Titre en anglais) : Use of Wireless Sensor Networks for operational safety and industrial prognosis

Keywords: Wireless Sensor Networks, Prognostic and Health Management, Diagnostic

Résumé: Une maintenance efficace d'un dispositif industriel ne peut être basée que sur la fiabilité et l'exactitude de données physiques captées sur ledit dispositif, à des fins de surveillance. Dans certains cas, le monitoring de tels systèmes industriels ou de zones à surveiller ne peut pas être assuré à l'aide de capteurs individuels ou filaires, du fait par exemple de problèmes d'accès ou de milieux hostiles. Les Réseaux de Capteurs Sans Fil (RCSF) sont alors une alternative. En raison de la nature des communications dans ces réseaux, et des caractéristiques des appareils composants ces derniers, un RCSF est à fort risque de pannes au niveau des capteurs, et dans ce cas la perte de diverses données est probable -ce qui peut s'avérer problématique pour le monitoring du dispositif. Pour étudier la pertinence des RCSF pour le processus dit de PHM (Prognostic and Health Management, utilisé pour déterminer le plan de maintenance d'un dispositif à surveiller), et l'impact des diverses stratégies déployées dans ces premiers sur ces derniers, nous avons proposé un premier algorithme de diagnostic efficace et l'avons utilisé dans un RCSF simulé pour en mesurer la performance (ce simulateur étant un programme que nous avons développé).

Nous avons alors proposé une démarche pour diagnostiquer l'état de systèmes physiques basée sur l'utilisation de la méthode dite des forêts aléatoires. Cette démarche repose sur deux phases : une première, hors ligne, et une seconde en ligne. Dans la phase hors ligne, l'algorithme des forêts aléatoires sélectionne les paramètres qui contiennent le plus d'information sur l'état du système. Ces paramètres sont utilisés, dans leur ordre d'importance, pour construire les arbres décisionnels qui constitueront la forêt. En injectant de l'aléatoire dans la base d'apprentissage, l'algorithme utilisera divers points de départ, et par la suite les arbres aussi seront aléatoires. Dans la phase en ligne, l'algorithme évalue l'état actuel du système en utilisant les données capteurs pour parcourir les arbres construits. Chaque arbre dans la forêt fournit une décision, et la classe finale est le résultat d'un vote majoritaire sur l'ensemble de la forêt. Quand les capteurs commencent à tomber en panne, certaines données associées à divers indicateurs de santé s'avèrent incomplètes ou sont perdues. Or, puisque les arbres ont des points de départ différents, l'absence de mesures pour un indicateur de santé ne conduit pas nécessairement à l'interruption du processus de prédiction de l'état global de santé du dispositif industriel : le processus de monitoring peut alors continuer. only be based on the reliability and accuracy of physical data captured on the device for monitoring purposes. In some cases, monitoring of such industrial systems or areas to be monitored cannot be carried out using individual or wired sensors, for example due to access problems or hostile environments. Wireless Sensor Networks (WSN) are then an alternative. Due to the nature of communications in these networks, and the characteristics of the devices that make up these networks, an WSNs is at high risk of sensor failures, and in this case the loss of various data is likely -which can be problematic for device monitoring. To study the relevance of WSNs for the so-called PHM (Prognostic and Health Management process, used to determine the maintenance plan of a device to be monitored), and the impact of the various strategies deployed in the latter, we proposed a first efficient diagnostic algorithm and used it in a simulated WSNs to measure its performance (this simulator being a program we developed).

We then proposed an approach to diagnose the condition of physical systems based on the use of the so-called random forest method. This approach is based on two phases: a first phase, offline, and a second phase online. In the offline phase, the random forest algorithm selects the parameters that contain the most information about the system's state. These parameters are used, in order of importance, to construct the decision trees that will make up the forest. By injecting randomness into the learning base, the algorithm will use various starting points, and then the trees too will be random. In the online phase, the algorithm evaluates the current state of the system by using the sensor data to scan the constructed trees. Each tree in the forest provides a decision, and the final class is the result of a majority vote on the whole forest. When the sensors begin to fail, some of the data associated with various health indicators are incomplete or lost. However, since trees have different starting points, the absence of measures for a health indicator does not necessarily lead to the interruption of the process of predicting the overall state of health of the industrial system: the monitoring process can then continue. DEDICATION * * This thesis is dedicated to my brother Abbas Farhat who was there for me during all the challenges that i faced in life and encouraged me. Also, I wouldn't be who i am today without the support of my supervisor Christophe Guyeux. * * I also dedicate this dissertation to my family who have helped me throughout the process. I will always appreciate all they have done for me. * * I want to thank my friends who led me to understand some of the most subtle challenges to our ability to thrive and always believing in me.

III Conclusion 119

I SCIENTIFIC BACKGROUND AND ISSUES

GENERAL INTRODUCTION

I n the last decades, and with the advancement of technology, industrial systems (in all its fields) became a necessity in the present day. Its importance lies in the services and facilities that it offers in our daily life. However, the dangers of industrial systems are no less than its importance because these systems are always subjected to failure. A failure might be irreversible or have undesirable outcomes with consequences varying from minor to severe. Therefore, the outcome of this failure might cause enormous losses both financially and casualties and causes service interruption. Moreover, system failure have also other dangers like degrading the quality of response, preventing the system from providing the intended output, and placing the clients' trust at risk. These reduces dramatically their equities and incomes, and also generate countless money loss and expensive fees. Not to mention endangering the environment in case of fire, explosions, gas leaks, etc. In Figure 1.1 we will show four of some of the major catastrophic events in history which occurred due to failure outcomes which lead to huge losses.

As Figure 1.1a shows, the Exxon Valdez spill is not considered a large one in comparison to the world's biggest oil spills, but it surely is a costly one due to the remote location of Prince William Sound (only accessible by boat and helicopter). 10.8 million gallons of oil was spilled on March 24, 1989, when the captain Joseph Hazelwood left the controls and the ship crashed into a Reef. The cleanup cost Exxon is $2.5 billion.

It is the worst off-shore oil disaster in the world. At some point, it was the single largest oil producer, where spewing out 317,000 barrels of oil per day occurred (see Figure 1.1b). It occurred on July 6, 1988. As a part of routine maintenance which was essential in preventing dangerous build-up of liquid gas, technicians were removing and checking safety valves when they forgot to replace one of the 100 identical valves. At 10 PM that same night, the world's most expensive oil rig accident was set in motion when a technician pressed a start button for the liquid gas pumps. In 2 hours, flames engulfed the 300 foot platform. Eventually it collapsed after killing 167 workers and resulting in $3.4 Billion worth of damages.

On January 28, 1986, the Space Shuttle Challenger was destroyed after 73 seconds from taking off due to a faulty O-ring as shown in Figure 1.1c. One of the joints failed to seal allowing pressurized gas to reach the outside. In turn, this caused the external tank to dump its load of liquid hydrogen that caused a massive explosion. Replacing the space shuttle cost $2 billion in 1986 ($4.5 billion in today's dollars). The cost of investigation, problem correction, and replacement of lost equipment cost $450 million from 1986-1987 ($1 Billion in today's dollars).

The costliest accident in history happened on April 26, 1986. It was the Chernobyl disas-19 ter which had been called the biggest socio-economic catastrophe in peacetime history (see Figure 1.1d). It contaminated 50% of the area of Ukraine. 1.7 million People were directly affected by the disaster and over 200,000 people had to be evacuated and resettled. The death toll was estimated to be 125,000 including people who died from cancer years later. The total cost is estimated to be roughly $200 Billion including cleanup, resettlement, and compensation to victims. A new steel shelter for the Chernobyl nuclear plant will cost $2 billion alone. Power plant operators who violated plant procedures and were ignorant of the safety requirements needed were the cause of the accident. Then, the failure outcomes in the industrial systems are very costly, and mostly these failures are due to unexpected technical errors, or due to lack of experience and responsibility from the workers which is needed to get the work done. From this context, it is very necessary to monitor the system, evaluate their health and diagnose them at any time, and then plan maintenance activities preferably to avoid disastrous failure results.

Maintenance is an important activity in industrial field. It is either performed to restore a machine/component, or to prevent it from breaking down. It aims at increasing system availability, readiness, and enhancing safety. Through time, different strategies have evolved in order to bring maintenance to its current state. This evolution was caused by the increasing demand of reliability in industry. Nowadays, industrial machines are required to avoid shutdowns while offering safety, reliability, availability, and all while reducing the costs.

Corrective maintenance is the first form of maintenance. In this strategy, actions are only taken when the system breaks and can no longer perform the intended tasks; actually, due to the interruption of production and time to repair activities, sudden shutdowns cost money and time in addition to client's trust and safety. Maintenance became a periodic activity for solving these problems. Domain experts depend on their knowledge and the observation of upcoming events to set time intervals so that the components are inspected and replaced if needed. Preventive maintenance (often called periodic) is performed regardless of the machine's condition which is considered the main drawback. But sometimes the machine can be in a healthy state so the maintenance will be unnecessary and will cost extra and avoidable fees. But even with periodic maintenance and inspections, random failures still occur. For that, in the early nineties, Condition Based Maintenance (CBM) was proposed and developed [START_REF] Heng | Rotating machinery prognostics: State of the art, challenges and opportunities[END_REF].

CBM is based on real-time observations. It is an on-line approach that assesses machine's health through condition measurements. CBM aims to increase the system reliability and availability, as any maintenance strategy, while reducing maintenance costs. This particular strategy has benefits which include avoiding unnecessary maintenance tasks and costs, as well as not interrupting the normal machine operations [START_REF] Heng | Rotating machinery prognostics: State of the art, challenges and opportunities[END_REF]. CBM decreases the number of maintenance operations and causes the influence of human error to be reduced. Predictive Maintenance (PM) is a new maintenance that has recently emerged. Based on the current condition, it predicts the system health in the future and defines the needed maintenance activities accordingly. In this way, if only a direct evidence is present that shows deterioration has actually occurred, the system is then taken out of service. This increases the efficiency of maintenance and productivity and decreases the maintenance support costs and logistics footprints. The evolution of maintenance strategies through time is summarized in Figure 1.2. Extra tasks are required for shifting from traditional maintenance strategies to CBM and PM. These tasks encompass data analysis and modeling, system surveillance, and decision making support system. This scientific approach is called Prognostics and Health Management (PHM).

Research in PHM field has gained and was given a great deal of attention. Prognostic models are developed in an attempt to predict the Remaining Useful Life (RUL) of machinery (or monitored area) before failure takes place. This is done according to the steps described in Figure 1.3.

Diagnostics aim for specifying and quantifying an actual failure while prognostics have the goal of anticipating failures. Prognostics consider the past events, in addition to the machine's current state, and operating conditions to estimate the RUL [START_REF] Andrew Ks Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF]. This estimation is done by studying the evolution of continuous measurements of parameters that need to be tracked in time for assessing health and diagnosing activity of the system under consideration. A number of sensor nodes usually gather these information. Sensory data are reported periodically to monitor the critical components. The data packets contain signals that correspond to measurements of monitoring parameters such as temperature, pressure, humidity, etc.

A monitored parameter has a fixed threshold. Once this threshold is reached, an alarm goes on indicating that a symptom of system deteriorating has been detected. And after that, a diagnosis of the state of the system is made, then the RUL is computed with an associated confidence limit and the maintenance activities are determined if needed.

Note that if the prediction model and the provided measurements are not accurate, the maintenance activity will possibly be performed either too soon or too late.

As far as we know, data collection in existent research works in PHM was performed either by means of individual sensors (just in simple applications), or via wired networks of sensor nodes (in all applications). The use of wired networks generated many drawbacks. The price of physical wires grows proportionally according to their length and weight. Then, the deployment of a Wireless Sensor Network (WSN) in some monitoring applications is mandatory rather than a choice because of the following reasons.

• It is easier sometimes to place wireless sensors in the monitoring area without the constraints of connecting them in wires because of accessibility restrictions.

• Some systems can be sensitive to extra weight, so by eliminating the wires, this can guarantee better performances and reduce extra costs.

• It can bring us closer to the data source that we need, comparing to when we use wires, and it will be easier to deploy more sensors in the monitoring process, thus getting more precise values.

None of the previous research work has questioned the issues of WSNs on the PHM. For that, and for precision and feasibility purposes in the PHM, in this thesis we will con-sider and study the case where sensors communicate their information within a Wireless Sensor Network.

The literature doesn't always consider the quality of the available data for the PHM process. The solutions that are proposed in the literature are based on the assumption of completeness and correctness of data. However, this assumption is far from real especially in the case of WSN monitoring. Therefore, in this thesis, we aimed for an algorithm that is able to adapt to the change of quality of the monitoring data. More specifically, we will propose in this thesis and use the random forest algorithm for diagnostics. Random forests usage (RF) is proposed for the industrial functioning diagnostics, more specifically in the terms of devices being monitored using a wireless sensor network (WSN). In the offline phase, the algorithm selects the relevant features for the best split in the decision tree, starting from the root until the leaf nodes are reached. This process will be repeated until the maximum number of trees is reached. Because it is based on random selections, the algorithm has the advantage of starting from different points (or distributions). For that, this can be beneficial in the online phase for mainly two reasons:

• Diagnostics can be performed considering the relevance of all features. The correspondent health class will be assigned according to a majority vote which reduces errors.

• The diagnostic process will not be interrupted if one feature is missing (due to sensor error) because other trees have a different starting point.

Unlike the conventional computer networks, WSNs are composed of a large number of sensor nodes with very limited and non-renewable energy. Moreover, because of the nature of communication in this network and characteristics of its devices, a WSN is at risk of failure. The accuracy and completeness of data that is going to be captured will be affected by this risk, and consequently PHM will be affected. Therefore to ensure that the data of the monitored area is as accurate as possible, we need to maintain the Quality of Service (QoS) of WSN for the longest possible time. WSN strategies that can be considered are: the topologies, coverage, deployment strategy, scheduling mechanism, density, security, data aggregation, frequency, packet transfer distance, battery, memory, etc. These strategies have a strong impact on the effectiveness of the network with time, consequently on the quality of the data that will be captured, and finally on PHM. Although both PHM and WSN strategies have been studied, as far as we know, none of the existing research work has considered the WSN strategies for PHM. For instance, in real life application, the monitored area might not be fully covered for several reasons like nodes failure, energy depletion, unadapted initial deployment, scheduling mechanisms, etc. Obviously, this flaw may lead to inaccurate and incomplete data, and if we do not take such issue into consideration while building a PHM process over a WSN, the results provided by diagnostic or prognostic may not be reliable. Therefore, in this thesis we will study the WSN strategies and their relation with prognostic and health management. We will focus on the impact of these strategies on the accuracy of the data captured by a wireless sensor network, and its consequences on the health diagnostic. To reach this goal, to evaluate both prognostic and health management with the WSN strategies, and to evaluate the importance of the random forest algorithm, we will use in WSN the random forest algorithm and five other diagnostic algorithms from literature. Namely Support Vector Machines (SVM), Naive Bayes (NB), Gradient Tree Boosting (GTB), Tree-Based Feature Selection (TBFS), and Nearest Neighbors (NN) methods.

Hence, a dependable WSN should be used in the monitoring process. Coverage is one of the most important measurements of WSNs quality of service, and it is closely related to the energy consumption of sensors [START_REF] Zhu | A survey on coverage and connectivity issues in wireless sensor networks[END_REF]. Thus, when designing WSNs, one of the most important challenges remains in the efficient use of these sensors, to increase the network's lifetime while guaranteeing high coverage rate. The coverage rate and the consumption energy are the two most important factors in WSN, and these two factors are obviously proportional to each other. Indeed, the coverage rate increases when the number of active sensors increases, but energy consumption increases accordingly. All of the research mentioned in the literature support the WSN by increasing the lifetime and maximizing the coverage rate of the network, thus making it more efficient with time. However, numerous problems can be emphasized in these solutions, and are often application oriented. There is always a priority for these solutions in which they are either oriented to maximize the coverage or to maximize the lifetime of the network. All the attempts tried to compromise between these factors were not efficient. Then, to solve this issue, we will propose in this thesis a reliable algorithm in graph theory for diagnostics and health management of the monitoring targets, by using the dominating sets proposed in [START_REF] Padma | A note on varieties of double domination[END_REF][START_REF] Atapour | On total dominating sets in graphs[END_REF]. The proposed algorithm is completely distributed: no centralized or intermediate control is needed. This is a hybridization of three variants of domination in the sensors-targets graph, namely the simple, double, and total ones, in a single algorithm, which makes the originality of our study. To illustrate the efficiency of this approach and its impact on the accuracy of diagnostics, we will use the six machine learning algorithms mentioned before that have been used to diagnose the area state. These learning processes are evaluated in a WSN context with the proposed algorithm. By relying on this algorithm, the data diagnostics (PHM) will be accurate for the longest time possible.

1.1/ DISSERTATION OUTLINE

The remainder of this thesis is organized as follows :

Chapter 2 presents an overview to show the advantages and disadvantages of each of wireless sensor networks and prognostic and health management. It also discusses the challenges of wireless sensor network monitoring in terms of industrial prognostics and health management. Finally, this chapter emphasizes what needs to be tackled in order to obtain good results.

Chapter 3 proposes the use of the random forest algorithm for diagnostic. The algorithm is used for diagnosing the state of an industrial device when there are incomplete monitoring parameters via a WSN. This method is confirmed by computer simulations while varying the data collection process.

Chapter 4 this chapter studies and shows the impact of WSN strategies on diagnostics/PHM, by relying on random forest algorithm and other algorithms in the literature. It also evaluates both prognostic and health management with the WSN strategies and the importance of the random forests algorithm suggested in Chapter 3.

Chapter 5 proposes effective and reliable distributed algorithm that relies on scheduling mechanism to increase the network lifetime. And also to increase the coverage rate to the maximum (it's able to compromise between these two factors) with the use of certain amount of active sensors in network.

Chapter 6 summarizes the work that was achieved and enumerates the contributions of this thesis. It also discusses the perspectives.

1.2/ PUBLICATIONS

SCIENTIFIC BACKGROUND

I n this chapter we will present an overview on wireless sensor networks and on prognostics and health management, to show the importance and characteristics of both, and their usefulness in real life. Furthermore, we will detail the links that can be established between these fields of research. And we will draw the challenges related to wireless sensor network based monitoring, which will lead to various solutions proposed in the contribution part of this manuscript.

2.1/ AN OVERVIEW ON WIRELESS SENSOR NETWORKS

In the last few years Wireless Sensor Networks (WSNs) have gained much attention in both public and research communities, caused by wealth of theoretical and practical challenges [START_REF] Ian F Akyildiz | A survey on wireless multimedia sensor networks[END_REF], and because they are expected to bring the interaction between humans, environment, and machines to a new paradigm. Even though it is an appealing topic which holds vision of an upcoming world, a huge gap still exists in the realizations of WSNs. This continuous research in WSNs has explored a variety of new applications enabled by larger scale networks of sensor nodes which are able to sense information from the environment, process the sensed data, and transmits it to the remote location [START_REF] Feng | System-architectures for sensor networks issues, alternatives, and directions[END_REF][START_REF] Raja | Sensor cubes: A modular, ultra-compact, power-aware platform for sensor networks[END_REF].

The wide growth of WSN is due to the improvement in low cost wireless communication and micro-electro-mechanical systems [START_REF] Ian F Akyildiz | A survey on sensor networks[END_REF]. These networks are wildly spreading due to their easy deployment and low cost. A while ago, transferring data from one unit to another requested expensive wiring. Currently, nodes can be placed almost anywhere, having no wiring contrasts, with any geographic or placement problems, and this is because of the minimal batteries feeding the nodes. Sensors have the advantage of being used even discreetly due to their small dimensions. These advantages permitted the deployment of WSN in various fields, such as: medicine, military, agriculture, environmental, home automation, etc. Generally, a WSN is composed of few base stations (one in most cases) and a large number of sensor nodes. The main role of these devices is to monitor physical or environmental conditions, and to cooperate to deliver the sensed data to a base station called the sink. An example of wireless sensor networks is shown in Figure 2.1.

WSN are composed by very small sensors with very limited and nonrenewable energy. In order to preserve this energy, network throughput has to be low. An issue lies which is, as all wireless sensor networks, WSN are not very secure. A node failure (or attack) can occur due to both limitation of energy, and random deployment in hostile and inaccessible areas [START_REF] Paul Walters | Wireless sensor network security: A survey. Security in distributed, grid, mobile[END_REF]. Moreover, they do not dispose with a predefined infrastructure and this fact emphasizes the importance of the chosen routing protocol. A reliable communication should be ensured by an adequate protocol. This reliability means reducing data loss, fastening communication, minimizing energy consumption, and other standards.

WSN are event-based systems that rely on the collective effort of several sensor nodes [START_REF] Özg Ür B Akan | Event-to-sink reliable transport in wireless sensor networks[END_REF]. This type of network tends to greatly increase the coverage rate for the area that should be monitored and also increase the accuracy of the information extracted from this area. The network extends the computational capability to reach the physical environments that are unreachable by the human beings. Generally, since sensor nodes function at a low frequency, they are not located far from each other. Because of that, there is a high possibility that many nodes sense the same data. This redundancy occurring will cause all the useful energy to be dissipated in vain if all of this redundant information was routed through the network.

In the next section, we will introduce a general overview of WSNs. This encompasses their advantages, and their drawbacks.

2.1.1/ SYSTEMS SOFTWARE FOR WSNS

A general-purpose operating system (OS) is an example of systems software. Due to the scarceness of resources and simplicity of applications, early WSNs did not include systems software. However, systems software is a requirement in complex applications because it facilitates the control of resources and increases the predictability of execution. Common interfaces provided by the software can hide the heterogeneity of platforms. Still, heavy computation and memory usage are the major disadvantages. The systems software for WSNs implement single node control and network-level distribution control. The low-level routines in a node are implemented by the single node control software, whereas application execution within several nodes is managed by the network-level distribution control.

The single node control operates on a physical node depicted in Figure 2.2. A processing unit consists of CPU, storage devices, and an optional memory controller for accessing the instruction memory of the main CPU. A sensing unit consists of sensors and an analog-to-digital converter (ADC). The communication with other sensor nodes is enabled by a transceiver unit. A power generator that harvests energy from environment can be used to extend a power unit. Other peripheral devices, depending on the application requirements, are attached to the node, like peripheral actuators for moving the node and location finding systems [START_REF] Ian F Akyildiz | A survey on sensor networks[END_REF]. The reference values in Figure 2.2 are the resources available in MICA2 mote [START_REF]Mica2 datasheet[END_REF]. The power consumption of a node when active is in order of mW and in order of µW when the node is in sleep. The power unit is typically an AA battery or similar energy source.

OS or virtual machine (VM) accomplishes the single node control. In the reference platform, OS is executed on the main CPU and it uses the same instruction and data memories as applications. OS implements services that include scheduling of tasks, Inter-Process Communication (IPC) between tasks, control of memory, and possible power control in terms of voltage scaling and component activation and inactivation. Interfaces to access and control peripherals are provided by OS. The interfaces are typically associated with layered software components with more sophisticated functionality, for example a network protocol stack.

Depending on the target application (mainly its goals and limits), a network setting is chosen, and the routing protocol will be developed accordingly. The protocol stack of sensor networks is composed of five different layers [START_REF] Ian F Akyildiz | A survey on sensor networks[END_REF], which are shown in Figure 2.3.

Physical layer:

it explains the ways of transmitting data packets after being converted into raw data bits which are suitable for transmission over the communication medium. Basically, this layer is responsible for frequency selection, signal detection, modulation, carrier frequency generation, and data encryption.

Data link layer:

it is charged of data stream multiplexing, data frame creation, medium access, and error control to provide reliable transmission. It is also in charge of the creation of the network infrastructure, transferring data, and sharing the communication resources fairly and efficiently between sensor nodes, for achieving good network performance in terms of energy consumption, network throughput, and delivery latency. It is also responsible for error control of transmission data.

Network layer:

it is responsible for routing the data from the source nodes until the sink. It also allows inter-networking with external networks, where the sink node can be used as a gateway. It controls and orders the system, forwards data packets, and takes charge of routing between intermediate routers. Also, it ensures internetworking with external networks.

4.

Transport layer: its responsibility is the end-to-end data delivery between sensor nodes and the sink. Traditional transport protocols cannot be applied directly to WSNs because of energy, computation, and storage constraints. Also, this layer provides other services like multiplexing, reliability, congestion avoidance, flow control, etc.

Application layer:

several protocols are included which perform various sensor applications like time synchronization, query dissemination, node localization, network security, etc. It can be defined as the user interface. It shows messages in a human recognizable and understandable format.

WSN are broadly spread because of their low cost, easy deployment, and their capacity to extract localized features. Currently, they can be found in nearly all monitoring applications. However, as far as we know, the literature of Prognostic and Health Management (PHM) does not report the use of this technology. Nevertheless, in some industrial applications, the WSN monitoring can be obligatory.

2.1.2/ SENSOR NETWORK APPLICATIONS

Sensor networks may consist of many different types of sensors such as seismic, low sampling rate magnetic, thermal, visual, infrared, acoustic and radar, which are capable of monitoring a wide variety of ambient conditions that include the following [START_REF] Estrin | Next century challenges: Scalable coordination in sensor networks[END_REF]:

• temperature,

• pressure,

• humidity,

• soil makeup,

• vehicular movement,

• noise levels,

• lightning condition,

• the presence or absence of certain kinds of objects,

• current characteristics such as speed, direction, and size of an object.

Sensor nodes can be used for detecting events, event ID, sensing location, and local control of actuators. The concepts of micro sensing and wireless connection of these nodes hold many promises in new application areas. The applications are categorized into, environmental, military, home, health, and other commercial areas. Expanding this classification with more categories is possible such as space exploration, chemical processing and disaster relief. [START_REF] Petriu | Sensor-based information appliances[END_REF]. These sensor nodes can interact with each other inside the domestic devices and with the external network via internet or satellite. They allow the management of home devices locally and remotely more easily by end users.

• Smart environment: two different perspectives can be found for the design of smart environment, i.e., human-centered and technology-centered [3]. For humancentered, a smart environment has to adapt to the needs of the end users in terms of input/output capabilities. For technology-centered, new hardware technologies, networking solutions, and middleware services have to be developed. A scenario of how sensor nodes can be used to create a smart environment is described in [START_REF] Herring | Component-based software systems for smart environments[END_REF].

The sensor nodes can be embedded into furniture and appliances, and they can communicate with each other and the room server. Communication can also occur between the room server and other room servers to learn about the services they offered e.g., printing, scanning, and faxing. These room servers and sensor nodes can be integrated with existing embedded devices to become self-organizing, self-regulated, and adaptive systems based on control theory models as described in [START_REF] Herring | Component-based software systems for smart environments[END_REF]. The "Residential Laboratory" at Georgia Institute of Technology set another example of smart environment [START_REF] Irfan | Ubiquitous sensing for smart and aware environments[END_REF]. Computing and sensing in this environment should be persistent, reliable, and transparent.

2.1.3/ SHORTCOMINGS OF A WSN

The purpose of designing WSNs is for efficient event detection. Consisting of a large number of sensor nodes deployed in a surveillance area, they are able to detect the occurrence of new events Efficiency is a necessity in such activities, which is hard to achieve with the constraints of WSNs. These limitations are detailed in the following.

