stage is the implementation of several sketches to allow stakeholders to focus on early solutions and select suitable ones for implementation in early prototypes. Then, they proceed to a primary implementation of the fully functional prototype and user tests to address the deficiencies left by the previous stage of co-conception in order to meet the user's needs. Finally, we find the deployment stage for a law scale on pilot sites to refine deployment procedures for later, resolve gaps and implement back-up strategies in case of a solution failure, including financial and legal safeguards for all the stakeholders. When working on technology, the living labs can be positioned between Technology Readiness Levels (TRL) 3 and 7, since they include the conception, validation, prototyping and pilot production phases. Thus, we will integrate in our research approach the presence of these new environments of co-creation and testing with users.

In fact, considering health services, these new spaces reinforce inter-relationship between actors and makes the patient a central actor who participates in the development of new services. We believe that this very particular organization is a good vector for boosting the culture of user centered innovation [START_REF] Hippel | Democratizing innovation: the evolving phenomenon of user innovation[END_REF] in healthcare. Indeed, as introduced before, if medicine will become predictive (use of big data for particular diseases), participative (long term patient pathways), preventive (anticipation of problems), personalized (adapted to individual profile) and patient driven (mediation for design of care plan with practitioners) [START_REF] Pingaud | System thinking in healthcare systems: The Connected Health Lab experience[END_REF], innovation in health will be a large breakthrough for society of the future. Clearly, each of these characteristics will contribute to a new need of managing innovation in health.

C. Applying UCD and Agile Methods in the Living Lab context

User involvement can take a variety of forms. From the One hand, some instances position the user as the main creator, in the case of lead users [START_REF] Hippel | Lead Users: A Source of Novel Product Concepts[END_REF] or open source communities. Others see participants operating as co-creators in practices such as design thinking [START_REF] Brown | Definitions of design thinking. Design Thinking: Thoughts by Tim Brown[END_REF]. From the other hand, participatory or User-Centered Design (UCD) treats users as passive subjects whose insights are captured and introduced in the innovation process, such as usability, human interaction [START_REF] Almirall | Mapping living labs in the landscape of innovation methodologies[END_REF]. As we aforementioned that Living Labs are situated between those limits of user involvement, it seems quite logical to apply some UCD and Agile practices in a Living Lab context. In fact, working with Agility improves collaboration and communication; it involves a local representative of the client (considered here as a user) to shorten the feedback loop between the development team and the end user [START_REF] Silva Da Silva | User-Centered Design and Agile Methods: A Systematic Review[END_REF]. We now identify some key concepts related to the UCD, which is an approach that aims to involve the users in a meaningful and appropriate way throughout a system's development [START_REF] Chamberlain | Towards a Framework for Integrating Agile Development and User-Centred Design[END_REF]. Concerning prototyping, [START_REF] Broschinsky | Using Persona with XP at LANDesk Software, an Avocent Company[END_REF] recommends the use of User Interface prototypes to bring the known customer requests into the discussion as quickly as possible. [START_REF] Brown | Stories, Sketches, and Lists: Developers and Interaction Designers Interacting Through Artefacts[END_REF] comments that sketches in addition to User Stories can be used as means of revealing errors, temporal information such task sequence, contextual information, etc. Concerning Users testing, as the name suggests, it involves the use of users testing for usability evaluation. Moreover, Users Stories should be integrated with scenario-based design and could be defined for the construction of prototypes [START_REF] Silva Da Silva | User-Centered Design and Agile Methods: A Systematic Review[END_REF].

UCD and Agile methods should be started in the early stages of the innovation process and applied all along. The Living Lab is a space that allows the implementation of these methods in a collaborative context from the stage of the idea to the stage of the result successfully put in the market. Indeed, this open innovation location has the faculty of simulation which strengthens the stakeholders' relationships to enhance the collaborative aspect.

III. RESEARCH APPROACH

With the same idea, digitalization will probably be one of the key point inside any strategy wanting to tackle these new medicine challenges. That's why, we try to conduct in our research approach a case study in the Connected Health Lab (CHL) of Castres (site web : isis.univ-jfc.fr). In fact, the CHL is an open innovation platform where user experience in ICT for Health (e-Health) is assessed. By users, we mean both patients and caregivers. The CHL recreates the environment of a patient pathway for simulation purposes to highlight all factors of resistance to open innovation. In an experience, the objective is either to detect the actual specific needs in terms of digital solutions or to evaluate the value of a digital solution at the prototype status under simulated conditions, including the level of change in practices related to its use. Fig. 2 shows a general overview of the CHL which constitutes our field of study. The CHL is composed of five rooms that represent different locations of the patient pathway. Three rooms are associated to classical homecare: home bedroom, general practitioner office, and drugstore. The other two rooms play the role of the hospital environment (reanimation room, bedroom). The living lab helps to assess the technology readiness. It gives a real alternative to experiments that are generally performed in vivo, in public healthcare institutions, which could not be considered as test beds for many reasons as discussed by [START_REF] Pingaud | System thinking in healthcare systems: The Connected Health Lab experience[END_REF]. For our research methodology, this living lab plays the role of a "Broker" location. In fact, it is a common resource in the collaborative space which ensures a coordination function between the various stakeholders. In addition, as we mentioned in the Fig. 1, it is useful at all stages of the lifecycle, particularly in prototyping and development.

At an engineering reference point of view, our challenge is to create a methodological reference framework with for a better steering of the innovation process, particularly within the scope of open innovation. To do this, we will observe activities performed in the CHL to build and improve our conceptual propositions starting from the field. We choose to adopt a modelbased approach [START_REF] Penide | Innovative process engineering: a generic model of the innovation process[END_REF] to design the reference framework. This framework will help us to define the data collection in order to position the Key Performance Indicators (KPI) that will build the dashboard to control the innovation process.

Knowing with accuracy the business level of the system, we will be able to explain the use of an Open Computer Aided Innovation tool at the software level, both levels being mapped to help stakeholders managing open innovation [START_REF] Flores | Open computer aided innovation to promote innovation in process engineering[END_REF].

IV. FINDINGS

A reference framework (Fig. 3) for open innovation engineering will characterize innovation with a model of key concepts, named meta model, and a life cycle (generic processes describing how to conduct and drive open innovation). A meta model defines the key concepts that can be used to describe a domain [START_REF] Kurpjuweit | Viewpoint-based Meta Model Engineering[END_REF]. So, we start by defining these key concepts first for innovation and then for open innovation. Fig. 4 is an extract of our meta model that shows few core concepts. In fact, our meta model contains a total of 71 concepts. We just want to summarize the context into which a living lab could be included when open innovation is depicted.

Then we defined the life cycle, i.e. classical innovation activities, based on business process models structured in three levels (management, operational and support levels). This results in a high level innovation process map that is relatively generic (Fig. 5). The major phases are in agreement with the Fig. 2. General overview of the Connected Health Lab [START_REF] Pingaud | System thinking in healthcare systems: The Connected Health Lab experience[END_REF] Fig. 3. The reference methodological framework approach adopted in our living lab. In fact, in the first stage of framing, we analyze the type of innovation (innovation by need, by opportunity, by technological market pull...) in order to evaluate the motivation to innovate and to delimit an idea with a real added value. Thereafter, we try to acquire the knowledge needed to implement the idea selected. Then, in order to evaluate the technical feasibility, we need to prototype and experiment the solution. Finally, we validate the solution in a real environment through a pilot development.

