Tom Druet

Huu Tinh Hoang

Bastien Chapuis

Emmanuel Moulin

QNDE2019-1234 PASSIVE GUIDED WAVE TOMOGRAPHY FOR PIPES INSPECTION

Keywords: passive methods, guided wave tomography, SHM

Structural Health Monitoring (SHM) consists in embedding sensors in a structure like aircraft fuselages, pipes or ship hulls in order to detect defects (for example cracks or corrosion in metallic materials or delamination in composite materials) before a serious damage occurs in the structure. Guided elastic waves emitted by a sensor and propagating to another one are often used as the physical way of detecting the defect. However, the implementation of SHM systems is restricted in many situations by the necessity to store or to harvest the electric energy necessary to emit the waves and also by the intrusiveness of the sensors. Guided wave tomography imaging is able to localize and quantify the severity of the defect when it comes to loss of thickness such as corrosion or erosion. However, it needs many sensors (generally piezoelectric -PZT-transducers) and it has a cost, particularly in terms of intrusiveness. Passive guided wave tomography, which uses ambient ultrasonic waves that are naturally present in the structure, is a promising method to design an SHM system with minimal power consumption and intrusiveness since no energy is required to emit the waves. In this paper the application of passive tomography to pipe inspection and comparison with classical "active" guided waves tomography approach are described.

INTRODUCTION

The maintenance has a prominent role from both cost point of view and security. The Structural Health Monitoring (SHM) consists in acquiring and analyzing data from sensors embedded in a structure in order to evaluate its health [START_REF]SAE international, Guidelines for Implementation of Structural Health Monitoring on Fixed Wing Aircraft[END_REF]. A physical way to interrogate large and thin structures such as aircraft fuselages, pipes or ship hulls can be offered by guided elastic waves emitted and received by a network of piezoelectric transducers (PZTs). Guided wave tomography algorithms are able to not only localize but also quantify the severity of defects consisting in a 1 Contact author: tom.druet@cea.fr loss of thickness such as corrosion or erosion. However, it needs many sensors (generally piezoelectric -PZT-transducers) and it has a cost, particularly in terms of intrusiveness.

An interesting possibility is to design a guided wave based SHM system with low intrusiveness thanks to the use of guided wave tomography with undersampled data. Indeed, less sensors are used compared to the optimal number but it is still possible to obtain a satisfactory image of the defect. Another interesting possibility which also answers to the electric energy consumption issue is to work only with receivers of elastic waves. To achieve this, passive methods based on elastic noise measurements can be used to retrieve the response between the sensors without using active (piezoelectric) sources [START_REF] Druet | Passive guided wave tomography for corrosion detection[END_REF] [START_REF] Druet | Passive guided waves measurements using fiber Bragg gratings sensors[END_REF]. We present here studies of passive guided wave tomography of pipes.

MATERIALS AND METHODS

In this section is presented the passive guided wave tomography. First the basis of the passive method used here is given and then the main steps of guided wave tomography are recalled.

Passive method

Guided wave tomography is commonly applied to active data: each sensor acts successively as a source whereas other sensors measure the propagated signals. In this work, the imaging is obtained from passive data, that is, the signal corresponding to each emitter-receiver couple is reconstructed thanks to the cross-correlation of the ambient noise measured simultaneously by every couple of sensors [START_REF] Lobkis | On the emergence of the Green's function in the correlations of a diffuse field[END_REF].

In the following, the ambient noise 𝜑 is a space-time stationary random field that is also delta correlated in space and time. Let's consider a plate as a 2D domain and two sensors placed at points 𝑥 and 𝑦. Let us denote 𝑉 𝑥 , respectively 𝑉 𝑦 , the ambient noise acquired by the sensor at 𝑥, respectively 𝑦, on the time interval [0, 𝑇], 𝑇 > 0. The cross-correlation 𝐶 𝑇 of these two signals is

where 𝐻 𝑥𝑦 is the impulse response of the system formed by the plate and the sensors, the sensor at 𝑥 acting as a source while the one at 𝑦 acts as a receiver. Note that if the domain is reciprocal, which is the case for our applications, then 𝐻 𝑥𝑦 = 𝐻 𝑦𝑥 . As the impulse response is causal, the positive time part of (3) corresponds to the impulse response for 𝑥 as a source and 𝑦 as a receiver and the negative part to the time-reversed impulse response for 𝑦 as a source and 𝑥 as a receiver.

Guided wave tomography

Once we have obtained passive signals from passive acquisition, we can apply guided wave tomography to these data.