2.1.3.1/ RESOURCES

What mostly limits the WSN capabilities is the available energy. The sensors are small sized devices, with tiny batteries as an energy supply. Moreover, the nodes are often deployed in hostile environments (mountains, enemy territory, etc.) where they cannot be recharged [START_REF] David W Carman | Constraints and approaches for distributed sensor network security (final)[END_REF].

Another important impact on the available energy for normal network tasks is the added security. Extra power is necessary for processing security functions (encryption, decryption, signing data, verifications), transmitting security related data (vectors for encryption/decryption), and securing storage (cryptographic key), which is critical for WSNs [START_REF] David W Carman | Constraints and approaches for distributed sensor network security (final)[END_REF][START_REF] Paul Walters | Wireless sensor network security: A survey. Security in distributed, grid, mobile[END_REF].

In addition to this, there is a very limited memory space deployed for a sensor node. The communication protocol and the security code share the storage space. The size of the latter has then to be limited to a minimum [START_REF] Paul Walters | Wireless sensor network security: A survey. Security in distributed, grid, mobile[END_REF].

There is also a limited buffering space in sensor nodes. With the increase in traffic flow towards the sink node, this will lead to packet loss. As a matter of fact, with new packets coming in, a sensor node cannot hold data packet for a long period. All the nodes will attempt to get rid of the old messages, in the case of a high traffic flow, to make space for the new ones by forwarding them to the next level. Therefore, as all sensor nodes tend to forward the captured data to the sink, the area around the sink tends to be quickly congested.

2.1.3.2/ COMMUNICATION

Wireless communication is known to be unreliable and it adds to the network's vulnerabilities. The absence of physical connections can result in:

• Channel errors: occurring at the recipient, wrong signals may arrive because of the noise in communication channels.

• Missing links: invalid or missing links between the sensors and consequent packet drop is caused by route updates, interference in the radio channels, energy exhaustion, etc.

• Communication latency: greater latency is achieved by both multi-hop routing and node processing. The time elapsed between packet sending and packet reception is the latency. Extra delays are taken into consideration when re-transmission is required for transmission errors. One of the major drawbacks of latency is that it makes synchronization among nodes hard to achieve.

• Network congestion: Dense packet exchange in the network reflects heavy traffic, and a concurrent access may present itself at certain regions. The Quality of service (QOS) of a node carrying so much data degrades as this leads to packet collision, packet loss, transmission delays, etc.

Most WSNs are deployed in harsh environment conditions and/or are exposed to adversary attacks. This can cause permanent (even irreversible) damage to the hardware because of the likelihood of physical attacks. The sensor nodes can be left unattended for a long period of time since the network is remotely managed. Therefore, detecting physical tampering or performing regular maintenance would be difficult, and thus, the network would remain unable to fulfill the intended tasks [START_REF] Paul Walters | Wireless sensor network security: A survey. Security in distributed, grid, mobile[END_REF].

Routing solutions in WSNs avoid central management point as it results in a single point of failure. This complicates the synchronization among nodes, and lowers packet delivery rate. Synchronization among nodes should be ensured which necessitates access to extra memory space to improve the communication protocol, consuming more resources.

2.2/ PROGNOSTICS AND HEALTH MANAGEMENT

Maintenance is an important activity in industrial field. It is carried out to either restore a machine/component, or prevent its break down. Its aim is to increase system availability, readiness, thus enhancing safety. Throughout time, vairous strategies have evolved and were developed to bring maintenance to its current state: condition-based and predictive maintenance. What caused this evolution was the increasing demand of reliability in industry. Prognostics and Health Management (PHM) is a tool to predict the Remaining Useful Life (RUL) of engineering assets and is the key process of condition-based and predictive maintenance. At the present days, industrial machines are needed to avoid shutdowns while offering safety and reliability [START_REF] Peng | Current status of machine prognostics in condition-based maintenance: a review[END_REF]. A great deal of attention was given to research in PHM field. Prognostic models are developed in an attempt to predict the RUL of machinery (or monitored area) before failure occurs. It is possible that the maintenance activity will be performed either too soon or too late, if the prediction model and the provided measurements are not accurate.

In Condition-Based Maintenance (CBM) and Predictive Maintenance (PM), the use of monitoring devices is for surveying the physical condition of the equipment. A maintenance activity should be performed when a certain level is reached (threshold) to ensure the continuity of the machine/system's normal functioning. CBM and PM require extra investments related to the monitoring equipment, in comparison to corrective and preventive maintenance. Nevertheless, it increases the system's availability and optimizes the service life, all while reducing downtime and avoiding unnecessary maintenance. Because of the advantages of Condition based maintenance in comparison to other strategies, its use is privileged. A comparison is summarized in prognostics, and decision making support [START_REF] Andrew Ks Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF]. This is done following the steps described in Figure 1.3, and explained in details in the following sections. ). CM data, on the other hand, contains measurements that are related to the condition of the machine (pressure measurements, environment data, etc.). These two types of information are of equal importance for health assessment, diagnostics, and prognostics. In Figure 2.4, a summary of data acquisition steps is given.

2.2.2/ DATA PROCESSING

Performing data cleaning is very important once the information is available and before building the degradation model, in order to enhance the results of health assessment, diagnostics, and prognostics. The importance of processing the data consists in providing signals that are robust against the variations that might affect the raw data since communicating channel might induce errors [START_REF] Russell | Artificial intelligence: A modern approach[END_REF]. Its aim is to isolate all the possible errors and avoid the so-called "garbage in, garbage out" problem. Authors of [START_REF] Bae | Study on condition based maintenance using on-line monitoring and prognostics suitable to a research reactor[END_REF] defined data processing as the collection and manipulation of items of data to produce meaningful raw data.

In practice, raw sensor signals can be complex, and data describing the degradation is not easy to read. Reported data can have a value type, a waveform type, or a multidimensional type. The last two types are very hard to exploit since they can contain noise. Data processing is considered an important step as it converts raw data into useful information. Prognostic literature have reported many processing techniques [START_REF] Niu | Intelligent condition monitoring and prognostics system based on data-fusion strategy[END_REF][START_REF] Da Tobon-Mejia | Cnc machine tool's wear diagnostic and prognostic by using dynamic bayesian networks[END_REF], like wavelet decomposition, data smoothing, data denoising, etc. Data processing can be divided into two main tasks: (1) pre-processing of sensor raw signals and (2) data analysis for more information extraction. The aim of this step is to improve the signals that are being received from the monitoring device, to provide a better understanding of the process that generated the data, to enhance the degradation model, and to render the computation more effective by reducing the measurement size.

The steps of a data processing system are given in Figure 2.5.

2.2.3/ HEALTH ASSESSMENT

Health assessment consists of determining the state of health of the system at a given time. For that, sensory data are reported periodically to monitor critical components. These data correspond to measurements of relevant parameters (moisture, pressure, temperature, etc.), and are useful in assessing the condition of the machine. Thresholds related to the monitored parameters are fixed. The system is considered to be in the corresponding state once a threshold is reached (See Figure 2.6). Health assessment task can be narrowed down to a classification problem, and there exist several ways to perform this task. For instance, human expertise can be the basis for identifying the class to which belongs a functioning profile. This needs a solid knowledge based on both experience and history of observations of the application domain. This knowledge will help in deriving rules relating an observation to its meaning and also can be explored to develop mathematical models describing the physics of the system under consideration. On-line measurements can be run through the developed model to identify the health state. This problem can also be solved by machine learning techniques. Offline observations will be used to train a model which will then be used on-line to determine the class of new observations.

2.2.4/ DIAGNOSTICS

After the fault has taken place, diagnostics is performed. It identifies the type of fault, its size, location, and cause. The aim of diagnostics is then to relate the cause to the effect.

It is an understanding of the relationship between what we observe and what happened before [START_REF] Jz Sikorska | Prognostic modelling options for remaining useful life estimation by industry[END_REF]. Figure 2.7 illustrates the successive steps of diagnostic process. 1. Fault detection is the process of reporting an anomaly in the system's behavior.

2.

Fault isolation is charged of determining and locating the cause (or source) of the problem. It identifies exactly which component is responsible of the failure.

3.

Fault identification aims at determining the current failure mode and how fast it can spread.

2.2.5/ PROGNOSTICS

The aim of diagnostics is to identify and quantify an actual failure, while the goal of prognostics is to anticipate failures. There exist several definitions concerning prognostics in the literature. We summarized some of them in Table 2.2.

Definition Authors Reference

Predicts how much time is left before a failure (or more) occurs, given the current machine condition and past operation profile. K.S. Jardine et al. [START_REF] Andrew Ks Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF] Indicates whether the structure, system, or component of interest can perform its function throughout its lifetime with reasonable assurance and, in case it cannot, to estimate the remaining useful life.

Zio and Di Maio

[126]

Estimation of the time before failure, or the remaining useful life, and the associated confidence value.

Tobon-Mejia et al. [START_REF] Da Tobon-Mejia | Cnc machine tool's wear diagnostic and prognostic by using dynamic bayesian networks[END_REF][START_REF] Alejandro Tobon-Mejia | A data-driven failure prognostics method based on mixture of gaussians hidden markov models[END_REF] Estimation of time to failure and risk for one or more existing and future failure modes. ISO 13381-1

[80]
Table 2.2: Some definitions of prognostics reported in the literature.

For estimating the RUL, prognostics consider past events, the machine's current state, and operating conditions. By inspecting the evolution of continuous measurements of parameters that need to be monitored in, this estimation is done in order to assess the machine's state. These parameters can be temperature, humidity, vibration, pressure, and so on. A monitored parameter has a fixed threshold. Once this threshold is reached, an alarm goes off indicating the detection of a symptom of system deterioration. The RUL is then computed with an associated confidence limit. The latter information illustrates to what point the predictions are trustworthy. There are two causes for the uncertainties of the RUL predictions: it is either threshold value of monitored parameter, or the RUL prediction itself. Figure 2.8 shows uncertainties concerning prediction of RUL. Collection of documented data covering the machinery and components under consideration is required for an efficient prognostic activity. All monitored parameters and descriptors need to be available with historical records of operations and events. For improving the prognostics results, failure identification and initial diagnostics are also obligatory [START_REF]Diagnostics of machines-prognostics part 1: General guidelines[END_REF].

2.2.6/ DECISION SUPPORT SYSTEM

The next step after performing prognostics and estimating RUL, is deciding the actions that need to be taken (repair, replacement, maintenance, oil changing, etc.). Decision making is a cognitive process. It consists of selecting an action among different possible scenarios, to produce a final choice. Figure 2.9 describes the general process. First of all, the objectives need to be established. An objective can be keeping component from failure until next inspection, reducing overall costs, or any other purpose a plant can be aimed for. Then, all the objectives are classified according to their priority and importance.

In order to answer the established objectives, alternative actions are developed, and the actions satisfying most of the objectives are selected. A decision needs to be made to select the appropriate action. This can be done by implementing a tool among different possibilities.

• Domain experts: very often, plants trust the advice that the engineers and experts in the domain provide them with. They are able to come up with good solutions and reveal the limitations related to a strategy.

• Eliminations: further solution is eliminating non-realistic solutions one by one, or comparing them in a pairwise manner. Lastly, the remaining option is selected.

• Analytic networks: these networks provide a hierarchy of the selected action with goals, alternatives, and consequences.

• Simulations: for visualizing the system's behavior under different conditions, many graphical tools exist. Simulations are a well-known tool for decision making support since they offer clarity and possibility to alter criteria while simulating.

2.2.7/ HUMAN MACHINE INTERFACE

The Human Machine Interface (HMI) is the user interface in a manufacturing or process control system. It is the space where the interactions between humans and machines take place. The goal of this interaction is allowing effective operation and controlling of the machine from the human end, while the machine simultaneously feeds back information that helps in the decision making process of the operator. HMI provides a graphic-based visualization of an industrial control and monitoring system. It is a software application that provides information to an operator or user about the state of a process, and to accept and implement the operators control instructions. To illustrate the analysis results comprehensively, the information is generally displayed in a graphic format (Graphical User Interface). Figure 2.10 gives an illustration of this interaction.

Fig. 2.10: Human machine interaction.

2.3/ INDUSTRIAL PROGNOSTICS AND HEALTH MANAGEMENT BY WSNS

The importance of reliability has increased in industry. It is a means to economic gain in addition to client trust. Over the past years, the research in prognostics has resulted in a variety of techniques and tools that offer plants the possibility to survey their systems, anticipate failures, and schedule maintenance activities. The existing tools differ from one to another as they have different advantages, disadvantages, complexities, etc. A great deal of attention was given to data-driven prognostic models because of their low cost and complexity, and easy deployment. First, the prediction model will acquire information about the monitored system, assess the current state, and then extrapolate the health state in the future.

Data gathering is usually based on the deployment of independent sensors. This choice is not always feasible. More than one parameter will be required for a complex system to be monitored, or one that spreads on a large area, and therefore as many sensors for the monitoring activity. The wiring may render a complicated process when connecting each of these sensors separately to the base station. As consequence, connecting the sensors in a network is privileged. There are several drawbacks for the usage of wired networks. Physical wires come with a price that grows proportionally according to their length and weight. For example, on an Airbus A380, we can measure 600 miles of cable weighing over 4 tons. Using wireless sensor networks (WSNs) would decrease the weight significantly, thus consumption of fuel, so the costs will be reduced. In general, by eliminating the difficulties that are related to wiring the network in-site, a WSN will render monitoring zones more accessible. Besides the reduction of the complexities of manufacturing, WSN reduces the associated hazards. As a matter of fact, physical wires cause the problem of signal point of failure. If the wire is damaged, all data transfer through it is paralyzed. In crucial applications, this can have dangerous outcomes, especially in targeted attacks. WSNs are mainly designed for surveillance purposes (military, medicine, agriculture, etc. [START_REF] Li | Survey on security in wireless sensor[END_REF]). Lately, a great deal of attention was given to WSN applications of monitoring in industry. Nowadays, they use sensor networks for maintenance scheduling in order to monitor their machinery. The sensors deployed to survey the system/component will provide data to assess the health, diagnose the system, and estimate the Remaining Useful Life (RUL). However, if there is inaccuracy in the data, the prediction which is based on it will be irrelevant. Before the network starts running, the dependability requirements and the WSN strategies (coverage, energy consumption, density, topology, deployment strategy, scheduling mechanism...) need to be considered and studied. Since real data is needed for good predictions, it is obvious that the first step should be ensuring a reliable source of information. Once the provided information are complete and correct, we will only need a robust health assessment model for good quality predictions. Thereby, they can provide accurate data for RUL prediction and maintenance scheduling. As shown in Figure 2.11, the acquired data will later be used in the PHM. Therefore its quality has a clear impact on the PHM model output. Even though many dependable solutions exist in WSN, these solutions are not always applicable. Since the computational capabilities of sensors are limited, the solutions are often application-oriented.

2.4/ THE CHALLENGES FACED

Although many models have been developed in the PHM research area, there are many aspects that still need deep studying in order to provide more diagnostics. How to use data fully? How to consider operating conditions in diagnostics and RUL prediction? How to allow multiple interactions while building a model? All these questions still need answers.

In the literature of PHM, it is very common that the causes of a failure are limited to the values of monitored parameters. Other factors seem to be neglected and overlooked even though they are responsible for the failures. Condition Monitoring (CM) data doesn't replace reliability data although it reflects online monitoring. In fact, CM data provide measurements informing about a single component state at a specific moment. A failure is a consequence of many factors (component age, different failing component, etc), and doesn't only consider a single parameter (temperature, humidity, etc).

Reliability data, informing about all these factors, give a bigger picture of the failing process. We are not neglecting the importance of CM data. But, while CM data provide information for short-term prediction, reliability data are able to extend these predictions until next maintenance window. What might limit the application of developed models to real machines is the complete neglecting of operating conditions, operating age, and interactions between failures. Operating conditions are never the same, they change all the time. If the model is not able to consider these changes, then it won't be able to produce a reliable estimation. Moreover, if we observe two similar components operating under similar conditions with different operating ages, we will notice that they won't fail at the same time. Operating age definitely has an influence on time of failure. Even a failure can accelerate or provoke another one.

Censored data is another issue to face when performing diagnostic. Many plants do not allow their system to run to failure. Often, replacement of components occurs before they actually fail. Consequently, the real time of failure is not kept record of. The performed preventive maintenance is mistaken for failure time, and RUL prediction is based upon that time. RUL value is critical for the maintenance scheduling. That is to say, the less accurate the prediction, the less reliable the maintenance schedule will be.

The reason behind the entire PHM process is the maintenance scheduling. Yet, once accomplished, the maintenance actions are not considered in the model. And generally, the related component is considered "as good as new". It is very essential to consider the effects of maintenance actions in the prediction model, at least for evaluating the model efficiency and studying the new failure behavior after the maintenance has been performed. What also drew our attention are the assumptions upon which the models rely.

As far as we know, none of the previous research work has questioned the dependability of WSN. It is generally assumed that:

• There is no constraint of energy consumption.

• Sensory data is available and there is no data loss.

• Sensor network is reliable.

• There is no fault in sensors.

However, these assumptions are far from a real life situation. The application of Wireless Sensor Networks (WSN) is very critical. The size of the sensor is very small so their batteries are very small with limited disposable energy. If this limitation is not considered by the communication in the network, the sensors will consume all the energy that they have quickly and be dropped. Consequently, the information can no longer circulate in the network. Nevertheless, an efficient energy WSN will not stop some nodes from being dropped. This means that the network has to be fault tolerant so that it can be able to pursue its functionalities in case any sudden events occur (sensor loss, interferences, etc). Moreover, like all wireless networks, WSN can be hacked. For that, data circulating in the network needs to be secured because of competitors and hackers that can steal information, change data, cause damage to the system, etc. Many research works have been done in WSN reliability field. Every application has its own features, and generalized solutions do not always solve the problem. In this thesis, we put the sensor under the spotlight and study the data acquisition more closely. Doing so, we face the incompleteness of data packets and their impact on the PHM process.

2.5/ CONCLUSION

Wireless sensor networks are known for their easy deployment, their low cost, and their capability to extract localized features. Unfortunately, their resources are limited and their computational capabilities are modest. And there is a constant threat to the network's reliability due to the nature of communication, packet loss, data alteration, transfer delays, and some other issues. The research in wireless sensor networks reveals some tools that can be put in place, to improve the quality of service. The adopted solutions considering the shortcomings of WSN are preferably application-oriented.

For optimizing the maintenance schedule for modern plants, prognostic and health management is considered an important tool. The economic benefit reflects an appropriate schedule. In the early steps of the process, it is important to focus on accuracy for ensuring that the developed degradation model guarantees the expected precision. Building a degradation model depends on key issues, such as model complexity, model strengths, and the amount of available information.

As far as we know, and for the data acquisition step, the current research works uses either independent sensors or sensors that are connected via physical wires. Using wires causes some complications for some industrial systems. Wiring might be very difficult and sometimes impossible, which means we should either adopt a wireless sensor network, or loose in terms of precision by placing the sensors slightly further from the target.

It is highly probable for data loss to occur in the case when the industrial system is monitored by a wireless sensor network, which will have an important impact on the quality of the predictions. In such a situation, the prognostic model is expected to maintain its robustness to the unpredictable lack of information. Keeping in mind the actual challenges of real-life applications, orientation of PHM needs to be in a way to meet the expectations of modern plants. To study the impact and efficacy of WSN in the process PHM, and to study the challenges that may face us, in the next chapter we will suggest a diagnostic algorithm that is efficient in literature to use and perform later in WSN. Using the suggested diagnostic algorithm in WSN is considered a solution for performing diagnostic with incomplete measurement at the processing unit (adapted with the characteristics of the WSN).

3

DIAGNOSIS VIA THE RANDOM FORESTS ALGORITHM D
iagnostic is considered a backbone in the prognostics and health management process. By identifying the current state of health of the system, the prediction of the remaining useful life begins. By relying on the sensor measurements, the values of the monitoring parameters will be used in the process of classification and differentiation between the different states that the system can be in. This task is called classification. It is a very important task to find predictive relationships from data, in many areas of information science. Initial discovery of relationships is usually done with a training set while a test set and validation set are used for evaluating whether the discovered relationships hold. More precisely, potential predictive relationships are discovered by a training set, and the strength and adequacy of a predictive relationship is assessed by a test set. This entire process takes success from the availability and correctness of the monitoring data. In this chapter, random forests are proposed for operating devices diagnostics in the presence of a variable number of features. In various contexts, like large or difficult-toaccess monitored areas, wired sensor networks providing features to achieve diagnostics are either very costly to use or totally impossible to spread out. Using a wireless sensor network can solve this problem, but this latter is more subjected to flaws and the probability of losing data packets is very high. Furthermore, the networks' topology often changes, leading to a variability in quality of coverage in the targeted area. Diagnostics at the sink level must take into consideration that both the number and the quality of the provided features are not constant, and that some politics like scheduling or data aggregation may be developed across the network. The aim of this chapter is (1) to show that random forests are relevant in this context, due to their flexibility and robustness, and (2) to provide first examples of use of this method for diagnostics based on data provided by a wireless sensor network.

3.1/ INTRODUCTION

In machine learning, classification refers to identifying the class to which a new observation belongs, on the basis of a training set and quantifiable observations, known as properties. In ensemble learning, the classifiers are combined to solve a particular computational intelligence problem. Many research works encourage adapting this solution to improve the performance of a model, or reduce the likelihood of selecting a weak classifier. For instance, Dietterich argued that averaging the classifiers' outputs guarantees a better performance than the worst classifier [START_REF] Thomas | An experimental comparison of three methods for constructing ensembles of decision trees: Bagging, boosting, and randomization[END_REF]. This claim was theoretically proven correct by Fumera and Roli [START_REF] Fumera | A theoretical and experimental analysis of linear combiners for multiple classifier systems[END_REF]. In addition to this, and under particular hypotheses, the fusion of multiple classifiers can improve the performance of the best individual classifier, if particular hypotheses are taken into consideration [START_REF] Tumer | Error correlation and error reduction in ensemble classifiers[END_REF].

Two of the early examples of ensemble classifiers are Boosting and Bagging. In Boosting algorithm [START_REF] Schapire | A brief introduction to boosting[END_REF], the distribution of the training set changes adaptively based on the errors generated by the previous classifiers. In fact, at each step, a higher degree of importance is accorded to the misclassified instances. At the end of the training, a weight is accorded to each classifier, regarding its individual performance, indicating its importance in the voting process. As for Bagging [START_REF] Breiman | Bagging predictors[END_REF], the distribution of the training set changes stochastically and equal votes are accorded to the classifiers. For both classifiers, the error rate decreases when the size of the committee increases.

In a comparison made by Tsymbal and Puuronen [START_REF] Tsymbal | Bagging and boosting with dynamic integration of classifiers[END_REF], it is shown that Bagging is more consistent but unable to take into account the heterogeneity of the instance space. In the highlight of this conclusion, the authors emphasize the importance of classifiers' integration. Combining various techniques can provide more accurate results as different classifiers will not behave in the same manner faced to some particularities in the training set. Nevertheless, if the classifiers give different results, a confusion may be induced [START_REF] Kanemoto | Diversity and integration of rotating machine health monitoring methods[END_REF]. It is not easy to ensure reasonable results while combining the classifiers. In this context, the use of random methods could be beneficial. Instead of combining different classifiers, a random method uses the same classifier over different distributions of the training set. A majority vote is then employed to identify the class.

In this thesis, the use of random forests (RF) is proposed for industrial functioning diagnostics, particularly in the context of devices being monitored using a wireless sensor network (WSN). A prerequisite in diagnostics is to consider that data provided by sensors are either flawless or simply noisy. However, deploying a wired sensor network on the monitored device is costly in some well-defined situations, specifically in large scale, moving, or hardly accessible areas to monitor. Such situations encompass nuclear power plants or any structure spread in deep water or in the desert. Wireless sensors can be considered in these cases, due to their low cost and easy deployment.

WSNs monitoring is somehow unique in the sense that sensors too are subjected to failures or energy exhaustion, leading to a change in the network topology. Thus, monitoring quality is variable too and it depends on both time and location on the device. Various strategies can be deployed on the network to achieve fault tolerance or to extend the WSN's lifetime, like nodes scheduling or data aggregation. However, the diagnostic processes must be compatible with these strategies, and with a device coverage of a changing quality. The objective of this research work is to show that RF achieve a good compromise in that situation, being compatible with a number of sensors which may be variable over time, some of them being susceptible to errors. More precisely, we will explain why random methods are relevant to achieve accurate diagnostics of an industrial device being monitored using a WSN. The functioning of RF will then be recalled and applied in the monitoring context. An algorithm will be provided, and an illustration on a simulated WSN will finally be detailed. This study differs from previous works as it addresses RF for industrial prognostics based on incomplete data provided by a WSN.

3.2/ RELATED WORK

Many research works have contributed in improving the classification's accuracy. For instance, tree ensembles use majority voting to identify the most popular class. They have the advantage of transforming weak classifiers into strong ones by combining their knowledge to reduce the error rate.

Usually, the growth of each tree is governed by random vectors sampled from the training set, and bagging is one of the early examples of this. In this method, each tree is grown by randomly selecting individuals from the training set without replacing them [START_REF] Breiman | Bagging predictors[END_REF]. The use of bagging can be motivated by three main reasons: (1) it enhances accuracy with the use of random features, (2) it gives ongoing estimates of the generalization error, strength, and correlation of combined trees, and (3) it is also good for unstable classifiers with large variance.

Meanwhile, Freund introduced the adaptive boosting algorithm Adaboost, which he defined as [START_REF] Freund | Experiments with a new boosting algorithm[END_REF]: "a deterministic algorithm that selects the weights on the training set for input to the next classifier based on the wrong classifications in the previous classifiers". The fact that the classifier focuses on correcting the errors at each new step remarkably improved the accuracy of classifications. Shortly after, in [START_REF] Amit | Shape quantization and recognition with randomized trees[END_REF] randomness was again used to grow the trees. The split was defined at each node by searching for the best random selection of features in the training set. Ho [START_REF] Tin | The random subspace method for constructing decision forests[END_REF] introduced the random subspace, in which he randomly selects a subset of vectors of features to grow each tree. Diettrich introduced the random split selection where at each node, a split is randomly selected among k best splits [START_REF] Thomas | An experimental comparison of three methods for constructing ensembles of decision trees: Bagging, boosting, and randomization[END_REF].

For these methods, and like bagging, a random vector sampled to grow a tree is completely independent from the previous vectors, but is generated with the same distribution.

Random split selection [START_REF] Thomas | An experimental comparison of three methods for constructing ensembles of decision trees: Bagging, boosting, and randomization[END_REF] and introducing random noise into the outputs [START_REF] Breiman | Using adaptive bagging to debias regressions[END_REF] both gave better results than bagging. Nevertheless, the algorithms implementing ways of re-weighting the training set, such as Adaboost [START_REF] Freund | Experiments with a new boosting algorithm[END_REF], outperform these two methods [START_REF] Breiman | Random forests[END_REF].

Therefore, Breiman combined the strengths of the methods detailed above into the random forest algorithm. In this method, individuals are randomly selected from the training set with replacement. At each node, a split is selected by reducing the dispersion generated by the previous step and consequently lowering the error rate. This algorithm is further detailed in Section 3.6.