As a matter of illustration, we introduce an example of a project in telemedicine [START_REF] Jordanova | Global Telemedicine and eHealth (A synopsis)[END_REF] for the elders that we have performed on the living lab in 2016. Telemedicine technology is developing and there is yet a real offer of technologies on the market. The experience was mainly on the developing operational phase of the map (Fig. 5) and it conveyed the concepts presented in the Fig. 4. The goal was to make an evaluation of practice changes to assess the level of maturity in the innovation when General Practitioners and patients are using the telemedicine terminal to operate. This project aims to promote the use of telemedicine, in parallel to the conventional practice of General Practitioner visits. The CHL team first provides a knowledge base in the form of a business process model at a more detailed level in order to have an explicit basis to think about practice assessment. The experience includes a characterization of changes in professional practices that has been associated with defined activities of the process model. By doing so, the context into which new practices born is more easily captured. Therefore, a comparison between the ex-ante and ex-post reality is possible. We show how changes has been characterized on Fig. 6 that describes the macro-process "prepare telemedicine consultation notice". The modelling language uses actigrams to represent inputs and outputs of activities. Activities are done by actors who are reference at the left side of a lane. From general point of view, the project identifies high level stakeholders linked by the associations summarized in Table I. Our observation on the experimental pilot was based on three axes of analysis: organization of the coordinated care path, performance of the coordinated care path and the steering of the coordinated care path. The first axis describes how to practice the telemedicine service and it is dictated by the generic process models that we have defined. On the second axis, that of performance, it seems important to us to consider the main expectations of patients in terms of value, namely the quality of care provided and the control of the risks associated with the care path. Finally, the performance of the coordinated care path is based on decision-making at a collective level and this immediately raises the question of the exchanges constructed by this collective. This structuring dimension of steering has been referred to as information sharing for coordination purposes. This repository of knowledge provided the basis for the collective reflection of the actors of this project. All the information gathered was a basis on which we were then able to study in order to construct proposals for indicators of changes in practices in the field of healthcare care services for the elderly.

V. CONCLUSION

Much has been written since the advent of the living labs a decade ago. Several articles, such as [START_REF] Niitamo | State-of-the-art and good practice in the field of living labs[END_REF], attest to the growing academic interest in living labs and its user driven innovation approach. This paper attempts to add a contribution in the scientific literature on the usefulness of living labs in the field of connected health in order to manage the open innovation. First, we provide a literature review on the open innovation concept. Then, we introduce the living labs as an open innovation tool, particularly in the health care field. Lacking an engineering global view to manage the open innovation process, we are study the development of a methodological reference framework adopting a model based approach. By doing so, we try to describe a problem through models, concepts and languages. In fact, we are building a meta model with a structure of two concentric spheres (a spherical core and a first concentric layer) as shown in Fig. 7. The core part represents the meta model of innovation and the peripheral part represents the open innovation one. Our first instantiation of the meta model still needs to be refined. This is a first perspective for the following of the study. In parallel, it will be closely related to the life cycle that we are defining, using the concepts of the meta model. To do so, we rely on a case study in the Connected Health Lab in order to improve our conceptual propositions. This is a second perspective for us. In addition, this paper introduces a case study performed in the living lab to better organize the care path in Telemedicine practice for the elderly [START_REF] Thompson | The ethical and policy implications of e-health and telemedicine: an ageing-focused review[END_REF]. This experiment will be helpful to illustrate the added value of the reference framework as an engineering foundation.

Fig. 1 .

 1 Fig. 1. The Living Lab Experiential Design (adapted from [20])

  Minimize transport activities for frailty people GP-PI Modification of economic model for TLM service payoff GP-TPS Risk assessment and quality improvement about medicine practice exercise TPS-PI Diffusion of technology for equal access to TLM services P = Patients, TLM = Telemedicine, TPS =TLM platform supplier, PI = Public Institution, GP = General Practitioner

Fig. 4 .

 4 Fig. 4. Extract of the meta model

TABLE I .

 I STAKEHOLDERS RELATIONSHIPS


Pierre Maho

Cyril Herrier

Thierry Livache

Pierre Comon

Simon Barthelmé

A calibrant-free drift compensation method for gas sensor arrays

Keywords: drift, correction method, gas sensors, electronic nose, calibrant-free

Gas sensors lack repeatability over time. They are affected by drift, the result of changes at the sensor level and in the environment. A solution is to design software methods that compensate for the drift. Existing methods are often based on calibration samples acquired at the start of each new measurement session. However, finding a good reference compound is a difficult task and generating calibration samples is time-consuming. We propose a model-based correction method which does not require any calibration sample over time, operating 'blindly'. In this study, we focus on the drift affecting electronic noses. To this end, we built a real data set acquired over 9 months in real-life conditions. By using the proposed method, we show that the drift is partly compensated, thus increasing the reliability of the electronic nose. Besides, we also show that the algorithm can easily adapt if the target compounds are not all sampled during every session.

Introduction

Drift is the result of several changes occurring at the sensor level and in the environment. These changes affect the repeatability of gas sensors over time [START_REF] Holmberg | Drift counteraction in odour recognition applications: lifelong calibration method[END_REF]. In other words, the response of these sensors towards a compound can change from one measurement session to another.

The causes of drift in gas phase are numerous and depend on the sensing and transduction mechanisms. Several authors have already detailed these causes and we refer the reader to these references for a complete introduction [START_REF] Holmberg | Drift Compensation, Standards, and Calibration Methods[END_REF][START_REF] Hierlemann | Higher-Order Chemical Sensing[END_REF]. To name a few examples, drift can be caused by variations in temperature and humidity between two sessions, by sensor poisoning, by sensor ageing, by memory effects, by changes in the sample matrix, etc... It is interesting to note that some components (e.g. temperature, humidity, memory effects, ...) of the drift can be avoided when the experiments are carried out in a controlled environment (see for instance the study of [START_REF] Vergara | Chemical gas sensor drift compensation using classifier ensembles[END_REF]). However, in the field, all these sources can co-occur, so their effects may not be separable. Consequently, all these causes are often treated as a single process.

In this article, we are interested in a particular type of instruments based on gas sensor arrays, namely electronic noses (eNoses). The technology is broadly inspired by mammalian olfaction [START_REF] Persaud | Analysis of discrimination mechanisms in the mammalian olfactory system using a model nose[END_REF] and is mainly based on a cross-sensitive chemical sensor array [START_REF] Albert | Cross-Reactive Chemical Sensor Arrays[END_REF]. An eNose aims at detecting, identifying and quantifying a broad variety of volatile compounds. In practice, this instrument is used in combination with pattern recognition algorithms for the identification part. Concretely, any application often starts by carrying out several measurements with the targeted compounds. We call this first measurement session 'Session 1'. Each gas sensor generates a time series reflecting the interaction between the sensor and the compound. This time series can be directly used for further analysis or identification [START_REF] Skov | A new approach for modelling sensor based data[END_REF] but a more common practice is to extract one or two features. One measurement leads then to a vector (by considering all the gas sensors) which is assumed to be characteristic of the compound and which is called a 'signature'. After Session 1, the database of signatures is used to train a classifier which will then predict the identity of future samples. eNoses are a good example to illustrate the issue of drift. Despite abundant proofs of concept in academic laboratories, eNoses are not a widespread technology used in everyday life conditions yet [START_REF] Marco | The need for external validation in machine olfaction: emphasis on health-related applications[END_REF][START_REF] Boeker | On 'Electronic Nose' methodology[END_REF]. According to some authors in the community, drift can be considered as one of the main factors explaining the small number of practical applications of these instruments [START_REF] Padilla | Drift compensation of gas sensor array data by Orthogonal Signal Correction[END_REF]. In the presence of drift between Session 1 and another session, say 'Session 2', the signatures of a given compound will be different. A direct consequence is that the classifier will not generalize from Session 1 to Session 2. In other words, it will confuse one compound for another, rendering the instrument useless.