Here, the used guided wave tomography algorithm is based on the original iterative Hybrid Algorithm for Robust Breast Ultrasound Tomography (HARBUT) [START_REF] Huthwaite | High-resolution guided wave tomography[END_REF] but adapted to the heavy constraints imposed by the SHM framework. The principal steps are recalled here after. The first step consists in performing a time-of-flight tomography which will be the initial guess of the following diffraction tomography steps. The idea is to update iteratively the map coefficients 𝑆 𝑖𝑗 with a correction 𝛿 𝑘 𝑆 𝑖𝑗 (𝑛) based on the difference between the data 𝜏 𝑘→𝑙 𝑑𝑒𝑓𝑒𝑐𝑡 and the computed time-of-flight 𝜏 𝑘→𝑙 (𝑛-1) on the (𝑛 -1) 𝑡ℎ map:

𝛿 𝑘 𝑆 𝑖𝑗 (𝑛) = 1 𝑁 𝑖𝑗𝑘 ∑ (𝑊 𝑖𝑗𝑘𝑙 𝜏 𝑘→𝑙 𝑑𝑒𝑓𝑒𝑐𝑡 -𝜏 𝑘→𝑙 (𝑛-1) 𝜆 𝑘→𝑙 ) , 𝑙 (1)
where 𝜆 𝑘→𝑙 is the trajectory's length, 𝑊 𝑖𝑗𝑘𝑙 is a weighting of the path and 𝑁 𝑖𝑗𝑘 is a normalization factor.

Once the time-of flight tomography image is computed, it becomes the first guess of an iterative diffraction tomography process. The image is then represented thanks to an object function 𝑂(𝑥), which reads:

𝑂(𝑥) = 𝑘 0 2 (( 𝑣 0 𝑣(𝑥) ) 2 -1), ( 1
)
where 𝑘 0 is the background wavenumber, 𝑣 0 the background phase velocity and 𝑣(𝑥) the phase velocity at position 𝑥, which is linked to the thickness thanks to the dispersion curves when the algorithm has converged. 𝑂(𝑥) can be calculated with diffraction tomography [START_REF] Huthwaite | Eliminating incident subtraction in diffraction tomography[END_REF]. The iterative version of HARBUT consists in computing iteratively the correction 𝑂 𝛿 (𝑥), which is added to the previous estimation 𝑂 𝑏 (𝑥) so that 𝑂(𝑥) = 𝑂 𝑏 (𝑥) + 𝑂 𝛿 (𝑥).

RESULTS AND DISCUSSION

In order to test the algorithm, numerical 3D elastodynamic simulations based on the Finite Element Method (FEM) were performed. The FEM code is internally developed in the NDE department of CEA-LIST [START_REF] Imperiale | A macro-element strategy based upon spectral finite elements and mortar elements for transient wave propagation modeling. Application to ultrasonic testing of laminate composite materials[END_REF]. The pipe and the defect simulated are presented in figure 1.

FIGURE 1: CONFIGURATION OF THE SIMULATION.

Two rings of 15 piezoelectric transducers are used to produce the signals. Then guided wave tomography algorithm is applied to this set of undersampled data. The result is presented on the figure 2. The lateral resolution is better than the axial one. This is a classic result when it comes to distributions of sensors that are not closed (limited-view imaging). This issue can be overcome with the virtual image space component iterative technique (VISCIT) [START_REF] Huthwaite | A new regularization technique for limited-view sound-speed imaging[END_REF]. We consider now the experimental case of a stainless steel pipe on which two rings of 15 piezoelectric transducers are glued using epoxy. The experimental setup is presented in figure 3.

First active acquisition is performed. Then, the guided wave tomography algorithm is applied to these active data. The image obtained is presented in figure 4. The defect considered is a removable magnet, which can be moved easily at different positions. It is an artificial defect that causes diffraction of guided waves, but is not representative of loss of thickness due to corrosion. That is why the image is presented in phase velocity and not in thickness. Finally, the passive acquisition case is considered. The source of noise considered here is a jet of compressed air which is applied on the intern surface of the pipe. Ten seconds of noise is acquired and then post-processed with the cross-correlation. Passive guided wave tomography is then applied to the passive signals and the result is presented in figure 5. The image obtained is slightly noisier but the position of the magnet is still well reconstructed.

FIGURE 5: EXPERIMENTAL PASSIVE CASE

CONCLUSION

Passive guided wave tomography is presented in this talk. To do so, first, simulation is used in order to test the algorithm, then the active case is studied experimentally and finally passive guided wave tomography is applied in the case of a noise produced by a jet of compressed air in a pipe.