3.3/ MACHINE LEARNING

Machine learning explores the study and construction of algorithms that can learn from and make predictions on data. These algorithms can function by building a model from example inputs, instead of following strictly static program instructions, to make datadriven predictions or decisions.

3.3.1/ REINFORCEMENT LEARNING

Reinforcement learning is a type of machine learning, and by that, also a branch of artificial intelligence. It is inspired by behaviorist psychology and related to how software agents are obliged to act in an environment for maximizing the cumulative reward. Reinforcement signal also known as simple reward feedback is a requirement for the agent to learn its behavior. This allows automatic determination of the ideal behavior within a specific context by machines and software agents, in order to maximize its performance.

The environment is typically formulated as a Markov Decision Process (MDP). In this problem, an agent according to his current state is supposed to decide the best action to select. The problem is known as a Markov Decision Process when this step is repeated. The main difference that lies between the classical techniques and reinforcement learning algorithms is that the latter do not need knowledge about the MDP and they target large MDPs where exact methods become unattainable.

What differentiate reinforcement learning from standard supervised learning is that correct input/output pairs are never presented, nor sub-optimal actions explicitly corrected. Furthermore, on-line performance was focused on, which involves finding a balance between exploration (of uncharted territory) and exploitation (of current knowledge).

There are many challenges in current Reinforcement Learning research. First of all, it is often too memory expensive to store values of each state, since the problems can be very complex. To solve this, it involves looking into value approximation techniques, such as Decision Trees or Neural Networks. Introducing these imperfect value estimations has many consequences, and research tries minimizing their impact on the quality of the solution. Moreover, problems are also generally very modular; similar behaviors reappear often, and modularity can be introduced to avoid learning everything all over again. Hierarchical approaches are common-place for this, but doing this automatically is proving a challenge. Finally, it is sometimes impossible to completely determine the current state because of limited perception. The performance of the algorithm is also affected by it, and most work has been done for compensating this Perceptual Aliasing.

3.3.2/ DEEP LEARNING

Deep machine learning is based on a set of algorithms that try to model high-level abstractions in data. It consists in learning representations of data, by which an observation can be represented in many ways. Research in this area tries to make better representations and creating models for learning these representations from large-scale unlabeled data. Improvements in neuroscience inspire some of the representations and are loosely based on interpretation of information processing and communication patterns in a nervous system, which tries to define a relationship between the stimulus and the neuronal responses and the relationship among the electrical activity of the neurons in the brain.

A main criticism of deep learning concerns the lack of theory surrounding many of the methods. Deep learning methods are frequently referred as black box, where most confirmations empirically done rather than theoretically.

3.3.3/ SUPERVISED LEARNING

Supervised learning aims at training a model, on the basis of a known set of data (inputs) and their known corresponding responses (outputs) that generate predictions to new observations. It is necessary that each input for the training set has a known corresponding output. For that, the data set is called labeled. For mapping new examples, a supervised learning algorithm is used that analyzes the training set and produces an inferred function. A best scenario will permit the algorithm accurately determine the class labels for unseen cases. This requires the learning algorithm to generalize from the training data to unseen situations. The main steps for a supervised learning are: (1) Preparing data, (2) choosing an algorithm, (3) fitting a model, (4) choosing a validation method, (5) Examining fit and updating until satisfactory results, and (6) using fitted models for predictions.

Usually, these algorithms are fast and accurate. When new data is given in input without knowing earlier the target, they are capable generalizing by giving a correct result. However, over-fitting is a common problem. Over-fitting occurs when the algorithm functions well on the training set but poorly with new observations. This means that the algorithm learned the data and not the underlying function. Another disadvantage of supervised learning is the computational complexity when the training uses large data sets.

3.3.4/ UNSUPERVISED LEARNING

The problem of unsupervised learning is the problem of finding a hidden structure in unlabeled data. Latent variables are assumed to cause all the observations, that is, the observations are assumed to be at the end of the causal chain. In practice, models for supervised learning often leave the probability for inputs undefined. As long as the inputs are available this model is not needed, but in case of absence in some of the input values, it is not possible to deduce anything about the outputs. Missing inputs cause no problem if the inputs are also modeled since they can be considered, as in unsupervised learning, latent variables.

Advantage of unsupervised learning is that it is possible to not provide the model with the correct results during the training. It can be used for clustering the input data in classes according to their statistical properties only, and the labeling can be accomplished even if the labels are only available for a small number of objects representative of the desired classes.

On the down side, there is no error or reward signal to evaluate a potential solution since the examples given to the learner are unlabeled. This distinguishes unsupervised learning from supervised learning and reinforcement learning.

Unsupervised learning is in a close relation with the problem of density estimation in statistics. However unsupervised learning also encircles many other techniques that seek to summarize and explain key features of the data. Many methods employed in unsupervised learning are based on data mining methods used to pre-process data.

3.3.5/ SEMI-SUPERVISED LEARNING

Semi-supervised learning lies between supervised learning (with completely labeled training data) and unsupervised learning (without any labeled training data) tasks and techniques and use unlabeled data for training, generally a small amount of labeled data with a large amount of unlabeled data. Many machine learning researchers have found that unlabeled data can produce reasonable development in learning accuracy, when used in conjunction with a small amount of labeled data. A proficient human agent or physical experiment is often required for obtaining labeled data for a learning problem. Thus, the cost associated with labeling process may make a fully labeled training impracticable, while acquisition of unlabeled data is relatively affordable. In such positions, semi-supervised learning can be of great practical value. In suchlike situations, semisupervised learning can have a great practical value. Semi-supervised learning is also of theoretical interest in machine learning and as a model for human learning.

3.4/ ENSEMBLE METHODS

Ensemble methods use multiple learning algorithms, in statistics and machine learning, to attain better predictive performance that could be obtained from any of the constituent learning algorithms. Unlike a statistical ensemble in statistical mechanics, which is usually infinite, a machine learning ensemble refers only to a concrete finite set of alternative models, but typically allows for much more flexible structure to exist among those alternatives.

Ensemble methods could be either homogeneous or heterogeneous. The same algorithm is used for the classification problem in homogeneous methods. It is the data set that is varied, as indicated in Figure 3.2. The algorithm is applied to each of the subsets and the final classifier is the combination of all the hypotheses. In a heterogeneous method, different algorithms are applied to the same training set, as illustrated in Figure 3.3.

If the training space is large in proportion to the number of examples, several hypotheses may be induced having the same performance. The algorithm is then forced to choose one which is probably not the best. This problem is solved by ensemble method by averaging a number of hypotheses, so the overall error is reduced. However, there are some constraints for ensemble methods. While a homogeneous method would need a reasonable size of data for the training, a heterogeneous one, because of different degrees of accuracy of the different algorithms, might cause some conflicts of fusion of results.

3.5/ OVERVIEW OF DIAGNOSTICS

With their constantly growing complexity, current industrial systems witness costly downtime and failures. Therefore, an efficient health assessment technique is mandatory. In fact, in order to avoid expensive shutdowns, maintenance activities are scheduled to pre-vent interruptions in system operation. In early frameworks, maintenance takes place either after a failure occurs (corrective maintenance), or according to predefined time intervals (periodic maintenance). Nevertheless, this still generates extra costs due to "too soon" or "too late" maintenance. Accordingly, considering the actual health state of the operating devices is important in the decision making process. Maintenance here becomes condition-based, and is only performed after the system being diagnosed in a certain health state.

Diagnostics is an understanding of the relationship between what we observe in the present and what happened in the past, by relating the cause to the effect. After a fault takes place, and once detected, an anomaly is reported in the system behavior. The fault is then isolated by determining and locating the cause (or source) of the problem. Doing so, the component responsible for the failure is identified and the extent of the current failure is measured. This activity should meet several requirements in order to be efficient [START_REF] Dash | Challenges in the industrial applications of fault diagnostic systems[END_REF]. these requirements are enumerated in the following.

• Early detection: in order to improve industrial systems' reliability, fault detection needs to be quick and accurate. Nevertheless, diagnostic systems need to find a reasonable trade-off between quick response and fault tolerance. In other words, an efficient diagnostic system should differentiate between normal and erroneous performances in the presence of a fault.

• Isolability: fault isolation is a very important step in the diagnostic process. It refers to the ability of a diagnostic system to determine the source of the fault and identify the responsible component. With the isolability attribute, the system should discriminate between different failures. When an anomaly is detected, a set of possible faults is generated. While the completeness aspect requires the actual faults to be a subset of the proposed set, resolution optimization necessitates that the set is as small as possible. A trade-off then needs to be found while respecting the accuracy of diagnostics.

• Robustness and resources: it is highly desirable that the diagnostic system would degrade gracefully rather than fail suddenly. For this finality, the system needs to be robust to noise and uncertainties. In addition to this, a trade-off between system performance and computational complexity is to be considered. For example, online diagnostics require low complexity and higher storage capacities.

• Faults identifiability: a diagnostics system is of no interest if it cannot distinguish between normal and abnormal behaviors. It is also crucial that not only the cause of every fault is identified, but also that new observations of malfunctioning would not be misclassified as a known fault or as normal behavior. While it is very common that a present fault leads to the generation of other faults, combining the effects of these faults is not that easy to achieve due to a possible non-linearity. On the other hand, modeling the faults separately may exhaust the resources in case of large processes.

• Clarity: when diagnostic models and human expertise are combined together, the decision making support is more reliable. Therefore, it is appreciated that the system explains how the fault was triggered and how it propagated, and keeps track on the cause/effect relationship. This can help the operator use their experience to evaluate the system and understand the decision making process.

• Adaptability: operating conditions, external inputs, and environmental conditions change all the time. Thus, to ensure relevant diagnostics at all levels, the system should adapt to changes and evolve in the presence of new information.

Existent diagnostic models have several limitations. Some of which are summarized in Table 3.1.

Diagnostic model Drawbacks

Markovian The degradation process can be considered as a stochastic process. The evolution of the degradation is a random variable that describes the different levels of the system's health state, from good condition to complete deterioration. The deterioration process is multistate and can be divided into two main categories [START_REF] Moghaddass | An integrated framework for online diagnostic and prognostic health monitoring using a multistate deterioration process[END_REF]:

1. Continuous-state space: the device is considered failed when the predefined threshold is reached.

2.

Discrete-state space: the degradation process is divided into a finite number of discrete levels.

As condition-based maintenance relies on reliable scheduling of maintenance activities, an understanding of the degradation process is required. For this finality, in this work, we consider the discrete-state space deterioration process.

3.6/ THE RANDOM FORESTS ALGORITHM

In this chapter, the use of Random Forest (RF) is proposed for industrial functioning diagnostics. The choice of RF was drove by a number of factors. First of all, the injected randomness in the algorithm gives the trees different starting points. In fact, when the sensor nodes start to fail, no certainty measurements will arrive at the base station for processing. When the corresponding monitored parameter is needed at the root of the decision tree, the health assessment or the diagnostics cannot be performed (due to lack of information). The injected randomness will give the process a continuity using other trees in the forest (trees that need another parameter). Second of all, the algorithm includes the feature selection step. In other algorithms used for diagnostic (the PHM process in general), the feature selection step needs to be done aside. The RF algorithm includes this step in the training phase. An example of a RF is given in Figure 3.4. The diagnostic is a key step for Remaining Useful Life (RUL) estimation. The machine/component's health state is identified according to the analysis and the predefined thresholds. Sensory data is periodically reported for monitoring critical components. This data corresponds to measurements of monitoring parameters and is useful in assessing the machine/ component's condition. There is a threshold for each monitoring parameter, that is once reached, the system is considered to be in the corresponding state.

The RF algorithm is mainly the combination of Bagging [START_REF] Breiman | Bagging predictors[END_REF] and random subspace [START_REF] Tin | The random subspace method for constructing decision forests[END_REF] algorithms, and was defined by Leo Breiman as "a combination of tree predictors such that each tree depends on the values of a random vector sampled independently and with the same distribution for all trees in the forest" [START_REF] Breiman | Random forests[END_REF]. This method resulted from a number of improvements in tree classifiers' accuracy.

This classifier maximizes the variance by injecting randomness in variable selection, and minimizes the bias by growing the tree to a maximum depth (no pruning). The steps of constructing the forest are detailed in Algorithm 1.

In a RF, the root of a tree i contains the instances from the training subset S i , sorted by their corresponding classes needs to be further developed (no pruning). For this purpose, at each node, the feature that guarantees the best split is selected as follows.

1. The information acquired by choosing a feature can be computed through:

1. The entropy of Shannon, which measures the quantity of information

Entropy(p) = - c k=1 P(k/p) × log(P(k/p)) (3.1)
where p is the number of examples associated to a position in the tree, c is the total number of classes, k/p denotes the fraction of examples associated to a position in the tree and labelled class k, P(k/p) is the proportion of elements labelled class k at a position p.

The Gini index, which measures the dispersion in a population

Gini(x) = 1 - c k=1 P(k/p) 2 (3.2)
where x is a random sample, c is the number of classes, k/p denotes the fraction of examples associated to a position in the tree and labelled class k, P(k/p) is the proportion of elements labelled class k at a position p.

2.

The best split is then chosen by computing the gain of information from growing the tree at given position, corresponding to each feature as follows:

Gain(p, t) = f (p) - n j=1 P j × f (p j ) (3.3)
where p corresponds to the position in the tree, t denotes the test at branch n, P j is the proportion of elements at position p and that go to position p j , f (p) corresponds to either Entropy(p) or Gini(p).

The feature that provides the higher Gain is selected to split the node.

The optimal training of a classification problem can be NP-hard. Tree ensembles have the advantage of running the algorithm from different starting points, and this can better approximate the near-optimal classifier.

In his paper, Leo Breiman discusses the accuracy of random Forests. In particular, he gave proof that the generalized error, although different from one application to another, always has an upper bound and so random forests converge [START_REF] Breiman | Random forests[END_REF].

The injected randomness can improve accuracy if it minimizes correlation while maintaining strength. The tree ensembles investigated by Breiman use either randomly selected inputs or a combination of inputs at each node to grow the tree. These methods have interesting characteristics as:

-Their accuracy is at least as good as Adaboost -They are relatively robust to outliers and noise -They are faster than bagging or boosting -They give internal estimates of error, strength, correlation, and variable importance -They are simple and the trees can be grown in parallel There are four different levels of diversity which were defined in [START_REF] Sharkey | Combining diverse neural nets[END_REF], level 1 being the best and level 4 the worst.

• Level 1: no more than one classifier is wrong for each pattern.

• Level 2: the majority voting is always correct.

• Level 3: at least one classifier is correct for each pattern.

• Level 4: all classifiers are wrong for some pattern.

RF can guarantee that at least level two is reached. In fact, a trained tree is only selected to contribute in the voting if it does better than random, i.e. the error rate generated by the corresponding tree has to be less than 0.5, or the tree will be dropped from the forest [START_REF] Breiman | Random forests[END_REF].

In [START_REF] Verikas | Mining data with random forests: A survey and results of new tests[END_REF] In this set of experiments, we consider that no level of correlation is introduced between the different features. Moreover, we suppose that at time t:

• Under normal conditions, temperature sensors follow a Gaussian law of parameter (20 × (1 + 0.005t), 1), while these parameters are mapped to [START_REF] Elghazel | Dependability of wireless sensor networks for industrial prognostics and health management[END_REF][START_REF]Mica2 datasheet[END_REF] in case of a malfunction of the industrial device. Finally, these sensors return the value 0 when they break down.

• The Gaussian parameters are (5 × (1 + 0.01t), 0.3) when both the industrial device and the pressure sensors are in normal conditions. The parameters are changed to (15, 1) in case of industrial failure, while the pressure sensors return 1 when they are themselves broken down.

• Finally, the 10 humidity sensors produce data following a Gaussian law of parameter (52.5 × (1 + 0.001t), 12.5) when they are sensing a well-functioning device. These parameters are set to [START_REF] Li | A survey on topology control in wireless sensor networks: Taxonomy, comparative study, and open issues[END_REF][START_REF] Amit | Shape quantization and recognition with randomized trees[END_REF] in case of device failure, while malfunctioning humidity sensors produce the value 0.

3.7.1.2/ SET OF EXPERIMENT 2

For this set, a linear correlation is injected between the studied features. A comprehensive study of correlation between features in maintenance prediction is given in [START_REF] Mosallam | Nonlinear relation mining for maintenance prediction[END_REF].

• Under normal conditions, temperature sensors follow a Gaussian law of parameter (20 × (1 + 0.005t), 1), while these parameters are mapped to [START_REF] Elghazel | Dependability of wireless sensor networks for industrial prognostics and health management[END_REF][START_REF]Mica2 datasheet[END_REF] in case of a malfunction of the industrial device. Finally, these sensors return the value 0 when they break down.

• When both the industrial device and the pressure sensors are in normal conditions, the value of pressure is computed as (x÷2+10), where x is the value of temperature.

The parameters are changed to (15, 1) in case of industrial failure, while the pressure sensors return 1 when they are themselves broken down.

• For a well-functioning device, the 10 humidity sensors produce data in the form of (x × 525 + 12). These parameters are set to [START_REF] Li | A survey on topology control in wireless sensor networks: Taxonomy, comparative study, and open issues[END_REF][START_REF] Amit | Shape quantization and recognition with randomized trees[END_REF] in case of device failure, while malfunctioning humidity sensors produce the value 0.

For both data sets, the probability that a failure occurs at time t follows a Bernoulli distribution of parameter t ÷ 35000.

Five levels of functioning are attributed to each category of sensors, depending on the abnormality of the sensed data. These levels are defined thanks to 4 thresholds, which are 22.9, 24.5, 26, and 28 degrees for the temperature (a temperature lower than 22.9°C is normal, while a sensed value larger than 28°C is highly related to a malfunctioning), 5.99, 6.4, 7.9, and 9 bars for the pressure parameter, and finally 68, 80, 92, and 95 percents for the humidity.

Data is generated as follows.

• For each time unit t = 1..100 during the industrial device monitoring, -For each category c (temperature, pressure, humidity) of sensors:

* For each sensor s belonging to category c:

• If s has not yet detected a device failure:

1. s picks a new data, according to the Gaussian law corresponding to a well-functioning device, which depends on both t and c, 2. a random draw from the exponential law detailed previously is realized, to determine if a breakdown occurs on the location where s is placed.

• Else s picks a new datum according to the Bernoulli distribution of a category c sensor observing a malfunctioning device.

The global failure level F t of a set of 110 sensed data produced by the wireless sensor network at a given time t is defined as follows. For each sensed datum d t i , i = 1..110, let f t i ∈ {1, .., 5} be the functioning level related to its category (pressure, temperature, or humidity). Then F t = max f t i | i = 1..110.

3.7.2/ RANDOM FOREST DESIGN

The random forest, constituted in this set of experiments by 100 trees, is defined as follows. For each tree T i , i = 1..100:

• A sample of 67% of dates τ 1 , . . . , τ 67 ∈ {1, ..., 100} is extracted

• The root of the tree T i is the tuple

{ j | F τ j = n, j = 1..67} n=1..5
, where X is the cardinality of the finite set X. Thus, its n-th coordinate corresponds to the number of times the device has been in the global failure n in this sample of observation dates.

• The category c having the largest Gain for the dates in the root node is selected.

The dates are divided into five sets depending on thresholds related to c. Then, 5 edges labeled by both c and failure levels l 0 i = {1, .., 5} are added to T i , as depicted in Figure 3 In other words, we only consider in this node a sub-sample of dates having their functioning level for category c equal to l 0 i , and we divide the sub-sample into 5 subsets, depending on their global functioning levels: the tuple is constituted by each cardinality of these subsets, see Figure 3.5.

• The process is continued, with: this vertex as a new root, the reduced set of observed dates, and the categories minus c. It is stopped when either all the categories have been regarded, or when tuple of the node has at least 4 components equal to 0.

3.7.3/ PROVIDING A DIAGNOSTIC ON A NEW SET OF OBSERVATIONS

Finally, given a new set of observations at a given time, the diagnostics of the industrial device is obtained as follows.

Let T be a tree in the forest. T will be visited starting from its root until reaching a leaf as described below.

1. All the edges connected to the root of T are labeled with the same category c, but with various failure levels. The selected edge e is the one whose labeled level of failure regarding c corresponds to the c-level of failure of the observations.

2.

If the obtained node n following edge e is a leaf, then the global level of failure of the observations according to T is the coordinate of the unique non zero component of the tuple. If not, the tree walk is continued at item 1 with node n as new root.

The global diagnostics for the given observation is a majority consensus of all the responses of all the trees in the forest.

3.7.4/ NUMERICAL SIMULATIONS

The training set is obtained by simulating 100 observations for 10 successive times, which results in 1000 instances. The resulting data base is then used to train 100 trees that will constitute the trained random forest. For each of the 100 performed simulations, we calculated the average number of errors in fault detection, produced by the trees in the forest. Figure 3.7 shows that this error rate remained below 15% through the simulation. This error rate includes both "too early" and "too late" detections. When certain sensor nodes stop functioning, this leads to a lack on information, which has an impact on the quality of predictions; this explains a sudden increase in the error rate with time. We can conclude from the low error rate in the absence of some data packets that increasing the number of trees in the RF helps improve the quality and accuracy of predictions.

As described in Section 3.7.1.2, a correlation was introduced between the features. Figure 3.8 shows the number of successful diagnostics when the number of tree estimators in the forest changes. As shown in this figure, the RF method guarantees a 60% success rate when the number of trees is limited to 5. As this number grows, the accuracy of the method increases to reach 80% when the number of trees is around 100. Comparing to the previous results, the correlation between the features helps decrease the uncertainties in diagnostics when the number of trees increases. The algorithm is able to understand the relationship between two features. Thus, when some values describing a feature are missing, the algorithm can deduct them from the available information about the rest features.

3.8/ CONCLUSION

Instead of using wired sensor networks for a diagnostics and health management method, it is possible to use wireless sensors. Such a use can be motivated by cost reasons or due to specific particularities of the monitored device. In the context of a changing number and quality of provided features, the use of random forests may be of interest. These random classifiers were recalled with details in this chapter, and the reason behind their use in the context of a wireless sensors network monitoring was explained. Finally, algorithms and first examples of use of these random forests for diagnostics using a wireless sensor network were provided. The simulation results showed that the algorithm guarantees a certain level of accuracy even when some data packets are missing.

The authors' intention is to compare various tools for diagnostics to the random forests, either when considering wireless sensor networks or wired ones. Comparisons will be carried out for both theoretical and practical aspects. The algorithm of random forests, for its part, will be extended to achieve prognostics and health management too. By relying on random forest algorithm and other algorithms in literature, in the next chapter we will study and show the impact of WSN strategies on diagnostics-PHM.

IMPACT OF WSNS STRATEGIES ON PROGNOSTICS AND HEALTH MANAGEMENT I n this chapter, we will use Wireless Sensor Network (WSN) techniques for monitoring an area under consideration, in order to diagnose its state in real time. What differentiates this type of network from the traditional computer ones is that it is composed by a large number of sensor nodes having very limited and almost nonrenewable energy. A key issue in designing such networks is energy conservation because once a sensor depletes its resources, it will be dropped from the network. This will lead to coverage hole and incomplete data arriving to the sink. Therefore, preserving the energy held by the nodes so that the network keeps running for as long as possible is a very important concern. If we achieve to improve the network lifetime and Quality of Service (QoS). Diagnosing the state of area will be more accurate for a longer time. One of the most important elements to achieve a QoS in WSN is the network coverage which is usually interpreted as how well the network can observe a given area. Obviously, if the coverage decreases over time, the diagnosis quality decreases accordingly. Various coverage strategies are thus proposed by the WSN community, in order to guarantee a certain coverage rate as long as possible, to reach a certain QoS that in turn will impact the diagnosis and prognostic quality. Various other strategies are in common use in WSN like data aggregation and scheduling, to preserve a QoS in wireless sensor networks, as long as possible. We argue that such strategies are not neutral if this network is used for prognostic and health management. Some politics may have a positive impact while other ones may blur the sensed data, like data aggregation or redundancy suppression, leading to erroneous diagnostics and/or prognostics. In this chapter, we will show and measure the impact of each WSN strategy on the resulting estimation of diagnostics, and therefore on Prognostic and Health Management (PHM). We emphasized several issues and studied various parameters related to these strategies that have a very important impact on the network, and therefore on data diagnostics over time. To reach this goal, to evaluate both prognostic and health management with the WSN strategies, and to evaluate the importance of the random forest algorithm suggested in Chapter 3, we have used in WSN the random forest algorithm and five other diagnostic algorithms from literature.

4.1/ INTRODUCTION

Modern industrial plants and areas for military, agricultural, health purposes, are always at risk of failure due to fire, robbery, attack, etc., which will be dangerous and costly since the costs of failure and system downtime are getting pricey. This is why, it is very necessary to evaluate their health and diagnose them at any time, and then plan maintenance activities to avoid disastrous failure results. Prognostic and Health Management (PHM) is a process allowing an advanced system to automatically test the area, diagnose it, isolate the failure, and try predicting the Remaining Useful Life (RUL) for this area before failure occurs [START_REF] Sun | Research and application of the prognostic and health management system[END_REF]. A maintenance scheduling is then determined and the area shutdown is prevented. But if the prediction model and the provided measurements are not accurate, the maintenance activity will not be done on time.

Online measurements of the operating conditions are required for assessing health and diagnosing activity of the area of interest, followed by RUL prediction. A number of sensor nodes usually gather these information. In this chapter, and as we have mentioned before, we consider the case where sensors communicate their information within a Wireless Sensor Network. Unlike the conventional computer networks, WSNs are composed of a large number of sensor nodes with very limited and non-renewable energy. Most of the time, they are deployed to capture the occurrence of possible events in hostile and inaccessible areas [START_REF] Yick | Wireless sensor network survey[END_REF]. However, there is a classical assumption in PHM that the monitored data are available and complete, which is not always true. Indeed, because of the nature of communication in this network and characteristics of its devices, a WSN is at risk of failure. The accuracy and completeness of data that is going to be captured will be affected by this risk, and consequently PHM will be affected [START_REF] Elghazel | Random forests for industrial device functioning diagnostics using wireless sensor networks[END_REF][START_REF] Elghazel | Prognostics and health management based on dependable wireless sensor networks[END_REF][START_REF] Elghazel | Dependable wireless sensor networks for prognostics and health management : a survey[END_REF]. Therefore to ensure that the data of the monitored area is as accurate as possible, we need to maintain the Quality of Service (QoS) of WSN for the longest possible time.

