Cécile Carra
email: cecile.carra@univ-artois.fr

Karim Tabia
email: karim.tabia@univ-artois.fr

Smart Home for Seniors: Opportunities and Challenges for AI

Keywords: Smart Home, Artificial Intelligence, Silver Economy

The world's population is aging. This raises crucial questions regarding the health care and well-being of these people. Technological solutions such as Smart homes for the elderly is one of the possible solutions. Smart homes are designed to help their residents perform their daily activities and improve their quality of life while preserving their privacy. These solutions are usually equipped with a set of software and hardware components such as sensors to monitor the living space and recognize the behaviors and activities of the resident. This makes it possible to detect anomalies, to inform about risk situations and to take appropriate measures. For the elderly, often living alone, this can improve their safety, monitor their health and inform their relatives, for example. The purpose of this paper is to present a quick and concise overview of Smart home solutions and a focus on opportunities and challenges for AI. In terms of current solutions, the focus will be on the functionalities that can be provided by this type of solutions and the technical challenges and limitations encountered in practice for the use and large scale deployment of these solutions. We will try to focus on a variety of tasks and issues that require the use of artificial intelligence for a truly intelligent Smart home. This position paper aims to first clarify the opportunities that these Smart home solutions represent for artificial intelligence, but also highlight the challenges and the technical limits. We also present the main results of a survey that we recently conducted with elderly people in Hauts-de-France in France. In doing so, we give some recommendations for the design and the deployment of Smart home on a large scale.

INTRODUCTION

The finding is well established: the world's population is aging. Indeed, most countries in the world are experiencing a growth in the number and proportion of older people in their population. According to data from World Population Prospects: Revision 2019 1 , by 2050, 16% of the world's population will be over 65, while in 2019 this proportion is 9%. By 2050, one in four people living in Europe and North America could be 65 years of age or older. In some countries, such as Japan, the United States of America or European countries, this phenomenon is mainly due to the arrival of the baby boomer generation at retirement age. In France, with the extension of the life span, the aging of the French population continues. People aged 65 and over represent 20% of the population. INSEE 2 's demographic forecasts predict that the proportion of people over 60 years old will 1 https://population.un.org/wpp/ 2 https://www.insee.fr/en/metadonnees/source/serie/ s1316 increase strongly until 2035 (their share should continue to increase, but more modestly afterwards). IN-SEE estimates that 4 million seniors would be in loss of autonomy in 2050. This problem affects all of the countries, but it has certain specificities in some regions, such as the fact that seniors are more vulnerable to poverty and health issues. There is then a real question of society regarding the care of this segment of the population.

Obviously, the needs in terms of health care and well-being of the elderly is a real challenge. It is also interesting economic opportunities, especially for AI. One of the solutions is of a technological nature and concerns Smart homes specially designed for the elderly. Indeed, the "industry" of Smart homes dedicated to the elderly has greatly developed in recent years to meet these needs still rising. Smart home technologies can bring real health and wellness benefits in addition to energy savings, comfort and safety. Solutions adapted to the needs of the elderly impose several constraints because of their state of health and, possibly, their loss of autonomy, loneliness for some of them and their relationship to technology. The ex-

(Preprint version)

isting Smart home technologies mainly concern home automation in the broad sense and is hardly adapted to the needs of the elderly. In addition, most existing solutions are expensive, complex to deploy and often require controlled and constrained environments. It is clear that lightweight, resource-efficient, inexpensive and scalable solutions are needed, which requires minimal effort of installation, configuration and control by users.

In general, a Smart home solution consists of a set of hardware components, communication systems and a set of application components (software) to control these solutions, set them up and possibly benefit from certain services that can be insured by service providers. They are still very basic functionalities and which only allow to centralize the control of certain connected devices and objects, and to make their use easier and achievable remotely. In the context of a Smart home for the elderly, at a minimum there must be intelligent functionalities to detect abnormal or atrisk situations, and possibly functionalities or services provided by health professionals or assistance. It is clear that the existing solutions are very limited at the technical level and several brakes must be lifted so that these solutions can be deployed and used in practice. For example, data security and interoperability of solutions are among the urgent challenges. Like any emerging field, we are witnessing a proliferation of supply in terms of more or less complete solutions. Hence the need for standardization and supervision of these solutions, particularly in the field of Smart homes for the elderly.

The aim of this position paper is to first highlight the opportunities that these Smart home solutions represent for artificial intelligence, but also highlight the challenges and the technical limits. We also present the main results of a survey that we recently conducted with elderly people in Hauts-de-France in France. The paper mainly addresses the technical aspects of Smart home solutions and does not claim that these solutions can address all the problems of home care for the elderly. In doing so, we give some recommendations for the design and the deployment of Smart homes on a large scale. This paper also does not really address the ethical and some societal aspects that are as important as the other aspects (technical, cost, efficiency, etc.) for the success of such technology.

HARDWARE AND SOFTWARE COMPONENTS

Smart home solutions have origins in several fields, mainly in computer science, robotics and electronics such as home automation, ubiquitous computing, embedded computing, telemedicine, gerontechnology, assistive robots and what is called "assistive living". Without being exhaustive, we find a whole range of components that can be classified into two categories.

Hardware Components

This category includes all acquisition devices (sensors), control devices (actuators) and data transmission devices.

• Sensors. These are components that measure certain parameters of interest such as temperature, presence, motion, pressure, humidity, brightness, sleep, heart rate, stress, etc. Some sensors such as heart rate sensors are worn or implanted directly into the body [START_REF] Pantelopoulos | A health prognosis wearable system with learning capabilities using nns[END_REF] while other sensors are generally placed, according to their nature, in certain areas of the living space. It should be noted at this level that some data may come from external sources, particularly when it comes to supplementing the data of the Smart home with ambient data such as the level of air pollution or the presence of air allergens.

• Actuators. These components allow to act on certain connected objects or devices for example to control the light, the heating or the shutters. These are usually switches, valves or motors that can be operated automatically or remotely.

• Means of Communication, Storage and Processing. These are all components or parts of components for temporarily storing certain data after acquisition, to transmit them to other components for long-term storage, to be processed or aggregated. Gateways and access points are used here to route data from the sensors to local units or servers that store and process such data. To communicate with a wide range of sensors, wireless and low-power technologies are generally used. Thus, we find Zigbee technology (IEEE 802.15.4 standard), WiFi (IEEE 802.11 standard) and Bluetooth (IEEE 802.15.1 standard or its variants).

Software Components

The hardware part is used to collect data of interest and serves as a medium for communication, storage and data processing. The application part serves, in turn, the processing of these data, their display to the user and the control of the solution. There are two main categories of applications: First, there are applications intended for the user, in this case the occupant of the Smart home, to give him certain information and enable him to control this solution. These are, in general, very basic features of data retrieval and management of devices and connected objects (for example, simple display of activity data). Some applications provide intelligent features such as fall detection or anomaly detection. Some solutions adopt a centralized architecture in the sense that the data is stored and processed locally on a dedicated server. Other solutions adopt a distributed architecture using remote services on the cloud, for example [START_REF] Li | Smart home: Architecture, technologies and systems[END_REF]. The reader interested in hardware aspects can find further details in [START_REF] Majumder | Smart homes for elderly healthcare-recent advances and research challenges[END_REF]

SMART FEATURES

While connected objects are able to collect many types of data with their sensors and possibly acquire additional data from other sources (such as data on air pollution or the presence of allergens that may to be provided in real time by certain websites), the collected data are often neither correlated nor used to infer relevant information or knowledge in order to carry out intelligent tasks or to offer relevant recommendations to the user [START_REF] Tabia | Data analytics and visualization for connected objects: A case study for sleep and physical activity trackers[END_REF][START_REF] Vallée | On personalization in iot[END_REF][START_REF] Le | Smart homes for older people: Positive aging in a digital world[END_REF]. For example, sleep quality is affected by a number of factors such as physical activity, stress, eating habits (such as drinking alcohol or caffeinated beverages and smoking) and environmental factors. It is then possible to analyze and correlate the data collected on these parameters in order to highlight the factors that have a positive or negative impact on the quality of the user's sleep [START_REF] Babaie | Sleep analytics and online selective anomaly detection[END_REF][START_REF] Tabia | Data analytics and visualization for connected objects: A case study for sleep and physical activity trackers[END_REF]. Of course, this requires the use of data analysis, data mining and artificial intelligence techniques to extract useful knowledge from the large amounts of data collected. We highlight in the following some smart features particularly relevant for the elderly.