  𝐶 𝑇 (𝑡, 𝑥, 𝑦) = 1 𝑇 ∫ 𝑉 𝑥 (𝜏)𝑉 𝑦 (𝑡 + 𝜏) d𝜏 𝑇 0 . (1) One can then show [4] that d 𝐶 𝑇 d𝑡 ∝ 𝐻 𝑥𝑦 (𝑡) -𝐻 𝑥𝑦 (-𝑡),

FIGURE 2 :

 2 FIGURE 2: GUIDED WAVE TOMOGRAPHY WITH SIMULATED DATA.

FIGURE 3 :

 3 FIGURE 3: EXPERIMENTAL SETUP.

FIGURE 4 :

 4 FIGURE 4: EXPERIMENTAL ACTIVE CASE


Shu Wu

Philippe Castagliola

Giovanni Celano

A Distribution-Free EWMA Control Chart for Monitoring Time-Between-Events-and-Amplitude Data

Keywords: Distribution-Free, Markov Chain, Statistical Process Monitoring, Time Between Events and Amplitude

During the recent years, many control charts have been developed for the simultaneous monitoring of the time interval T between successive occurrences of an event E and its magnitude X. All these TBEA (Time Between Events and Amplitude) control charts assume a known distribution for the random variables T and X (like, for instance, the normal, lognormal, gamma or Weibull distributions). But, in practice, as it is rather difficult to know the actual distribution of the time T or the amplitude X, proposing a distribution-free approach could be a way to overcome this "distribution choice" dilemma. For this reason, we propose in this paper a distribution-free upper-sided EWMA (Exponentially Weighted Moving Average) type control chart, for simultaneously monitoring the time interval T and the magnitude X of an event. In order to investigate the performance of this control chart and obtain its run length properties, we also develop a specific method called "continuousify" which, coupled with a classical Markov chain technique, allows to obtain reliable and replicable results. A numerical comparison shows that our distributionfree EWMA TBEA chart performs as the parametric Shewhart TBEA chart, but without the need to pre-specify any distribution. An illustrative example obtained from a French forest fire database is also provided to show the implementation of the proposed distribution-free EWMA TBEA control chart.

Introduction

Control charts are undeniably the most powerful tools in SPM (Statistical Process Monitoring) for improving the quality and the productivity of productions. But, during the recent years, several techniques based on control charts have also been developed to monitor processes in non-manufacturing sectors, such as in the health-care sector (like diseases, see Woodall [START_REF] Woodall | The Use of Control Charts in Health-Care and Public-Health Surveillance[END_REF]), the meteorological sector (like extreme weather or climate events, see Vučijak et al. [START_REF] Vučijak | Climate Change Assessment using Statistical Process Control Methods[END_REF]) or the geological sector (like earthquakes or volcanic eruptions, see Santiago and Smith [START_REF] Santiago | Control Charts based on the Exponential Distribution: Adapting Runs Rules for the t Chart[END_REF], Ho [START_REF] Ho | Statistical Control Chart for Regime Identification in Volcanic Time Series[END_REF]). In general, when a particular negative event E is of interest, two important characteristics should be recorded: the time T between two consecutive occurrences of this event and its amplitude X. In CONTACT Philippe Castagliola. Email: philippe.castagliola@univ-nantes.fr most situations, a decrease in T and/or an increase in X will result in a negative, hazardous or disastrous consequence and, therefore, it has to be monitored with dedicated control charts called TBEA (Time Between Events and Amplitude) control charts.

It is worth recalling that Calvin [START_REF] Calvin | Quality Control Techniques for "Zero Defect[END_REF] firstly proposed to monitor the cumulative number of conforming items between two nonconforming ones for improving the performance of traditional attribute control charts for monitoring high-quality processes. Lucas [START_REF] Lucas | Counted Data CUSUM's[END_REF] and Vardeman and Ray [START_REF] Vardeman | Average Run Lengths for CUSUM Schemes when Observations are Exponentially Distributed[END_REF] were at the origin of TBE (Time Between Events) control charts, as they were the first to propose new control charts with the idea of monitoring TBEs data for monitoring high-quality processes. Since then, several TBE control charts (both for phases I and II) have been proposed in the literature. For instance, the TBE exponential chart has been studied by Chan et al. [START_REF] Chan | Cumulative Quantity Control Charts for Monitoring Production Processes[END_REF] and Xie et al. [START_REF] Xie | Some Effective Control Chart Procedures for Reliability Monitoring[END_REF]. Bourke [START_REF] Bourke | The Geometric CUSUM Chart with Sampling Inspection for Monitoring Fraction Defective[END_REF] developed a geometric CUSUM chart for monitoring TBE data. Gan [START_REF] Gan | Exact Run Length Distributions for One-Sided Exponential CUSUM Schemes[END_REF], Borror et al. [START_REF] Borror | Robustness of the Time Between Events CUSUM[END_REF] and Shafae et al. [START_REF] Shafae | Cumulative Sum Control Charts for Monitoring Weibull-Distributed Time Between Events[END_REF] investigated an exponential TBE CUSUM (cumulative sum) control chart. A design procedure for TBE control charts with runs rules has been proposed by Cheng and Chen [START_REF] Cheng | An ARL-Unbiased Design of Time-Between-Events Control Charts with Runs Rules[END_REF]. Qu et al. [START_REF] Qu | Time-Between-Event Control Charts for Sampling Inspection[END_REF] studied some TBE control charts for sampling inspection. Readers can also refer to Shamsuzzaman et al. [START_REF] Shamsuzzaman | Integrated Control Chart System for Time-Between-Events Monitoring in a Multistage Manufacturing System[END_REF], Zhang et al. [START_REF] Zhang | Economic Design of Time-Between-Events Control Chart System[END_REF], Zhang et al. [START_REF] Zhang | Design and Application of Exponential Chart for Monitoring Time-Between-Events Data under Random Process Shift[END_REF], Fang et al. [START_REF] Fang | Monitoring of Time-Between-Events with a Generalized Group Runs Control Chart[END_REF].