WSN strategies that can be considered are: the topologies [START_REF] Li | A survey on topology issues in wireless sensor network[END_REF][START_REF] Ye | An energy-efficient mac protocol for wireless sensor networks[END_REF][START_REF] Li | A survey on topology control in wireless sensor networks: Taxonomy, comparative study, and open issues[END_REF], coverage [START_REF] Fan | Coverage problem in wireless sensor network: A survey[END_REF][START_REF] Tc He | Coverage analyses of plane target in sensor networks based on clifford algebra[END_REF], deployment strategy [START_REF] Patil | Issues of connectivity and coverage in wireless sensor networks[END_REF][START_REF] Chang | Deployment strategies for wireless sensor networks[END_REF], scheduling mechanism [START_REF] Wang | A survey of energy-efficient scheduling mechanisms in sensor networks[END_REF][START_REF] Tc He | Coverage analyses of plane target in sensor networks based on clifford algebra[END_REF][START_REF] Tian | A node scheduling scheme for energy conservation in large wireless sensor networks[END_REF][START_REF] Tian | Location and calculation-free node-scheduling schemes in large wireless sensor networks[END_REF][START_REF] Hefeeda | Energy-efficient protocol for deterministic and probabilistic coverage in sensor networks[END_REF], density [4], security [START_REF] Khan Pathan | Security in wireless sensor networks: issues and challenges[END_REF][START_REF] Hammad | Security for wsn based on elliptic curve cryptography[END_REF], data aggregation [START_REF] Krishnamachari | The impact of data aggregation in wireless sensor networks[END_REF][START_REF] Yuan | Data density correlation degree clustering method for data aggregation in wsn[END_REF][START_REF] Km Uma Maheswari | Data aggregation in wireless sensor networks[END_REF], packet transfer distance, battery, memory, etc. These strategies have a strong impact on the effectiveness of the network with time, consequently on the quality of the data that will be captured in the monitored area, and finally on PHM. Coverage, for instance, is in a close relation with energy consumption of sensors [START_REF] Zhu | A survey on coverage and connectivity issues in wireless sensor networks[END_REF]: the challenge remains in the efficient use of sensors to increase the lifetime of the network while maximizing the coverage. For this reason, before deployment, a prior study of the network is needed which is the only way to ensure that the area is well covered, leading to the reception of accurate data. Although both PHM and WSN strategies have been studied, as far as we know, none of the existing research work has considered the WSN strategies for PHM. For instance, in real life application, the monitored area might not be fully covered for several reasons like nodes failure, energy depletion, unadapted initial deployment, scheduling mechanisms, etc. Obviously, this flaw may lead to inaccurate and incomplete data, and if we do not take such issue into consideration while building a PHM process over a WSN, the results provided by diagnostic or prognostic may not be reliable.

In this chapter, we study the WSN strategies and their relation with prognostic and health management. We focus on the impact of these strategies on the accuracy of the data captured by a wireless sensor network, and its consequences on the health diagnostic of the monitored area. Diagnostic is a very important part in PHM, in which the determina-tion of RUL begins by identifying the current health state of the system. Our objective is to show that usual diagnostic processes that perform well in classical data provided by a well deployed wired network of sensors, may face a dramatic decrease of performances in the case where data are obtained via a WSN, due to the diversity and variation of WSN strategies usual in such networks. For that, we used six machine learning algorithms to diagnose the area state, namely the so-called Support Vector Machines (SVM), Naive Bayes (NB), Random Forests (RF), Gradient Tree Boosting (GTB), Tree-Based Feature Selection (TBFS), and Nearest Neighbors (NN) methods. Then we studied the behavior of these algorithms when particular issues, inherent to the perfectible nature of WSNs, are present. We focused in this chapter on several important strategies that affect the coverage in WSN, that is, the ones resulting from usual energy consumption of sensors, scheduling mechanisms of sensors, and the problems related to the density and deployment of sensors frequently reported in the wireless sensor network community. In addition, we study the topologies in WSN and its impact on the diagnostics. We used four different types of topology (the most used ones in WSN) which are: distributed, hierarchical, centralized, and decentralized topologies in order to show and study several parameters and issues for each type of topology like density, security, data aggregation, frequency, packet transfer distance, and energy consumption that have an important impact on the diagnostics over time, and therefore on PHM. This research work presents the issues occurring usually in WSNs strategies that are relevant to consider for diagnostics.

4.2/ COVERAGE ISSUES IN WIRELESS SENSOR NETWORKS

WSNs are event-based systems that rely on the collective effort of several sensor or micro-sensor nodes [START_REF] Özg Ür B Akan | Event-to-sink reliable transport in wireless sensor networks[END_REF]. This type of network tends to greatly increase the coverage rate for the area that should be monitored and also to increase the accuracy of the information extracted from this area. A sensor node is a tiny device that has the capability to sense new events, compute the sensed values, and communicate information. WSNs are composed by very small sensors with very limited and non renewable energy that can be deployed when monitoring physical and environmental phenomena such as temperature, vibrations, light, humidity, etc [START_REF] David W Carman | Constraints and approaches for distributed sensor network security (final)[END_REF]. For this energy to be preserved, network throughput has to be low. Another issue is that, as all wireless networks, WSN are not very secure. Both energy limitation and random deployment in hostile and inaccessible areas can cause a node failure (or attack) [START_REF] Paul Walters | Wireless sensor network security: A survey. Security in distributed, grid, mobile[END_REF][START_REF] Fan | Coverage problem in wireless sensor network: A survey[END_REF]. Various characteristics are reported for such kind of networks, they are recalled in what follows.

Due to the nature of communication in this network and to the characteristics of its devices, a WSN is at risk of failure so this will have an effect on the accuracy of the data that will be captured. Therefore we need to maintain the Quality of Service (QoS) of WSN for the longest possible time to ensure the accuracy of the data of the monitored area. Sensor nodes have a short radio range and they collaborate to cover a given surveillance area. The coverage problem arises as: how to ensure that, at any time, any zone in the network is covered by at least one sensor node [START_REF] Tian | Connectivity maintenance and coverage preservation in wireless sensor networks. ad hoc networks journal[END_REF]. One of the most important measurements of WSN QoS is coverage, which is in a close relation with energy consumption of sensors. A basic and important function of WSN is to monitor areas or targets for a long period, such as fire monitoring and environment detection. And a critical issue in the WSN applications is the coverage problem because sensors are often deployed in remote or inaccessible environments or are spread in an arbitrary manner [START_REF] Akbari | An adaptive energy-efficient area coverage algorithm for wireless sensor networks[END_REF][START_REF] Li | A study on one-dimensional k-coverage problem in wireless sensor networks[END_REF]. Therefore, 68CHAPTER 4. IMPACT OF WSNS STRATEGIES ON PROGNOSTICS AND HEALTH MANAGEMENT the challenge remains in the efficient use of these sensors to increase the lifetime of the network while maximizing the coverage. Indeed, in the WSNs community, coverage is one of the most active areas of research, in which the measurement of how well and for how long the sensors are able to observe the physical space usually defines the coverage problem.

For many years, a lot of works have been dedicated to the coverage-related issues in WSNs since it is a fundamental problem [START_REF] Kantaros | Distributed communication-aware coverage control by mobile sensor networks[END_REF][START_REF] Habib | A unified framework for k-coverage and data collection in heterogeneous wireless sensor networks[END_REF][START_REF] Tian | Deployment and reallocation in mobile survivability-heterogeneous wireless sensor networks for barrier coverage[END_REF]2,[START_REF] Pham | Risk-based adaptive scheduling in randomly deployed video sensor networks for critical surveillance applications[END_REF][START_REF] Jacques M Bahi | Lowcost monitoring and intruders detection using wireless video sensor networks[END_REF]. There is limited energy resource in each sensor node, so this makes energy conserving of the sensors, and prolonging the network lifetime while maximizing the coverage of areas or targets, an important and difficult issue in the applications of WSN. Indeed energy is a very critical resource and must be used very sparingly. Many coverage algorithms were found in recent years, but problems still exist in them [START_REF] Zorbas | B {GOP}: An adaptive algorithm for coverage problems in wireless sensor networks[END_REF][START_REF] Wang | Integrated coverage and connectivity configuration in wireless sensor networks[END_REF][START_REF] Hefeeda | Energy-efficient protocol for deterministic and probabilistic coverage in sensor networks[END_REF][START_REF] Gaurav S Kasbekar | Lifetime and coverage guarantees through distributed coordinate-free sensor activation[END_REF]. These algorithms are often based on the subject to be covered (area versus discrete points), sensor deployment mechanism (random versus deterministic) as well as other WSN properties (e.g., minimum energy consumption and network connectivity) [START_REF] Liang | A survey of coverage problems in wireless sensor networks[END_REF].

Regarding energy preservation problem in WSNs, authors in [START_REF] Zhu | A survey on coverage and connectivity issues in wireless sensor networks[END_REF] studied three different approaches, all maintaining the initial coverage QoS. The first approach focuses on optimizing coverage deployment strategy, while the second one consists of planning a scheduling of active sensors that enables other sensors to go into a sleep mode. Finally, the third approach is adjusting the sensing range of sensors for energy conservation. Moreover, recent works show that an efficient density control in high density sensor networks saves significant amounts of energy. Increasing the number of sensors in network makes them closer to each other in the area, which will facilitates the communication between them and reduce the packet transfer distance. This will reduce the energy consumption of sensors, therefore lifetime and coverage rate in WSN will increase. Thus the coverage rate and network lifetime are greatly related to number of sensors in the monitoring area [4,[START_REF] Silva | Reliability and availability evaluation of wireless sensor networks for industrial applications[END_REF][START_REF] Taherkordi | Dependability considerations in wireless sensor networks applications[END_REF][START_REF] Elghazel | Dependability of wireless sensor networks for industrial prognostics and health management[END_REF].

Based on the researches done before, there are three types of problems related to coverage, which are: area, point, and barrier coverage [START_REF] Cardei | Coverage in wireless sensor networks[END_REF][START_REF] Liang | A survey of coverage problems in wireless sensor networks[END_REF], as shown in Figure 4.1.

• Area coverage: this is the most popular coverage problem in WSNs, which has been widely studied for many years. The main objective of the sensor network is to cover or monitor an area (or region), i.e., each point in the area should be covered, and the network lifetime should be maximized. Figure 4.1a shows an example of a random deployment of sensors to cover a given parallelogram-shaped area [START_REF] Kantaros | Distributed communication-aware coverage control by mobile sensor networks[END_REF][START_REF] Pham | Risk-based adaptive scheduling in randomly deployed video sensor networks for critical surveillance applications[END_REF].

• Point coverage: the objective in this problem is to cover a set of points (targets). It considers how to maximize the network lifetime such that all the objectives in the monitored area are covered. Figure 4.1b gives an example of monitoring the discrete targets in a WSN, the black nodes form the set of active sensors.

• Barrier coverage: barrier coverage problem is detecting the probability of a moving object found when crossing the deployment region of WSN. Its goal is to minimize the probability of undetected penetration through the barrier.

4.3/ TOPOLOGIES ISSUES IN WIRELESS SENSOR NETWORKS

In wireless sensor networks, the connectivity of the network is established via radio transmission between sensors. For two sensors to be able to communicate, they must be within some critical range of each other, as transmission capability is finite. And the packet transfer distance between these sensors in this range depends on the topology used in WSN. A network is connected if any node can communicate with any other node, possibly using intermediate nodes as relays. The variability of this connectivity is due to node failures, introduction of additional nodes, variations in sensor location, which requires the adaptability of underlying network structures and operations. Since sensors may be spread in an arbitrary manner, one of the fundamental issues that arises in sensor networks in addition to coverage is thus the connectivity. In order to ensure connectivity and data accuracy in addition to coverage, WSNs use redundant coverage where multiple sensor nodes cover the same physical location. Therefore, coverage may vary across the network. A solution to save energy in the network rises on finding scheduling mechanisms. The objective of such mechanisms is to activate or deactivate redundant nodes while keeping as much as possible a dense coverage and thus ensuring connectivity.

Another metric to save energy in sensor networks is to reduce the amount of data collected and transmitted via the network. Data gathering in WSNs can be either periodic or event-driven [START_REF] Hareb | Energy-efficient data aggregation and transfer in periodic sensor networks[END_REF]. In periodic applications [START_REF] Makhoul | Residual energy-based adaptive data collection approach for periodic sensor networks[END_REF][START_REF] Makhoul | An adaptive scheme for data collection and aggregation in periodic sensor networks[END_REF], data is gathered periodically, while in event-driven applications gathering depends on the occurrence of some events. In both cases, the goal of aggregation operations is to reduce energy dissipation by holding packets for as long as possible in intermediate nodes. All packets will be combined together before being forwarded in the network. It is obvious to see that a decrease in energy consumption leads to an increase in the overall delay, and vice versa. A reliable solution would aim at finding an acceptable trade off between energy consumption and delay in WSNs [START_REF] Hareb | An enhanced K-means and ANOVA-based clustering approach for similarity aggregation in underwater wireless sensor networks[END_REF][START_REF] Soonmok Kwon | Dinamic timeout for data aggregation in wireless sensor netwoks[END_REF]. All of these elements are finally related to the network topologies.

WSNs can be either heterogeneous or homogeneous [START_REF] Li | Survey on security in wireless sensor[END_REF]. In the latter, all nodes have the same role and characteristics. In the former, nodes have different roles: some nodes simply sense and forward information while others aggregate data, manage their area, perform computations, etc. Consequently, some of the nodes can be equipped with higher energy, longer radio range, etc. Several WSN topologies were used in existing monitoring applications, but all of them revolved around four different types (or models) of topologies which are: distributed, hierarchical, centralized, and decentralized ones. They are recalled hereafter.

• Distributed topology: in distributed topologies, there is no management of the network by the central node (or a region of it). They consist of a collection of nodes having equal roles. Therefore, no aspect of hierarchy is considered. No prior infrastructure is imposed before the network starts running; each node discovers its surrounding area and decides which node(s) to communicate with. This decision usually relies on the radio range and the transfer distance. Distributed topologies render the network's maintenance an easy task: if a node fails, its neighbors, within their sensing range, will establish new links with other nodes, and the network will continue to work normally.

• Hierarchical topology: the organization of sensor nodes can be in several levels, making a hierarchical topology (or a tree topology). Level 0 is represented by the root and there is no level above. From two adjacent levels, sensor nodes are connected in an end to end manner. The hierarchical model can be seen as three different layers: (1) the core layer (the root), which is enhanced for availability and performance, (2) the distribution layer, which implements policies and forwards messages, and (3) the access layer (the leaf nodes) that represents the access point to the network. Scalability is the advantage of Hierarchical WSN. The network is more manageable and the task of isolating and detecting faults is simplified due to the presence of different levels.

• Centralized topology: it is one of the easiest topologies to design and implement (also called star topology). All the sensor nodes have a simple task which is sensing new information and forwarding it to a central node where all the data processing will be proceeded with. One of the major problems of this topology is that it presents a single point of failure. The whole network will become paralyzed if a problem occurs at the central node: the data packet cannot be forwarded nor processed when a new event is detected.

• Decentralized topology: decentralized topologies are considered as a combination of the distributed and the centralized topologies. The network is divided into regions (or clusters) which are locally managed by a central node (called the Cluster Head CH). This topology offers a reasonable settlement between energy consumption and Quality of Service (QoS). In this type of topology, there is a reduction of congestion problem and the network no longer has a single point of failure.

4.4/ BENEFITS OF WSN IN PHM

Reliability is necessary in industry, or for any monitored area in general. This is the means to economic gain as well as client trust. For the past years, research in prognostics resulted in variety of tools and techniques that offer the possibility for plants to survey their systems, anticipate failures, and schedule maintenance activities. WSNs are mainly designed for surveillance purposes. They can be deployed in many fields such as military, automotive, agriculture, medicine, and so on [START_REF] Li | Survey on security in wireless sensor[END_REF]. Recently, a great deal of attention was given to WSN applications by industry. These sensor networks are used to monitor the machinery for maintenance scheduling. Furthermore, data will be provided by the sensors deployed to survey the system/component in order to assess the health, diagnose the system, and estimate the RUL. However, inaccuracy in the data will cause the prediction based on it to be irrelevant.

WSN strategies evoked previously have important impact on the accuracy of data and therefore have an important impact on PHM [START_REF] Bahi | Resiliency in distributed sensor networks for prognostics and health management of the monitoring targets[END_REF][START_REF] Elghazel | Dependability of wireless sensor networks for industrial prognostics and health management[END_REF]. Then, before the network starts running, studying the strategies in details in WSNs needs to be considered. Thereby, they can provide accurate data for RUL prediction and maintenance scheduling. As stated previously, our aim is to reveal the impact of such strategies on the accuracy of the captured data from the monitored area and therefore on PHM, using description and numerical simulations. In general, the determination of how well the sensing field is monitored or tracked by sensors is referred to as the coverage problem. The important issue in WSNs is to satisfy the requirements of the coverage and to extend the network lifetime. Many coverage algorithms have been proposed in recent years. But numerous problems can be emphasized in them, as the sensors have restricted computational capabilities, limited batteries, and the solutions are often application-oriented. Indeed, since good predictions rely on real data, it is certain that the first step to be done in the research is ensuring a reliable source of information. -There is no constraint of energy consumption.

-Sensory data is available.

-Sensor network is reliable.

We indicated in Table 4.1 the relationship between issues of coverage in WSNs and their impact on the sensors data and then on PHM. We then explained each point found in this table.

Issues of coverage in WSN Impact on PHM -If the surveillance area is completely covered at a certain time (uniform coverage)

-The captured data will be complete at that time, so the PHM will be accurate -If a coverage hole is observed in the network, then the area is not completely covered -The PHM will be imprecise if a certain event occurred where the coverage hole is found -Sensor deployment strategy (manually or randomly) -The deployment of sensors can cause coverage holes, and incomplete data for an accurate PHM, when the monitoring area is unknown and possibly hazardous -The performance of coverage algorithms is very important for the network's usability and effectiveness -The data captured by WSN will be complete depending on the efficiency and performance of the coverage algorithms, that will cause precise PHM and vice versa -If WSNs ensure the whole area coverage, so in the long term the sensors will die (no energy in sensors) -In the long term, the sensors data will be incomplete for PHM reliability -In the surveillance area, if the density of the sensors increases -The captured data is larger in a time interval, so the PHM will be more accurate -In order to attain reliability in WSN, sensing coverage and sensing level need to be considered -WSN captures reliable data, then the process of PHM will be more precise and vice versa -The models of WSN have an important impact on quality of data and energy consumption -Impact on PHM varies and depends on the model and place of failure in WSN Table 4.1: Potential links between issues of coverage in WSN and their impact on PHM.

4.4.1/ COMPLETE COVERAGE

Monitoring areas or targets for a long period, such as battlefield, fire monitoring and environmental detection is considered a basic and important function of WSN. The useful data in the monitored areas is collected and sent to the base station or the sink [START_REF] Akbari | An adaptive energy-efficient area coverage algorithm for wireless sensor networks[END_REF]. So if the surveillance area is completely covered by sensors at a certain time, the captured data will be complete, then the PHM will be accurate.

4.4.2/ COVERAGE HOLE

How much a region is covered, or in this case monitored, can be defined as the coverage rate. The form of network can dynamically change over time depending on the geographic region, inter-node separations, residual battery power, static and moving obstacles, presence of noise, and other factors that may cause a coverage hole. If a certain event occurred where the coverage hole is found, and with more events being undetected, the outputs of the network will not be reliable [START_REF] Patil | Issues of connectivity and coverage in wireless sensor networks[END_REF][START_REF] Akbari | An adaptive energy-efficient area coverage algorithm for wireless sensor networks[END_REF], so PHM will not be reliable.

4.4.3/ DEPLOYMENT STRATEGY

One important criterion for deploying an efficient sensor network is by finding optimal node placement strategies. Nodes can either be placed manually at predetermined locations or randomly such as sprinkling nodes from an aircraft. The sensors are randomly scattered in most practical situations. Then the Maximizing of coverage and maintaining a lower cost of deployment is a challenge. However, due to severe resource constraints and hostile environmental conditions, it is difficult to find a random deployment strategy that minimizes cost, reduces computation and communication, is resilient to node failures, and provides a high degree of area coverage [START_REF] Patil | Issues of connectivity and coverage in wireless sensor networks[END_REF][START_REF] Chang | Deployment strategies for wireless sensor networks[END_REF]. This will cause incomplete data and imprecise PHM.

4.4.4/ COVERAGE ALGORITHMS

Different coverage algorithms have been suggested in recent years in WSNs (centralized, or distributed and localized). The coverage algorithms are often based on the subject to be covered (area versus discrete points), sensor deployment mechanism (random or deterministic) along with other WSN properties (e.g., minimum energy consumption and network connectivity [START_REF] Liang | A survey of coverage problems in wireless sensor networks[END_REF]). Due to the limited energy resource in each sensor node, sensors need to be used in efficient manner in order to increase the lifetime of the network and maximize the coverage. There are three different approaches to the problem of conserving energy in WSNs and all of the approaches must keep the initial coverage QoS (see Section 4.2). Research related to WSNs is still an open field, where there is no best way to address a problem and no mistake-free solutions. Many research works have been done in this domain. But every application has its own features and the generalized solutions do not always solve the problem. Increasing in the efficiency and performance of coverage algorithms lead to the increase in data and PHM accuracy and vice versa.

4.4.5/ SENSORS ON LONG TERM

The energy of sensors is limited and impossible to be replaced. Important issues exist in the application of WSN like the conservation of this energy and the prolongation of the network lifetime while at the same time guaranteeing the coverage of areas or targets. We can conclude in all cases, that the nodes in the long term will exhaust their energy supply (the sensors are dead), which will cause a coverage hole leading to incomplete data for PHM.

4.4.6/ DENSITY

The density control in high density sensor networks saves significant amounts of energy. Therefore the coverage rate and network lifetime will increase if the number of sensors in the monitoring area increases [4]. So the captured data will be much larger for a longer time and the PHM will be more accurate.

4.4.7/ RELIABILITY

A reliable network is a network that is able to continuously deliver a correct service and to avoid failures that are more frequent or more severe. One of the most important factors in WSNs is the coverage rate. In order to attain reliability in WSNs (by fault prevention, removal, forecasting, and fault tolerance) [START_REF] John | An introduction to computing system dependability[END_REF][START_REF] Taherkordi | Dependability considerations in wireless sensor networks applications[END_REF][START_REF] Silva | Reliability and availability evaluation of wireless sensor networks for industrial applications[END_REF][START_REF] Ian F Akyildiz | A survey on sensor networks[END_REF][START_REF] Dd Geeta | Fault tolerance in wireless sensor network using hand-off and dynamic power adjustment approach[END_REF][START_REF] Ilyas | Handbook of sensor networks: compact wireless and wired sensing systems[END_REF], sensing coverage and sensing level need to be considered [START_REF] Choi | Short paper[END_REF]. Then the WSNs capture reliable and complete data causing the PHM to be more precise and vice versa.

4.4.8/ WSN MODELS

The most used WSN models (topologies) are hierarchical, distributed, centralized, or decentralized. These models have an important impact on energy consumption, security codes, and quality of data. Different impact on PHM depends on the model of WSN and the place of failure. Several solutions to preserve the network's energy have been investigated [START_REF] Van Dam | An adaptive energy-efficient mac protocol for wireless sensor networks[END_REF][START_REF] Ye | An energy-efficient mac protocol for wireless sensor networks[END_REF], and they include the study of the topology. For example the centralized topology may result in extra costs related to energy consumption and security codes [START_REF] Li | Survey on security in wireless sensor[END_REF]. If a problem occurs at the central node, the whole network becomes paralyzed.

4.5/ EXPERIMENTAL PROTOCOL

4.5.1/ WIRELESS SENSOR NETWORK SIMULATION

In order to simulate a WSN for monitoring the area under consideration and to show the impact of WSN strategies on the PHM, we used three types of sensing fields: temperature, pressure, and humidity. The number and parameters of each type of sensor depends on the type of study, as indicated in Section 4.6. Each sensor type captures specific data depending on the operating age t, and we consider that no level of correlation is introduced between the different features:

• Under normal conditions, temperature sensors follow a Gaussian law of parameter (20 × (1 + 0.005t), 1). In case of a malfunction of the area in the range of this sensor, these parameters are mapped to (350, 20). Finally, these sensors return the value 2 when they break down.

• The pressure sensors produce data following a Gaussian law of parameter (5 × (1 + 0.01t), 0.3) when they are sensing a well-functioning area. The parameters changed to (20, 2.5) in case of area failure in the location where the sensor is placed, as long as the pressure sensors return 1 when they are broken down.

• The Gaussian parameters are (52.5 × (1 + 0.001t), 12.5) when both the area and the humidity sensors are in normal conditions. These parameters are set to (80, 10) in case of area failure in the range of this sensor, whereas malfunctioning humidity sensors produce the value 3.

Each sensor follows a Poisson process (Pp) of parameter (200 × (1 -0.01t) + 0.01), to determine if a breakdown occurs in the location where the sensor is placed. Subsequently all of these sensors execute the Algorithm 2.

Algorithm 2: Sensor algorithm if Pp < 1 then the area and the sensors are in normal conditions else if 1 ≤ Pp < 100 then the area is in failure (in the range of this sensor) else the sensor is broken down end if end if Each category of sensors has its own constant threshold, depending on the abnormality of the sensed data. If the data captured by the sensor in a specific category exceeded the threshold, this indicates that a symptom of system deteriorating has been detected. Then a diagnostic study aims at specifying and quantifying an actual failure (whether it failed or not). In this chapter, we used six algorithms for diagnosis which are mentioned in Section 4.5.2. Many applications of WSN exist like area monitoring, industrial monitoring, health care monitoring, environmental/earth sensing, etc. These applications have their own features and the threshold is related to those features. In this chapter, we chosen to simulate a WSN to monitor an area, to study later the strategies of WSN and their impact on PHM. Finally, we consider threshold values as follows: 26 degrees for temperature, 7 bars in pressure, and 80 percents of humidity.

The deployment strategy (manually or randomly) of sensors [START_REF] Patil | Issues of connectivity and coverage in wireless sensor networks[END_REF], the adjustment of the coverage radius of sensors [START_REF] Zhu | A survey on coverage and connectivity issues in wireless sensor networks[END_REF], and the coverage in WSN (as we explained in Section 4.2) have an important impact on the accuracy of the data captured by WSN that will be used in PHM process. In order to study the impact of such strategies of WSN on PHM, in our simulation we consider the following hypotheses:

• Most of the times, the area to be monitored is hazardous and hard to access because of the difficulty in its geographical area like monitoring the forests, oceans, military zones, etc. Therefore we used random deployment for area monitoring.

• The region to be monitored is a rectangle of area A = L × W, such that L and W are the length and width of the region respectively. The area of the coverage range of a sensor is mostly related to the area of the monitored region. Therefore we consider that the coverage area is set to be equal to 1% of the total area of the region. Subsequently, the coverage radius will be R c = 1/10 × √ A/π. And we consider that the radio radius R r equals double the coverage radius (R r = 2R c ).

• We considered that at time t = 0 (when the WSN starts working after the deployment of sensors) the area is fully covered by the sensors used in networks to monitor this area.

4.5.2/ MACHINE LEARNING ALGORITHMS

In this chapter, we focused on studying the diagnostic system (area monitoring) to examine its state: if failure is present or not at a certain time depending on the data captured by the sensors. As we mentioned before, diagnostic is a very important part in the PHM and the determining of RUL begins with identifying the system current state of health. For that, we use this step to evaluate the PHM, and observe how the process is affected by the strategies of WSNs.

The research in PHM is very broad and the authors working in this domain use several algorithms in order to perform the diagnostic of the system state. These methods (machine learning algorithms) are explained in Chapter 3. Machine learning displays a detailed study about the system and from it, an algorithm is built. These algorithms can be operated by building a model from example inputs, in order for the algorithm to be able to diagnose or take decision for new data.