Activity Recognition

The goal here is to recognize the current activity of the person occupying the Smart home. In general, this involves associating a flow of data from sensors (such as motion sensor data, presence data or video data) with a class or subset of predefined classes such as Sleeping, Cooking, Eating , Reading, Telephoning, etc. Activity recognition results can be used to keep the elderly family or carers informed, for example, or to detect abnormalities or situations that are dangerous and that may require intervention. At the level of activity recognition systems, we find essentially rule-based systems, probabilistic models [START_REF] Lu | Activity recognition in smart homes[END_REF], classifiers and techniques of machine learning and computer vision [START_REF] Skocir | Activity detection in smart home environment[END_REF][START_REF] Kim | Human activity recognition and pattern discovery[END_REF][START_REF] Chen | Activity Recognition: Approaches, Practices and Trends[END_REF].

Anomaly Detection

The goal here is to analyze activity data (such as physical activity, sleep data, or physiological data) to detect abnormalities. By anomaly, we mean a significant deviation from the norm or the habits of the user. Simply put, an analysis of a person's history of sleep schedules and durations may reveal patterns and sometimes abnormal or unusual behaviors. Often, statistical and machine learning models are used to associate a degree of "normality" with a data flow. Once an anomaly is detected, an alert can be raised to deal with this situation. By way of illustration, several approaches are based on the analysis of time series of sleep data to detect anomalies that may be due to stress or illness [START_REF] Babaie | Sleep analytics and online selective anomaly detection[END_REF][START_REF] Sfar | Activity recognition for anomalous situation detection[END_REF][START_REF] Ye | Anomaly event detection in temporal sensor network data of intelligent environments[END_REF].

Detection of Risk Situations

This type of functionality is essential for people living alone and it aims to detect falls, depressive states, dehydration, taking abnormal medication to name a few examples. Several technologies are needed to quickly detect such situations as the analysis of sounds or videos, computer vision, sensors and dedicated portable devices as well as ground vibration sensors.

Virtual Assistants

Smart or virtual assistants are particularly useful in the context of the Smart home for the elderly who have several needs that can be met by these assistants. These include searching for information on the Internet, calendaring features, or notepad. A Smart assistant can also take care of performing some basic tasks such as making a phone call, sending an email or SMS. It can also be used as a voice interface for home automation or to locate certain objects in the house. Regarding the solutions and projects specifically dedicated to the elderly, we can mention the European project Alfred whose objective is to design and develop an interactive assistant for independent living and active aging, or the project DALIA aiming to be an integrated and personalized solution, based on a personal virtual assistant, which helps older adults to live at home independently.

Alert Management

Alert management is a problem that does not directly concern the user but rather the structures and service providers that are supposed to intervene to support the alerts raised. Effective alert management requires prioritization, filtering, false alarm detection and alert correlation mechanisms.

Recommendation

Several other smart features can be added and integrated into a Smart home solution. Recommendation features may be considered [START_REF] Mshali | A Survey on Health Monitoring Systems for Health Smart Homes[END_REF][START_REF] Vallée | On personalization in iot[END_REF] (for example, recommending physical activity if the system experiences a lack of activity for a period of time or recommending a reduction in heating for a certain period of time for a better sleep). Moreover, a virtual assistant can offer games to maintain better cognitive abilities [START_REF] Hao | Survey on serious game for elderly[END_REF]. This type of functionality will eventually require additional data and other tools and will impact the costs and reliability of these solutions. Among the most successful projects in terms of intelligent features, there is the American project CASAS [START_REF] Cook | Casas: A smart home in a box[END_REF] where sensor data is collected via a computer network and stored in a database that a smart program uses to learn and infer useful knowledge about habitual behaviors and activities. Such models are subsequently used for anomaly detection or provide recommendations. Based on this information, a Smart home can select and automate actions that meet the goals of the Smart home application. In addition, the CASAS architecture facilitates the development and implementation of future smart home technologies by providing a lightweight, easy-to-install design that delivers smart, out-of-thebox functionality requiring minimal installation and configuration efforts.

MAIN TECHNICAL CHALLENGES

This section focuses on the current challenges that need to be addressed for the use and adoption of Smart home solutions in practice. There are, in fact, four big issues now:

Data Security

Given the nature of the data collected, it is clear that privacy considerations and sensitive data in general are critical to the success of a Smart home solution.

In this context, security is to be taken broadly, including the security of collection, storage, the processing chain, the communication of information and the security of the devices themselves [START_REF] Jacobsson | A risk analysis of a smart home automation system[END_REF].

Interoperability

This aspect concerns the lack of interoperability in order to share, for example, some data provided by different vendors or solution vendors. This allows, among other things, to build a solution by integrating components from different suppliers. At present, interoperability in Smart home technologies and associated applications is unsatisfactory due to the use of different technologies, incompatible communication protocols, conflicts of interest, types and heterogeneous data formats [START_REF] Asad | Interoperability in iot based smart home: A review[END_REF]. Improving interoperability requires the use of standardization, the development of data exchange and encoding protocols.

Industrialization

This aspect concerns the transition from the prototype stage to reliable solutions that can be used in practice. This involves mastering several criteria such as low costs, low maintenance, non-intrusiveness and security of solutions, their energy savings and the care of the real needs of the elderly. It is necessary to find a good compromise between all the conflicting criteria (security, cost, ease of use and installation/configuration, Smart features, etc.).

Personalization

One of the challenges facing the use of Smart home solutions and more generally IoT is personalization [START_REF] Vallée | On personalization in iot[END_REF]. This latter aims to provide services tailored specifically for the user needs and preferences. However, user effort are often needed to pro-vide some information that cannot be automatically sensed. For example, if the food intake is provided directly by the user, then because it is time-consuming and perhaps boring, this information will not be provided appropriately by the user. Same problems hold for taking into account user's preferences and constraints.

In addition to the challenges mentioned above, every aspect of Smart home solutions has its own challenges. In particular, the main weaknesses in the detection of anomalies concern the difficulty of distinguishing between abnormalities and normal changes, false positive rates that are too high, lack of diagnosis and cold start problem (period preceding data collection representing the normal and usual habits and behaviors of the user). Indeed, the detection of anomalies can be based either on a specification or definition of normal behaviors, or based on the data of normal behaviors and situations. The problem is that it is very difficult to define normal behavior. The difference between what is considered normal and abnormal behavior is often very vague. In the case of anomaly detection in a Smart home solution, the human behavior is not regular and changes over time. On the other hand, what is abnormal for one person may be quite normal for another.

In activity recognition, there are first limitations related to the low level of reliability of the tools used (for example, the recall and accuracy of prediction techniques are still too weak), the low account of the context, history and also the difficulty of distinguishing between the activities of several people in the same living space. For Smart assistants, the main challenges are, of course, privacy and confidentiality of data, personalization [START_REF] Vallée | On personalization in iot[END_REF] and taking into account the context and specificities and particularities of each user, and finally, the interaction and so-called senior-friendly features.

WHAT ABOUT THE ELDERLY, THEIR NEEDS AND THEIR EXPECTATIONS?

If the finding concerning the aging of the population worldwide is shared and confirmed statistically from year to year, there is not really any case study or surveys concerning the sociology of the elderly and their relationship to technology, nor their perceptions and needs for Smart homes or connected objects. As part of a project, we conducted a survey with elderly people in the Hauts-de-France region in France. Here are the main elements concerning this survey.

The study has the following objectives :

• know about the practices that seniors have of the new technologies of information and communication;

• understand the knowledge and use, if any, that they have connected objects that contribute to the quality of life in the home; and

• collect their expectations on this issue.