A 'naive' solution would be to briefly retrain the classifier before using it for Session 2. Obviously, this solution is not desirable at all since each experiment is timeconsuming. Therefore, the solution adopted in the literature is generally to design methods that compensate for the drift. All these methods assume what we call the Fundamental Hypothesis: that all the individual drifts of each targeted compound must be related in some way (in the simplest case, equal). In other words, if each compound drifts along a random direction that is different for each targeted compound, then there is nothing one can do and all methods will fail.

Over the past two decades, many different correction methods have been put forward in the literature. These methods differ on the amount of information that is assumed to be available.

A classical approach is to assume the existence of calibration samples of one or several reference compounds during each measurement session. The drift of the reference compounds must be related in some way to the drift of the targeted compounds [START_REF] Haugen | A calibration method for handling the temporal drift of solid state gas-sensors[END_REF]. The calibration samples are then used to estimate the drift directions. We call this scenario: Calibrant Scenario (CS). In this article, we discuss 'Principal Component Analysis -Component Correction' (PCA-CC) from [START_REF] Artursson | Drift correction for gas sensors using multivariate methods[END_REF], which is one of the most popular methods in the field [START_REF] Marco | Signal and data processing for machine olfaction and chemical sensing: a review[END_REF].

However, acquiring new samples before each new measurement session is time-consuming. In addition, finding reference compounds is not an easy task and requires a lot of upstream work [START_REF] Kalivas | Calibration maintenance and transfer using tikhonov regularization approaches[END_REF], which must be repeated anytime an application with new target compounds is designed. Consequently, two other approaches have been considered in the field to relax the assumption that calibration samples are available.

The first one assumes that the drift has been captured by training data. In a classification task, a training set is always required and can be acquired during a single measurement session. Here, one assumes that the training set has been acquired across multiple sessions over time. If a long enough time has elapsed between sessions, then the training set must contain drift information. So this group of methods extracts the drift directions directly from the training set and assumes that these directions will no longer evolve for subsequent sessions. We call this scenario: Multi-Session Scenario (MSS). Hereafter, we discuss 'Orthogonal Signal Correction' (OSC) from [START_REF] Padilla | Drift compensation of gas sensor array data by Orthogonal Signal Correction[END_REF] and 'Common Principal Component Analysis -Component Correction' (CPCA-CC) from [START_REF] Ziyatdinov | Drift compensation of gas sensor array data by common principal component analysis[END_REF].

The second class of methods assume that the only labelled data available comes from the training set, specifically a single measurement session (in which drift is not present). In fact, MSS can be attractive since it does not require any reference compound but it requires a long experimental time. For instance, the results of OSC [START_REF] Padilla | Drift compensation of gas sensor array data by Orthogonal Signal Correction[END_REF] were obtained with data acquired over 15 days (the number of measurement sessions is not detailed). For the results of CPCA-CC, the authors used at least one month of recordings [START_REF] Ziyatdinov | Drift compensation of gas sensor array data by common principal component analysis[END_REF]. This long experimental time has some limitations, so that other authors try another more appealing approach: they simply assume that there is no labelled data for estimating the drift. So the directions of the drift between two different sessions must be estimated 'blindly'. We call this scenario: Blind Scenario (BS). The methods dealing with the BS often rely on a preliminary classification step. The classification step identifies the samples and helps therefore to estimate the drift directions. Further in this article, we discuss the method of Di [START_REF] Di Carlo | Drift correction methods for gas chemical sensors in artificial olfaction systems: techniques and challenges[END_REF].

We refer the reader to [START_REF] Marco | Signal and data processing for machine olfaction and chemical sensing: a review[END_REF] or [START_REF] Rudnitskaya | Calibration Update and Drift Correction for Electronic Noses and Tongues[END_REF] for a more exhaustive review of existing methods, which include for instance transfer learning approaches [START_REF] Nikzad-Langerodi | Domain adaptation for regression under beer-lambert's law[END_REF].

The three scenarios clearly differ in difficulty, since we do not have the same level of information about the data from one scenario to another. By increasing order of dif-ficulty and by decreasing order of experimental cost, we can rank the scenarios as: Calibrant, Multi-Session and Blind. A simplified representation of the three scenarios is reported in Figure 1. The Blind Scenario is the least restrictive in practice but is also the least explored in the literature.

In this article, we propose a new correction method for the Blind Scenario based on a drift model: each class drifts along a common direction weighted by a coefficient depending on the class. This method is called 'Expectation-Maximisation Component Correction' (EMC 2 ). In the next section, we develop the theory and the proposed algorithm. We also present a way to initialise EMC 2 . The method is first tested on a simulated data set and then on a real data set that we built. This data set has been generated over 9 months, through 15 different sessions and shows a clear drift. The ultimate goal of this paper is to be able to correctly classify sessions 2 to 15 using only Session 1 for the training of the classifier (and without knowing any labels in sessions 2 to 15).

Theory

We assume that we have acquired N 1 samples (in total) of R different compounds during Session 1, all labelled. During Session 2, Q ≤ R compounds among the R target compounds are acquired and the N 2 samples are unlabelled. The instrument boards P chemical sensors.

A scalar is noted a, a vector a and a matrix A.

Drift model

Let g(x|r, t, c r (t)) be the probability density of the signature x ∈ R P of the compound r, measured at time t, at concentration c r (t). In this article we are interested in compounds classification, so variations in concentration could cause problems for identification. If a classifier learns to differentiate A from B in a given range of concentration, it is unlikely that it will generalize out of the range since the signatures will not be the same (they depend on the concentration). The signature is often normalised to remove or reduce variations due to changes in concentration. In this article, we normalise each signature by dividing by its L 2 -norm. We assume that x has been normalised and we note x the concentration-free version of x. We note f (x|r, t) the probability density of the normalised signature x, and we define as class r all the normalised signatures of the compound r.

Under the Fundamental Hypothesis, the drift of all compounds is assumed to be related. Here, we assume that the distribution of each compound r is simply shifted by a common vector d(t) ∈ R P weighted by a coefficient α r (t) ∈ R (depending on the compound). Mathematically, it means that there exists a density f 0 (.|r) such that f (x|r, t) = f 0 (x + α r (t)d(t)|r). We assume that d(t = 0) = 0, which means that the training set (Session 1) defines the starting position. We can define the following drift model for a point x(t) at time t:

x(t) ∼ f 0 (x + α r (t)d(t)|r) (1)
Finally, we assume that the drift is a slow-varying process, which is a commonly used assumption [START_REF] Hierlemann | Higher-Order Chemical Sensing[END_REF]. This means that for a short time window ∆ t , we have d(t+∆ t ) ≈ d(t) and α(t+∆ t ) ≈ α(t). So, if Session 2 starts at t = T 2 and lasts only for few hours, we assume that each distribution r has been shifted by α r (T 2 )d(T 2 ). In other words, noting x 1 a sample from Session 1 (training session) and x 2 a sample from Session 2 (testing session):

x 2 ∼ f 0 (x 1 + α r (T 2 )d(T 2 )|r) (2)
A simplified representation of the drift model is reported in Figure 2.