We have chosen six machine learning algorithms to diagnose the system, which were used previously by several authors in the literature in order to evaluate their interest for PHM. We will then evaluate these six diagnostic algorithms in a WSN strategies situation. As stated previously, these algorithms are: The Random Forests (RF) proposed in Chapter 3, Support Vector Machine (SVM) [START_REF] Galar | Remaining useful life estimation using time trajectory tracking and support vector machines[END_REF], Naive Bayes (NB) [START_REF] Zhang | The optimality of naive bayes[END_REF][START_REF] Ng | A naive bayes model for robust remaining useful life prediction of lithium-ion battery[END_REF], Gradient Tree Boosting (GTB) [START_REF] Peter | Boosting algorithms: Regularization, prediction and model fitting[END_REF], Tree-Based Feature Selection (TBFS) [START_REF] V Sugumaran | Feature selection using decision tree and classification through proximal support vector machine for fault diagnostics of roller bearing[END_REF], and Nearest Neighbors (NN) [START_REF] Mcroberts | Estimating forest attribute parameters for small areas using nearest neighbors techniques[END_REF]. These algorithms can be summarized as follows:

• Support Vector Machine: SVM is a learning technique that Vladimir Vapnik developed. In machine learning, SVMs are learning models that are supervised and are associated with learning algorithms that analyze data and recognize pattern, used to classify and regress pattern. A SVM training algorithm builds a model that attributes new examples in one category or the other relying on a certain set of data learning, each marked for belonging to one or two categories, making it a non-probabilistic binary linear classifier.

• Naive Bayes: they are direct acyclic graphs which are a synthesis of probability and graph theory that illustrate random variables and probabilistic inter-dependencies.

They are constituted by a set of nodes that represents different states and directed edges that describe the transition probability between these states.

• Gradient Tree Boosting: GTB was introduced by Leo Breiman. It is a machine learning technique for problems related to regression and classification that produces a prediction model in the form of an entity of weak prediction models, typically decision trees. Based on the errors generated by the previous classifier, the distribution of the training set varies adaptively for each tree. In this study, we took GTB composed of 100 trees then the majority vote (by these 100 trees) is used to identify the class.

• Tree-Based Feature Selection: in machine learning and statistics, feature selection, also known as variable selection, attribute selection, or variable subset selec-tion, is the process of selecting a subset of relevant features (variables, predictors) for use in model construction. Feature selection techniques are used for four reasons: (1) simplification of models to make them easier to interpret by researchers/users, (2) shorter training times, (3) to avoid the curse of dimensionality, (4) to enhance generalization by reducing overfitting (formally, reduction of variance). The central premise when using a feature selection technique is that the data contains many features that are either redundant or irrelevant, and can thus be removed without incurring much loss of information. Redundant or irrelevant features are two distinct notions, since one relevant feature may be redundant in the presence of another relevant feature with which it is strongly correlated.

• Nearest Neighbors: NN is a non-parametric method used for classification. The input consists of the k closest training examples in the feature space. The output is a class membership. An object is classified by a majority vote of its neighbors, with the object being assigned to the class most common among its k nearest neighbors (k is a positive integer, typically small). If k = 1, then the object is simply assigned to the class of that single nearest neighbor. NN is a type of instance-based learning, or lazy learning, where the function is only approximated locally and all computation is deferred until classification. The NN algorithm is among the simplest of all machine learning algorithms. NN can be useful to assign weight to the contributions of the neighbors, so that the nearer neighbors contribute more to the average than the more distant ones. For example, a common weighting scheme consists in giving each neighbor a weight of 1/d, where d is the distance to the neighbor.

Finally we need a large and reliable data set in order to train these algorithms. And so, we can later diagnose the system (area monitoring) from the new data that will be captured by WSN. For that, we take data consisting of N lines, each line is composed by T temperature data, P data of pressure, and H data of humidity to train these algorithms.

All of these data are generated in the way mentioned in Section 4.5.1 (same type of data that will be captured by WSN during area monitoring).

4.6/ SIMULATION RESULTS

In this section and by relying on our simulation, we will show the impact of various WSN strategies on diagnostics from several studies that we performed. These studies are shown and explained in the remainder of this chapter.

4.6.1/ IMPACT OF SCHEDULING MECHANISM IN WSN ON DIAGNOSTICS

To study the effects of scheduling mechanism on diagnostics, we simulated a WSN composed of 200 sensors, sensing respectively the levels of temperature (70 sensors), pressure [START_REF] Li | A survey on topology control in wireless sensor networks: Taxonomy, comparative study, and open issues[END_REF], and humidity (60 sensors). We considered that the WSN model is decentralized, for that we used 16 Cluster Heads to simulate the network on the area under consideration.

The decision or diagnosis that is given by the algorithm is always related to the data learning of these algorithms. For example if the data learning is incomplete and it received a complete data to diagnose it, the error rate will be larger than if the data learning was complete and it received a complete data and vice versa. So, in brief, data learning as well as the data that will be used to diagnose the monitored area, determines the accuracy of the algorithm. For that, we used two types of data (complete and incomplete data) to train these algorithms, and to study for each data type, the variation of the error rate for each algorithm, with the variation of percentage of active sensors in WSN (scheduling).

The first study that was done in scheduling was to simulate a WSN in which all nodes are working at the same time: the percentage of active sensors is 100 %. Like each real WSN, the sensors on the long term will die (because of energy consumption) or break down (due to various causes as the operating age). This simulation was repeated 20 times on two different types of data learning for algorithms: complete data learning and incomplete data learning as shown in for SVM, [START_REF] Amit | Shape quantization and recognition with randomized trees[END_REF][START_REF] David W Carman | Constraints and approaches for distributed sensor network security (final)[END_REF] for NB, [START_REF] Bahi | Resiliency in distributed sensor networks for prognostics and health management of the monitoring targets[END_REF][START_REF] Carle | Preserving area coverage in wireless sensor networks by using surface coverage relay dominating sets[END_REF] for RF, [START_REF] Cho | Energy efficient protocols for low duty cycle wireless microsensor networks[END_REF][START_REF] Hefeeda | Energy-efficient protocol for deterministic and probabilistic coverage in sensor networks[END_REF] for GTB, [START_REF] Bai | Deploying wireless sensors to achieve both coverage and connectivity[END_REF][START_REF] Carle | Preserving area coverage in wireless sensor networks by using surface coverage relay dominating sets[END_REF] for TBFS, and [START_REF] Cerpa | Ascent: Adaptive self-configuring sensor networks topologies[END_REF][START_REF] Estrin | Next century challenges: Scalable coordination in sensor networks[END_REF] for NN.

The difference between Figure 4.3 and Figure 4.4 is that Figure 4.4 shows the results of algorithms training with incomplete data. During these 20 simulations, the error intervals (in %) of these algorithms changed to become as follows: [START_REF] Ian F Akyildiz | A survey on sensor networks[END_REF][START_REF] Carle | Preserving area coverage in wireless sensor networks by using surface coverage relay dominating sets[END_REF], [0, 14], [START_REF] Hammad | Security for wsn based on elliptic curve cryptography[END_REF][START_REF] Breiman | Bagging predictors[END_REF], [0, 8], [2,[START_REF] Amit | Shape quantization and recognition with randomized trees[END_REF], and [START_REF] Ian F Akyildiz | A survey on wireless multimedia sensor networks[END_REF][START_REF] Breiman | Random forests[END_REF] for SVM, NB, RF, GTB, TBFS, and NN respectively.

As depicted in the two figures, the error rate in on what we mentioned at the beginning of this section, the accuracy of the algorithm is related to data learning and the data captured by WSN to be treated. We can conclude that the data captured by the WSN in each simulation is incomplete, because simply the sensors on long term are dead or broken down, and these facts lead to not covered places in the area (coverage hole) and therefore to incomplete data. We conclude from this study that the WSN on the long term will capture incomplete data and this will have an important impact on the diagnosis of the area state. Therefore the battery consumption of sensors in WSN is a very important strategy that has an impact on the work of networks over time, and taking it into consideration through our study is very important.

Scheduling mechanism strategy is one of the best solutions to preserve the energy of sensors for a longer time. Let us deepen the study of this strategy in WSN and its impact on diagnostics, in which the goal is to evaluate the variation of error rate with respect to the variation of the percentage of the active sensors, as shown in Figure 4.5 and Figure 4.6 with complete and incomplete data learning respectively. Note that in this strategy, choosing the active sensor in WSN at a certain time is considered random. Each point in these figures is an average of error rates of a given algorithm on 20 simulations (for a certain percentage of active sensors in WSN). That is, for instance, the percentage of errors found in The difference between Figure 4.5 and Figure 4.6 is that, in 4.6, we have studied the variation of error rate of the algorithm with respect to the variation of active sensors, in case where the training of these algorithms has been performed on incomplete data. As shown in Figure 4.6, which is similar to Figure 4.5, the error rates of the algorithms is decreasing as the percentage of active sensors in WSN is increasing. The percentage of errors (in case where only 10 % of the sensors are working) is equal to: 56 % (SVM algorithm), 52 % (NB), 53 % (RF), 48 % (GTB), 50 % (TBFS), and 54 % (NN). These percentages decrease to become 14.7 %, 5.4 %, 10.7 %, 2.6 %, 4 %, and 12.4 %, for SVM, NB, RF, GTB, TBFS, and NN algorithms respectively (when all sensors are active).

We deduce from these two figures that the error rate in Figure 4.5 is evolving in a way larger than the one in Figure 4.6, according to the modification of the number of active sensors. So, the error rate for each percentage in Figure 4.5 is larger than the one in Figure 4.6. Consequently the data captured by WSN is incomplete, despite the percentage of working sensors in WSN: incomplete data learning with incomplete captured data produces error rates lower than if the data learning was achieved with incomplete captured data, and vice versa. By relying on the change of curves in these two figures, we conclude that, as the percentage of active sensors in WSN increases, the coverage rate in the area monitoring increases accordingly, and so the data captured by WSN will fi- nally be more accurate and precise for diagnostics. For that, the strategy of scheduling in WSN is very important and has a large impact on the efficiency of the network. Then this strategy must be taken into consideration in our study of networks, and we should rely on effective algorithms capable of using the sensors in an efficient manner, in order to increase the lifetime of the network and maximize the coverage over time.

4.6.2/ IMPACT OF DENSITY-DEPLOYMENT OF SENSORS ON DIAGNOSTICS

In order to study the consequences of WSN strategies based on density-deployment and coverage, we now consider a network composed of 600 sensors, sensing respectively the levels of temperature (200 sensors), pressure (200 sensors), and humidity (200 sensors). And since WSN model is decentralized (like the study in Section 4.6.1), we used 48 Cluster Heads to simulate a WSN on the area under consideration.

In the first study, we simulated all the active sensors in the WSN at the same time with complete data learning of algorithms (for all the monitored area), in order to study the impact of the density of sensors on diagnostics. This study is like the one that was done in Section 4.6.1 (shown in Figure 4.3), but now we increase the number of sensors in WSN to be three times larger, therefore the density in WSN is improved. This simulation has been repeated 20 times just like the preceding studies, in order to evaluate the impact of the parameter we modified (number of sensors in WSN) on diagnostics, as shown in shown in this figure, during these 20 simulations, each algorithm shown a specific error interval as follows: [START_REF] Bahi | Resiliency in distributed sensor networks for prognostics and health management of the monitoring targets[END_REF][START_REF] Carle | Preserving area coverage in wireless sensor networks by using surface coverage relay dominating sets[END_REF] for SVM, [START_REF] Hammad | Security for wsn based on elliptic curve cryptography[END_REF][START_REF] Jacques M Bahi | Lowcost monitoring and intruders detection using wireless video sensor networks[END_REF] for NB, [START_REF] Bae | Study on condition based maintenance using on-line monitoring and prognostics suitable to a research reactor[END_REF][START_REF] Breiman | Bagging predictors[END_REF] for RF, [2,[START_REF] Özg Ür B Akan | Event-to-sink reliable transport in wireless sensor networks[END_REF] for GTB, [5,[START_REF] Amit | Shape quantization and recognition with randomized trees[END_REF] for TBFS, and [START_REF] Bae | Study on condition based maintenance using on-line monitoring and prognostics suitable to a research reactor[END_REF][START_REF] Breiman | Random forests[END_REF] for NN. In comparison with Figure 4.3, the error rates of the six algorithms in Figure 4.7 have decreased compared to the rates obtained in Figure 4.3.

From this comparison, we can verify the obvious fact that the density in WSN must be taken into consideration: density is a very important strategy, and has a great impact on coverage rate in an area, and therefore on the accuracy of the data with the time that will be captured by WSN, and used in diagnostics.

We used the result obtained in Figure 4.7 as a starting point, in which we studied later the impact of diagnostics with the variation of the location of this density of sensors at the length of area monitoring (asymmetrical density on abscissas). We evaluated two different cases in this section. In the first one, we considered that the sensors are distributed uniformly and the area is fully covered, and we studied how error rates evolve with the modification of data learning algorithms at the length of monitored area (asymmetrical density on abscissas) as shown in Figure 4.9. In the other (opposite) case, we considered complete data learning in which the data include all the information about the monitored area (we can call it uniform data learning), and we studied the variation of error rate of algorithms with the variation of the location of density of sensors in WSN at the length of area monitoring (asymmetrical density on abscissas) as shown in Figure 4.10. It is worth mentioning that Figure 4.8a is an example about uniform distribution (asymmetrical density on abscissa is 0) and Figure 4.8b is related to non-uniform distribution (in this case, the asymmetrical density on abscissa is 50). Each point in the Figures 4.9 and 4.10 is an average for the error rates of a certain algorithm on 20 simulations: the error rates of the algorithms in these two figures, in case where the asymmetrical density on the abscissa Figure 4.9 shows the variation of error rate for the six considered algorithms, in case where the sensors are uniformly distributed in a fully covered area, with the variation of data learning of algorithms at the length of the monitored area (asymmetrical density on abscissas). It is worth mentioning that data learning is related to each region in the monitored area. For example if the data is complete, this means that the data learning includes information about all of the monitored area (asymmetrical density on abscissa is 0), as in Figure 4.8a. Conversely, Figure 4.8b illustrates the case where the learning process is realized when asymmetrical density on abscissa is 50, which means that the data learning includes information about only half of the area. As shown in Figure 4.9, as the asymmetrical density on abscissas increases (learning nodes are moved to the right of the area), the error rate of these algorithms increases until reaching (for an asymmetry of 50 %): 60 % (SVM), 54 % (NB), 50 % (RF), 58 % (GTB), 52 % (TBFS), and 59 % (NN). These values must be compared to the case where there is no asymmetrical density, in which the error rates are 18 % for SVM, 10 % for NB, 16 % for RF, 4 % for GTB, 7 % for TBFS, and 17 % for NN respectively. Based on these simulations, we can conclude that the density level is an important strategy, and has a great impact on coverage, and therefore on diagnostics. Additionally, the location of nodes (uniformity in the density) is of importance in both the learning stage, and in the testing one (area monitoring after learning). These two elements have a real impact on diagnostic and must be taken into consideration. Our experiments have illustrated that enlarging the uniformity of the node distribution improves the diagnosis quality of the monitored area, as a better coverage leads to a more accurate monitoring of the area. Since data learning is important for the accuracy of the algorithms in diagnosing, we studied the variation of data learning of the area that we need to monitor and we noticed that accuracy of diagnosis increases when the data learning contained all the information about the area. Finally, the best diagnostic happens when the data learning is complete and contains all the area monitoring. And also the distribution of sensors in an area must be uniform in a way that the area is covered to the maximum, as shown in Figure 4.9 and 4.10 in case where the asymmetrical density on the abscissa is 0, and this is what we have clarified before in Section 4.4.

4.6.3/ IMPACT OF TOPOLOGIES ON DIAGNOSTICS

In order to illustrate the impact of topology strategies on the quality of data over time, and on the diagnostic of the state of the monitored area, we simulated four different topologies: decentralized topology, distributed topology, hierarchical topology, and centralized topology.

4.6.3.1/ DECENTRALIZED TOPOLOGY

In order to study the impact of decentralized topology on the diagnostics, we took WSN composed of 300 sensors, sensing respectively the levels of temperature (100 sensors), pressure (100), and humidity (100 sensors). We consider that the nodes are grouped into 30 clusters, each cluster being managed by a leader called cluster head (CH) or aggregator. The sensors capture the data from the area and send it to the CH, the latter aggregates the data and sends it to another CH or to the sink. In this study, we consider that the data aggregation at the CH happens as follows:

S -1 i=0 D c i /S (4.1)
where D is the data sent from the sensor to the aggregator, c is the type of sensor (temperature, pressure, or humidity), and S is the number of data that will be aggregated each time (for example, every 3 data from a certain type which are sent to CH from sensors, undergo aggregation).

The scenario of this topology is shown in Figure 4.11, the deployment of sensors is random, and the distribution and partition of CH on sensors follows K-means clustering method. Each sensor sends data to its CH. The latter, after aggregating these data, sends the aggregation to the closest CH on the condition that this CH is closer to the sink. If no CH meets this requirement, it will send it directly to the sink as shown in sensors in this cluster find other closest clusters to be in. In addition, CHs communicating with this inactive CH change their routes to the closest active CH to the sink. What is worth mentioning is that the black circles are the active sensors, the white circles are the inactive sensors, the black hexagons are the active CH, the white hexagons are the inactive CH, and finally the crossed circle is the sink. Fig. 4.12: Error rate in diagnostics if the topology is decentralized with the variation of time.

As mentioned before, the topology may be dynamic, the sensors or CHs on the long term will die (because of energy consumption) or break down (due to various causes as the operating age). Figure 4.12 shows the variation of error rate for the six considered algorithms, in the case where the topology is decentralized, with the variation of time t (operating age). Each point in this figure is an average of error rates of a given algorithm on 20 simulations (for a certain t). As shown in the figure, during t = 0..60

(if 0 ≤ t ≤ 60),
each algorithm has a specific error interval (in %) as follows: [START_REF] Carle | Preserving area coverage in wireless sensor networks by using surface coverage relay dominating sets[END_REF][START_REF] Cho | Energy efficient protocols for low duty cycle wireless microsensor networks[END_REF] for SVM, [START_REF] Bahi | Resiliency in distributed sensor networks for prognostics and health management of the monitoring targets[END_REF][START_REF] Breiman | Random forests[END_REF] for NB, [START_REF] Bai | Deploying wireless sensors to achieve both coverage and connectivity[END_REF][START_REF] Breiman | Random forests[END_REF] for RF, [START_REF] Ian F Akyildiz | A survey on wireless multimedia sensor networks[END_REF][START_REF] Atapour | On total dominating sets in graphs[END_REF] for GTB, [START_REF] Atapour | On total dominating sets in graphs[END_REF][START_REF] Bhardwaj | Upper bounds on the lifetime of sensor networks[END_REF] for TBFS, and [START_REF] Breiman | Random forests[END_REF][START_REF] David W Carman | Constraints and approaches for distributed sensor network security (final)[END_REF] for NN. After that (if t > 60) the error rate for each algorithm increased significantly at these intervals to reach at t = 70, 44 % for SVM, 24 % for NB, 22 % for RF, 36 % for GTB, 23 % for TBFS, and 40 % for NN. This shows that at this time the sensors and CH in WSN are dying or breaking down, and this fact leads to the presence of uncovered places in the area (coverage hole) and therefore incomplete data for diagnostics. Then, when the WSN exceeds t = 60, the error rate of algorithms increases as time increases, to reach 91 % at t = 100 if the algorithm is SVM, 89 % for NB, 88 % for RF, 90 % for GTB, 89 % for TBFS, and 90 % for NN (approximately the whole network is inactive). Note that the error rate in this simulation (decentralized topology) is related to the method of data aggregation. Then, the error rates start to increase over time, as numerous sensors in the network emptied their batteries, to reach the following rates: 70 % at t = 100 if the algorithm is SVM, 64 % for NB, 58 % for RF, 66 % for GTB, 60 % for TBFS, and finally 68 % for NN. We can remark that, at this time and with 30 CHs, the whole network became inactive in our previous simulation (see Fig. 4.12). We can deduce from these simulations that the difference between obtained results is due to two causes. The first reason, which is the most important one, is related to the increase of cluster heads (and thus of clusters) in the network. Indeed, the works that a CH must realize (receiving data from sensors, aggregation, and transmission) is now divided on more CHs, leading to a decrease of the energy consumption for each cluster head. The preservation of energy in each CH consequently leads to the increase of the network's lifetime, and to better diagnostic results in terms of error rates (see Figure 4.12). The other cause is that the increase in number of CHs in the network decreases the distance between the sensors and their associated CH. Hence the consumption of energy of sensors decreased accordingly, which leads to a lifetime increase. To determine which of these two causes has a greater impact on topologies and diagnostics with time, the number of CH in the network was taken as 30 (as the study shown in Figure 4.12), but now we increased the density of sensors in order to decrease the distance between the sensors and the CH and between the sensors themselves. In this study, we used 400 sensor nodes (100 more sensors) sensing respectively the levels of temperature (140 sensors), pressure (130 sensors), and humidity (130 sensors) and the result is shown in Figure 4.14. The error rate for the six algorithms in this figure varied significantly from what was shown in Figure 4.12 (with 300 sensors), but is opposite to what is shown in Figure 4.13. The error rate of diagnostics with 400 sensors remained constant (along the intervals determined before) during t = 0..50, while with 300 sensors, the error rate remained constant during t = 0..60. After that, the error rate of the algorithms started to increase with time and this shows that the sensors in WSN started dying, to reach 91 % at t = 90 if the algorithm is SVM, 90 % if NB, 89 % if RF, 91 % if GTB, 89 % if TBFS, and 90 % if NN (the whole network is inactive). Recall that with 300 sensors, the whole network become inactive at t = 100 as shown in Figure 4.12.

Even though the distance between the sensors and CH decreased after increasing the density of the sensors, the lifetime of WSN decreased. This is because of a very important reason which should be taken into consideration, that is, the number of sensors in each cluster in WSN (with decentralized topology) that are locally managed by CH. Normally, as the number of sensors in the clusters increases, the work of the CHs also increases, and therefore consumption of energy will also increase. Figure 4.14 explained this conclusion when we increased the number of sensors in the network, and we noticed the variation of the lifetime from previous studies. Finally, we can conclude the importance of dividing WSN to the largest possible number of clusters in order to divide the work on a larger number of CH, and therefore reduce energy consumption. And to get an accurate data from the area monitoring, the area should be fully covered for the longest achievable period, and here the density of sensors in WSN plays an important role. Consequently, we should offer a reasonable trade off between the number of sensors in WSN and between the largest possible number of clusters in this network, to get the longest lifetime and therefore an accurate data for diagnostics for a longer period.

4.6.3.2/ DISTRIBUTED TOPOLOGY

In order to study this topology and its impact on diagnostics, we consider a WSN composed of 300 sensors, sensing respectively the levels of temperature (100 sensors), pressure (100 sensors), and humidity (100 sensors).

The scenario of distributed topology is shown in Figure 4. [START_REF] Jacques M Bahi | Lowcost monitoring and intruders detection using wireless video sensor networks[END_REF], where all sensor nodes in the network have the same role and importance; i.e., there is no aggregation role, no clusters, and no CHs. Data packets are forwarded in a hop-by-hop manner. Each sensor is able to discover its neighbors within a radio radius of 2R c (R c is the coverage radius).

We assume that every node can access information about its neighbors, including their locations. The nodes choose neighbors to communicate with, and the latter should be closest to the sink within the sender's radio range. If the sensor is closest to the sink, the sensor will then send it directly to the sink. Figure 4.16 presents the variation of error rate for the six considered algorithms in the case of distributed topology, with the variation of time t (operating age). Each point in this figure is an average of error rates of a given algorithm on 20 simulations (for a certain t).

As shown in the figure, if 0 ≤ t ≤ 40, each algorithm has a specific error interval (in %) as follows: [START_REF] Bahi | Resiliency in distributed sensor networks for prognostics and health management of the monitoring targets[END_REF][START_REF] Breiman | Random forests[END_REF] for SVM, [START_REF] Özg Ür B Akan | Event-to-sink reliable transport in wireless sensor networks[END_REF][START_REF] Amit | Shape quantization and recognition with randomized trees[END_REF] for NB, [START_REF] Özg Ür B Akan | Event-to-sink reliable transport in wireless sensor networks[END_REF][START_REF] Atapour | On total dominating sets in graphs[END_REF] for RF, [2,[START_REF] Hammad | Security for wsn based on elliptic curve cryptography[END_REF] for GTB, [4,[START_REF] Ian F Akyildiz | A survey on wireless multimedia sensor networks[END_REF] for TBFS, and Fig. 4.16: Error rate in diagnostics if the topology is distributed with the variation of the time.

[ [START_REF] Atapour | On total dominating sets in graphs[END_REF][START_REF] Bhardwaj | Upper bounds on the lifetime of sensor networks[END_REF] for NN. After that (if t > 40) the error rate for each algorithm increased significantly at these intervals to reach, at t = 50, 40 % for SVM, 20 % for NB, 16 % for RF, 30 % for GTB, 18 % for TBFS, and 36 % for NN. This shows that at this time the sensors in WSN are dying or breaking down, and this fact leads to the presence of uncovered places in the area (coverage hole) and therefore incomplete data for diagnostics. Then, when the WSN exceeds t = 40, the error rate of algorithms increases with time to reach 90 % at t = 90 if the algorithm is SVM, 89 % if NB, 88 % if RF, 90 % if GTB, 89 % if TBFS, and 91 % if NN (approximately the whole network is inactive).

At each data transfer, the energy of a sender is reduced regarding its distance from the recipient. So packet transfer distance is one of the most important issues in topologies, and play an important role in the variation of lifetime of network and therefore on diagnostics with time. In order to study the importance and impact of transfer distance strategy in network, we simulated a WSN with distributed topology but now in a scenario different from what was used in previous study (shown in Figure 4.16). In this simulation, each sensor in WSN chooses the closest neighbor to communicate with. At all times, if the sensor is closest to the sink, it will directly communicate with the sink. After certain time t = x, the sensors may become inactive and the routes always change in function of the closest neighbors, the result of this study is shown in Figure 4.17. The error rate for the six algorithms in this figure varied significantly from what is shown in Figure 4.16. When we reduced the transfer distance in the topology, the error rate of diagnostic remained constant (along the same intervals mentioned in previous study) during t = 0..60, while in a previous study, the error rate remained constant during t = 0..40. After that the error rate of algorithms started to increase over time, to reach 62 % at t = 100 if the algorithm is SVM, 58 % for NB, 50 % for RF, 59 % for GTB, 54 % for TBFS, and 60 % for NN. What is worth mentioning is that in previous study, the whole network became inactive at t = 90, as shown in Figure 4.16.

Thus the lifetime of WSN increased significantly when we decreased the transfer distance between sensors, and the network remains active for a longer time, and therefore the sink continues to receive information from the monitored area for a longer time. By relying on this study and on the comparison between these results (Figure 4.16 and 4.17), we can conclude to the importance of transfer distance strategy in WSN on lifetime, and thus on diagnostics with time.