In some applications, it is clearly important to monitor the time between events but also the amplitudes associated with these events. Recent enhancements to the statistical monitoring of an event E, not only quantified by its time T between two consecutive events but also by its amplitude X, have been introduced in the literature and they have been called TBEA (Time Between Events and Amplitude) control charts. Wu et al. [START_REF] Wu | A Control Scheme for Monitoring the Frequency and Magnitude of an Event[END_REF] were the first to propose a combined T /X control chart based on a T chart to monitor the time interval and on a X chart to monitor the amplitude. From that moment, several other TBEA control charts have been developed, see Wu et al. [START_REF] Wu | A Single Control Chart for Monitoring the Frequency and Magnitude of an Event[END_REF], Wu et al. [START_REF] Wu | A Cumulative Sum Scheme for Monitoring Frequency and Size of an Event[END_REF], Qu et al. [START_REF] Qu | A Cusum Scheme for Event Monitoring[END_REF], Qu et al. [START_REF] Qu | A Cusum Chart for Detecting the Intensity Ratio of Negative Events[END_REF], Ali and Pievatolo [START_REF] Ali | Time and Magnitude Monitoring based on the Renewal Reward Process[END_REF], Rahali et al. [START_REF] Rahali | Evaluation of Shewhart Time-Between-Events-and-Amplitude Control Charts for Several Distributions[END_REF].

All existing methods mentioned above for monitoring TBEA data are parametric, i.e. they assume that the distributions of both T and X are perfectly known. However, in many practical situations, the distributions of these random variables are unknown or their parameters cannot be correctly estimated by means of a Phase I retrospective study. This has been studied by Qiu [START_REF] Qiu | Introduction to Statistical Process Control[END_REF], who has shown that using parametric control charts is not a reliable solution when the validity of the distribution is in question. In the specific case of monitoring TBEA data, Rahali et al. [START_REF] Rahali | Evaluation of Shewhart Time-Between-Events-and-Amplitude Control Charts for Several Distributions[END_REF] investigated the use of parametric approaches in order to monitor "fires in forests". They experienced difficulties in selecting the most suitable distribution for the time T and the amplitude X due the limited number of Phase I data and the small differences in terms of the statistic (Kolmogorov-Smirnov) measuring the quality of the fit (making several distributions being a possible candidate, such as the gamma, lognormal, normal and Weibull distributions). It is well known that a fitting error on the distribution of observations can result in a poor in-control performance of the control chart. To overcome this problem, distribution-free control charts have been investigated in the literature. Among the most recent ones, we can cite Celano et al. [START_REF] Celano | The Performance of the Shewhart Sign Control Chart for Finite Horizon Processes[END_REF] who investigated the statistical performance of a Shewhart sign control chart in a process with a finite production horizon, Abid et al. [START_REF] Abid | Use of Ranked Set Sampling in Nonparametric Control Charts[END_REF] who proposed a nonparametric EWMA control chart based on the sign test using RSS (Ranked Set Sampling), Castagliola et al. [START_REF] Castagliola | An EWMA-Type Sign Chart with Exact Run Length Properties[END_REF] who proposed a new Phase II EWMA-type chart for count data based on the sign statistic, Abid et al. [START_REF] Abid | An Efficient Nonparametric EWMA Wilcoxon Signed-Rank Chart for Monitoring Location[END_REF] who introduced a nonparametric EWMA control chart based on the Wilcoxon signed-rank statistic using ranked set sampling, Abid et al. [START_REF] Abid | Investigating the Impact of Ranked Set Sampling in Nonparametric CUSUM Control Charts[END_REF][START_REF] Abid | On Designing a New Cumulative Sum Wilcoxon Signed Rank Chart for Monitoring Process Location[END_REF] who suggested nonparametric CUSUM sign and Wilcoxon signed-rank control charts for monitoring and detecting possible deviations from the process mean using ranked set sampling. Interested readers can find a comprehensive discussion in the very recent review of Chakraborti and Graham [START_REF] Chakraborti | Nonparametric (Distribution-free) Control Charts: an Updated Overview and Some Results[END_REF] who discuss many Phase I and Phase II distribution-free control charts and give some suggestions for next research directions. Practical guidelines for the distribution-free control charts implementation can be found in the books of Qiu [START_REF] Qiu | Introduction to Statistical Process Control[END_REF] and Chakraborti and Graham [START_REF] Chakraborti | Nonparametric Statistical Process Control[END_REF].