In the case where Session 2 lasts longer than a few hours, it is possible that the slow-varying assumption will be violated. However, a straightforward solution is to divide Session 2 into smaller sessions (overlapping or not).

We can notice that the proposed model could be improved by taking into account that L 2 -normalized signatures belong to the unit P -sphere. In this case, the transfer caused by the drift could be more properly modeled as a circular shift. Here, we implicitly assume that all the signatures are close enough together in the high dimension space to suppose that they belong (approximately) to an Euclidean space of dimension P -1 (the tangent hyperplane).

The goal of the next section is to estimate the R coefficients α r (T 2 ) and the direction d(T 2 ) ∈ R P , which characterize the drift between Session 1 and Session 2. This task would have been trivial if the labels ℓ(T 2 ) ∈ R N2 of Session 2 were known, but they need also to be estimated during the compensation process (since the weight α r depends on the compound). This estimation of the labels makes the task more challenging.

To lighten the notation, we remove the time index

T 2 of Session 2. So, α r (T 2 ) is noted α r , d(T 2 ) is noted d, etc...

Expectation-Maximisation Component Correction

Before going into details, we need an additional assumption: a necessary condition for identifiability. A model is said to be identifiable if the parameters of the model can be theoretically estimated uniquely. In our case, the model assumes that each class is shifted along a common direction whose amplitude depends on the class. If there is a single class during Session 2, then we can estimate R different translations moving the class in Session 2 to each class in Session 1, thus correcting the drift. However, which one of the R translations is the true one? It would be difficult to answer this question1 . This is a matter of identifiability, and to be able to correctly estimate the parameters, we assume that we are not in this particular case: in Session 2, we have sampled at least two different (unknown) classes, so 2 ≤ Q ≤ R.

Expectation-Maximisation (EM) algorithm is an iterative method for finding maximum-likelihood solutions of statistical models that have "latent variables" [START_REF] Dempster | Maximum Likelihood from Incomplete Data via the EM Algorithm[END_REF][START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF]. Latent variables are variables that cannot be directly observed. In our case, the latent variables are the labels ℓ ∈ R N2 of the data X ∈ R N2×P of Session 2 and the parameters are the drift direction d ∈ R P and the weights α ∈ R R . The main difficulty is that we cannot estimate ℓ without the knowledge of d and α, and we cannot estimate d and α without the knowledge of ℓ. To solve this issue, an EM iteration proceeds in two steps: the Expectation step and the Maximisation step. The Expectation step will estimate the latent variables by considering that the parameters are known and the Maximisation step will do the reverse. These two steps are then alternated until convergence.

To derive the algorithm, we assume that each compound r is drawn from a Gaussian distribution N (µ r , S r ), with µ r ∈ R P the class centroid and S r ∈ R P ×P the class covariance matrix. µ r and S r are estimated from the data X 1 ∈ R N1×P of Session 1. We define as π r (π the corresponding vector of size R) the probability that a sample of Session 2 belongs to the class r.

During the Expectation step, d, α and π have been already estimated so we can estimate the a posteriori probability that a sample x n ∈ R P of Session 2 belongs to class r:

p(ℓ n = r|x n , d, α, π) = π r N (x n , µ r + α r d, S r ) R k=1 π k N (x n , µ k + α k d, S k ) (3)
Equation ( 3) is classical in EM algorithms for mixture estimation and the reader is referred to [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF] for details.

Algorithm 1: Expectation-Maximisation Component Correc- tion (EMC 2 ) Data: X1 ∈ R N 1 ×P , ℓ1 ∈ R N 1 , X ∈ R N 2 ×P Initialise d(0) , α(0) , π(0) while d(i) unchanged do // Expectation step ∀n ∈ 1, N2 , ∀r ∈ 1, R , ρ(i+1) nr = π(i) r N (xn, µ r + α(i) r d(i) , Sr) k π(i) k N (xn, µ k + α(i) k d(i) , S k ) // Maximisation step ∀r ∈ 1, R , π(i+1) r = n ρ(i+1) nr N2 d(i+1) = Λ -1 n,k ρ(i+1) nk α(i) k S -1 k (xn -µ k ) with Λ = k ( α(i) k ) 2 S -1 k n ρ(i+1) nk d(i+1) = d(i+1) ∥ d(i+1) ∥ 2 ∀r ∈ 1, R , if π(i+1) r ≥ 1 N 2 then α(i+1) r = n ρ(i+1) nr d(i+1) T S -1 r (xn -µ r ) d(i+1) T S -1 r d(i+1) n ρ(i+1) nr else end end // Drift correction ∀n ∈ 1, N2 , ln = arg max r p(ℓn = r|xn, d, α, π) ∀n ∈ 1, N2 , xn,cor = xn -αl n d ρnr = p(ℓn = r|xn, d, α, π)
During the Maximisation step, we estimate each parameter by maximising the likelihood. The likelihood is the probability of observing X given the parameters d, α and π. For the estimation, we simply write the derivatives with respect to each parameter and set them to zero. The derivations are given in the Appendix. There is an obvious indetermination in our model: a scale ambiguity between α and d. To remove this ambiguity, we simply normalise d after its estimation.

After convergence (typically, less than 20 iterations), we have estimated the parameters d, α and π. The drift of each class r is assumed to be along d but the amplitude α r of this shift depends on the class. Consequently, we need to estimate the label ℓ n of the sample x n to correct its drift. To that end, we take the class which maximises the a posteriori probability:

ln = arg max r p(ℓ n = r|x n , d, α, π) (4)
Once we have estimated the label ℓ n of the sample x n , we can, optionally, correct it:

x n,cor = x n -αl n d (5)
It is interesting to note that our EM-based algorithm removes one component from x n which is αl n d. It is reminiscent of some methods from the literature called Component Correction (e.g. PCA-CC from [START_REF] Artursson | Drift correction for gas sensors using multivariate methods[END_REF]). Therefore we call this method: Expectation- ). We first select random directions from the centroids of X 1 to some random samples of X. Then, for each candidate direction, we orthogonalise X 1 and X against the direction. In the subspace, we classify X ⊥ with a k-NN trained on X 1,⊥ . Finally we compute a score w as the trace of the within-class covariance matrix (Eq. ( 6)) in the subspace taking both sessions. Bad directions are assumed to have a greater value w. Here, d 2 is a better initial direction than d 1 . Numbers (e.g. ➀) stand for the order of the different steps of the initialisation procedure.