4.6.3.3/ HIERARCHICAL TOPOLOGY

As what has been mentioned in Section 4.3, sensor nodes can be organized in several levels, making a hierarchical topology. The sensor nodes are organized in a tree hierarchy from the sink (being the root of a tree), until sensor nodes having no descendants (leaf nodes). In order to study the impact of hierarchical topology on diagnosis, and to compare this topology to other ones, we took WSN composed of 300 sensors (leaf nodes), sensing respectively the levels of temperature (100 sensors), pressure (100 sensors), and humidity (100 sensors). These sensors are considered as the access layer in this topology (third layer). We considered 30 nodes playing the role of the second layer in topology (the distribution layer), which implements policies and forward messages. These nodes are in charge to build the links between the leaf nodes towards the sink (core layer). Each of these 30 sensors has the same battery supply of leaf nodes; conversely, in a decentralized topology, CHs have received an extra supply, therefore the batteries last longer and we finally obtain more data for diagnostics. , each algorithm has a specific error interval (in %) as follows: [START_REF] Jacques M Bahi | Lowcost monitoring and intruders detection using wireless video sensor networks[END_REF][START_REF] Breiman | Bagging predictors[END_REF] for SVM, [START_REF] Ian F Akyildiz | A survey on wireless multimedia sensor networks[END_REF][START_REF] Amit | Shape quantization and recognition with randomized trees[END_REF] for NB, [START_REF] Ian F Akyildiz | A survey on wireless multimedia sensor networks[END_REF][START_REF] Atapour | On total dominating sets in graphs[END_REF] for RF, [4,[START_REF] Hammad | Security for wsn based on elliptic curve cryptography[END_REF] for GTB, [4,[START_REF] Ian F Akyildiz | A survey on wireless multimedia sensor networks[END_REF] for TBFS, and [START_REF] Atapour | On total dominating sets in graphs[END_REF][START_REF] Jacques M Bahi | Lowcost monitoring and intruders detection using wireless video sensor networks[END_REF] for NN. These intervals are approximately the same where the topology is distributed (where the whole network is active), and this is because in these two topologies, there is no data aggregation. Then, for t > 20, the error rate for each algorithm increased significantly at these intervals to reach, at t = 30, 40 % for SVM, 20 % for NB, 16 % for RF, 30 % for GTB, 18 % for TBFS, and 36 % for NN. Such results show that at this time the sensors in WSN (in access or distribution layer) are dying or breaking down, and this fact leads to the presence of uncovered places in the area (coverage hole) and therefore incomplete data for diagnostics. Then, when the WSN exceeds t = 20, the error rate of algorithms increases with time, to reach What is worth to mention is that the batteries of the nodes in the distribution layer are like the ones in access layer, therefore the lifetime of WSN will increase more if the batteries in distribution layer were bigger. This difference is due to two basic reasons: first, the increase in number of nodes in distribution layer will divide the work, from receiving data from access layer and transmission of this data to the sink on more nodes, therefore the energy consumption of each node in this layer will decrease which will increase the lifetime. The other reason is the increase in number of nodes in distribution layer reduced the distance between these nodes and the nodes in access layer. To determine which of these two causes has the greatest impact on topologies and diagnostics over time, the number of sensors in distribution layer in the network was fixed to 30 (as the study shown in Figure 4. [START_REF] Breiman | Using adaptive bagging to debias regressions[END_REF]. We increased the density of sensors in access layer (leaf nodes) in order to decrease the distance between the sensors in this layer and the sensors in distribution layer, and between the sensors themselves. In this study, we used 400 leaf nodes (100 more sensors) sensing respectively the levels of temperature (140 sensors), pressure (130 sensors) and humidity (130 sensors) and the result is shown in The lifetime of WSN decreased even though the distance between the nodes in the second layer (distribution layer) and the nodes in the third layer (access layer) decreased after increasing the density. This is for the same reason that we studied in decentralized topology, which is the number of leaf nodes in access layer, by which each group of them is locally managed by parent node (in distribution layer). Normally, as the number of leaf nodes increases, the work of parent nodes also increases, and therefore consumption of energy will also increase. Figure 4.21 explained this conclusion when we increased the number of nodes in the network, and we noticed the variation of the lifetime from previous studies. Finally, we can conclude the importance of dividing WSN to the largest possible number of clusters, in order to divide the work of the network on largest possible number of parent nodes, so the energy consumption will be divided and reduced. Conversely, in order to get accurate data, the monitored area should be fully covered and for the longest possible time and here lies the importance of sensor density in WSN. So with hierarchical topology, we should offer a reasonable trade off between the number of sensors in access layer and between the largest possible number of parent node in this network, to get the longest lifetime and hence accurate data for diagnostics for a longer period.

4.6.3.4/ CENTRALIZED TOPOLOGY

In order to study the impact of this topology on diagnosis, we consider a WSN composed of 300 sensors, sensing respectively the levels of temperature (100 sensors), pressure (100 sensors), and humidity (100 sensors). In centralized topology, all the sensor nodes have the simple task of sensing new information and forwarding it to a central node, where all the data processing is done as shown in Figure 4.22a. In this topology, we can notice that, after t = x, the nodes that exhaust their energy first are the farthest from the sink. This is due to the long distance of packet transfer as shown in We remark, in Figure 4.16 (with distributed topology), the noticeable variation of error rate of the algorithms with time from what is shown in Figure 4.12 (with decentralized topology). We can note that in Figure 4.16 the sensors, after t = 40, started dying or breaking down, and the whole network became inactive at t = 90, while in Figure 4.12, the sensors or CH started dying or breaking down after t = 60, and the whole network became inactive at t = 100. Moreover we can notice that in Figure 4.12, during t = 0..60 (the whole network is active) the error rate is evolving in a way larger than the one in Figure 4.16 during t = 0.. 40 (idem). From this study and based on this comparison we can conclude that the lifetime of the networks with decentralized topology is greater than if it were a distributed one, which is due to the fact that the data aggregation reduces the number of packet transfer, and therefore it further reduces the overall energy consumption in the network. But the error rate of diagnosis is greatly related to the data aggregation method (if the topology is decentralized) because data aggregation always reduces the data accuracy that will be used for diagnosis, and this is shown and clarified in these two figures where the whole network is active.

We had a different scene in Figure 4.19 (with hierarchical topology) because the variation of error rate of the algorithms varied with time in a significant way from what is shown in Figure 4.12 and 4.16 (with decentralized and distributed topology). We notice that, in Figure 4.19, the sensors after t = 20 started dying or breaking down, while in previous studies, the sensors became inactive after this time. Based on this study, we can conclude that the lifetime of the network with hierarchical topology is smaller than if it were with a decentralized or distributed topology (the network lifetime defined as time until the first node dies). Furthermore, we note that the whole network with hierarchical topology became inactive at t = 100 when considering the decentralized topology, while with distributed one the whole network became inactive at t = 90. Based on these results, we can conclude to the importance, when deploying a wireless sensor network for diagnostics purposes, of dividing the WSN in area monitoring into regions which are locally managed by a central node (or parent node).

If we suppose that the network lifetime can alternatively be defined as the time until the first node dies, then by relying on the curve modifications, we can conclude that the lifetime of the network with centralized topology is smaller than if it were a decentralized, distributed, or hierarchical one. Moreover, the whole network with hierarchical and decentralized topology became inactive at t = 100, while with distributed and centralized topology the whole network became inactive at t = 90. Based on these four results, we confirm what we mentioned before about the importance of dividing the WSN in area monitored into regions, which are locally managed by a parent node (the network remains active for a longer time, therefore the sink continues to receive information from the monitoring area for a longer time). Based on this work, we were able to notice the importance and impact of each type of topology in WSN on diagnostics with the increase of operating age of WSN, and focus on several issues related to these types of topologies.

4.6.4/ ALGORITHM COMPLEXITIES

The last aspect to investigate in the choice of machine learning algorithms in a PHM based on data provided by a WSN is the algorithm complexities. This complexity varies according to the methods used for data classification, to learn the machine, and to perform diagnosis or to take decision from new data captured by the network. Indeed, efficiency of the machine learning algorithms depends both on its accuracy in diagnostic and on its complexity. By relying on our simulation, we studied the complexities for all the algorithms mentioned before, and the results are shown in

4.7/ CONCLUSION

The WSNs provide PHM with a new way of distributed data collection and wireless transmission for diagnosing the state of an area and to be informed if it is in failure or not. WSNs strategies are important factors for achieving QoS in WSNs application. In this chapter, we explained the relation between WSN strategies and their impact on area diagnostic, and therefore on PHM. We mentioned and studied several important strategies in WSN that have an important influence on QoS of WSN, and we proved from this study that they also have an important impact on diagnostics.

In this chapter, we studied the variation of the accuracy of diagnosing for six algorithms, and by relying on this variation we determined and focused on several issues related to these strategies. By relying on our simulation, we notice that if the captured data is incomplete, the diagnostic by the random forests algorithm will be more accurate than the others. And if the data becomes more accurate, the diagnostic by the Gradient Tree Boosting will be the best. Then by relying on this study, we can conclude that Random forests algorithm is more efficient than the rest of the algorithms, due to the low complexities of the algorithms. And more importantly because the RF agrees with the type of work of WSN and its characteristics, since the presence of shortcomings of a WSN always cause incomplete data for the diagnostics (PHM).

From this chapter, we were able to conclude that the data captured by the WSN on long term is incomplete, because simply the sensors become inactive for several reasons, most importantly energy consumption. Therefore the battery consumption of sensors in WSN is a very important strategy that has an impact on the work of networks with time, and taking it into consideration through our study is very important. Scheduling mechanism is one of the best solutions to preserve the energy of sensors for a longer time. For that, we studied this strategy and we proved that as the percentage of active sensors in WSN is decreasing (because of scheduling of sensors), the accuracy of area diagnosis will decrease due to the reduction of the coverage rate of WSN, so the accuracy of the captured data that will be used to diagnose will decrease. In a second step we studied the impact of the density of sensors and we shown that if the density of sensors increases, the accuracy of the area state will increase, and this shows that the coverage rate will increase. Then, the data accuracy will increase with the increase of number of sensors in monitored area. Additionally, the location of nodes (uniformity in the density) is of importance in both the learning stage, and in the testing one (area monitoring after learning). These two elements have a real impact on diagnostic and must be taken into consideration.

Moreover, we studied the topology effects in WSN through four different topologies, each one of them belonging to a certain type as follows: distributed, hierarchical, centralized, and decentralized topology. From this work and by relying on these topologies, we were able to prove that topologies have a great impact on the accuracy of the data and therefore on PHM, and this impact varies according to the type and parameters of topologies in WSN. In the next chapter, we will propose and study effective and reliable algorithm that relies on scheduling mechanism to increase the network lifetime, and also to increase the coverage rate to the maximum with the use of certain amount of active sensors in network (density). By relying on this algorithm, the data diagnostics will be accurate for the longest time possible.

DISTRIBUTED MINIMAL DOMINATING

SENSOR-TARGETS FOR DIAGNOSTICS

A s already discussed in this manuscript, one of the most important issues of the QoS in WSN is the area coverage. This is interpreted as how well the network can cover and observe a given area. In order to save energy, many studies have been proposed to schedule redundant sensor nodes such that only a subset of sensor nodes that meets the coverage and connectivity requirement needs to be active at a time. However, none of the articles in the literature address the impact of coverage on a diagnostic based on data provided by a WSN. In this chapter, we propose a fully distributed algorithm based on a theory of domination in graphs and we study its impact on diagnostics by using the six machine learning algorithms previously introduced. The complexity of the proposed algorithm is analyzed and its correctness is proven. Experimental results demonstrate that our algorithm, despite its lower complexity, outperforms both its direct competitor, the Probabilistic Coverage Protocol PCP [START_REF] Hefeeda | Energy-efficient protocol for deterministic and probabilistic coverage in sensor networks[END_REF] and the optimistic solution (which we call BaseLine) in several aspects, including number of activated sensors, convergence time, total energy consumed, network lifetime, coverage rate, and its ability to achieve a good compromise between these factors. We show that the use of this algorithm on a WSN will provide more accurate data for a longer duration, leading to a more reliable diagnostics process.

5.1/ INTRODUCTION

A dependable WSN should be used in the monitoring process. As established in a previous chapter, coverage is one of the most important measurements of WSNs quality of service, and it is closely related to the energy consumption of sensors [START_REF] Zhu | A survey on coverage and connectivity issues in wireless sensor networks[END_REF]. Thus, when designing WSNs, one of the most important challenges remains in the efficient use of these sensors, to increase the network's lifetime while guaranteeing high coverage rate.

Actually, there is a proportionality between coverage rate and energy consumption. Indeed, the coverage rate increases when the number of active sensors increases, but energy consumption increases accordingly. All the research works in the WSN literature focus on either the network lifetime or the coverage rate: they either target to maximize the lifetime or the coverage, while all the attempts that tried to find a compromise between these factors are indeed not efficient. For instance, in the context of extending the network's lifetime, a possible solution is to maintain only a number of sensor nodes as low as possible in an active mode [START_REF] He | Leveraging prediction to improve the coverage of wireless sensor networks[END_REF][START_REF] He | Energyefficient capture of stochastic events under periodic network coverage and coordinated sleep[END_REF][START_REF] Gaurav S Kasbekar | Lifetime and coverage guarantees through distributed coordinate-free sensor activation[END_REF]. Although this solution obviously solves any energetic issues, it failed to achieve a large coverage rate compatible with this efficient energy consumption.

In this chapter, we propose a reliable algorithm for diagnostics and health management of the monitoring targets. The proposed algorithm is completely distributed: no centralized or intermediate control is needed. This is a hybridization of three variants of domination in the sensors-targets graph, namely the simple, double, and total ones, in a single algorithm, which makes the originality of our study. To illustrate the efficiency of this approach and its impact on the accuracy of diagnostics, six machine learning algorithms (mentioned before) have been used to diagnose the area state. These learning processes are evaluated in a WSN context with the proposed algorithm.

The first work conducted in this chapter in terms of experimental study has been to compare the algorithm in the three aforementioned cases (simple, double, and total domination) with the optimistic BaseLine solution, in order to evaluate first the efficiency of the distributed algorithm in a WSN and at both lifetime and coverage rate level of the network. The BaseLine solution considered in the experimental study is the fault free version of the algorithm, which is defined as the protocol assuming that the system is completely safe (this is the case when all sensors in WSN are active at the same time). In other words, the BaseLine is the case where both the coverage and energy consumption are maximal. In a second part of this chapter, we achieved a comparison with an accurate and a well known algorithm called Probabilistic Coverage Protocol (further denoted as PCP) that has been published in [START_REF] Hefeeda | Energy-efficient protocol for deterministic and probabilistic coverage in sensor networks[END_REF], to prove the importance and efficiency of the proposed algorithm in WSN. Moreover, the impact on the coverage rate and on energy consumption has been studied, together with the effects on diagnostics. Furthermore, a very important factor in WSN is node density, which has a direct impact on both lifetime and coverage, and therefore on the data accuracy used for the diagnostics. Consequently, this factor may have an impact on the effectiveness of our proposal. To show and evaluate this, we evaluated our algorithm in a simulated WSN having 300 and 600 sensors. These simulations show that the proposed algorithm can increase the coverage rate while decreasing the energy consumption of the network significantly, if we compare it with BaseLine and the PCP: our algorithm achieves a good compromise between these two important factors, leading to more accurate data provided by the WSN and during a longer time, and thus to more reliable diagnostics.

5.2/ ENERGY EFFICIENCY OF COVERAGE PROTOCOLS IN WSN

Reducing energy consumption to extend the network lifetime while maximizing the coverage rate is one of the most important challenges in designing WSNs. These two factors are obviously proportional to each other, so as coverage rate increases, energy consumption increases accordingly. All the proposals suggested by authors in the literature attempt to find a compromise solution between these two factors, to ensure maximal coverage under a minimal consumption constraint. Such proposals are recalled hereafter.

Authors in [START_REF] Zou | A distributed coverage-and connectivitycentric technique for selecting active nodes in wireless sensor networks[END_REF] state that the set of active nodes selected by a connected dominating set (CDS) provides full coverage and connectivity. Keeping only a minimal number of sensors active and putting others into low-powered sleep mode, while the active sensors can maintain the communication connectivity and cover the target region completely, is one of the promising approaches to preserve system energy.

In [START_REF] Tian | A coverage-preserving node scheduling scheme for large wireless sensor networks[END_REF], authors proposed an energy-efficient node scheduling based coverage mechanism. The "off-duty eligibility rule" determines whether a node's sensing area is included in its neighbors' sensing area. The nodes investigate the off-duty eligibility rule in the self-scheduling phase. Eligible nodes turn off their communication and sensing units, whereas all other nodes will perform sensing tasks in the sensing phase.

A distributed node scheduling mechanism is presented in [START_REF] Tian | A node scheduling scheme for energy conservation in large wireless sensor networks[END_REF], which can run on each sensor node. The possibility of one node to become a redundant one can be analyzed according to the relationship of the positions of different nodes. If a node is a redundant one, it moves to sleep. The algorithm assesses the redundant nodes by considering only the neighbors in its sensing range, while there are still other redundant nodes in active mode. Thus, the performance of this mechanism needs to be enhanced.

The reference [START_REF] Tian | Location and calculation-free node-scheduling schemes in large wireless sensor networks[END_REF] has considered the mechanism proposed in [START_REF] Tian | A node scheduling scheme for energy conservation in large wireless sensor networks[END_REF]. In this article, authors want to know precisely the position information of nodes. Then, supposing that the area does not require a complete coverage, they proposed three node scheduling algorithms in [START_REF] Tian | Location and calculation-free node-scheduling schemes in large wireless sensor networks[END_REF]: a first one based on the number of neighbors, a second scheduling algorithm based on nearest neighbor, and a final one based on probability.

A centralized k-decision algorithm is provided in [START_REF] Huang | The coverage problem in a wireless sensor network[END_REF][START_REF] Huang | The coverage problem in threedimensional wireless sensor networks[END_REF]. It only needs to detect whether arbitrary nodes in the perimeter of the sensing disk can be covered by other k nodes. Then, the mechanism can determine if the entire region is k-covered. Unfortunately, the algorithm is centralized with high complexity, in particular for large-scale WSN applications.

The mechanism presented in [START_REF] Wang | Integrated coverage and connectivity configuration in wireless sensor networks[END_REF] is a k-coverage protocol, which ensures that if any point within the intersection area of sensing disks can be k-covered, then the entire region also can be k-covered. However, the mechanism does not consider the contribution value of the network, when a node changes from the sleep state to the active one. This leads to have a network coverage of low efficiency and nodes of larger redundancy degree.

In [START_REF] Carle | Energy-efficient area monitoring for sensor networks[END_REF][START_REF] Carle | Preserving area coverage in wireless sensor networks by using surface coverage relay dominating sets[END_REF], the authors propose a distributed connected coverage algorithm. The method aims to minimize the number of active nodes in order to save energy and extend the network lifetime. The basic idea of this algorithm is to construct a dominating set of network. Through the periodic reconstruction of the dominating set, the mechanism can effectively extend the network lifetime.

Authors of [START_REF] Slijepcevic | Power efficient organization of wireless sensor networks[END_REF] construct continuous disjoint sets. At each time, sensor nodes that can cover as many targets as possible are selected to form the sets. Although the mechanism in [START_REF] Slijepcevic | Power efficient organization of wireless sensor networks[END_REF] can ensure that all targets are covered in the network, the mechanism has a very stringent constraint: at each time, the constructed sets must be disjoint.

The authors in [START_REF] Kar | Node placement for connected coverage in sensor networks[END_REF] discuss how to maximize the network lifetime when the deployment environment of the network is safe and controllable, and they guarantee that all the network targets have been covered and that the network is connected. The work presented in [START_REF] Tc He | Coverage analyses of plane target in sensor networks based on clifford algebra[END_REF] assumes that the network is homogeneous, i.e., the sensing radius of the nodes in the network are all equal, and the communication radius of nodes is equal to the sensing radius. Then, the paper proposes a polynomial approximate algorithm, based on minimum spanning tree, which improves the performance of the network.

In [START_REF] Liu | Analysis for multicoverage problem in wireless sensor networks[END_REF], the k-target coverage problem in wireless sensor networks has been studied, and an algorithm that does not require location information of the nodes has been presented. Based on a mathematical model presented in [START_REF] Liu | Analysis for multicoverage problem in wireless sensor networks[END_REF] and on the ratio of node's sensing radius to monitor a given area, the algorithm is able to know the number of sensor nodes that are needed to cover the monitoring area. The algorithm could greatly reduce the cost of hardware in the network, and it can improve the QoS of the network. in [START_REF] Hefeeda | Energy-efficient protocol for deterministic and probabilistic coverage in sensor networks[END_REF], a general coverage algorithm, which also considers the network connectivity, has been presented. The proposal, called Probabilistic Coverage Protocol (PCP), works for the common disk sensing model as well as for probabilistic sensing model. To support the latter, authors have introduced the notion of probabilistic coverage of a target area with a given threshold θ, which means that an area is considered to be covered if the probability of sensing an event occurring at any point in the area is at least θ. They prove the correctness of the protocol and provide bounds on its convergence time and message complexity.

All of aforementioned research works support the WSN by increasing the lifetime of the network while maximizing coverage. However, various problems can be emphasized in these solutions, which are most of the time very application-oriented. Indeed, these solutions always have a priority: they are either oriented to maximize the coverage or to maximize the lifetime of the network. And all the attempts to optimize the trade-offs between these objectives are not efficient. This is why, in this chapter, we present an efficient distributed minimal dominating sensor-targets to improve the accuracy of the prognostic and health management of the monitoring targets. This study differs from previous works for the following reasons:

• We orchestrate the trade-offs relation between energy and coverage optimization and the accuracy of the diagnostic-prognostic of the monitoring targets.

• Unlike earlier methods, we use a new concept of domination in graphs [START_REF] Padma | A note on varieties of double domination[END_REF][START_REF] Atapour | On total dominating sets in graphs[END_REF], to achieve significant energy/coverage enhancement and therefore to increase the reliability of the diagnostics and prognostics.

• We provide provable guarantees for the election process of working nodes for the monitoring targets.

• Our algorithm can start from any initial configuration with arbitrary number of awake nodes, and reach a legitimate stable situation in a finite number of moves or steps.

In order to compare our algorithm to PCP [START_REF] Hefeeda | Energy-efficient protocol for deterministic and probabilistic coverage in sensor networks[END_REF] protocol, which is the closest work to the one presented in this chapter, we briefly outline hereafter the main features of this algorithm.

5.3/ PROBABILISTIC COVERAGE PROTOCOL

The idea of PCP is to activate a subset of deployed sensors to form an approximate triangular lattice over the area to be covered. PCP works in rounds of R seconds each. R is chosen to be much smaller than the average lifetime of sensors. In the beginning of each round, all nodes start running PCP independent of each other, then a number of messages will be exchanged between nodes to determine active/sleep nodes. PCP refers to the distance between the vertices of the triangular lattice as the maximum separation between working nodes, and it is denoted by s. The value of s is computed from the sensing range r s of sensor nodes. In the disk sensing model, the maximum separation is set to √ 3r s as it has been shown in [START_REF] Bai | Deploying wireless sensors to achieve both coverage and connectivity[END_REF].

Under the exponential sensing model, to ensure that the probability of sensing at the least-covered point is at least the coverage threshold parameter θ, the authors compute the maximum separation, s, as:

√ 3        r s - ln 1- 3 √ 1-θ α       
, where α is a factor that describes how fast the sensing capacity decays with distance. From this equation, it is clear that if we set α = ∞, then the exponential sensing model reduces to the disk sensing model.

Unlike our algorithm, PCP protocol assumes nodes to have positioning informations. This helps to construct a triangular lattice allowing high performance for the coverage issue. Conversely, the algorithm we develop is completely distributed and does not need to know the positions of any sensor node in the network.

5.4/ DOMINATING SETS

5.4.1/ SIMPLE DOMINATING SETS

In graph theory, a dominating set for a graph G = (V, E) is a subset S of V such that every vertex not in S is adjacent to at least one member of S . Note that every graph G = (V, E) has a dominating set namely V. For example in Figure 5.1 a graph having minimal dominating sets of cardinality three (the set {1,3,5}), four (the set {3,6,7,8}), and five (the set {2,4,6,7,8}).

5.4.2/ DOUBLE DOMINATING SETS

A dominating set is said to be double dominating set if every vertex in V -S is adjacent to at least two vertices in S . The minimum cardinality taken over all, the minimal double Note that for any connected graph G, γ dd (G) ≤ n, and if G is a graph with no isolated vertices, then the complement V -S of every minimal double dominating set S is a double dominating set. For example in Figure 5.2, let S ={1, 3, 5}; V -S = {2, 4, 6, 7}. Hence both S and V -S are dominating sets for the graph in Figure 5.2. In this case of the graph in Figure 5.3, any dominating set must contain the isolated vertices 8 and 9. Hence, if S is a dominating set then V -S is not a dominating set.

5.4.3/ TOTAL DOMINATING SETS

A set S of vertices in a graph G(V, E) is called a dominating set if every vertex v ∈ V is either an element of S or is adjacent to an element of S . A set S of vertices in a graph G(V, E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S . The total domination number of a graph G denoted by γ t (G) is the minimum cardinality of a total dominating set in G. Clearly γ(G) ≤ γ t (G), also it has been proved that γ t (G) ≤ 2γ(G) . For example, in the Figure 5.4, the graph has minimal total dominating sets of cardinality four S = {1, 2, 4, 5}; γ t (G) = 4. Proof. A direct consequence of all previous lemmas. The exact count of execution rounds is 2n + 4.

5.6/ SIMULATION RESULTS

The proposed algorithm is completely distributed, no centralized or intermediate control is necessary. It is a hybridization of three variants of domination (simple, double, and total) in a single algorithm with distributed computation, i.e., the minimal dominating set is calculated in a fully distributed way on the graph of sensors-targets, which makes the originality of our study.

By relying on our simulation explained in details in Section 4.5, we used six algorithms for diagnosis which were mentioned before to show and study the impact of this algorithm on the quality of service of the WSN, and the accuracy of the data produced during the network time service. And, therefore, its impact on the accuracy of diagnosing the state of the monitored area.