These three objectives should make it possible to characterize the relationship of seniors with ICTs and connected objects and to bring out useful analysis for the question of the acceptability and usability of connected home technologies. Data were collected in 2019 by questionnaires, completed face-to-face and at home (with some exceptions). 96 questionnaires were exploited and processed using Sphinx Campus3 . Respondents have the following characteristics: 53.8% in the age group 65 -74 and 46.2% over 75; 63.4% women and 36.6% men; 81.7% retired, having been mostly employed or workers; 60.2% having a monthly income between 800 and 1500 euros, 33.3% having an income of more than 1500 euros and 6.5% having less than 800 euros. The household consists of an average of 1.7 persons and 2.4 children. 75.3% of the respondents are tenants of their home.

ICT Equipment and Uses

In more than 59% of cases, the household has two or more equipments (Internet, computer, Smartphone or tablet) but more than 29% of respondents say they have none. They are 42.6% to use Internet daily, 27.2% to connect once or twice a week, and 29.4% to connect more rarely or never. The most common uses of the Internet in the last 12 months are in order of importance: searches on the Internet (49%), the exchange of short text messages or photos (45%), exchange e-mails ( 42%). The three main obstacles for respondents are (in order of importance): the difficulty of using the Internet (67.7%), the insufficient protection of personal data on the Internet(51%), the uselessness of Internet for everyday life (42.7%). It is essentially (in order of importance): tensiometers, scales, bracelets or watches measuring physical activity and surveillance cameras. Their use is daily for 47.6% of respondents with connected objects; 33.3% report using them occasionally and 19% rarely or never. The reasons given for non-use refer first of all to the feeling that they did not constitute an aid in everyday life and that these objects were too restrictive. The two main reasons that led respondents to use a connected object the first time are in order of importance, a health problem (47.6%) and then the search for comfort (28.6%). The recommendation to buy a connected object comes first and foremost from the family (71.4%), then from the doctor (42.9%) and finally friends (19%).

Connected Objects

Needs and Expectations

The three main types of expectations for connected objects that emerge are, in order of importance: improve daily life (79.2%); prevent accidents in life (79.2%); motivate to do physical activities to stay healthy (38.5%). The three main types of advice expected to equip with a connected object are in order of importance : its use (83.3%), its functionalities (75%), its health (71,.9%). The family for 55.2% of the respondents, the professionals of the connected objects for 54.2%, then, further the health professionals represent the people perceived as the best able to give advices for the choice of a connected object.

The communication of measurements made by connected objects would be more natural to his doctor for 85.4% of respondents, then to the family (49%).

In the case where respondents could benefit from a Smart home, 69.8% say they would be very embarrassed to be filmed or recorded continuously. The three main barriers to the purchase or use of new products or services are in order of importance: the fear of not knowing how to use them (66.7%); the price / remainder too high (66.7%); and far behind, the possible malfunctions (37.5%). The following is the cost / utility ratio, the confidentiality of the data and the lack of information on the product or the service.

A Differentiated Relationship to ICTs and Connected Objects

The sex of the respondents, and even more so their age, the socio-professional category, and just as much the use of the computer tool and the Internet as part of a professional activity, play a significant role in the equipment, use and vigilance concerning the protec-tion of digital personal data. The 65-74 year olds, the men, the highest socio-professional situation, the professional use of ICT lead to a higher rate in terms of equipment, use and acceptability. There is a concern about the protection of personal data on the . It is most likely this population that will experiment first Smart home solutions, provided that the issue of protection of personal data finds a satisfactory answer. Overall, 47.9% of respondents believe that the development of connected objects will allow better monitoring of the health of each. For 25%, it will rather represent an additional constraint that will not change the health of everyone. The remaining 25% say they do not know. These last results show, following those that have been mentioned above, that, on the one hand, the usefulness of connected objects in the quality of life at home remains to be demonstrated, and that, on the other hand, the usability must be a priority issue to allow for their dissemination. Nearly half of the respondents, however, are already convinced of their interest in their health. But the problem of data security is to be addressed, which will help, in part, to promote the acceptability of connected home technologies. It will still be necessary to overcome the feeling of intrusion and intrusiveness associated with the data collection on which the operation of Smart homes rests.

CONCLUSIONS

Smart home solutions are still an emerging market where there is a lot of innovation and it will take time to get enough perspective to identify the real needs, develop effective solutions and advance on the field of standardization, interoperability and security of the solutions. In addition, a range of services from certain players and operators must be developed so that Smart home solutions really meet the needs of the elderly. Currently, there is low acceptance of users of Smart home solutions [START_REF] Portet | Design and evaluation of a smart home voice interface for the elderly: Acceptability and objection aspects[END_REF] in general.

The majority of the solutions are complex and still at the stage of proof-of-concept or prototype. This is, in part, due to the fact that no application or service is really "essential", their costs are high, the offer from partners, installers and service providers is very limited. To this can be added the shortage of skills (for example, installers), lack of standards or their divergence and the fact that there is still no real crossapplication integration provider today. In the future, it is likely that there will be a change in lifestyles and the relationship to technology will change in the direction of gradual adoption and according to the needs and contexts of these technological solutions. Indeed, on the one hand the Internet is more and more in-dispensable and we are witnessing the ever-growing importance of entertainment and security needs (as for video surveillance). On the other hand, there is a lot of progress in the convergence and connectivity of everyday objects (such as the telephone, Internet, television, household appliances, remote monitoring ...). The decreasing costs of technology and technological improvements and availability (due to the rise of wireless solutions and the improvement of broadband penetration) will no doubt end up attracting many users. Finally, the increased cooperation of companies and consortia for better interoperability is likely to encourage the integration of different Smart home technologies.

Software solution for surveys and data analysis.

ACKNOWLEDGEMENTS

This work benefited from a funding " Bonus Qualité Recherche 2019 -Smart home et seniors: Prototypage d'une solution smart home et expérimentation" of Artois University, France.


Sebastien Chaigne

Guillaume Cardouat

Julien Louradour

Fanny Vaillant

Sabine Charron

Frederic Sacher

Thomas Ducret

Romain Guinamard

Edward Vigmond

Thomas Hof

Transient Receptor Potential Vanilloid 4 channel participates in mouse ventricular electrical activity

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


James Ledoux
email: james.ledoux@insa-rennes.fr

Filtering and the EM-algorithm for the Markovian Arrival Process

Keywords: Hidden Markov process, Point process, Innovations method, Zakai's filter, Queuing theory, Software reliability

Filtering and the EM-algorithm for the Markovian arrival process

Introduction

A major issue in stochastic modeling is to calibrate the models from data. Here, we focus on a class of Markovian models known as the Markovian Arrival Processes (MAP) (e.g. see [START_REF] Neuts | Structured Stochastic Matrices of M/G/1 Type and Their Applications[END_REF], [START_REF] Asmussen | Matrix-analytic models and their analysis[END_REF]). An MAP is formally defined as a bivariate Markov process (N t , X t ) t≥0 where (N t ) t≥0 is a counting process of "arrivals", (X t ) t≥0 is a Markov process with a finite state space, say {1, . . . , n}, and the transition probabilities of (N t , X t ) t≥0 satisfy the following additivity property: for any k = 0, 1, i, j = 1, . . . , n, m ∈ N and 0 ≤ s < t

P{N t = k + m, X t = j | N s = m, X s = i} = P{N t -N s = k, X t = j | X s = i}.

1

The other transition probabilities are zero. The property above implies that the entries of the generator A of (N t , X t ) t≥0 satisfy similar conditions: for k = 0, 1, i, j = 1, . . . , n and m ∈ N A (k + m, j), (m, i) = A (k, j), (0, i) := D k (j, i)

and the other entries are zero. The equalities above define two n × n-matrices D 0 , D 1 . The non-negative numbers D 0 (j, i), D 1 (j, i) (j = i) represent the rates at which (X t ) t≥0 jumps from state i to j with no arrival and one arrival respectively. The non-negative entry D 1 (i, i) is the rate at which one arrival occurs, (X t ) t≥0 staying in state i. Note that the Markov process (X t ) t≥0 has the generator Q = D 0 + D 1 . Listing the state space N × {1, . . . , n} in lexicographic order, the generator of an MAP has the form

A =       D 0 0 • • • D 1 D 0 . . . 0 D 1 . . . . . . . . . . . .      
.