Algorithm 2: Heuristic for initialising EMC 2 Data:

X1 ∈ R N 1 ×P , ℓ1 ∈ R N 1 , X ∈ R N 2 ×P Input: A ∈ N, k ∈ N Compute the centroids µ r (X1) from X1 Select randomly A samples X A ∈ R A×P in X Compute all the B = A × R possible directions with unit-norm between each sample in X A and each centroid µ r (X1) for i in 1 : B do X 1,⊥ = X1(I P -did T i ) X ⊥ = X(I P -did T i ) ℓi = k-NN(X 1,⊥ , ℓ1, X ⊥ , k)
Merge X 1,⊥ and X ⊥ as X⊥ , ℓ1 and ℓi as ľ

Compute the score wi with ( X⊥ , ľ) as the trace of Eq. ( 6)

end I = arg min i wi ℓ = ℓ I , d (0) = d I , α (0) r = (µ r (X) -µ r (X1)) T d (0) , π (0) r = #{ℓ I =r} N Maximisation Component Correction, abbreviated EMC 2 .
The algorithm is reported in Algorithm 1.

Initialisation

The results of EM algorithms depend strongly on the starting point [START_REF] Mclachlan | The EM algorithm and extensions[END_REF]. Poor initial parameters d(0) , α(0) , π(0) will likely lead to a bad local maximum and as a result to a poor drift correction. Therefore, we propose a heuristic to correctly initialise EMC 2 .

The main idea is that a simple orthogonalisation against d will superimpose X 1 and X [START_REF] Boulet | Pretreatments by means of orthogonal projections[END_REF] and information about d must be contained in some pairwise directions between some samples in X 1 (here, we use the centroids µ r (X 1 ) since we know the labels ℓ 1 ) and some samples in X.

We start by selecting random directions going from each centroid µ r (X 1 ) of X 1 to A random samples of X (integer A is fixed beforehand). For each random direction, we project X and X 1 in the subspace orthogonal to the candidate direction. These versions of X and X 1 are called X ⊥ and X 1,⊥ . A good direction would then super-impose X ⊥ and X 1,⊥ , meaning that we would have only R different clusters. A bad direction would likely create additional clusters in the subspace. To check if X ⊥ and X 1,⊥ are superimposed or not, a good figure of merit is the within-class covariance matrix. Note X⊥ the combination of the two datasets X ⊥ and X 1,⊥ :

S w = R r=1 n∈r (x n,⊥ -µ r ( X⊥ ))(x n,⊥ -µ r ( X⊥ )) T (6)
with µ r ( X⊥ ) the centroid of the class r in the orthogonal subspace, computed from X⊥ .

S w evaluates how far from each other the samples belonging to a same class are. The greater the within-class covariance is, the more spread-out the classes are. To have only a single number, we take w = Tr(S w ). The lower w is, the better X ⊥ and X 1,⊥ are superimposed.

However, the within-class covariance matrix needs labels and we do not have any label for X ⊥ (the Session 2 projected in the orthogonal subspace). Therefore, the final step is a classification step. For that, we simply apply a k-NN in the orthogonal subspace of the candidate direction to estimate the labels ℓ of X ⊥ based on (X 1,⊥ , ℓ 1 ). The initialisation procedure is described in Algorithm 2 and a graphical representation is reported in Figure 3.

As a remark, we assume here that there actually is a significant drift in the data. If there is no drift, applying a drift correction may lead to worse results (this is true for every method of drift correction). In our case, the initialization will lead to a random direction and this random direction can unfortunately be correlated with a discriminative direction. The risk is then that two clusters may be merged together after correction. To overcome this issue, a drift detection method needs to be developed to decide to apply or not a drift correction method.

Code availability

R code for Algorithms 1 and 2 as well as the other methods used in this paper are available at: https:// github.com/mahopie/eNoseDrift.git. Simulation code is also provided.

Data sets

Artificial data set

For the generation of the artificial data set, we reuse and complicate the simulation settings proposed by Di Carlo et al. (2011).

A simulation starts with the generation of a training set X 1 ∈ R N1×P with no drift and with R = 6 compounds, N 1 = 60 and P = 4. Each class is Gaussian with parameters µ r ∼ N (0 P , β2 2P I P ) and S r = I P . Then, we generate data with drift according to the model in Equation ( 1) and only Q ≤ R compounds are simulated (for the same training set, we test from

Q = R to Q = 2).
Our model only implies that the drift is additive, slowvarying and along a direction d(t) independent of the compound (only the weight factor is). So, d(t) is not constrained to be constant or linear over time, and can be rather complex. To introduce a non-linearity in the drift over time, we simulate the direction d(t) with an arbitrary non-linear function:

d(t) = t h d 0 + t h sin( t h d 0 )
, sin is applied component-wise, d 0 ∼ U(0, 1) is a random direction (normalised) and h is a hyperparameter (h = 30). The weights α r ∼ U(0.5, 3) are also random and the class probabilities π r are set to 1 Q . So, ∀r ∈ 1, Q , ∀t ∈ 0, t max :

x r (t) ∼ N (µ r + α r t h (d 0 + sin( t h d 0 )), I P ) (7)
t max is the duration of the experiment and is set to 251.

The simulations are random so the generated classes could be hardly separable. In this case, we may not evaluate only the drift correction but also the initial discrimination between classes. To prevent this issue from occuring, we discard simulations for which any pairwise distance between centroids is lower than β 2 [START_REF] Di Carlo | Increasing pattern recognition accuracy for chemical sensing by evolutionary based drift compensation[END_REF]. The simulations (starting from the generation of Session 1) are repeated 100 times.

Real data set

A part of the data set presented in this paper has been already introduced in [START_REF] Maho | Olfactive robot for gas discrimination over several months using a new optoelectronic nose[END_REF].

Electronic nose. The optoelectronic nose used in this study is the commercial version of the one described in [START_REF] Brenet | Highly-Selective Optoelectronic Nose Based on Surface Plasmon Resonance Imaging for Sensing Volatile Organic Compounds[END_REF]. The instrument is provided by the company Aryballe. More details about the working principle of the instrument are given in [START_REF] Brenet | Highly-Selective Optoelectronic Nose Based on Surface Plasmon Resonance Imaging for Sensing Volatile Organic Compounds[END_REF] or in [START_REF] Maho | Reliable chiral recognition with an optoelectronic nose[END_REF]. In this article, the instrument uses 26 different chemical sensors which are replicated 2 or 3 times on the surface, leading to an array of 63 gas sensors. This electronic nose is rather new compared to other existing eNoses (e.g. metal-oxide based sensors) but it has Experimental setup. We built a robotic setup to study drift in realistic conditions. Three VOC sources of pure compounds (Geranyl acetate, Citral and β-Pinene) are disseminated in an open environment. This environment is scanned by a robot which carries the eNose. By following a predefined path, the robot goes over the VOC sources and the eNose detects them. The path is repeated multiple times to obtain a substantial data set at each measurement session. The setup and the response (after the baseline subtraction) of the instrument after one lap are reported in Figure 4. Temperature and humidity are not controlled during the experiments (temperature variations belong to [20.4°C, 34.5°C] and humidity variations to [14.2%, 40.3%]) and compounds concentration is neither controlled, nor measured.

To have a better idea of the experiments, we encourage the reader to watch the video in footnote 2 . In this video, a larger setup is used but the idea remains the same.