In the following sections, we assess the impact of our algorithm on the quality of data captured by the WSN over time, and therefore its consequence on the accuracy of diagnostic. We first remark that density is a very important factor in such networks, which has an effective impact on both lifetime and coverage. Indeed, the average distance between sensors decreases when density increases. This affects the packet transfer distance on the one hand, and the number of neighbors for each sensor on the other hand. Thus, to take this fact under consideration, we performed simulations using two densities of WSNs: 300 and 600 sensors respectively. Figure 5.6 shows the variation of error rate over time for the six considered algorithms, in the case where the proposal is applied in simple domination mode. Each point in the figures represents the average of error rates on 20 simulations. As shown in Figure 5.6a (with 300 sensors), during the time-steps t = 1..120, each algorithm exhibits a specific but quite constant error rate. However, after the time-step t = 120, the error rate increases significantly for each algorithm. Indeed, the number of sensor nodes that "die" due to an empty battery (or fail-stop failure) becomes non negligible, leading to uncovered targets in the area (i.e., coverage holes) and therefore to incomplete data for diagnostics. More precisely, after t = 120 errors increase as time goes up to finally reach a rate of 91 % at t = 200 in the support vector machine, 90 % for NB, 89 % for RF, 90 % for GTB, 89 % for TBFS, and 91 % for NN. Finally, at time-step 200, almost all the network is inactive. This phenomenon is less sensitive in the case of 600 sensors, as depicted in Figure 5.6b, even if a similar observation can be outlined about the importance of density in wireless sensor networks based PHM. Figure 5.7 also presents the variation of error rate for the six considered algorithms, but now in the case where the proposal is in double domination mode. When considering 300 sensors, we can see in Figure 5.7a that each algorithm has a specific error interval during the time-steps t = 1..100, which is a bit better than in the previous simple domination case. However, the increasing of this rate due to empty batteries starts earlier than in Figure 5.6, and the networks shut down 10% before than in the simple domination case. A similar observation stands for, mutatis mutandis, the network with 600 sensor nodes, as shown in Figure 5.7b. Considering the total domination mode, Figure 5.8 reveal that the changes are in the same direction than for the double domination rate, but they are more pronounced. For instance, in Figure 5.8a, the error rate is approximately equal to 10% during the half part of the simulation time-step, while it was in average equal to 30% and 20%, for simple and double domination respectively. However, this good performance on error rates impacts the network lifetime, and nodes rapidly start to die after the time-step t = 80 (120 and 100 iterations for simple and double domination resp.). The time where nodes start to die can however be shifted when the density of the network is increased.

In all simulations, we can observe that neural networks and nearest neighbors have the worst behavior over time. Random forests have the lowest error rate in general, except in the total domination case, where its error rate is not improved when compared with the double domination mode. The best results have been obtained in general using the gradient tree boosting, no matter the number of nodes. However, this good performance has a cost, and the network is a bit more impacted than RF, TBFS, or NB, by the death of nodes.

5.6.1/ COMPARISON WITH THE OPTIMISTIC BaseLine SOLUTION

The BaseLine solution considered in the experimental study is the fault free version of the algorithm, which is defined as the protocol assuming that the system is completely safe (which is the case when all sensors in WSN are active at the same time). In other words, the BaseLine is the case when the coverage and energy consumption are maximal.

Obtained results are shown in Figure 5.9. As expected, no matter the considered machine learning algorithm: diagnostic error rates start to increase significantly after time-steps t = 60 or t = 70, depending on the number of sensors, if we compare it with our proposal. Furthermore, the increase is more fast, the whole network becoming inactive at timesteps t = 110 and t = 120 respectively. This is a consequence of the large number of active sensors in network. Focusing on the error rate value, and during t = 0..60 and t = 0..70 (where the whole network is active with 300 and 600 sensor respectively), the BaseLine performs slightly as well as our proposal, namely in the total domination mode. But we achieve better performances by increasing the lifetime service of the network up to 33%. This allows sensor networks to gracefully degrade in performance instead of failing unpredictably.

We now compare our proposed algorithm with the optimistic BaseLine case, in order to evaluate its efficiency in WSN at both the coverage level and the network lifetime, and therefore on diagnostics. We use the Gradient Tree Boosting for the sake of readability and its performance compared to other machine learning algorithms as shown above.

The best diagnostic accuracies are obtained with the BaseLine and the total domination as pointed in Figure 5.10, the first one being slightly better than our proposal using the GTB algorithm. However, the slight improvement of this rate is very costly in terms of network lifetime, if we compare again with the total domination: our proposal shuts down almost 50% later than the BaseLine. Generally speaking, the WSN shuts down faster when using the BaseLine, for instance before the beginning of the performance decrease for the single domination mode. This latter succeeds the most to prolong the lifetime of the WSN. The double domination, for its part, achieves a good compromise performance between network lifetime and diagnostic accuracy. This is due to the reasonable minimal number of necessary active sensors in the network at the same time.

Similar results have been obtained for both energy consumption and coverage, as respectively depicted in Figures 5.11 of diagnostic accuracy in the BaseLine case is paid a high price in terms of energy consumption: this latter being larger in the BaseLine, each sensor empties its battery more rapidly, which therefore reduce the network lifetime. This reduction has a direct impact on the coverage, as can be verified in Figure 5.12. Another remark to point out is the decreasing speed of the energy, which is very large in the BaseLine case, no matter the number of sensors. In comparison, this decrease is more regular and less pronounced for our proposal, as can be seen in Figure 5.11. Indeed, Figure 5.11 explains the results obtained in Figure 5.12. The network having the highest energy consumption has a larger coverage rate, and this coverage rate varies over time according to the network state. Comparisons between our proposed algorithm with its three modes on the one hand, and the BaseLine on the other hand, illustrate the good performance of our proposal, as we are able to reduce the energy consumption while preserving at the same time a good coverage rate. This has a direct impact on diagnostics accuracy. To put it in a nutshell, as both the network lifetime and coverage are improved, a low diagnostic error rate has been obtained for a longer duration.

5.6.2/ COMPARISON WITH THE PCP ALGORITHM

Let us now compare the proposal with the Probabilistic Coverage Protocol [START_REF] Hefeeda | Energy-efficient protocol for deterministic and probabilistic coverage in sensor networks[END_REF] (presented in section 5.3) in terms of coverage rate, energy consumption, and then on diagnostics.

The same simulation protocol used in the previous section has been followed. Figure 5.13 presents the error rate variation when considering the PCP with the 6 machine learning algorithms studied in this chapter. Again, two stages can be emphasized: a normal running stage of the WSN that leads to an acceptable error rate for the diagnosis, and a second stage where nodes begin to empty their batteries. Like in our proposal and for the BaseLine case, we obtain a good performance for the random forests and the worst case for the support vector machine. The gradient tree boosting, for its part, achieves a median performance compared to the five other machine learning algorithms.

During the first half part of the experiments, the diagnostic using data provided by the network that embeds the PCP algorithm has presented an error rate in average equal to 22%, no matter the machine learning algorithm and the network density. This error rate is better than our proposal in the single domination mode, but it is a little less than the double domination one. This is explained by the fact that the increase of the error rates starts earlier with the PCP, and the network is totally quiet earlier too. Furthermore, both the BaseLine and the total domination outperform largely the PCP algorithm, if we focus only on the error rate during the normal running stage of the network. The duration of this normal running stage is finally almost the same between PCP and our total domination mode, while it is lower for the BaseLine (and larger for both the single and double domina- Like in the previous section, we decided to compare more deeply the behaviors of the PCP and of our proposals, in terms of error rates, energy consumption, and coverage rate. The gradient tree boosting machine learning has been selected again, for the sake of its performance. The evolution of error rates over time are provided in Figure 5.14, in which we can see that the PCP performance is not as good as our 3 proposals. Indeed, if we except the single domination, error rates in diagnostics are larger for the PCP. Furthermore, the nodes obviously emptied their batteries earlier in PCP than for our proposals, no matter the density of the network.

The energy consumption of sensors when the probabilistic coverage protocol is deployed in the WSN is really better than when the BaseLine is considered, see Figure 5.15. The PCP algorithm has in general a similar energy consumption than our algorithm, being between the total domination mode (the most consuming one) and the two other modes. This is the case too for the coverage criterion, in which the two algorithms have a similar behavior in the normal working stage, while our proposals tend to preserve more the coverage over time, when the effects of empty batteries can be emphasized. This is illustrated in Figure 5.16b.

We can conclude this study by the good performance of our proposal when compared with the state-of-the-art PCP algorithm, as the simple domination is able to greatly reduce the energy consumption of the network, while the total domination has a larger coverage rate if compared with the PCP. The double domination, for its part, achieves a better compromise between these two criteria. At each time, data captured from the network will be more reliable for a longer time, leading therefore to a more accurate diagnostics.

Let us finally perform a last numerical simulation, namely on convergence time. This latter is the time needed by the algorithm to determine active/sleep nodes, and it is desired to be as small as possible [START_REF] Hefeeda | Energy-efficient protocol for deterministic and probabilistic coverage in sensor networks[END_REF]. To shorten the convergence time, more sensors should be put in sleep or active mode faster, and this leads to save more energy in the network. Indeed, as the number of sensors in the wait state decreases in our algorithm, the convergence time will decrease. Therefore, the algorithm has to avoid this state as possible. To study this metric is then very important to show the efficiency of the proposed algorithm. This latter has been compared with PCP, and obtained results are shown in Figure 5.17. We can see that this convergence time is in general larger for the PCP than for our proposals. This is explained by the low time complexity of the proposed algorithm to compute the minimal dominating sensor-targets as proven in Section 5.5.1.

5.7/ CONCLUSION

In this chapter, we proposed and evaluated a fully distributed algorithm in graph theory by using the dominating sets. We studied and simulated the proposed algorithm in the three cases in WSN, to show in each case its impact on the work and effectiveness of the WSN with time, and therefore on the diagnostics data by using six diagnostic algorithms, and also its impact on the process PHM.

The first study conducted was a comparison of the proposed algorithm with the BaseLine, because in this case the coverage rate and energy consumption are maximal. From this study we can conclude that the lifetime of network in simple domination is greater than that if it were in total domination, while the accuracy of diagnostics and the coverage rate in the latter is more important if it were in simple domination. The algorithm in double domination achieved a compromise between these two factors. However, with BaseLine the lifetime of the network is smaller and the accuracy of diagnostic is greater. When comparing with the PCP, we have concluded that: (1) lifetime with the simple domination is greater when we used PCP in the network, while lifetime with PCP is greater when we used total domination in the network.

(2) The coverage rate with PCP is greater than if the algorithm is in simple domination, while for the distributed algorithms in total domination, the coverage rate in the network increased from when we used the PCP. (3) The proposed algorithm in double domination was able to achieve a higher coverage rate and greatly reduce the energy consumption in WSN, from when we used the PCP in the network.

Then with the double domination, we were able to achieve a compromise between the simple and total domination at the level of energy consumption and the coverage rate. And finally (4) the convergence time if we used PCP is changing with time in a way greater than the one if we used our algorithm in the three cases. Therefore, by relying on this study, we can deduce the importance, efficacy and priority of the distributed algorithm from PCP at the level of energy consumption and coverage rate, and its importance in achieving a compromise between these factors, moreover its significance at the level of convergence time.

6

CONCLUSION AND PERSPECTIVES 6.1/ CONCLUSION Prognostics and health management (PHM) has become a very important tool in modern industry (area monitoring). It is the basis of condition-based maintenance and predictive maintenance. PHM helps increase availability, reliability, and safety by predicting the remaining useful life of engineering assets. The goal of this activity is to avoid unnecessary shutdowns by planning maintenance before the system faces a failure. By studying the evolution of continuous measurements of parameters of the system under consideration, this process is done. Relevant parameters are determined and monitored. Their values, crossed with the corresponding state, will be used to build a degradation model for the system under consideration. Means of independent sensors usually perform this monitoring activity. Sensors are most likely connected through physical wires in case of complex systems (ones that have different types of parameters to monitor) or systems spreading on a large area (monitoring needs to be done in different spots). The use of wired networks for some applications causes several drawbacks like difficulty of access, high costs, single point of failure, etc. A solution for these problems is the use of Wireless sensor networks (WSN) because of their fault tolerance, low cost, and easy deployment.

As far as we know, for the monitoring activity, existent research works in the prognostics and health management field used either sensors that are connected with physical wires, or individual sensors. Since sometimes the use of wireless sensor networks is a requirement rather than a choice, we challenged this possibility. Therefore, we addressed some challenges and issues in this thesis which weren't questioned before primarily the completeness of data used in the PHM process. So, to diagnose the state of an area, in this thesis, the PHM was provided by the WSNs with a new way of distributed data collection and wireless transmission. Unfortunately, it is not that easy to solve the problems. WSN are vulnerable, prone to failures, and data packets are very likely to be lost in the transfer process. Some techniques need to be put in place before deployment for ensuring the reliability of the network and reducing the amount of the lost information.

All the research that existed in literature, considered that prognostics and health management approaches are based on the assumption of completeness of data. However, these assumptions don't reflect life situations in any way since they are not adapted with the characteristics of the WSN. The captured data may become incomplete with time because of the presence of several shortcomings in this network. Then, we challenged this belief in this thesis, and placed ourselves in the case where the monitoring data is incomplete. We proposed the usage of the random forest algorithm for the diagnostic step. We 121 forced the sensors to randomly fail by injecting random errors in the network. We proved the efficiency and importance of the algorithm by counting on our simulation (is the first examples of use a diagnostic method based on data provided by a wireless sensor network), and we also proved that the algorithm is capable of adapting to the changes in the quality of the monitoring conditions. Thus, to further make the quality of diagnostics better, we looked at data loss from another perspective. Reliability mechanisms are unable to guarantee to total elimination of data loss. For that, we benefited from the particularity of the random forest algorithm. Thanks to its different starting points, the algorithm is given more precision and a useful variation by the random distribution aspect. The algorithm was able to recover quickly from missing data packet at the processing step since the trees in the forest, regarding the initial position, are different. With an injected error in the network, we were able to successfully assess the system's diagnostic. This error aims for forcing the nodes to fail and therefore reducing the amount of information at the processing unit.

After we suggested an important diagnostic algorithm (random forests) which can adapt with the changes occurring in WSN, we headed with our study to understand the reason of these changes occurring in WSN with time, which causes incomplete diagnostics-PHM data. For that, by relying on the random forest and five other diagnostic algorithms from literature, we explained in this thesis the relation between WSN strategies and their impact on area diagnostic, and therefore on PHM. We mentioned and studied several important strategies in WSN that have an important influence on QoS of WSN, and we proved from this thesis that they also have an important impact on data diagnostics with time.

Therefore from this study and by relying on our simulation, we were able to conclude and show several important items related to WSN strategies and PHM, and several issues that should be taken into consideration:

• The importance and efficacy of random forests algorithm in WSN in comparison with other algorithms that exist in literature.

• The data diagnostic captured by the WSN on long term is incomplete (energy consumption of sensors, breaking down of sensors, etc).

• Using the scheduling mechanism in WSN in an efficient way to increase the lifetime to the maximal, while conserving the high coverage rate (the data diagnostics-PHM will be accurate for the longest time possible).

• The importance of density of sensors by increasing the coverage rate and lifetime in WSN, therefore its impact on data with time.

• The deployment of sensors in the area monitoring (in the learning and testing stage) have a real impact on diagnostic.

• The topologies have a great impact on the accuracy of the data and therefore on PHM, and this impact varies according to the type and parameters of topologies in WSN (density, packet transfer distance, the method of data aggregation, dividing the WSN in area monitoring into regions, etc).

The coverage rate and the energy consumption in WSN are the two most important objectives in WSN. Therefore, the accuracy of diagnostics is achieved by the efficient use of sensors in WSN, to increase the network's lifetime while guaranteeing high coverage rate (by achieving a compromise solution between these two factors). All the attempts done in literature to find a suitable trade-off between these factors were not efficient. For that, to solve this problem, we proposed in this thesis a distributed efficient algorithm which is able to achieve a good compromise between the coverage rate and lifetime (energy consumption). This leads to increase of data accuracy and therefore to enhance the reliability of PHM for a longer time. Then, in this thesis, we proposed and evaluated a fully distributed algorithm for diagnostics and health management. The proposed algorithm is a hybridization of three variants of domination, namely the simple, double, and total ones, in a single algorithm with a distributed computation. We theoretically studied and numerically simulated it, to show for each variant its impact on the effectiveness of the WSN over time, and on the diagnostics provided by six machine learning methods (the random forests and the five other diagnostic algorithms). Our proposal has been compared with both the optimistic solution (BaseLine) and its direct competitor (the Probabilistic Coverage Protocol PCP), and at each time we outperformed theme in terms of energy consumption, coverage, and diagnostics error rate.

6.2/ PERSPECTIVES

Using the wireless sensor networks for prognostics and health management is considered a challenging area of study. Getting a reliable data PHM for the longest time possible remains the greatest challenge and needs lots of research. These research always aim to improve the quality of health assessment, diagnostics, prognostics, and decision making. We will show several issues that need to be taken into consideration in the future work for a more reliable WSN-PHM:

1. We will use and experiment many diagnostic methods from other classifying approaches in WSN. The random forests algorithm is from data-driven models (approach), so using other algorithms from different classification like Knowledgebased models, hybrid models, and physical models allow us to know which is the better approach and which fits with the characteristics of WSN, and allow us to broadly evaluate the random forests algorithm.

2.

Most of the existing research work is limited to the use of condition monitoring (CM) data. Event data can be combined with CM data and kept record of, to increase the accuracy of health assessment, diagnostics, prognostics, and decision making.

CM data only provides quantitative information about specific monitored parameters. On the other hand, event data provides indications about repair actions, system breaking down, maintenance activities, etc. The model can better represent the behavior of the system (or component) and its degradation model by gaining knowledge about all events as they take place.

3.

Mostly, the training is done off-line, and the time dedicated to data collection could be limited in some cases. Industrials withhold some information due to confidentiality issues in other cases. However both situations mean that the data that is collected for the training set does not necessarily reflect all possible scenarios that could possibly be encountered. For this reason, updating the model online would be interesting in a way that rectifies the gap between the predicted class/RUL and the observed class/RUL.

4.

One of the reasons why wireless networks are not used for industrial monitoring is because they are easily hacked. A security code can be implemented to protect the networks from attacks. Thus, data will then be protected from misrouting, alterations, and falsification. Consequently, data at the base station will be more accurate, and thereby the predictions also.

5.

We intend to investigate other types of algorithms in WSN, like the centralized and hierarchical ones, to study their relevance for PHM on data provided by a WSN. We will compare them with our distributed method.

6. Finally we have to design our proposal on a physical network that monitors a real area, to observe in practice the real impact of the WSN on PHM on the one hand, and to compare with the results that we obtained by our numerical simulations on the other hand.

R ésum é :

Une maintenance efficace d'un dispositif industriel ne peut être bas ée que sur la fiabilit é et l'exactitude de donn ées physiques capt ées sur ledit dispositif, à des fins de surveillance. Dans certains cas, le monitoring de tels syst èmes industriels ou de zones à surveiller ne peut pas être assur é à l'aide de capteurs individuels ou filaires, du fait par exemple de probl èmes d'acc ès ou de milieux hostiles. Les R éseaux de Capteurs Sans Fil (RCSF) sont alors une alternative. En raison de la nature des communications, et des caract éristiques des appareils composants dans ces r éseaux, un RCSF est à fort risque de pannes au niveau des capteurs, et dans ce cas la perte de diverses donn ées est probable ce qui peut s'av érer probl ématique pour le monitoring du dispositif. Pour étudier la pertinence des RCSF pour le processus dit de PHM (Prognostic and Health Management), utilis é pour d éterminer le plan de maintenance d'un dispositif à surveiller), et l'impact des diverses strat égies d éploy ées dans ces premiers sur ces derniers, nous avons propos é un premier algorithme de diagnostic efficace et l'avons utilis é dans un RCSF simul é pour en mesurer la performance (ce simulateur étant un programme que nous avons d évelopp é). Nous avons alors propos é une d émarche pour diagnostiquer l' état de syst èmes physiques bas ée sur l'utilisation de la m éthode dite des for êts al éatoires. Cette d émarche repose sur deux phases: une premi ère, hors ligne, et une seconde en ligne. Dans la phase hors ligne, l'algorithme des for êts al éatoires s électionne les param ètres qui contiennent le plus d'information sur l' état du syst ème. Ces param ètres sont utilis és, dans leur ordre d'importance, pour construire les arbres d écisionnels qui constitueront la for êt. En injectant de l'al éatoire dans la base d'apprentissage, l'algorithme utilisera divers points de d épart, et par la suite les arbres aussi seront al éatoires. Dans la phase en ligne, l'algorithme évalue l' état actuel du syst ème en utilisant les donn ées capteurs pour parcourir les arbres construits. Chaque arbre dans la for êt fournit une d écision, et la classe finale est le r ésultat d'un vote majoritaire sur l'ensemble de la for êt. Quand les capteurs commencent à tomber en panne, certaines donn ées associ ées à divers indicateurs de sant é s'av èrent incompl ètes ou sont perdues. Or, puisque les arbres ont des points de d épart diff érents, l'absence de mesures pour un indicateur de sant é ne conduit pas n écessairement à l'interruption du processus de pr édiction de l' état global de sant é du dispositif industriel. Les diverses strat égies couramment implant ées dans les RCSF ont un impact important sur l'efficacit é du r éseau au cours du temps. Et, par cons équent, sur la qualit é des donn ées qui seront captur ées, et enfin sur le pronostic et la gestion de l' état de sant é du dispositif. C'est pourquoi, dans cette th èse, nous avons aussi étudi é en profondeur ces strat égies, et leurs effets sur le PHM. Nous avons mis en évidence divers probl èmes et nous avons évalu é de nombreux param ètres li és à ces strat égies, qui tous ont un impact plus ou moins important sur le r éseau. Et, par cons équent, sur le diagnostic au cours du temps. Pour atteindre cet objectif, pour évaluer la pronostic et gestion de la sant é avec les strat égies de RCSF, et pour évaluer l'importance de l'algorithme de for êt al éatoire, nous avons utilis é dans les RCSF l'algorithme de for êt al éatoire et cinq autres algorithmes de diagnostic de la litt érature: Support Vector Machines, Naive Bayes, Gradient Tree Boosting, Tree-Based Feature Selection, and Nearest Neighbors methods. Un probl ème cl é dans la conception de ces r éseaux est la conservation d' énergie: une fois qu'un capteur a épuis é ses ressources, il est perdu pour le r éseau. Une telle perte peut entraîner un probl ème de couverture, et donc des donn ées incompl ètes. Ainsi, le maintien de la qualit é de service (QoS) d'un r éseau de capteurs sans fil pendant une longue p ériode est tr ès important afin d'assurer des donn ées pr écises. Nous avons finalement propos é un algorithme enti èrement distribu é bas é sur la th éorie de domination dans les graphiques, et nous avons étudi é son impact sur le diagnostic en utilisant les six algorithmes mentionn és ci-dessus. Nous avons alors montr é que l'utilisation de cet algorithme dans un RCSF fournira des donn ées plus fiables pour une dur ée plus longue, ce qui conduira à un processus de diagnostic plus solide.

Mots-cl és : R éseaux de capteurs sans fil, Pronostic et gestion de l' état de sant é d'un dispositif industriel, Diagnostic, For êts al éatoires, Strategies de RCSF, Algorithme distribu é.

  Titre : titre (en français) : Utilisation de réseaux de capteurs sans fil pour la sécurité opérationnelle et le pronostic industriel Mots clés : Réseaux de capteurs sans fil, Pronostic et gestion de l'état de santé d'un dispositif industriel, Diagnostic.

CONTENTS 5 . 4 . 3

 543 Total dominating sets . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 The proposed algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Correctness and convergence proofs . . . . . . . . . . . . . . . . . 5.6 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Comparison with the optimistic BaseLine solution . . . . . . . . . . . 5.6.2 Comparison with the PCP algorithm . . . . . . . . . . . . . . . . . . 5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  (a) Exxon Valdez -$2.5 Billion. (b) Piper Alpha Oil Rig -$3.4 Billion. (c) Challenger Explosion -$5.5 Billion. (d) Chernobyl -$200 Billion.

Figure 1 . 1 :

 11 Figure 1.1: Catastrophes due to failure outcomes.

Figure 1 . 2 :

 12 Figure 1.2: History of maintenance strategies.

Figure 1 . 3 :

 13 Figure 1.3: General steps of PHM.

Fig. 2 . 1 :

 21 Fig. 2.1: Wireless Sensor Networks monitoring.

Fig. 2 . 2 :

 22 Fig. 2.2: Reference hardware platform architecture of a sensor node.

Fig. 2 . 3 :

 23 Fig. 2.3: Protocol stack for WSN.

2. 2

 2 .1/ DATA ACQUISITION Prognostics and Health Management (PHM) requires information about the targeted physical assets. The sensors which are placed on/around the critical components acquire the data. Later on, the developed algorithm will be fed stored data as inputs for health assessment, diagnostics, and/or prognostics. Here we can observe the significance of the quality of the data as all of the next steps rely on these measurements. The gathered information can contain either event-data or Condition Monitoring (CM) data. Event-data reveal what happened, what were the causes, and what was done (repair, breakdown, installation, etc.

Fig. 2 . 4 :

 24 Fig. 2.4: Data acquisition system.

Fig. 2 . 5 :

 25 Fig. 2.5: Data processing system.

Fig. 2 . 6 :

 26 Fig. 2.6: Health assessment process.

Fig. 2 . 7 :

 27 Fig. 2.7: Diagnostics different steps.

Fig. 2 . 8 :

 28 Fig. 2.8: An illustration of RUL with uncertainties.

Fig. 2 . 9 :

 29 Fig. 2.9: Decision support system.

Fig. 2 .

 2 Fig. 2.11: PHM steps with WSN monitoring.

Figure 3 .

 3 [START_REF]Mica2 datasheet[END_REF] show different groups of machine learning algorithms.

Fig. 3 . 1 :

 31 Fig. 3.1: Machine learning algorithms.

Fig. 3 . 2 :

 32 Fig. 3.2: Data set diversification in homogeneous ensemble methods.

Fig. 3 . 3 :

 33 Fig. 3.3: Algorithm diversification in heterogeneous ensemble methods.

Fig. 3 . 4 :

 34 Fig. 3.4: An illustration of the random forest.

. 5 . 5 .Fig. 3 . 5 :

 5535 Fig. 3.5: Example of a tree in the random forest.

Fig. 3 . 6 :

 36 Fig. 3.6: Delay in failure detection with respect to the number of simulations.

Figure 3 .

 3 Figure3.6 presents the delay between the time the system enters a failure mode and the time of its detection. This is done in the absence of correlations between the different features. The 0 time value of delay, the negative values, and positive value refer to in time predictions, early predictions and late predictions of failures, respectively. The plotted values are the average result per number of simulations which varies from 1 to 100. With time, sensor nodes start to fail in order to simulate missing data packets. As a result, the RF algorithm was able to detect 54% of the failures either in time or before their occurrence.

Fig. 3 . 7 :

 37 Fig. 3.7: Error rate in diagnostics with respect to the number of simulations.

Fig. 3 . 8 :

 38 Fig. 3.8: Number of successful diagnostics with respect to the number of trees.

Figure 4 .

 4 1c shows an example of a general barrier coverage problem where the start and end points of the path are selected from bottom and top boundary lines of the area. The selection of the path depends on the objective.

Fig. 4 . 1 :

 41 Fig. 4.1: Types of coverage problems.

Fig. 4 . 2 :

 42 Fig. 4.2: PHM steps with insufficient coverage in WSN.

Figure 4 . 3 andFig. 4 . 3 :

 4343 Fig. 4.3: Error rate in diagnostics in the case of complete learning and the percentage of active sensors is 100 % with respect to the number of simulations.

Figure 4 .