(1)

The so-called Markov Modulated Markov Process is the special instance of an MAP obtained in setting

D 1 := Diag(λ(i)) D 0 := Q -Diag(λ(i))
where Diag(λ(i)) is a diagonal matrix with ith diagonal entry λ(i) and Q is the generator of the Markov process (X t ) t≥0 . In this model, the arrival instants constitute a Poisson process with intensity λ(i) during a sojourn time of (X t ) t≥0 in state i. The main properties of such a class of processes may be found in [START_REF] Fischer | The Markov-modulated Poisson process (MMPP) cookbook[END_REF], for instance.

MAPs have been introduced in queuing theory in order to consider Markovian input streams for queuing systems with non-independent inter-arrival durations. The Markov property allows to deal with analytically tractable models. This class of models has gained widespread use in stochastic modeling of communication systems, in reliability for systems and many other applications areas (e.g see [START_REF] Neuts | Matrix-analytic methods in queueing theory[END_REF] for an extensive bibliography). Our motivation for dealing with MAPs originates in the software reliability modeling using an "architecture-based" approach. Indeed, a standard model in this context was provided by [START_REF] Littlewood | A reliability model for systems with Markov structure[END_REF]. It has inspired most other works (see [START_REF] Goseva-Popstojanova | Architecture-based approach to reliability assessment of software systems[END_REF] for a recent survey). The failure process associated with such a kind of models turns to be a point process associated with a specific MAP. Thus, in our context of software reliability modeling, (N t ) t≥0 is the counting process of failures and (X t ) t≥0 is interpreted to be a Markovian model of the flow of control between the modules of a software.

Although of widespread use for more than twenty years now, the statistical analysis and specifically fitting of MAPs to data, has been discussed only recently. This is a major issue, especially in software reliability modeling where, to the best of our knowledge, little has been done for the statistical estimation of the parameters of the architecture-based models. Since an MAP is specified by the matrices D 0 , D 1 , the fitting of MAPs to data requires the estimation of the non-negative parameter vector θ = {D k (j, i), k = 0, 1 i, j = 1, . . . , n}.

(2)

The only available data is the observation of arrivals. In that perspective, the process (N t , X t ) t≥0 can be thought of as a partially observed Markov process or a hidden Markov process. The observed process is the counting process of arrivals (N t ) t≥0 and the state or hidden process is the finite Markov process (X t ) t≥0 . The EM-algorithm is a standard way to estimate the parameters of hidden Markov processes. Specifically, it has been used by [START_REF] Rydén | An EM-algorithm for estimation in Markov-modulated Poisson process[END_REF] for the Markov Modulated Poisson Process, by [START_REF] Asmussen | Phase-type distributions and related point processes: Fitting and recent advances[END_REF] for the Phase-Type distributions, by [START_REF] Breuer | An EM algorithm for batch arrival processes and its comparison to a simpler estimation procedure[END_REF], [START_REF] Klemm | Modeling IP traffic using the Batch Markovian Arrival Process[END_REF] for general MAPs. The numerical experiments reported in their studies show that the EM-algorithm works well in general. All these works use the standard forward-backward principle which is based on data processing in batch. Due to the backward pass through the data, the storage cost is linear in the number of observations. [START_REF] Elliott | Hidden Markov Models[END_REF] has proposed a filter-based EM-algorithm where the standard forward-backward form of the E-step of the algorithm is replaced by a single forward pass procedure. The main advantages are that "on-line" estimation is allowed and the storage cost does not depend on the number of observations so that very large data sets can be processed. To implement the filter-based approach, finite-dimensional recursive filters for various statistics related to MAPs must be found (if there exist). The aim of this paper is to provide such finite-dimensional filters.

The paper is organized as follows. In Subsection 2.1, the basic material on stochastic calculus needed throughout the paper is introduced. Next, the EM-algorithm and the forward-backward strategy are discussed for MAPs. In Subsection 2.3, the filters associated with the statistics of interest for using the EM-algorithm are shown to be the solutions of non-linear stochastic differential equations (SDEs). In Subsection 2.4, the so-called unnormalized/Zakai filters are proved to be the solution of linear SDEs. These SDEs allow a recursive computation of filters involved in the re-estimation formulas of the parameters in (2). They are the main contributions of the paper. The forward-backward and filter-based strategies are briefly compared in Subsection 2.5. Concluding comments are reported in Section 3.

Finite-dimensional filters

Main notation and convention

• Vectors are column vectors. Row vectors are obtained by means of the transpose operator (•) ⊤ . The ith component of any vector v is denoted by v(i).

For any pair of vectors

u, v ∈ R n , u, v = u ⊤ v, is the usual scalar product in R n .
1 is a n-dimensional vector with each entry equals to one.

• For any right-continuous with left-hand limits (rcll) function t → f t , the left-hand limit at t of f is denoted by f t-and ∆f t := f tf t-for t > 0 is the jump of the function at time t. We set ∆f 0 := f 0 .

• The state space of the Markov process (X t ) t≥0 is assumed to be X := {e i , i = 1, . . . , n}, where e i is the ith vector of the canonical basis of R n . With this convention, the indicator function 1 {Xt=e i } of the set {X t = e i } is the ith component of vector X t , that is X t , e i . In other words, for any time t, the vector X t is the n-dimensional vector

X t = 1 {Xt=e i } n i=1
. Therefore, the sum of the components of vector X t , 1, X t , is equal to 1 for any time t.

• All processes are assumed to be defined on the same probability space (Ω, F, P). The internal filtrations of processes (N t ) t≥0 and (N t , X t ) t≥0 are denoted by F N := (F N t ) t≥0 and F := (F t ) t≥0 respectively, where F N t := σ(N s , s ≤ t) and F t := σ(N s , X s s ≤ t). These filtrations are assumed to be complete, that is F 0 , F N 0 contain all the sets of P-probability zero of F.

• For any integrable F-adapted random process (Z t ) t≥0 , the conditional expectation

E[Z t | F N t ]
is denoted by Z t and ( Z t ) t≥0 is called the filter associated with the process (Z t ) t≥0 .

Basic material on the observed/hidden processes

In this paper, we are concerned with the following basic statistics of an MAP

L 1,ji t := 0<s≤t ∆N s X s , e j X s-, e i = t 0 X s , e j X s-, e i dN s j = i L 0,ji t := 0<s≤t (1 -∆N s ) X s , e j X s-, e i O (i) t := t 0 X s-, e i ds.
(3) For i = j, the first -second -statistic is the number of jumps of (X t ) t≥0 from state e i to state e j coming with one -no -jump of the counting process over the interval ]0, t]. L 1,ji t can also be thought of as the number of arrivals coming with a jump of (X t ) t≥0 from state e i to e j over the interval ]0, t]. L 1,ii t is the number of arrivals up to time t, (X t ) t≥0 staying in state e i at each arrival instant. The third statistic is the sojourn time of (X t ) t≥0 in the state e i in the interval [0, t]. This family of statistics is introduced because their filters appear in the re-estimation formulas (10) used for estimating the parameters of an MAP with the EM-algorithm.