But, as far as we know, no research has been conducted so far on proposing a distribution-free control chart for monitoring TBEA data. This will be the main and first goal of this paper where a new upper-sided distribution-free EWMA control chart for monitoring TBEA data will be introduced. As evaluating Run Length related values (ARL, SDRL, ...) for an EWMA scheme based on discrete data is a challenging problem, the second goal of this paper will consist in proposing a dedicated method, called "continuousify", which allows reliable and replicable results to be obtained.

The structure of this paper is as follows. In Section 2, the distribution-free statistic to be monitored as well as the "continuousify" technique used for computing the Run Length properties are both introduced. Then, in Section 3 the optimal design of the proposed distribution-free chart is presented. In Section 4 a comparison with parametric Shewhart TBEA charts proposed in Rahali et al. [START_REF] Rahali | Evaluation of Shewhart Time-Between-Events-and-Amplitude Control Charts for Several Distributions[END_REF] is performed. A real case example implementing the new upper-sided distribution-free EWMA TBEA control chart is presented in Section 5 and conclusions and future researches are given in Section 6.

A distribution-free EWMA TBEA control chart

Let D 0 = 0, D 1 , D 2 , . . . be the dates of occurrence of a specific negative event E, let T 1 = D 1 -D 0 , T 2 = D 2 -D 1 , . . . be the time intervals between two consecutive occurrences of the event E and let X 1 , X 2 , . . . be the corresponding magnitudes of this event occurring at times D 1 , D 2 , . . . and assumed to be independent of T 1 , T 2 , . . . (see Figure 1). It must be noted that D 0 = 0 is the date of a "virtual" event which has no amplitude associated with.

Let F T (t|θ T ) and F X (x|θ X ) be the unknown continuous c.d.f. (cumulative distribution functions) of T i and X i , i = 1, 2, . . ., where θ T and θ X are known α-quantiles, respectively. More precisely, when the process is in-control, we have θ T = θ T0 , θ X = θ X0 and, when the process is out-of-control, we have θ T = θ T1 , θ X = θ X1 . Without loss of generality, we will consider in this paper that θ T and θ X are the median values (i.e. the 0.5-quantiles) of T i and X i , respectively. Other α-quantiles can be considered based on the investigated event's severity of consequences.

Let p T = P(T i > θ T0 |θ T ) = 1 -F T (θ T0 |θ T ) and p X = P(X i > θ X0 |θ X ) = 1 -F X (θ X0 |θ X ), i = 1,
2, . . ., be the probabilities that T i and X i are larger than θ T0 and θ X0 assuming that the actual median values are θ T and θ X , respectively. If the process is in-control, we have p T = p T0 = 1 -F T (θ T0 |θ T0 ) = 0.5, p X = p X0 = 1 -F X (θ X0 |θ X0 ) = 0.5 and, when the process is out-of-control, we have

D 0 = 0 D 1 D 2 D 3 • • • • • • • • • T 1 T 2 T 3 X 1 X 2 X 3 Figure 1.
Times of occurrence D i , time intervals T i and amplitudes X i of a negative event E

p T = p T1 = 1 -F T (θ T0 |θ T1 ), p X = p X1 = 1 -F X (θ X0 |θ X1 ).
Let us define the statistics ST i and SX i , for i = 1, 2, . . . as

ST i = sign(T i -θ T0 ), SX i = sign(X i -θ X0 ),
where sign(x) = -1 if x < 0 and sign(x) = +1 if x > 0. Because T i and X i are assumed to be continuous random variables, the unlikely case x = 0 will not be considered in the definition of the charting statistic. Nevertheless, as sometimes ties may occur in practice (for example, as a consequence of the selected time unit for T i ), we will explain in the "Illustrative example" section how this situation can be handled by practitioners without significantly affecting the final result. In order to simultaneously monitor, in a distribution-free way, the time interval T i between consecutive occurrences of the event E and its magnitude X i , we suggest to define the statistic S i , for i = 1, 2, . . . as

S i = SX i -ST i 2 .
By definition, we have S i ∈ {-1, 0, +1} and, more precisely, we have:

• S i = -1 when the process is in an acceptable situation, i.e. when T i increases (ST i = +1) and, at the same time, X i decreases (SX i = -1). • S i = +1 when the process is in an unacceptable situation, i.e. when T i decreases (ST i = -1) and, at the same time, X i increases (SX i = +1). • S i = 0 when the process is in an intermediate situation, i.e. when both T i and X i increase or when both T i and X i decrease.

It is easy to prove that the p.m.f. (probability mass function)

f Si (s|p T , p X ) = P(S i = s|p T , p X ) of S i is equal to f Si (s|p T , p X ) =        p T q X if s = -1 p T p X + q T q X if s = 0 q T p X if s = +1 0 if s ∈ {-1, 0, 1}
, where q T = 1p T , q X = 1p X and its c.d.f. F Si (s|p T , p X ) = P(S i ≤ s|p T , p X ) is equal to

F Si (s|p T , p X ) =        0 if s ∈ (-∞, -1) p T q X if s ∈ [-1, 0) p T + q T q X if s ∈ [0, 1) 1 if s ∈ [1, +∞)
.