Data set. The responses of the gas sensors look like peaks due to the very short time injection (∼ 1 sec). Each peak corresponds to one experiment with one of the 3 com- pounds. A segmentation procedure is applied to extract these peaks and is detailed in another paper (see [START_REF] Maho | Real-time gas recognition and gas unmixing in robot applications[END_REF]). Note that the resulting time series are uncommonly short, compared to the standard for e-Noses, where measurements typically take a few minutes. Each peak is integrated for each chemical sensor. Therefore, each peak gives a signature x. The replicas are averaged to reduce dimensionality, so P = 26 and x ∈ R 26 . A session usually lasts for few hours so some compounds run out at the end of the experiment and injected concentration decreases. To get rid of the variations in concentration, we normalise each signature individually by dividing it by its ℓ 2 -norm. The normalised signature is noted x. One lap gives one signature for each one of the R = 3 compounds, so after a certain number of laps we have a data matrix X ∈ R N ×P . X corresponds to only one measurement session lasting for few hours. Since we perform several measurement sessions over time, we note X i ∈ R Ni×P the data acquired during Session i.

In total, we have acquired 15 different sessions over 9 months, corresponding to a total of ∼2000 samples. The measurement sessions differ on several experimental parameters: the amount of liquid solution left in the gas sources, the environmental conditions (temperature, humidity), the running order, the robot speed and the number of laps. Some of these experimental parameters and the recording dates are reported in Table 1.

To add another difficulty, all the sessions have not been generated with the same setup. Most of the sessions have been generated with a setup containing only the 3 target compounds (see Figure 4). However, some of the sessions have been generated with a similar, but larger, setup containing 12 different compounds in 24 different cups, including the studied compounds 2 . For these sessions, we extracted from the data the 3 compounds of interest. In Table 1, the column R indicates the number of compounds actually present in the session.

In Table 1, we also report some classification results (SVM column). They correspond to the ability of the eNose to tell the 3 compounds apart when the training and the testing are performed in the same session: this corresponds to the 'naive' solution explained in Introduction and we call them intra-session scores. They have been obtained after a dimensionality reduction with PCA using 5 components and after a 5-fold cross validation using a linear SVM. Even if the overall performance of the eNose on these 3 compounds is rather stable over time, we can note that some scores can be lower than the score obtained during Session 1. This can be an effect of the change in some experimental parameters (temperature, humidity, amount of liquid, presence of other compounds in the room, etc...). If one wants to use only Session 1 to predict the future sessions, we will see in Section 4.2 that a drift between the sessions appeared.

The data set is available upon request.

Results and discussion

Artificial data set

In Figure 5a (left), we present a random realization of the performed simulations. In order to cope with the non-linearity that we introduced, we segment X as a series of non-overlapping windows of size ∆ t = 9. For one simulation, it corresponds to a total of 28 time windows. Each time window is considered as a measurement session of 9 × Q samples (Session i) which needs to be drift corrected. We recall that Q, the number of compounds in Session i, varies from 2 to R = 6.

We compare the proposed method to 2 other methods: PCA-CC from [START_REF] Artursson | Drift correction for gas sensors using multivariate methods[END_REF] working under the Calibrant Scenario and the method of Di Carlo and Falasconi (2012) working under the Blind Scenario. The methods considering the Multi-Session Scenario are not used here since the first sessions will not contain enough information about the future directions (d(t) is rather complex).

For PCA-CC, the calibration samples are accumulated over time (for Session i we use the calibration samples from Sessions 1 → i) and the number of components to remove is set to the minimal number of components required to explain at least 80% of the variance of X cal (the matrix containing the calibration samples). For the classification, the corrected samples are classified with a linear SVM learnt from Session 1. Compound 1 is used as the reference.

EMC 2 and Di Carlo's method are applied session-bysession. These two algorithms return automatically labels. For Di Carlo's methods we use linear SVM and Mahalanobis distance (see Di [START_REF] Di Carlo | Drift correction methods for gas chemical sensors in artificial olfaction systems: techniques and challenges[END_REF]). We initialise EMC 2 by using Algorithm 2. In Figure 5a (right), we report the result using EMC 2 for a random realization.

In Figure 5b, we report the classification results. The column labelled "Raw" corresponds to the results with no drift correction.

EMC 2 clearly outperforms PCA-CC while the latter requires much more information (calibration samples). We notice that both methods are stable regarding the number of compounds present Q, whatever the performance. Parameter Q mostly influences the variance of the classification rate: the lower Q is, the more a misclassification is influencing the score. The performance of PCA-CC is surprisingly quite poor. We explain that by the small number of chemical sensors that we simulated (P = 4) in addition to the complexity of the drift. Even if [START_REF] Artursson | Drift correction for gas sensors using multivariate methods[END_REF] claimed that PCA-CC can compensate for a nonlinear drift by removing more and more components, we noticed that this method tends to remove discriminative directions. In fact, due to the small number of chemical sensors simulated here, it is more likely that the Principal directions of X cal (the matrix containing the calibration samples) carry information about class discrimination. To support this remark, we made other simulations with the same settings, but we increased the dimension P of the space to 6. In this case, the average score of PCA-CC reached 82 % (averaged over Q, not reported in Figure 5b).

EMC 2 also outperforms the method of Di Carlo et al. It is interesting to note that the performance of the method of Di Carlo et al. is influenced by Q: the lower Q, the better. We explain that by the fact that their method makes use of the correction factor of Session i as initialisation for Session i + 1. So, this method will tend to accumulate errors and errors will likely increase with Q. In fact, the classification rate tends to considerably decrease over time (starting from 100% for Session 2 to 70% for Session 28). EMC 2 does not suffer from this drawback since each session (or time window) is treated separately. So, if a session is badly corrected, it will not influence the next session.

Real data set

We recall that each signature x ∈ R 26 is obtained after averaging the replicas and after normalising it by its ℓ 2 -norm. To reduce dimensionality, we use PCA and we project the data of each Session i onto the 5 first Principal directions of the data of Session 1 (and only Session 1). In other words, we use only the knowledge of Session 1 to reduce the number of dimensions. For PCA-CC, CPCA-CC and OSC, the correction is applied before reducing dimensionality (it leads to better results). For EMC 2 and Di Carlo's method, we first reduce dimensionality since some matrices of dimension P ×P need to be estimated from the data (covariance matrices for EMC 2 and correction factors for Di Carlo's method).

Results for Q = 3. In Figure 6a (left), we report the projection of the entire data set onto the 2 first Principal directions of the data from Session 1. From this representation, the drift between Session 1 and the subsequent sessions is clear. If we try to predict Sessions 2 to 15 by training on Session 1 then the score reaches the chance level (33%) after Session 3 (purple line in Figure 6b).

To evaluate the correction methods, we train a linear SVM using only the data from Session 1 and we predict the corrected data from Sessions 2 to 15 (after dimensionality reduction down to 5). EMC 2 and Di Carlo's method directly return the labels ℓ i of Session i so we use them for computing the scores. However, we can also use any other classifier after correcting the data using EMC 2 . For all methods, we remove only one component for the comparison to be fair.