The basic material below can be found in [START_REF] Bremaud | Point Processes and Queues[END_REF], [START_REF] Klebaner | Introduction to Stochastic Calculus with Applications[END_REF] for instance. We report here a F-semi-martingale (or Doob-Meyer here) representation of the counting processes (N t ) t≥0 , (L 0,ji t ) t≥0 , (L 1,ji t ) t≥0 . It is clear from their definition that (N t ) t≥0 , (L 0,ji t ) t≥0 , (L 1,ji t ) t≥0 are counters of specific transitions in (N t , X t ) t≥0 . Now, the F-semi-martingale decomposition of the number of transitions N t (y, x) of (N t , X t ) t≥0 from state x to state y at time t is known to be

N t (y, x) = t 0 A(y, x) 1 {(N s-,X s-)=x} ds + M t (y, x)
where (M t (y, x)) t≥0 is a F-martingale and A is the generator of (N t , X t ) t≥0 . Then, it is easily deduced from the special structure of A (see ( 1)) that the F-semi-martingale decomposition of the counting processes above are

N t = t 0 λ s ds + M t with λ s := 1, D 1 X s- (4) L k,ji t = t 0 D k (j, i) X s-, e i ds + M L k,ji t k = 0, 1 (5)
where j = i for k = 0, and (M) t≥0 , (M L k,ji ) t≥0 are F-martingales. The process (λ t ) t>0 is the so-called stochastic intensity of (N t ) t≥0 with respect to the filtration F.

The basic F-semi-martingale decomposition of the Markov process (X t ) t≥0 is

X t = X 0 + t 0 QX s-ds + M X t (6)
where (M X t ) t≥0 is a F-martingale (and a martingale with respect to the internal filtration of (X t ) t≥0 as well). We recognize in ( 4) and ( 6) a standard representation of a continuoustime hidden Markov process, with (X t ) t≥0 as state process and (N t ) t≥0 as observed process. The observation and state "noises" (M t ) t≥0 , (M X t ) t≥0 are correlated here. The following results will be used throughout the proofs.

(R1) The stochastic integrals in this paper are Lebesgue-Stieltjes integrals. We report here the product rule for two rcll processes (Z

(1) t ) t≥0 , (Z (2)
t ) t≥0 having finite variations on the bounded intervals, i.e. having locally finite variations,

Z (1) t Z (2) t = Z (1) 0 Z (2) 0 + t 0 Z (1) s-dZ (2) s + t 0 Z (1) s-dZ (2) s + 0<s≤t ∆Z (1) s ∆Z (2) s = Z (1) 0 Z (2) 0 + t 0 Z (1) s-dZ (2) s + t 0 Z (1) s dZ (2) s .
Since any rcll process (Z t ) t≥0 has finitely many jumps on finite intervals for almost all ω ∈ Ω, the following standard Lebesgue integrals (R2) For any integrable process (Z t ) t≥0 such that

t 0 E |Z s | ds < +∞, we have that t 0 Z s ds - t 0 Z s ds defines a F N -martingale.
(R3) Let (H t ) t≥0 be a F-predictable process and (M t ) t≥0 be a F-martingale of integrable variation on any bounded intervals. If

t 0 E |Z s | |dMs| < +∞ for any t, then ( t 0 H s dM s ) t≥0 is a F-martingale.
Note that any left-continuous F-adapted process is F-predictable. The same statement holds replacing everywhere the filtration F by

F N . (R4) For any F-martingale (M t ) t≥0 , ( M t ) t≥0 is a F N -martingale.

The EM-algorithm and the forward-backward strategy

We briefly describe the EM-algorithm for our continuous-time hidden Markov model. We refer to [START_REF] Breuer | An EM algorithm for batch arrival processes and its comparison to a simpler estimation procedure[END_REF], [START_REF] Klemm | Modeling IP traffic using the Batch Markovian Arrival Process[END_REF] for full details. For a fixed parameter vector θ as defined by ( 2), the underlying probability measure and the associated expectation are denoted by P θ and E θ respectively. X 0 or its probability distribution x 0 is assumed to be known. The observed data are supposed to be the arrival times {t 0 , . . . , t K }, where we set t 0 := 0. Without loss of generality, we assume in this subsection that t := t K . The likelihood of the observed data is under

P θ l(θ, N ) := 1 ⊤ 1 l=K D 1 exp D 0 (t l -t l-1 ) x 0 (7)
and is called the observed/incomplete data likelihood. Now, suppose that the complete data {N s , X s , s ≤ t} are available. Then, the complete data likelihood function is, under

P θ , L(θ; N, X) := n i,j=1, D 1 (j, i) L 1,ji t n i,j=1,j =i D 0 (j, i) L 0,ji t n i=1 e D 0 (i,i)O (i) t n i=1 x 0 , e i X 0 ,e i . (8)
It can be shown that a new estimate θ := { D k (j, i), i, j = 1, . . . , n; k = 0, 1} satisfying l( θ, N ) ≥ l(θ, N ), is obtained as a result of the two following steps.

1. E-step.

Compute the so-called pseudo-log-likelihood

Q(• | θ) defined by Q(θ * | θ) := E θ log L(θ * ; N, X) | F N t with θ * := {D * k (j, i), i, j = 1, . . . , n; k = 0, 1}. It is easily seen from (8) that Q(θ * | θ) = n i,j=1,j =i log D * 0 (j, i) L 0,ji t + n i,j=1 log D * 1 (j, i) L 1,ji t + n i=1 D * 0 (i, i) O (i) t + K (9)
where K is a constant that does not depend on the parameters, and

k = 0, 1, L k,ji t := E θ [L k,ji t | F N t ], O (i) t := E θ [O (i) t | F N t ]
are the filters associated with the statistics defined in (3).

2. M-step.

Determine θ maximizing the function in ( 9) under the constraints that n j=1 (D * 0 (j, i)+ D * 1 (j, i) = 0, i = 1, . . . n. Using the Lagrange multipliers method, it is shown that, for i, j = 1, . . . , n,

D 1 (j, i) = L 1,ji t O (i) t and D 0 (j, i) = L 0,ji t O (i) t with i = j. (10)
Therefore, the idea is to pick up an initial value θ (0) for the parameter and to iterate 1-2 as long as a stopping criterion is not satisfied. As a result, we obtain a sequence (θ (m) ) m≥0 of estimates corresponding to non-decreasing values of the observed likelihood function (with equality iff θ (m+1) = θ (m) under mild conditions). Note that the zero entries of D k s are preserved by the procedure above.

Remark 1 We mention that formulas (10) are intuitively supported by the fact that, canceling the conditional expectation operation, we retrieve the estimators that we would obtained applying the standard Maximum Likelihood method to the complete data likelihood (8), that is if the complete data were observed.

Remark 2 In software reliability context, a priori estimates for θ using procedures reported in Goseva-Popstojanova and Trivedi (2001) can be obtained. They are based on data collected at earlier phases of the software life cycle (validation phases, integration tests,. . . ). These estimates might appear to be rough estimates especially when the software is in operation. This motivates a re-estimation of the parameters by EM when failure data are collected during the operational life of the software.

Remark 3 In this paper, the number of hidden states n is assumed to be known. This is adequate, for example, in speech recognition where the hidden states are the elements of a finite alphabet, in architecture-based software reliability modeling where the hidden states are the modules of a piece of software. However, in many applications, this is not the case. A generic situation is when a partially observed model is used as a statistical model for fitting to empirical time series. Estimation of the number of hidden states is known to be a hard problem. It is not intended to address this fundamental issue here. In the context of hidden Markov chains, the so-called "order estimation problem" is surveyed in (Ephraim and Merhav 2002, Section VIII). Recent progress in this direction is reported in [START_REF] Boucheron | An information theoretic perspective on order estimation[END_REF]. We refer the reader to these papers and the references therein for details. This iterative procedure requires the computation of the conditional expectations in (10). For hidden Markov models, the "forward-backward " formulation of the EMalgorithm -also referred to as the Baum-Welch formulation -is the standard way to do it. This is what is done in the previously mentioned works. The basic idea is to write these conditional expectations using Fubini's theorem as

L 1,ji t = t 0 P{∆N s = 1, X s-= e i , X s = e j | F N t }ds O (i) t = t 0 P{X s = e i | F N t }ds L 0,ji t = t 0 P{∆N s = 0, X s-= e i , X s = e j | F N t }ds
so that the conditional probabilities under the integral sign must be evaluated. The probabilities P{X s = e i | F N t }, s < t are known as the state smoothers and are computed according the "forward-backward" strategy. The other conditional probabilities are deduced from these state smoothers. The final form of the EM-algorithm is given by Figure 1 (see [START_REF] Klemm | Modeling IP traffic using the Batch Markovian Arrival Process[END_REF]).