In practice, it is actually always possible to define and implement an EWMA TBEA type control chart directly monitoring the statistic S i using an equation like

Z i = λS i + (1 -λ)Z i-1
, where λ ∈ [0, 1] is some smoothing parameter to be fixed and Z 0 = 0. The problem of this approach is that, because of the discrete nature of the random variable S i , it is impossible to accurately compute (using Markov chain or integral equation methods, for instance) the run length properties (average run length ARL and standard deviation of the run length SDRL) of such a control chart and, therefore, it is impossible to tune the chart parameters in order to obtain a predefined in-control performance. If, for instance, the Markov chain approach, (as detailed hereafter), is used in order to compute the ARL or the SDRL, the results will i) heavily fluctuate depending on the value of the selected number m of subintervals and ii) not exhibit any monotonic convergence when m increases, making useless such an approach. This point will be highlighted at the end of this section. Of course, it is always possible to obtain these values using simulations but, even in this case, if it is quite easy to compute small ARL or SDRL values with some precision, it becomes just impossible to obtain reliable results when these values become very large.

Since the Markov chain and integral equation methods give good results in the case of continuous random variables, (and more particularly in the case of the normal distribution, which is an unbounded one), we therefore suggest to transform the discrete random variable S i into a new continuous one, denoted as S * i , (say that we "continuousify" the random variable S i ), and to monitor it using a traditional EWMA scheme. We suggest to define the statistic S * i as a mixture of 3 normal random variables Y i,-1 ∼ Nor(-1, σ), Y i,0 ∼ Nor(0, σ) and Y i,+1 ∼ Nor(+1, σ), with weights w -1 = p T q X , w 0 = p T p X + q T q X and w +1 = q T p X (corresponding to the probabilities f Si (s|p T , p X ), s ∈ {-1, 0, +1}), respectively, i.e.

S * i =    Y i,-1 if S i = -1, Y i,0 if S i = 0, Y i,+1 if S i = +1.
Concretely speaking, this means that if, at i = 1, 2, . . ., we have S i = s ∈ {-1, 0, +1} then, in order to obtain S * i , we just have to generate a Nor(s, σ) random number to "continuousify" the random variable S i . The fact that random numbers have to be generated does not imply that the Run Length properties (ARL, SDRL, ...) are obtained using simulations. As shown below, the Run Length properties of the uppersided distribution-free EWMA TBEA control chart are obtained with an exact Markov chain based method. But, in order to use this approach, it is necessary to assume that the discrete random variables S i have been transformed into continuous ones. This is why random numbers are generated. The parameter σ > 0 has to be fixed and, as it will be shown later, its value does not significantly affect the performance of the control chart as long it is neither too small nor too large. Since S * i is defined as a mixture of normal distributions, its c.d.f.

F S * i (s|p T , p X ) = P(S * i ≤ s|p T , p X ) is equal to F S * i (s|p T , p X ) =p T q X F Nor (s| -1, σ) + (p T p X + q T q X )F Nor (s|0, σ) + q T p X F Nor (s| + 1, σ) (1)
where F Nor (s|µ, σ) is the c.d.f. of the normal Nor(µ, σ) distribution. As an example, we plotted in Figure 2 the p.d.f. of S * i when the process is in-control, (i.e. the weights are w -1 = 0.25, w 0 = 0.5 and w +1 = 0.25), for σ ∈ {0.1, 0.125, 0.15, 0.2}. We suggest to only investigate this range of values as it seems that when σ < 0.1 the "peaks" around {-1, 0, +1} become too sharp and when σ > 0.2 these "peaks" become too smooth. It is not difficult to demonstrate that the expectance E(S * i ) and variance The in-and out-of-control c.d.f., expectance and variance of S * i can be simply obtained by replacing, in the previous equations, p T and p X by either p T0 and p X0 or p T1 and p X1 , respectively. In particular, if the process is in-control, we have p T0 = q T0 = 0.5, p X0 = q X0 = 0.5 and the expectance and variance of S * i simplify to E(S * i ) = 0 and V(S * i ) = σ 2 + 0.5.

V(S * i ) of S * i are equal to E(S * i ) = p X -p T , V(S * i ) = σ 2 + p T q T + p X q X . σ = 0.1 σ = 0.125 0 0.5 1 1.5 2
As it is more important to detect an increase in S i or S * i (in order to avoid more damages or injuries/costs, for instance) rather than a decrease, we suggest to define the following upper-sided EWMA TBEA control chart based on the statistic

Z * i = max(0, λS * i + (1 -λ)Z * i-1 ), (2)
with the following upper asymptotic control limit UCL defined as

UCL = E(S * i ) =0 +K λ 2 -λ × V(S * i ) = √ σ 2 +0.5 = K λ(σ 2 + 0.5) 2 -λ , (3)
where λ ∈ [0, 1] and K > 0 are the control chart parameters to be fixed and the initial value Z * 0 = 0.