For the methods working under the Multi-Session Scenario, we use only the two first sessions. In fact, this parameter can greatly influence the results and is particularly difficult to tune. In practice, we cannot know early how Raw data Corrected data 57% 98%

• • • • • • • Compound 1 2 3 4 5 6 Session 1 Session i • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • session 28 session 19 Q Raw PCA-CC Di Carlo EMC 2 et al. 6 64±10 62±16 80±15 98±2 5 65±13 60±18 85±12 98±2 4 65±15 61±21 89±14 97±2 3 65±17 60±23 93±10 97±3 2 64±21 59±27 95±11 89±9 (a) (b)

Principal direction 1 Principal direction 2

Raw data

Corrected data

• • • • Compound Session 1 Session i • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • -4 -2 0 2 -2 0 2 4 Citral Geranyl acetate -Pinene Principal direction 1 Days after Session 1 Classification rate (%) PCA-CC EMC 2 OSC CPCA-CC Di Carlo Raw • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • -4 -2 0 2 4 -2 0 2 4
(a) Projection of the data onto the 2 first Principal directions of Session 1 (b) Classification rates over time many sessions we will need to correctly evaluate the drift directions. So we apply here the principle: the least, the better.

Intra Calibrant Multi-Session Blind • • • • • • • •• • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 25
In Figure 6a (right), we report a projection of corrected data using EMC 2 . In Figure 6b, we report the classification results over time (or over sessions) for all the methods considered in this paper. These results (excluding Session 1) are summarized in Table 2.

Clearly, the method using calibration samples (PCA-CC) outperforms the others. We observed that the accumulation of the calibration samples over time (for processing Session i, we take all the calibration samples from Sessions 1 to i) helps the method to find better directions for the drift. If we no longer accumulate the samples but use only the calibration samples from Sessions 1 and i, then the averaged score goes down to 70% and becomes equivalent to the one obtained by our method (not reported in Table 2). The accumulation will help if the direction is stable over time but can also decrease the performance in case of a direction change (which is hard to predict in practice). In addition, the results that we reported in Table 2 for PCA-CC depend greatly on the choice of the reference compound. We chose Citral because it leads to the best results. However, in practice, we do not know the labels of the subsequent sessions so it is quite hard to tell which compound will be the best reference in the end. In fact, if we take β-Pinene as reference, then the average score goes down to 63% (not reported in Table 2). These two remarks reveal that this kind of methods depends strongly on experimental decisions which are hard to make.

Methods working under the Blind Scenario do not suffer from these decisions which influence the overall performance. However, they can be less robust. Table 2 shows that the proposed method, EMC 2 , performs fairly well without requiring any labels from the subsequent sessions. It performs even better than the Multi-Session based methods. The results validate both the proposed algorithm and its heuristic for initialisation. In Figure 7a, we report the drift directions of each compound individually (colored arrows) and the common direction estimated with EMC 2 (black arrow). We can note that the assumption according to which each class drifts along a common direction seems appropriate. In Figure 7b, we report the class weights α r (dashed: ground-truth, solid: EMC 2 estimation). We observe that the factors are different for each class and vary over time, as expected by the proposed model. It is noteworthy that the estimation of the weights of β-pinene by EMC 2 fits perfectly with the groundtruth. However, the algorithm performs less well for the weights of Citral and Geranyl acetate. This may be due to the overlap between the two clusters which leads to some misclassifications in our correction procedure. In fact, we observed that most of the misclassifications comes from the Citral cluster which is confused by the classifier with Geranyl acetate. These errors then result in an underestimation of the weights.

The poor performance of CPCA-CC and OSC can be attributed to the small number of training sessions that we have used. The 2 first sessions may not contain enough drift information about the subsequent sessions. In this case, performance can be increased by considering more sessions but it is again a matter of experimental decisions. This remark highlights their main drawback: in practice, nobody knows how many training sessions are required for the future.

The last method used, namely Di Carlo's method, has the same performance than the Raw method, the method which does not apply any correction. In fact, this method fails to adapt when the data shows a rather 'discontinuous' drift (in comparison to the 'continuous' impression of the simulated drift in Figure 5a left). In other words, this method fails when a significant drift is observed between two different sessions. This method would likely perform better if sessions had been closer together in time. However, the timing of the sessions is often not controlled in the field and methods have to cope with large changes.

Results for Q = 2. In practice, a training set with the R target compounds will be generated during Session 1. However, it is not realistic to expect that the R compounds will be measured every time. It is more realistic to expect only Q < R compounds. The identity of these Q compounds is obviously unknown, which makes the task of drift correction even more tricky for the methods under the Blind Scenario.

In our data set, all the sessions measure the same number R = 3 of compounds. However, we can artificially remove from the subsequent sessions one compound out of the R compounds, leading to Q = 2 compounds. Only the first session (and the 2 first sessions for methods con- sidering the MSS) keeps R = 3 compounds for the training. Drift correction methods are then applied according to the same pipelines. For methods requiring calibration samples, one compound out of the Q = 2 compounds is randomly selected as the calibrant for all the sessions. For selecting which compound is removed from all the testing sessions, the solution is simple: in separated experiments, each one of the R = 3 compounds is selected in turn. Table 3 reports the classification results, averaged over sessions (excluding Session 1). The ranking is more or less unchanged compared to Table 2. The method considering the CS still outperforms the others. However, when Q = 2, 50% of the information is known by this group of methods (information from calibration samples), so the significance of their performance must be moderated. More surprisingly, the proposed method EMC 2 performs fairly well, meaning that the method is able to adapt to a class disappearance during Sessions 2 to 15. Concerning the methods CPCA-CC and OSC, their global performance is poor which is mainly due to the data set where β-Pinene is removed. In fact, most of the good classifications of these methods come from this cluster, and by removing it, we remove the good classifications.

Conclusion

In this paper, we proposed a correction method to compensate for the drift affecting gas sensor arrays. In the literature, most drift-correction methods to assume the Calibrant Scenario or the Multi-Session Scenario. However, we saw that their results depend on experimental decisions which are hard to make in practice: which reference compound to take? Do we keep the calibration samples of the previous sessions? How many training sessions do we need for correctly estimating the drift?

As an alternative, the methods working under the Blind Scenario are free of these considerations. By using only the labels available during the first measurement session, they are by far the least restrictive in practice. This contribution proposed a new correction technique for this group of methods which has been little explored in the literature. It is based on a drift model and has been validated on artificial and real data sets. The method can easily adapt to abrupt changes in the drift direction which could occur over time and does not require that all the classes are sampled during each new measurement session (i.e. some classes may be absent in some sessions).

However, drift is still an open issue and various perspectives of this work can be explored by further studies. First, our method has compensated only partly for the drift so there is still some margin for improving the results. Second, we studied here the drift correction for compounds classification. Another main task is the quantification of the volatile compounds. It is still unclear what would give the correction for this task. Finally, a possible goal by using gas sensor arrays is to use one instrument to generate a data base for training a classifier, and then generalize to other instruments deployed in the field. In this case, another drift will appear, an inter-instrument drift (see for instance the study of [START_REF] Fonollosa | Calibration transfer and drift counteraction in chemical sensor arrays using Direct Standardization[END_REF]). In fact, the manufacturing process of gas sensors is rarely perfectly reproducible. This means that a classifier which learns from one instrument may not directly generalize to another instrument. In this article, we focused only on the intra-instrument drift. The solution proposed may generalize to inter-instrument drift.

Appendix

By assuming independent and identically distributed samples, the likelihood is defined as: To conclude, inject λ = -N 2 in Equation ( 12) and multiply both sides by π k :

∀k ∈ 1, R , πk = n ρ nk N 2 (13) Equation ( 13) is classical in EM algorithms for mixture estimation (see [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF]) while Equations ( 10) and ( 11) are derived from the model that we proposed.