Bearing in mind the filter-based approach pioneered by [START_REF] Elliott | Hidden Markov Models[END_REF], it is expected that the conditional expectations in (10) are solution of finite-dimensional recursive equations, that is finite-dimensional filters exist. The main contribution of the paper is to provide such finite-dimensional filters in Theorem 4 and finite-dimensional unnormalized/Zakai filters in Theorem 6. It is clear from the second and third lines of the algorithm in Figure 1 that a forward and a backward pass through the data are required. In contrast, a single pass -a forward pass -through the data set is needed for the filter-based approach. We get back to the comparison of the two strategies in Subsection 2.5.

Stochastic differentials equations for the filters

In fact, we show that the filters defined, for every t, by

O (i) X t := E O (i) t X t | F N t and L k,ji X t := E L k,ji t X t | F N t , k = 0
, 1 turn to be finite-dimensional. Then, since the sum of the components of vector X t is equal to 1 for any t, the filters that we are interested in, are obtained as follows

O (i) t = 1, O (i) X t , and L k,ji t = 1, L k,ji X t .
We know from (4) that the F-stochastic intensity of the counting process (N t ) t≥0 is given, for t > 0 by λ t = 1, D 1 X t-. Then, it can be deduced from [START_REF] Bremaud | Point Processes and Queues[END_REF] that the F N -stochastic intensity ( λ t ) t>0 of (N t ) t≥0 is, for t > 0,

λ t := 1, D 1 X t-, (11)
f 0 (x) := exp(D 0 x) and f 1 (x) := D 1 exp(D 0 x); for l = 1, . . . , K, ∆t l := t lt l-1 with t 0 := 0.

Forward.

α 0 := x 0 , c 0 := 1, for l = 1, . . . , K, α l := f 1 (∆t l )α l-1 , c l := 1 ⊤ α l
Backward. β ⊤ K+1 := 1 ⊤ , and for l = K, . . . , 1 β ⊤ l := β ⊤ l+1 f 1 (∆t l ) For i, j = 1, . . . , n: L 0,ji 0 := 0, L 1,ji 0 := 0 and for l = 1, . . . , K:

L 0,ji l := L 0,ji l-1 + β ⊤ l+1 t l t l-1 f 1 (t l -s) e j D 0 (j, i) e ⊤ i f 0 (s -t l-1 ) ds α l-1 L 1,ji l := L 1,ji l-1 + β ⊤ l+1 e j D 1 (j, i) e ⊤ i f 0 (∆t l )α l-1 O (i) l := O (i) l-1 + β ⊤ l+1 t l t l-1 f 1 (t l -s)e j e ⊤ i f 0 (s -t l-1 ) ds α l-1 O (i) t K = O (i) K c K L 0,ji t K = L 0,ji K c K L 1,ji t K = L 1,ji K c K Comment.
Once normalized to 1, the forward quantities α l , l = 0, . . . , K give the state filter at times t 0 , . . . , t K

X t l = E[X t l | F N t l ] = (P{X t l = e i | F N t l }) n i=1 = α l c l l = 0, . . . , K.
Comment. The conditional probability P{X t l = e i | F N t K } for l = 0, . . . , K -1 are given by

P{X t l = e i | F N t K } = β ⊤ l e i e ⊤ i α l n j=1 β ⊤ l e j e ⊤ j α l
Figure 1: Forward-Backward implementation of the filters that is the process defined for t ≥ 0 by

N t := N t - t 0 λ s ds (12)
is a F N -martingale. (N t ) t≥0 is called the innovations martingale.

Theorem 4 Let us consider the F N -stochastic intensity ( λ t ) t>0 of (N t ) t≥0 and the innovations martingale (N t ) t≥0 defined in ( 11) and ( 12) respectively.

1. Filter for the state. We have for any t ≥ 0

X t = X 0 + t 0 Q X s-ds + t 0 D 1 X s--X s-λ s λ s dN s . (13a)
2. Filter for the sojourn time in e i . We have for any t ≥ 0

O (i) X t = t 0 Q O (i) X s-+ X s-, e i ds e i ds + t 0 D 1 O (i) X s--O (i) X s-λ s λ s dN s . (13b)
3. Filter for the numbers of specific jumps of the MAP. We have for any t ≥ 0

L 0,ji X t = t 0 [Q L 0,ji X s-+ D 0 (j, i) X s-, e i e j ]ds + t 0 D 1 L 0,ji X s--L 0,ji X s-λ s λ s dN s (13c) L 1,ji X t = t 0 Q L 1,ji X s-+ D 1 (j, i) X s-, e i e j ds + t 0 D 1 (j, i) X s-, e i e j + D 1 L 1,ji X s--L 1,ji X s-λ s λ s dN s . (13d)
We mention that equation (13a) turns to be one of the main ingredient of the proof in [START_REF] Gravereaux | Poisson approximation for some point processes in reliability[END_REF] of the (optimal) convergence rate of an MAP to a Poisson process when the arrivals are rare. We see that the stochastic differential equations in Theorem 4 are non-linear, so that the numerical procedure used for solving these equations has to be carefully implemented. For instance, we recall that negative probabilities may be obtained as a result of the numerical computation of the state filter of a Markov process observed in an additive Brownian noise -also called Wonham's filter -with an Euler-Maruyama discrete-time approximation (see (Kloeden et al. 1994, Chap 6)). The purpose of the next subsection is to derive linear SDEs from which the filters may be obtained. Such linear SDEs are more suited to numerical computation.

Remark 5 A filter for the number of transitions N X,ji t of (X t ) t≥0 from state e i to e j (i = j) up to time t can be derived as those of Theorem 4. Indeed, we obtain an SDE for N X,ji X t := E N X,ji t X t | F N t from Theorem 4 using the fact that N X,ji X t = L 0,ji X t + L 1,ji X t for i = j. Then, the sum of he components of the vector N X,ji X t gives the filter associated with N X,ji t . Since D 0 + D 1 = Q, the transition rates of (X t ) t≥0 can be estimated by the EM-algorithm using

Q 1 (j, i) = N X,ji t O (i) t
as re-estimation formula.

Proof. A proof of (13a) may be found in [START_REF] Gravereaux | Poisson approximation for some point processes in reliability[END_REF]. In the sequel, (M t ) t≥0 -( M t ) t≥0 -will denote a generic F-martingale -F N -martingale. The proof of ( 13b) is as follows. The product rule (R1) with the fact that (O 6),( 3) and ( R3). ( 15)

(i) t ) t≥0 has no jump give O (i) t X t = t 0 O (i) s-dX s + t 0 X s-dO (i) s (14) = t 0 QO (i) s-X s-ds + t 0 X s-, e i e i ds + M t from (
Taking the conditional expectation with respect to F N t on each side of the equation above, and using (R2) and (R4), we obtain

O (i) X t = t 0 Q O (i) X s-ds + t 0 X s-, e i e i ds + M t . ( 16
)
The integral representation of F N -martingales says us that (e.g. see [START_REF] Bremaud | Point Processes and Queues[END_REF])

M t = t 0 G (i) s dN s (17)
where (G

(i)
t ) t≥0 is a F N -predictable process which is called the innovations gain. Thus, the proof will be complete if we show that

G (i) s = D 1 O (i) X s--O (i) X s-λ s λ s . ( 18
)
The product N t × O (i) X t has the following form from the product rule (R1)

N t O (i) X t = t 0 N s-d O (i) X s + t 0 O (i) X s-dN s + 0<s≤t ∆N s ∆ O (i) X s . Since ∆ O (i) X s = G (i) s ∆N s = G (i)
s ∆N s from ( 16),( 17),( 12) and (∆N s ) 2 = ∆N s , the last term in the right-hand side of the inequality above is

t 0 G (i)
s dN s . Then, we deduce from ( 12),( 16) and (R3) that

N t O (i) X t = t 0 N s-[Q O (i) X s-+ X s-, e i e i ] ds + M t + t 0 O (i) X s-λ s ds + t 0 G (i) s λ s ds + M t . (19)
Next, the product