In order to obtain the zero-state ARL and SDRL of the proposed distribution-free upper-sided EWMA TBEA control chart, we suggest to use the standard approach proposed by Brook and Evans [START_REF] Brook | An Approach to the Probability Distribution of CUSUM Run Length[END_REF], which assumes that the behavior of this control chart can be well represented by a discrete-time Markov chain with m + 2 states, where states i = 0, 1, . . . , m are transient and state m + 1 is an absorbing one. The transition probability matrix P of this discrete-time Markov chain is

P =   Q r 0 1   =        Q 0,0 Q 0,1 • • • Q 0,m r 0 Q 1,0 Q 1,1 • • • Q 1,m r 1 . . . . . . . . . . . . Q m,0 Q m,1 • • • Q m,m r m 0 0 • • • 0 1       
, where Q is the (m + 1, m + 1) matrix of transient probabilities, where 0 = (0, 0, . . . , 0) and where the (m + 1, 1) vector r satisfies r = 1 -Q1 (i.e. row probabilities must sum to 1) with 1 = (1, 1, . . . , 1) . The transient states i = 1, . . . , m are obtained by dividing the interval [0, UCL] into m subintervals of width 2∆, where ∆ = UCL 2m . By definition, the midpoint of the i-th subinterval (representing state i) is equal to H i = (2i -1)∆. The transient state i = 0 corresponds to the "restart state" feature of our chart (due to the presence of the max(. . . ) in ( 2)). This state is represented by the value H 0 = 0. Concerning the proposed upper-sided EWMA TBEA control chart, it can be easily proven that the generic element Q i,j , i = 0, 1, . . . , m, of the matrix Q is equal to

• if j = 0, Q i,0 = F S * i - (1 -λ)H i λ p T , p X , (4)
• if j = 1, 2, . . . , m, Q i,j = F S * i H j + ∆ -(1 -λ)H i λ p T , p X -F S * i H j -∆ -(1 -λ)H i λ p T , p X (5)
Let q = (q 0 , q 1 , . . . , q m ) be the (m + 1, 1) vector of initial probabilities associated with the m + 1 transient states. In our case, we assume q = (1, 0, . . . , 0) , i.e. the initial state corresponds to the "restart state". When the number m of subintervals is sufficiently large (say m = 300), this finite approach provides an effective method that allows the ARL and SDRL to be accurately evaluated using the following classical formulas

ARL(p X , p T ) = q (I -Q) -1 1, (6)
SDRL(p X , p T ) = 2q (I -Q) -2 Q1 + ARL(1 -ARL). (7)
In order to clearly illustrate, for the proposed upper-sided EWMA TBEA control chart, the difference between using or not the suggested "continuousify" technique, we present in Table 1 the ARL values obtained for several combinations of (p X , p T ), m ∈ {100, 120, . . . , 400} and σ = 0.125. In Table 1, we also provide ARL values obtained by simulations (last row of Table 1). Based on Table 1, the following conclusions can be drawn:

• when the "continuousify" technique is not used, (see the left side of Table 1, denoted as "without continuousify"), the ARL values obtained using the Markov chain method have a large variability with m; furthermore, they do not show any visible monotonic convergence when m increases. The worst case is for (p X = 0.7, p T = 0.4) for which some ARL values are even negative! This phenomenon is known to happen even in the case of continuous random variables when the smoothing parameter λ is too small. In this case, the Markov chain approach does not converge and provide meaningless (i.e. either negative or too large) ARL values. The fact that the random variables are discrete makes this phenomenon even stronger due to the fact that the probabilities in ( 4) and ( 5) are not necessarily continuous / smooth. For the remaining combinations (p X , p T ) the fluctuation is noticeable with a particularity for m = 260 which gives (for some unclear reason) larger ARL values, if compared to the others. • when the "continuousify" technique is used, (see the right side of Table ). This is logical as the control limits "with continuousify" are a bit larger than those "without continuousify" due to the extra term σ > 0 in (3).

Numerical analysis

The goal of this section is twofold:

(1) obtaining optimal values (λ * , K * ) for the upper-sided EWMA TBEA control chart parameters (λ, K) as to minimize the out-of-control ARL(λ * , K * , σ, p T , p X ) for p T = 0.5 and p X = 0.5 under the constraint ARL(λ * , K * , σ, 0.5, 0.5) = ARL 0 , where ARL 0 is a predefined value for the in-control ARL; (2) demonstrating that the choice of the parameter σ does not significantly impact the out-of-control performance of this chart as long as this value is not too small nor too large.

The optimal values for (λ * , K * ) are listed in Table 2 with the corresponding out-of-Table 1. ARL for the distribution-free EWMA TBEA chart computed with and without the "continuousify" technique.