Figure 1 :

 1 Figure 1: Simplified representation of the different scenarios that we proposed to classify the different state-of-the-art methods. Symbols and class contours stand for the session index. Colored points correspond to labelled samples while white points correspond to unlabelled samples. Left figure: calibration samples are measured during each session (Calibrant Scenario), here A is the calibrant. Middle figure: the training set includes several measurement sessions over time, and we assume that it contains drift information (Multi-Session Scenario). Right figure: the training set includes only Session 1 and there is no other labelled samples (Blind Scenario).

Figure 2 :

 2 Figure 2: The drift model that we assume. Each class is translated along a common direction weighted by a coefficient depending on the class.

Figure 3 :

 3 Figure3: Graphical representation of the initialisation procedure for EMC 2 (A = 4, R = 3). We first select random directions from the centroids of X 1 to some random samples of X. Then, for each candidate direction, we orthogonalise X 1 and X against the direction. In the subspace, we classify X ⊥ with a k-NN trained on X 1,⊥ . Finally we compute a score w as the trace of the within-class covariance matrix (Eq. (6)) in the subspace taking both sessions. Bad directions are assumed to have a greater value w. Here, d 2 is a better initial direction than d 1 . Numbers (e.g. ➀) stand for the order of the different steps of the initialisation procedure.

Figure 4 :

 4 Figure 4: Top: robotic setup. Bottom: response of the instrument after one lap. Each peak corresponds to the passage of the robot over a gas source.

Figure 5 :

 5 Figure 5: (a) A random realization (Q = 3). We highlighted by ellipses samples from Session 19, and by squares samples from Session 28 (only for classes 2 & 3, but the same applies to class 1). The indicated scores correspond to classification rates by training on Session 1 (the ten first points). Left figure corresponds to raw data (without any correction) and right figure corresponds to data corrected using EMC 2 . (b) Classification results (%) ± standard deviation, over the 100 simulations. Raw column corresponds to the data with no drift correction.

Figure 7 :

 7 Figure 7: Validation of the drift model. (a) Drift directions over sessions of each compound individually (colored arrows) and the common direction estimated by EMC 2 . Directions are represented along the 2 first Principal directions, and from Session 1 to Session i. (b) True class weights (dashed lines) over time and estimated weights using EMC 2 (solid lines).

③π

  to work with the log-likelihood H:H(d, α, π|X) = N2 n=1 ln( R k=1 π k N (x n , µ k + α k d, S k )) (9) We note N (k) = N (x n , µ k + α k d, S k ) and ρ nk = p(ℓ n = k|x n , d, α, π). -µ k -αk d) T S -1 k (x n -µ k -αk d)) -µ k -α k d) T S -1 k (x n -µ k -α k d)) dT S -1 k (xn-µ k )-α k dT S Estimation of π. Since vector π contains probabilities, R k=1 π k = 1.To incorporate this constraint in the optimisation, we build the Lagrangian Γ(π, λ):Γ(π, λ) = H(d, α, π|X) + λ( k λ = 0 ⇐⇒ λ = -N 2

Table 1 :

 1 Information about the data sets studied in this section. NA values are due to the absence of the sensors at the recording date. R column corresponds to the number of compounds used in the experiment, if R = 12 then we extract from the session the 3 target compounds. RH stands for Relative Humidity. SVM column corresponds to the cross-validated classification rate (repeated 10 times) when we train and test on the same session (mean ± standard deviation).

		Date	Ni	R	T (°C)	RH (%)	SVM (%)
	1	06-07-2018	110	3	NA	NA	98 ± 1
	2	14-09-2018	112	3	NA	NA	94 ± 2
	3	21-09-2018	93	3	[31.6, 34.5]	[27.0, 36.0]	90 ± 1
	4	04-10-2018	165	3	[25.1, 29.6]	[30.1, 39.0]	93 ± 1
	5	16-10-2018	71	3	[28.0, 28.9]	[34.6, 37.5]	92 ± 3
	6	09-11-2018	142	3	[23.3, 28.2]	[31.4, 37.5]	81 ± 1
	7	23-11-2018	132	12	[20.4, 27.6]	[26.6, 40.0]	89 ± 1
	8	04-12-2018	148	3	[23.5, 27.1]	[34.7, 40.3]	91 ± 1
	9	14-12-2018	173	3	[24.4, 27.8]	[18.6, 23.7]	91 ± 1
	10	20-12-2018	161	12	[26.2, 27.5]	[24.1, 29.3]	76 ± 1
	11	09-01-2019	75	12	[25.1, 27.5]	[20.3, 24.7]	71 ± 3
	12	11-01-2019	71	3	[25.6, 27.3]	[14.2, 16]	89 ± 2
	13	06-02-2019	164	3	[24.2, 27.5]	[17.2, 21.3]	91 ± 1
	14	13-02-2019	73	12	[24.4, 27.0]	[17.6, 23.3]	87 ± 1
	15	21-03-2019	256	3	[24.4, 27.2]	[18.1, 22.7]	94 ± 0
		9 months	1934		[20.4, 34.5]	[14.2, 40.3]	[71, 98]

  Left figure corresponds to the uncorrected data: circles stand for Session 1 and squares for Sessions 2 to 15. All the sessions are projected onto the 2 first Principal directions (PCA) of the data from Session 1. Right figure corresponds to the data after applying EMC 2 session-by-session. (b) Classification rates over time for all the methods considered in this paper. Intra is when we train and test on Session i. Raw method corresponds to the uncorrected data (training on Session 1 and testing on Session i).

			100
			75
			50
			0	100	200
	Figure 6: (a) Method	Scenario Score (%)
	PCA-CC	CS	82.1
	EMC 2	BS	69.5
	CPCA-CC	MSS	62.8
	OSC	MSS	59.8
	Raw	-	39.9
	Di Carlo	BS	39
	Intra-session -	87.9
	Table 2: Ranked averaged scores over sessions. Intra-session method
	is when we train and test on Session i (5-fold cross-validation pro-
	cedure). Raw method corresponds to the uncorrected data (training
	on Session 1 and testing on Session i).

Table 3 :

 3 Average scores over sessions when one compound is removed from the subsequent sessions. Results are ranked according to the global average (over sessions and compound out).

	Method	Scenario		Compound out	Mean
			Citral Geranyl	β-Pinene
	PCA-CC	CS	81	92	75	82
	EMC2	BS	92	57	70	73
	OSC	MSS	78	59	34	57
	CPCA-CC	MSS	72	59	28	53
	Di Carlo	BS	56	47	47	50
	Raw	-	54	55	6	39
	Intra-session -	90	99	91	93

We could use the second order moments assuming that they are different for each class, but this could be easily violated in practice.

Link to the explanatory video: https://drive.google.com/file/ d/1AddmCYFwQcHtG1aTksn8zyC-0e7CDzbU/view

Acknowledgments

This work was supported by the FUI-WISE AAP21 Minalogic project, funded by BPI France. This work has been partially supported by MIAI at Univ. Grenoble Alpes (ANR-19-P3IA-0003), chair on detection of pollutants, by an ANR grant to Simon Barthelmé (GenGP, ANR-16-CE23-0008) and by the ANR-18-CE42-0012 OBP-optinose project.