N t × O (i)
t X t is rewritten from the rule product (R1) and the fact that (O (i) t ) t≥0 has no jump as 15) and (R3). Let us compute the last integral in the right-hand side of equality above. Since j X s , e j = k X s-, e k = 1, and ∆N s X s , e j X s-, e k = ∆L 1,jk s we find that 5) and (R3)

N t O (i) t X t = t 0 N s-d(O (i) X) s + t 0 O (i) s-X s dN s = t 0 N s-[QO (i) s-X s-+ X s-, e i e i ] ds + t 0 O (i) s-X s dN s + M t from (
t 0 O (i) s-X s dN s = 0<s≤t O (i) s-X s ∆N s = j e j k t 0 O (i) s-dL 1,jk s = t 0 O (i) s- j e j k D 1 (j, k) X s-, e k ds + M t from (
= t 0 O (i) s-D 1 X s-ds + M t .
Then, we deduce from the last equality that

N t O (i) t X t = t 0 N s-[QO (i) s-X s-+ X s-, e i e i ] ds + t 0 D 1 O (i) s-X s-ds + M t .
Taking the conditional expectation on both sides of the previous formula and using (R2) and (R4), a second decomposition of the semi-martingale

(N t O (i) X t ) t≥0 is N t O (i) X t = t 0 N s-[Q O (i) X s-+ X s-, e i e i ] ds + t 0 D 1 O (i) X s-ds + M t . ( 20
)
We know that the locally finite variations part of the decomposition of a special semimartingale is unique. Then, we identify the corresponding terms in the decompositions ( 19) and ( 20), that is the Lebesgue integrals. The expression (18) of the gain G (i) follows easily.

The formulas (13c,13d) are shown in the same way. We only provide the main steps of the computation for ( L 1,ji X t ) t≥0 . The product rule (R1) and formulas (4)-( 6) allow us to write

L 1,ji t X t = t 0 [QL 1,ji s-X s-ds + D 1 (j, i) X s-, e i e j ] ds + M t .
Taking the conditional expectation with respect to F N t on both sides of the equations above, using (R2),(R4) and the representation theorem of the F N -martingales, we obtain

L 1,ji X t = t 0 [Q L 1,ji X s-+ D 1 (j, i) X s-, e i e j ] ds + t 0 G (L) s (dN s -λ s ds) (21)
where G (L) is the innovations gain.

A first F N -representation of the semi-martingale N t L 1,ji X t is obtained from (R1),( 21), ( 4) and (R3)

N t L 1,ji X t = t 0 N s-[Q L 1,ji X s-+ D 1 (j, i) X s-, e i e j ] ds + t 0 L 1,ji X s-λ s ds + t 0 G (L) s λ s ds + M t . (22)
Next, the product N t × (L 1,ji t X t ) may be rewritten using (R1) as

N t L 1,ji t X t = t 0 N s-[QL 1,ji s-X s-+ D 1 (j, i) X s-, e i e j ] ds + t 0 D 1 (j, i) X s-, e i ds e j + t 0 D 1 L 1,ji s-X s-ds + M t .
Conditioning with respect to F N t on both sides of the previous formula and using (R2) and (R4) lead to a second decomposition of the special semi-martingale

(N t × L 1,ji t X t ) t≥0 N t L 1,ji X t = t 0 N s-[Q L 1,ji X s-+ D 1 (j, i) X s-, e i e j ] ds + t 0 D 1 (j, i) X s-, e i e j ds + t 0 D 1 L 1,ji X s-ds + M t . (23)
The final form of the gain G (L) is obtained by identifying the terms with locally finite variations in the decompositions ( 22) and ( 23). The innovation form of the filter L 1,ji X t in (13d) is deduced by replacing the gain G (L) in (21).

Zakai filters

In this part of the paper, we derive the so-called Zakai filters associated with the statistics in (3). A technique of change of measure is used. We recall some basic facts which are borrowed from (Bremaud 1981, Chap VI) for instance. Assume that

µ(i) := 1, D 1 e i > 0, i = 1, . . . , n.
Then, we have 0

< min i µ(i) ≤ λ t = 1, D 1 X t-≤ i µ(i) so that 1 i µ(i) ≤ 1 λ t ≤ 1 min i µ(i) . (24)
Let us consider the likelihood ratio

L t := exp t 0 ln 1 λ s dN s + t 0 (λ s -1) ds = 0<s≤t ( 1 λ s ) ∆Ns exp t 0 (λ s -1) ds
which is a solution of the equation

L t = 1 + t 0 L s-( 1 λ s -1) (dN s -λ s ds).
Then, (L t ) t≥0 is a (P, F)-martingale from ( 24) and (R3). A new probability measure P 0 on (Ω, ∨ t≥0 F t ) is defined by dP 0 dP Ft := L t .

It results from Girsanov's theorem that, under P 0 , (N t ) t≥0 is a F-homogeneous Poisson process with intensity λ s ≡ 1. Therefore, the process (n t ) t≥0 defined by

n t := N t -t (25)
is a (P 0 , F)-martingale and must be thought of as the innovations martingale associated with (N t ) t≥0 under the new probability measure P 0 . Next, set

L t := 1 L t . (26)
(L t ) t≥0 is a solution of the equation

L t = 1 + t 0 L s-(λ s -1) dn s (27)
and is a (P 0 , F)-martingale. We know that P ≪ P 0 and dP dP 0 Ft = L t .

Under the mild condition stated at the beginning of this subsection, P and P 0 are equivalent probability measures. The expectation with respect to P 0 is denoted by E 0 [•]. Any F N -conditional expectation, under P, of an integrable F-measurable random variable, may be computed from Bayes formula (e.g. see [START_REF] Elliott | Hidden Markov Models[END_REF])

E[Z | F N t ] = E 0 [ZL t | F N t ] E 0 [L t | F N t ] P a.s. ( 28
)
The above equation allows us to derive the Zakai filters

E 0 [ZL t | F N t ] by the formula E 0 [ZL t | F N t ] = E[Z | F N t ] E 0 [L t | F N t ]. (29)
The following final fact is used to derive the Zakai filters associated with the statistics defined in (3). It is known from (Bremaud 1981, R 8, p 174) that L 0 t := E 0 [L t | F N t ] satisfies the equation

L 0 t = 1 + t 0 L 0 s-( λ s -1) dn s (30)
where λ t is the F N -stochastic intensity of (N t ) t≥0 given in (11).

Theorem 6 The processes (n t ) t≥0 and (L t ) t≥0 are defined in ( 25) and ( 26) respectively. For any F-adapted integrable process (Z t ) t≥0 , σ(Z t ) denotes the conditional expectation

E 0 [Z t L t | F N t ]. 1.
Zakai filter for the state. We have for any t ≥ 0

σ(X t ) = X 0 + t 0 Qσ(X s-) ds + t 0 (D 1 -I)σ(X s-) dn s . (31a)
2. Zakai filter for the sojourn time in e i . We have for any t ≥ 0

σ(O (i) t X t ) = t 0 Qσ(O (i) s-X s-) + σ(X s-), e i e i ds + t 0 (D 1 -I)σ(O (i) s-X s-) dn s . (31b)
3. Zakai filter for the numbers of specific jumps for the MAP. We have for any t ≥ 0

σ(L 0,ji t X t ) = t 0
Qσ(L 0,ji s-X s-)+D 0 (j, i) σ(X s-), e i e j ds+ t 0 (D 1 -I)σ(L 0,ji s-X s-) dn s (31c)

σ(L 1,ji t X t ) = t 0 Qσ(L 1,ji s-X s-) + D 1 (j, i) σ(X s-), e i e j ds + t 0 (D 1 -I)σ(L 1,ji s-X s-) + D 1 (j, i) σ(X s-
), e i e j dn s .