 4 Figure 4.3 indicates the variation of error rate for the six algorithms mentioned before, and during 20 simulations, in the particular case where the algorithms have complete data learning and all the sensors in WSN are active. As shown in the figure, during these 20 simulations, each algorithm presented a specific error interval (in %) as follows:[START_REF] Bulusu | Scalable coordination for wireless sensor networks: self-configuring localization systems[END_REF][START_REF] Hareb | Energy-efficient data aggregation and transfer in periodic sensor networks[END_REF] for SVM,[START_REF] Amit | Shape quantization and recognition with randomized trees[END_REF][START_REF] David W Carman | Constraints and approaches for distributed sensor network security (final)[END_REF] for NB,[START_REF] Bahi | Resiliency in distributed sensor networks for prognostics and health management of the monitoring targets[END_REF][START_REF] Carle | Preserving area coverage in wireless sensor networks by using surface coverage relay dominating sets[END_REF] for RF,[START_REF] Cho | Energy efficient protocols for low duty cycle wireless microsensor networks[END_REF][START_REF] Hefeeda | Energy-efficient protocol for deterministic and probabilistic coverage in sensor networks[END_REF] for GTB,[START_REF] Bai | Deploying wireless sensors to achieve both coverage and connectivity[END_REF][START_REF] Carle | Preserving area coverage in wireless sensor networks by using surface coverage relay dominating sets[END_REF] for TBFS, and[START_REF] Cerpa | Ascent: Adaptive self-configuring sensor networks topologies[END_REF][START_REF] Estrin | Next century challenges: Scalable coordination in sensor networks[END_REF] for NN.

Figure 4 . 3 (Fig. 4 . 4 :

 4344 Fig. 4.4: Error rate in diagnostics in the case of incomplete learning and the percentage of active sensors is 100 % with respect to the number of simulations.

Figure 4 . 5 and

 45 Figure 4.6 in case where all sensors were working (100 %) are equal to the average of the percentage of errors found in Figure 4.3 and Figure 4.4 respectively.

Figure 4 .Fig. 4 . 5 :

 445 Figure 4.5 contains the variation of error rate for six algorithms in case of complete learning with the variation of the percentage of the active sensors in WSN. As shown in the figure, as the percentage of active sensors in WSN decreases, the error rate of these

Fig. 4 . 6 :

 46 Fig. 4.6: Error rate in diagnostics in case of incomplete learning with the variation of the percentage of active sensors.

Figure 4 . 7 .

 47

Figure 4 .Fig. 4 . 7 :

 447 Figure 4.7 presents the error rate variations of our six algorithms during 20 simulations. As

Fig. 4 . 8 :

 48 Fig. 4.8: asymmetrical density on abscissa.

Fig. 4 . 9 :

 49 Fig. 4.9: Error rate in diagnostics with the variation of data learning in a way of asymmetrical density on the abscissas (length of area monitoring).

Fig. 4 . 10 :

 410 Fig. 4.10: Error rate in diagnostics with the variation of area coverage by sensors in a way of asymmetrical density on the abscissas (length of area monitoring).

Figure 4 .

 4 Figure 4.10, for its part, is the result of the variation of error rate for six algorithms in case of complete data learning (the data includes information about all of the area monitoring), with the variation of the location of density of sensors in WSN at the length of area (asymmetrical density on abscissas). Figure4.8a is when the uniform distribution of sensors in area monitoring (asymmetrical density on abscissa is 0): area is fully covered, while Figure4.8b explains one case, if the distribution is non-uniform (in case asymmetrical density on abscissa is 50). As shown in Figure4.10, as the asymmetrical density on abscissas decreases (sensors in WSN are more uniformly distributed), the error rate of these algorithms decreases until the error rate of the algorithm (only in the case if the asymmetrical density on abscissa is 0) equals 18 % for SVM, 10 % for NB, 16 % for RF, 4 % for GTB, 7 % for TBFS, and 17 % for NN. But if the asymmetrical density on the abscissa is 50, the error rate will be equal to 86 % for SVM algorithm, 82 % for NB, 74 % for RF, 84 % for GTB, 76 % for TBFS, and 85 % for NN.

Figure 4 .

 4 Figure 4.10, for its part, is the result of the variation of error rate for six algorithms in case of complete data learning (the data includes information about all of the area monitoring), with the variation of the location of density of sensors in WSN at the length of area (asymmetrical density on abscissas). Figure4.8a is when the uniform distribution of sensors in area monitoring (asymmetrical density on abscissa is 0): area is fully covered, while Figure4.8b explains one case, if the distribution is non-uniform (in case asymmetrical density on abscissa is 50). As shown in Figure4.10, as the asymmetrical density on abscissas decreases (sensors in WSN are more uniformly distributed), the error rate of these algorithms decreases until the error rate of the algorithm (only in the case if the asymmetrical density on abscissa is 0) equals 18 % for SVM, 10 % for NB, 16 % for RF, 4 % for GTB, 7 % for TBFS, and 17 % for NN. But if the asymmetrical density on the abscissa is 50, the error rate will be equal to 86 % for SVM algorithm, 82 % for NB, 74 % for RF, 84 % for GTB, 76 % for TBFS, and 85 % for NN.

  Figure 4.11a. After time t = x, the CH and sensors may become inactive for several reasons, most importantly energy consumption or activity scheduling. If a CH became inactive, (a) Sensors network at time t = 0. (b) Sensors network at time t = x.

Fig. 4 . 11 :

 411 Fig. 4.11: Scenario of decentralized topology.

Fig. 4 . 13 :

 413 Fig. 4.13: Decentralized topology with 2 × CH.

Figure 4 .

 4 Figure 4.13 shows the result of WSN with decentralized topology under the same conditions and parameters that we explained in previous result (Figure 4.12), but here we used 60 CHs instead of 30 CHs. With these parameters, the error rate of the six algorithms varied in a significant way from what is shown in Figure 4.12 (with 30 CHs). More precisely, the error rate of the algorithms remained constant (along the intervals determined before) for t = 0..70, while in our previous simulation, it remained constant until t = 60.Then, the error rates start to increase over time, as numerous sensors in the network emptied their batteries, to reach the following rates: 70 % at t = 100 if the algorithm is SVM, 64 % for NB, 58 % for RF, 66 % for GTB, 60 % for TBFS, and finally 68 % for NN. We can remark that, at this time and with 30 CHs, the whole network became inactive in our previous simulation (seeFig. 4.12). We can deduce from these simulations that the difference between obtained results is due to two causes. The first reason, which is the most important one, is related to the increase of cluster heads (and thus of clusters) in the network. Indeed, the works that a CH must realize (receiving data from sensors,

Fig. 4 . 14 :

 414 Fig. 4.14: Decentralized topology with +100 sensors.

  (a) Sensors network at time t = 0. (b) Sensors network at time t = x.

Fig. 4 . 15 :

 415 Fig. 4.15: Scenario of distributed topology.

Fig. 4 . 17 :

 417 Fig. 4.17: Distributed topology with closest transfer distance between sensors.

  (a) Sensors network at time t = 0. (b) Sensors network at time t = x.

Fig. 4 . 18 :

 418 Fig. 4.18: Scenario of hierarchical topology.

Fig. 4 . 19 :

 419 Fig. 4.19: Error rate in diagnostics if the topology is hierarchical with the variation of time.

Figure 4 .

 4 Figure 4.19 indicates the variation of error rate for the six considered algorithms under the same conditions of the previous studies, but here with a hierarchical topology. Again, each point in this figure is an average of error rates of a given algorithm on 20 simulations (for a certain t). As shown in the figure, if 0 ≤ t ≤ 20, each algorithm has a specific error interval (in %) as follows:[START_REF] Jacques M Bahi | Lowcost monitoring and intruders detection using wireless video sensor networks[END_REF][START_REF] Breiman | Bagging predictors[END_REF] for SVM,[START_REF] Ian F Akyildiz | A survey on wireless multimedia sensor networks[END_REF][START_REF] Amit | Shape quantization and recognition with randomized trees[END_REF] for NB,[START_REF] Ian F Akyildiz | A survey on wireless multimedia sensor networks[END_REF][START_REF] Atapour | On total dominating sets in graphs[END_REF] for RF,[4,[START_REF] Hammad | Security for wsn based on elliptic curve cryptography[END_REF] for GTB,[4,[START_REF] Ian F Akyildiz | A survey on wireless multimedia sensor networks[END_REF] for TBFS, and[START_REF] Atapour | On total dominating sets in graphs[END_REF][START_REF] Jacques M Bahi | Lowcost monitoring and intruders detection using wireless video sensor networks[END_REF] for NN. These intervals are approximately the same where the

Fig. 4 . 20 :

 420 Fig. 4.20: Hierarchical topology with 2× sensors in distribution layer.

Figure 4 .

 4 Figure 4.20 shows the result of WSN with hierarchical topology under the same conditions and parameters that we explained in previous result (shown in Figure4.[START_REF] Breiman | Using adaptive bagging to debias regressions[END_REF], but here we used 60 nodes in distribution layer instead of 30 nodes. With these parameters, the error rate of the six algorithms varied significantly from what is shown in Figure4.19 (with 30 nodes in distribution layer). In addition, the algorithms remained constant (along the intervals mentioned before) during t = 0..30, while in a previous study, the error rate remained constant during t = 0..20. After that, the error rate of the algorithms started to increase over time, and this shows that sensors in WSN started dying, to reach 91 % at t = 100 if the algorithm is SVM, 90 % for NB, 89 % for RF, 90 % for GTB, 89 % for TBFS, and 91 % for NN. Thus, when we used 30 more nodes in distribution layer, the lifetime of the network increased 10 × t, and this is what we obtained in Figure4.19 and 4.20. What is worth to mention is that the batteries of the nodes in the distribution layer are like the ones in access layer, therefore the lifetime of WSN will increase more if the batteries in distribution layer were bigger. This difference is due to two basic reasons: first, the increase in number of nodes in distribution layer will divide the work, from receiving data

Fig. 4 . 21 :

 421 Fig. 4.21: Hierarchical topology with +100 sensors in access layer.

Figure 4 . 21 .

 421 The error rate for the six algorithms in this figure varied differently from what was shown in Figure 4.19 (with 300 leaf nodes), and it is opposite to what is shown in Figure 4.20. These error rates of diagnostics remained constant (along the intervals determined before) during t = 0..20, after that the error rate of the algorithms started increasing with time, to reach 90 % at t = 80 if the algorithm is SVM, 89 % if NB, 89 % if RF, 90 % if GTB, 89 % if TBFS, and 90 % if NN (the whole network is inactive). With 300 leaf nodes, the whole network become inactive at t = 100 as shown in Figure 4.19.

Figure 4 .

 4 22b. The black and white circles are the active and inactive sensors respectively, and the crossed circle is the sink.(a) Sensors network at time t = 0. (b) Sensors network at time t = x.

Fig. 4 . 22 :

 422 Fig. 4.22: Scenario of centralized topology.

Figure 4 .Fig. 4 . 23 :

 4423 Figure 4.[START_REF] Cardei | Coverage in wireless sensor networks[END_REF] shows the variation of error rate for the six considered algorithms in the case of a centralized topology. Each point in this figure is an average of error rates of a given algorithm on 20 simulations. As we can see, at t = 0 (when the WSN starts working), each algorithm has a specific error rate (in %) as follows: 18 % for SVM, 10 % for NB, 12 % for RF, 5 % for GTB, 7 % for TBFS, and 16 % for NN. During the work of the network, with time, the sensors that are located farthest from the sink start dying first because they consume more energy than the others, and this is due to the long distance of packet transfer. For that, at t = 10 the error rate increased in a noticeable way to become 52 % with SVM, 35 % with NB, 28 % with RF, 44 % with GTB, 32 % with TBFS, and finally 48 % with NN. This is a proof that the data became incomplete for diagnostic, as the regions far from the sink are no longer covered by sensors. Then, when the WSN exceeds t = 10, the error rate of algorithms increases with time to reach 91 % at t = 90 if the algorithm is SVM, 89 % for NB, 88 % for RF, 90 % for GTB, 89 % for TBFS, and 90 % for NN.

  , 4.16, 4.19, and 4.23, for decentralized, distributed, hierarchical, and centralized topology respectively. The aim of this comparison is to focus on several elements or issues related to topologies that have a great impact on diagnostics.

104CHAPTER 5 .

 5 DISTRIBUTED MINIMAL DOMINATING SENSOR-TARGETS FOR DIAGNOSTICS

Fig. 5 . 1 :

 51 Fig. 5.1: Minimal dominating sets.

106CHAPTER 5 .

 5 DISTRIBUTED MINIMAL DOMINATING SENSOR-TARGETS FOR DIAGNOSTICS dominating set is called double domination number and is denoted by γ dd (G). For example in the Figure 5.2 the graph having minimal double dominating sets of cardinality three S = {1, 3, 5}; γ dd (G) = 3.

Fig. 5 . 2 :

 52 Fig. 5.2: Minimal double dominating sets.

Fig. 5 . 3 :

 53 Fig. 5.3: Graph with isolated vertices.

Fig. 5 . 4 :

 54 Fig. 5.4: Minimal total dominating sets.

  (a) With 300 sensors. (b) With 600 sensors.

Fig. 5 . 6 :

 56 Fig. 5.6: Diagnostics error rate over time with the simple domination algorithm.

Fig. 5 . 7 :

 57 Fig. 5.7: Diagnostics error rate over time with the double domination algorithm.

112CHAPTER 5 .

 5 DISTRIBUTED MINIMAL DOMINATING SENSOR-TARGETS FOR DIAGNOSTICS (a) With 300 sensors. (b) With 600 sensors.

Fig. 5 . 8 :

 58 Fig. 5.8: Diagnostics error rate over time with the total domination algorithm.

  (a) With 300 sensors. (b) With 600 sensors.

Fig. 5 . 9 :

 59 Fig. 5.9: Diagnostics error rate over time using BaseLine on a WSN.

Fig. 5 . 10 :

 510 Fig. 5.10: Error rate using the Gradient Tree Boosting with our three proposals and with BaseLine.

  and 5.12, which are not surprising. The slight improvement 114CHAPTER 5. DISTRIBUTED MINIMAL DOMINATING SENSOR-TARGETS FOR DIAGNOSTICS (a) With 300 sensors. (b) With 600 sensors.

Fig. 5 .

 5 Fig. 5.11: Energy consumption of sensors in battery unit u, for our three proposals and with BaseLine.

  (a) With 300 sensors. (b) With 600 sensors.

Fig. 5 . 12 :

 512 Fig. 5.12: Coverage rate of sensors (%) with time in WSN with the three cases in the proposed algorithm and with BaseLine.

  (a) With 300 sensors. (b) With 600 sensors.

Fig. 5 . 13 :

 513 Fig. 5.13: Diagnostic error rate with Probabilistic Coverage Protocol on a WSN.

116CHAPTER 5 .

 5 DISTRIBUTED MINIMAL DOMINATING SENSOR-TARGETS FOR DIAGNOSTICS tion modes). To sum up, the probabilistic coverage protocol is poorer in average than our proposals, according to this first set of experiments that consider the six aforementioned machine learning algorithms. (a) With 300 sensors. (b) With 600 sensors.

Fig. 5 . 14 :

 514 Fig. 5.14: Error rate using the Gradient Tree Boosting, with the PCP and our proposals.

  (a) With 300 sensors. (b) With 600 sensors.

Fig. 5 .

 5 Fig. 5.15: Energy consumption of sensors (in battery unit) between PCP and the proposals.

Fig. 5 . 16 :

 516 Fig. 5.16: Coverage rate of sensors between the proposals and the PCP.

  (a) With 300 sensors. (b) With 600 sensors.

Fig. 5 .

 5 Fig. 5.17: Convergence time in the three cases in the proposed algorithm and with PCP.

  Catastrophes due to failure outcomes. . . . . . . . . . . . . . . . . . . . . . 1.2 History of maintenance strategies. . . . . . . . . . . . . . . . . . . . . . . . 1.3 General steps of PHM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An illustration of the random forest. . . . . . . . . . . . . . . . . . . . . . . . 3.5 Example of a tree in the random forest. . . . . . . . . . . . . . . . . . . . . 3.6 Delay in failure detection with respect to the number of simulations. . . . . . 3.7 Error rate in diagnostics with respect to the number of simulations. . . . . . 3.8 Number of successful diagnostics with respect to the number of trees. . . . 4.1 Types of coverage problems. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 PHM steps with insufficient coverage in WSN. . . . . . . . . . . . . . . . . .

		LIST OF FIGURES
	1.1
	Bibliographie	134

6 Conclusion and Perspectives 121 6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Wireless Sensor Networks monitoring. . . . . . . . . . . . . . . . . . . . . . 2.2 Reference hardware platform architecture of a sensor node. . . . . . . . . . 2.3 Protocol stack for WSN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Data acquisition system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Data processing system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Health assessment process. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Diagnostics different steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 An illustration of RUL with uncertainties. . . . . . . . . . . . . . . . . . . . . 2.9 Decision support system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Human machine interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 PHM steps with WSN monitoring. . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Machine learning algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Data set diversification in homogeneous ensemble methods. . . . . . . . . 3.3 Algorithm diversification in heterogeneous ensemble methods. . . . . . . . 3.4 4.3 Error rate in diagnostics in the case of complete learning and the percentage of active sensors is 100 % with respect to the number of simulations. . 4.4 Error rate in diagnostics in the case of incomplete learning and the percentage of active sensors is 100 % with respect to the number of simulations. ABBREVIATIONS WSN . . . . . Wireless Sensor Network OS . . . . . . . Operating System CPU . . . . . . Central Processing Unit ADC . . . . . . Analog-to-Digital Converter VM . . . . . . . Virtual Machine IPC . . . . . . . Inter-Process Communication QOS . . . . . . Quality of Service PHM . . . . . . Prognostics and Health Management RUL . . . . . . Remaining Useful Life CBM . . . . . . Condition-Based Maintenance PM . . . . . . . Predictive Maintenance CM . . . . . . . Condition Monitoring HMI . . . . . . . Human Machine Interface MDP . . . . . . Markov Decision Process MLP . . . . . . Multi-Layer Perceptron RVM . . . . . . Relevance Vector Machine SVM . . . . . . Support Vector Machines RF . . . . . . . . Random Forests NB . . . . . . . Naive Bayes GTB . . . . . . Gradient Tree Boosting TBFS . . . . . Tree-Based Feature Selection NN . . . . . . . Nearest Neighbors CH . . . . . . . Cluster Head Pp . . . . . . . . Poisson process PCP . . . . . . Probabilistic Coverage Protocol CDS . . . . . . Connected Dominating Set

  Impacts of wireless sensor networks strategies and topologies on prognostics and health management. Ahmad Farhat, Christophe Guyeux, Abdallah Makhoul, Ali Jaber, Rami Tawil, and Abbas Hijazi. Journal of Intelligent Manufacturing (JIMS).

	1.2.1/ PUBLISHED PAPERS
	1.

2. On the topology effects in wireless sensor networks based prognostics and health management. In 19th IEEE International Conference on Computational Science and Engineering (CSE 2016), pages 1-9. IEEE, IEEE, aug 2016. Ahmad Farhat, Abdallah Makhoul, Christophe Guyeux, Rami Tawil, Ali Jaber, and Abbas Hijazi. 3. Random Forests for Industrial Device Functioning Diagnostics Using Wireless Sensor Networks. IEEE AEROSPACE CONFERENCE, 2015. Big Sky, Montana, USA (2015, Pages 1-9). Wiem Elghazel, Jacques Bahi, Ahmad Farhat, Christophe Guyeux, Mourad Hakem, Kamal Medjaher, Noureddine Zerhouni. 1.2.2/ SUBMITTED PAPERS 4. Distributed Minimal Dominating Sensor-Targets for Diagnostics and Health Management. Ahmad Farhat, Christophe Guyeux, Mohammed Haddad, Mourad Hakem, and Abdallah Makhoul. Submitted to IEEE International Conference on Pervasive Computing and Communications. 5. A study on the Impact of Network Topology on Wireless Sensor Networks Performance based on Prognostics and Health Management. Ahmad Farhat, Christophe Guyeux, Abdallah Makhoul, Ali Jaber, Rami Tawil, and Abbas Hijazi. Submitted to Journal of Ad Hoc & Sensor Wireless Networks (AHSWN). 6. On the coverage effects in wireless sensor networks based prognostic and health management. Ahmad Farhat, Christophe Guyeux, Abdallah Makhoul, Ali Jaber and Rami Tawil. Submitted to International Journal of Sensor Networks (IJSNET).

Table 2

 2

	.1.

Table 3 .

 3

	process -Aging is not considered
		-Different stages of degradation process cannot
		be accounted for
		-Large volume of data is required for the training
		-The assumptions are not always practical
	Bayesian networks -Prior transitions are not considered
		-Complete reliance on accurate thresholds
		-Many state transitions are needed for efficient
		results
		-Unable to predict unanticipated states
	Neural networks	-Significant amount of data for the training
		-Retraining is necessary with every change of
		conditions
		-Pre-processing is needed to reduce inputs
	Fuzzy systems	-Increasing complexity with every new entry
		-Domain experts are required
		-Results are as good as the developers' under-
		standing

1: Limitations of diagnostic models.

  91 % at t = 100 if the algorithm is SVM, 90 % if NB, 89 % if RF, 90 % if GTB, 89 % if TBFS, and 90 % if NN. In other words, approximately the whole network is inactive.

Table 4 .

 4 2.

	Diagnostic algorithm	Real-time performances (s) t = 1 t = 10 t = 100
	Support Vector Machine	0.27	2.5	23
	Naive Bayes	0.21	1.9	18
	Gradient Tree Boosting	0.3	2.8	26
	Tree-Based Feature Selection	0.25	2.3	21
	Nearest Neighbors	0.18	1.7	15.5
	Random forests	0.22	2	19

Table 4 .

 4 2: The calculation time of diagnostic algorithms (second) with respect to the operating age (t).

Table 4 .

 4 2 shows the calculation time of each diagnostic algorithm, in seconds and for three kinds of iteration numbers (t = 1, 10, and 100 iterations). This is the time taken by a given simulated method to perform the complete diagnostic process. For instance, a Support Vector Machine used for PHM on WSN takes 0.27s. to perform one iteration over the network, while 23s. are needed for t = 100 iterations. By relying on the results shown in Table4.2, we can notice that the Nearest Neighbors has a lower complexity than the other algorithms. It is followed by Naive Bayes, Random forests, Tree-Based Feature Selection, Support Vector Machine, while the most complex tool is the Gradient Tree Boosting.What is worth mentioning is that diagnosis accuracy and complexity are two objectives that cannot be reached together (related to the type of data learning and data captured, whether it is complete or not). For instance, Nearest Neighbors outperforms the other methods in terms of complexity, but it has in general a worse error rate than the other algorithms in the various situations investigated previously. Support Vector Machine, for its part, appears as a bad choice when dealing with these two objectives. Conversely, Random Forests have a reasonable complexity, while exhibiting a low error rate during our simulation. This method appears as a good compromise for prognostic and health management based on data provided by a wireless sensor network.

Je remercie également tous les personnels de laboratoire Femto-St, du d épartement DISC, et aussi les personnels de l'Universit é Libanaise pour l'excellente ambiance de travail, les échanges et l'aide pr écieuse apport ée tout au long de la r éalisation de ces travaux de recherche, d'encadrement et d'enseignement.Enfin, je remercie ma famille et mes amis pour leur soutien ind éfectible tout au long de mon parcours et lors de la pr éparation et r édaction de ce m émoire. Je remercie tout particuli èrement mon fr ère Abbas, m'amie Mira, et mes parents pour leur soutien et leurs encouragements quotidiens. Sans eux, je n'aurais sans doute pas accompli les travaux pr ésent és dans ce m émoire.

(a) Uniform distribution of sensors on an area (the asymmetrical density on abscissa is 0).(b) Non-uniform distribution of sensors on an area (in case asymmetrical density on abscissa is 50).

% for RF, 5 % for GTB, 7 % for TBFS, and 16 % for NN. During the work of the network, with time, the sensors that are located farthest from the sink start dying first because they consume more energy than the others, and this is due to the long distance of packet transfer. For that, at t = 10 the error rate increased in a noticeable way to become 52 % with SVM, 35 % with NB, 28 % with RF, 44 % with GTB, 32 % with TBFS, and finally 48 % with NN. This is a proof that the data became incomplete for diagnostic, as the regions far from the sink are no longer covered by sensors. Then, when the WSN exceeds t = 10, the error rate of algorithms increases with time to reach 91 % at t = 90 if the algorithm is SVM, 89 % for NB, 88 % for RF, 90 % for GTB, 89 % for TBFS, and 90 % for NN.

ACKNOWLEDGEMENTS

Je remercie tout particuli èrement l' équipe encadrante: le Professeur Christophe GUYEUX, le Professeur Abbas HIJAZI, le Monsieur Mourad Hakem et le Monsieur Ali JABER pour m'avoir propos é de travailler avec eux sur la th ématique du Prognostics and Health Management using Wireless Sensor Networks . Je les remercie pour leur confiance, leur soutien, leurs conseils. Ce travail n'aurait pas pu être accompli sans leurs critiques constructives.

5.5/ THE PROPOSED ALGORITHM

Before getting into details of our algorithm, we introduce some definitions and notations: let G = (V; E) be the graph modeling the sensor network, with |V| = n and |E| = m. We assume sensor node identifiers to be unique. We recall that sensor node identifier is unique if and only if i.Id j.Id holds for each i, j ∈ V(i j). A sensor node can be in one of these three states: IN, OUT, or WAIT.

The algorithm is presented in an Event-driven programming way. It is composed of a set of rules, each handling a particular event that could happen on target or sensor nodes.

Constants

5.5.1/ CORRECTNESS AND CONVERGENCE PROOFS

In our model, we assume a fair fully distributed scheduler, i.e., every node execute activated rules right upon its event happening. We also assume the initial configuration to be arbitrary. Moreover, if a node is activated by two or more rules, it first executes the one with lower sequence number. Lemma 5.5.1 . Once the algorithm terminated, there is no node in the WAIT state.

Proof. If we suppose the opposite, there will be at least one node either activated for rule [R3] or [R5]. A contradiction with algorithm's termination. Lemma 5.5.2 . Once the algorithm terminated, the set formed by nodes with state = IN forms a (total/double) dominating set according to the specification.

Proof. If it is not the case, there will be at least one node activated for rule [R1]. A contradiction.

Lemma 5.5.3 . Once the algorithm terminated, the set formed by nodes with state = IN is minimal.

Proof. If it is not the case, there will be at least one node activated for rule [R4] because it is an unnecessary dominator. A contradiction. Lemma 5.5.4 . After the two first rounds, all nodes will have their variables set to correct values according to the algorithm.

Proof. Rule [R1] is responsible to update Need and Leave variables, all other rules update the state variable. After two execution rounds, all activated nodes will have the possibility to execute [R1] (if activated) and another rule modifying the state variable. Lemma 5.5.5 . After the third round, all nodes will be dominated.

Proof. By previous lemma, all not dominated nodes will have their Need variables set to Lacking. Thus all neighbors of a not dominated node will be activated to join the dominated set by rule [R2]. Lemma 5.5.6 . After the fourth round, at least one unnecessary dominator node will leave the dominating set every two rounds.

Proof. At the fourth round, all unnecessary dominators will have their state variables set to WAIT . Then, nodes will elect the one to leave the set with their Leave pointer. This process will need one round. Once election finished, at least the node with minimum ID and having its state set to WAIT will be activated for rule [R5] to leave the dominating set.

Theorem 5.5.7 . The algorithm constructs a minimal (total/double/simple) dominating set within O(n) distributed execution rounds.