(31d)

Proof. Let G (i)
s be the innovations gain defined in (18). We find from the product rule (R1) and using ( 30),( 13b) and (R3), that

L 0 t O (i) X t = t 0 L 0 s-d O (i) X s + t 0 O (i) X s-d L 0 s + 0<s≤t ∆ L 0 s ∆ O (i) X s = t 0 L 0 s-Q O (i) X s-+ X s-, e i ds + t 0 L 0 s-G (i) s (dN s -λ s ds) + t 0 O (i) X s-L 0 s-( λ s -1) dn s + t 0 G (i) s L 0 s-( λ s -1) dN s = t 0 L 0 s-Q O (i) X s-+ X s-, e i ds + t 0 L 0 s-G (i) s λ s dn s + t 0 O (i) X s-L 0 s-( λ s -1)dn s .
Since we know from (18) that G (i)

s λ s = D 1 O (i) X s--O (i)
X s-λ s , we obtain after some simplifications that

L 0 t O (i) X t = t 0 Q L 0 s-O (i) X s-+ L 0 s-X s-, e i ds + t 0 (D 1 -I) L 0 s-O (i) X s-dn s .
Using ( 29) and the notation introduced in Theorem 6, the equation above has the final form (31b). The other formulas are obtained in a quite similar way. SDEs in Theorem 6 are standard linear ordinary differential equations (ODE) between two jumps of (N t ) t≥0 . Therefore, a basic way to deal with the equations (31a-31d) is to integrate the linear ODEs over the interval of time between two jumps and to update the solution at the endpoint of the interval. For instance, the state filter σ(X t ) is solution of

d dt q t = (Q -D 1 + I)q t = (D 0 + I)q t , q t l-1 := σ(X t l-1 )
in the interval [t l-1 , t l [, where (t l ) l∈N is the sequence of times of jump of (N t ) t≥0 (t 0 := 0). Then, the solution at time of jump t l is updated as follows ∆σ(X

t l ) = (D 1 -I)σ(X t l -) =⇒ σ(X t l ) = D 1 σ(X t l -).
In this special case, it is easily seen that (31a) has an explicit solution given for t > 0 by

σ(X t ) = exp(t) exp D 0 (t -t Nt ) D 1 exp D 0 (t Nt -t Nt-1 ) • • • D 1 exp D 0 t 1 X 0
and the vector of conditional probabilities X t is for t > 0

X t = exp D 0 (t -t Nt ) D 1 exp D 0 (t Nt -t Nt-1 ) • • • D 1 exp D 0 t 1 X 0 1 ⊤ exp D 0 (t -t Nt ) D 1 exp D 0 (t Nt -t Nt-1 ) • • • D 1 exp D 0 t 1 X 0 .
Assume that K jumps of (N t ) t≥0 have been observed at times 0 < t 1 < • • • < t K and set t 0 := 0. The solutions of (31a-31d) may be computed on the grid Π := {0, t 1 , . . . , t K } from the recursive formulas in Figure 2. We are now in position to discuss the comparison between the two procedures reported in Figures 1,2.

Comments on the comparison of the forward-backward and filter-based strategies

The two procedures involve the computation of the matrix exponential functions f 0 , f 1 as well as their integrals. The uniformization method is known to be efficient for computing transient measures of continuous-time Markov processes involving matrix exponentials. This method is based on the following decomposition of the matrix exponential of a subgenerator D 0 (i.e. i = j D 0 (i, j) ≥ 0 and (1 ⊤ D 0 )(i) ≤ 0 for any i).

exp(D

0 t) = +∞ k=0 exp(-ut) u k k! P k with P := I + 1 u D 0 , and u > max i (-D 0 (i, i)).
f 0 (x) := exp(D 0 x) and f 1 (x) := D 1 exp(D 0 x); for l = 1, . . . , K, ∆t l := t lt l-1 with t 0 := 0.

for l = 1, . . . , K σ(X t l ) = f 1 (∆t l ) σ(X t l-1 ) σ(L 0,ji t l X t l ) = f 1 (∆t l )σ(L 0,ji t l-1 X t l-1 ) + t l t l-1 f 1 (t l -s)e j D 0 (j, i) e ⊤ i f 0 (s -t l-1 ) ds σ(X t l-1 ) σ(L 1,ji t l X t l ) = f 1 (∆t l )σ(L 1,ji t l-1 X t l-1 ) + e j D 1 (j, i) e ⊤ i f 0 (∆t l )σ(X t l-1 ) σ(O (i) t l X t l ) = f 1 (∆t l ) σ(O (i) t l-1 X t l-1 ) + t l t l-1 f 1 (t l -s)e i e ⊤ i f 0 (s -t l-1 ) ds σ(X t l-1 )
Comment. The factor exp(∆t l ) is omitted in the equations above, because the estimates at a fixed instant of D 0 , D 1 from (10) only require the knowledge of the filters up to a constant. The numerical interests are in the facts that (1) matrix P is a non-negative (primitive) matrix, so that only non-negative real numbers are involved in the computations, (2) a robust computation of the Poisson probabilities may be carried out and (3) the level of truncation of the series above may be a priori controlled. We refer to [START_REF] Stewart | Introduction to the Numerical Solution of Markov chains[END_REF] for complete details. Now we turn to the differences between the two procedures. The weakness of the filter-based approach is its computational cost. Indeed, we have to deal with a specific recursive equation for each statistic involved in the re-estimation formulas (10). So that, the computational cost is O(n 4 ) in the number of parameters to be estimated. The corresponding cost for the forward-backward strategy is only O(n 2 ). However, when fitting data to an MAP model, the experiments clearly show that the number n of states to be considered for the hidden Markov process (X t ) t≥0 should be relatively small (n is not found to be greater than 3 in [START_REF] Klemm | Modeling IP traffic using the Batch Markovian Arrival Process[END_REF]). Then, it can be expected in some practical situations than the difference of computational complexity will be not too large.

It is clear from Figure 1 that the sequence of vectors α l , l = 0, . . . , K provided by the forward recursion have to be stored before performing the backward recursion. Therefore, the storage cost is O(K) in the number of observed data. Now, the storage cost for the recursive procedure in Figure 2 does not depend on the number K of observations. We only have to store the previous estimate of each statistics. From this point of view, the filter-based approach must be preferred to the forward-backward one when processing large data-sets. This is the case in the experiments reported in [START_REF] Klemm | Modeling IP traffic using the Batch Markovian Arrival Process[END_REF], where trace files of 200, 000 arrivals are considered. Thus, a recursive implementation of the EM-algorithm has the advantage that the number of observations is not fixed so that on-line estimation may be considered, and this number may be as large as possible. Finally, the various filters in the filter-based scheme are decoupled and are suitable for parallel implementation on a multiprocessor system.

Large experiments on the use of the EM-algorithm have been carried out in [START_REF] Lang | Parameter estimation for phase-type distributions[END_REF], [START_REF] Asmussen | Fitting phase-type distributions via the EM-algorithm[END_REF], [START_REF] Rydén | An EM-algorithm for estimation in Markov-modulated Poisson process[END_REF], [START_REF] Klemm | Modeling IP traffic using the Batch Markovian Arrival Process[END_REF] with the forward-backward strategy. Since the differences between the two procedures are mainly on the order of complexity, here it is not intended to provide further numerical experiments. The appealing properties and main drawbacks of the EM-algorithm in estimation problem are well-known. The implementation is relatively easy, the algorithm is robust but its convergence -if convergence takes place -is slow. We refer to [START_REF] Wu | On the convergence properties of the EM algorithm[END_REF] for a discussion on the convergence rate of EM.

Conclusion

When only the counting process is observed in a Markovian Arrival Process, parameter estimation may be carried out with the help of the EM-algorithm. Then, filters for various statistics associated with this model must be computed. With a view to implementing the filter-based strategy popularized by Elliott, the filters and an unnormalized/Zakai form of the filters are shown to be the solution of stochastic differential equations. As a result, we obtain recursive computational procedures for filtering. The case of batch arrivals, that is of BMAPs, may easily be included in the discussion. We just have to consider the observation process (N t ) t≥0 as a multivariate point process of arrivals.

Figure 2 :

 2 Figure 2: Recursive implementation of the filters