"without continuousify" "with continuousify" (σ = 0.125)

m p T = 0.4 p T = 0.3 p T = 0.2 p T = 0.1 p T = 0.4 p T = 0.3 p T = 0.2 p T = 0.1 p X = 0.7
p X = 0.8 p X = 0.9 p X = 0.6 p X = 0.7 p X = 0.8 p X = 0.9 p X = 0. control values of (ARL, SDRL) for p T ∈ {0.1, 0.2, . . . , 0.4} (as we are only interested in a decrease in T ), p X ∈ {0.5, 0.6, . . . , 0.9} (as we are only interested in an increase in X), for four possible choices for σ ∈ {0.1, 0.125, 0.15, 0.2} and assuming ARL 0 = 370.4.

For instance, in Table 2, when σ = 0.125, p T = 0.4 and p X = 0.6 the optimal chart parameters are (λ * , K * ) = (0.025, 2.174) and the corresponding values for the out-ofcontrol (ARL, SDRL) are ARL = 51.11 and SDRL = 32.63. From Table 2 we can draw the following conclusions:

• No matter the value of σ, when p T = p X = 0.5 we exactly obtain ARL = ARL 0 = 370.4 (as expected). In this case, it exists an infinite number of couples (λ * , K * ) exactly satisfying the constraint ARL = ARL 0 = 370.4. These couples are denoted with "(-,-)". It has to be noted that without the "continuousify" technique used in this paper, it would have been impossible to exactly obtain ARL = ARL 0 = 370.4 due to the discrete nature of the random variable S i . • No matter the value of σ, the out-of-control ARL values monotonically decrease when the values of p T decrease and/or the values of p X increase. Due to the symmetry of ST i and SX i in the definition of the random variable S i , the performance of the distribution-free upper-sided EWMA TBEA chart is the same for any combination of (p T = α T , p X = α X ) or (p T = 1-α X , p X = 1-α T ) where α T and α T are two probabilities in [0, 1]. For this reason, only the lower side of each table is presented, being the upper side immediately be derived by symmetry. For example, if σ = 0.125, the optimal parameters (λ * , K * ) and corresponding out-of-control ARL and SDRL for p T = 0.4 and p X = 0.7 are the same as the ones for p T = 0.

Comparative studies

The goal of this section is to compare the proposed upper-sided distribution-free EWMA TBEA chart with the three parametric Shewhart type control charts introduced in Rahali et al. [START_REF] Rahali | Evaluation of Shewhart Time-Between-Events-and-Amplitude Control Charts for Several Distributions[END_REF] based on statistics Z 1 , Z 2 and Z 3 . It is important to note that these statistics depend on standardized versions X = X µX 0 and T = T µT 0 of X and T , respectively, where µ X0 and µ T0 are the in-control mean values for X and T . The 2-parameters distributions considered in Rahali et al. [START_REF] Rahali | Evaluation of Shewhart Time-Between-Events-and-Amplitude Control Charts for Several Distributions[END_REF] were i) the gamma, lognormal, normal and Weibull distributions for the amplitude X and ii) the gamma, lognormal and Weibull distributions for the time between events T , leading to a combination of 11 scenarios. For more details concerning the definition of statistics Z 1 , Z 2 and Z 3 and the parametrization of these distributions, do refer to Rahali et al. [START_REF] Rahali | Evaluation of Shewhart Time-Between-Events-and-Amplitude Control Charts for Several Distributions[END_REF]. In this paper, we will only investigate two scenarios:

• Scenario #1: a Normal distribution for X with in-control mean µ X0 = 10 and standard-deviation σ X0 = 1 and a gamma distribution for T with in-control mean µ T0 = 10 and standard-deviation σ T0 = 2, i.e. X ∼ Nor(10, 1) and T ∼ Gam(25, 0.4). • Scenario #2: a Normal distribution for X with in-control mean µ X0 = 10 and standard-deviation σ X0 = 2 and a Weibull distribution for T with in-control mean µ T0 = 10 and standard-deviation σ T0 = 1, i.e. X ∼ Nor(10, 2) and T ∼ Wei(12.1534, 10.4304).

In this parametric framework, when an upper shift is occurring, it can be due to: i) either a mean shift in the amplitude X from µ X0 to µ X1 = δ X µ X0 where δ X ≥ 1 is the parameter quantifying the change in the amplitude, ii) or a mean shift in the time T from µ T0 to µ T1 = δ T µ T0 where δ T ≤ 1 is the parameter quantifying the change in the time, ii) or also a change in both the amplitude X from µ X0 to µ X1 = δ X µ X0 and the time T from µ T0 to µ T1 = δ T µ T0 .

As it is usually very difficult to know the actual values of δ X and δ T , we will use the Expected Average Run Length (EARL) criterion and, more particularly:

• the EARL X for X (assuming δ T = 1) defined as:

EARL X = δX ∈ΩX f δX (δ X )ARL(δ X , 1)
• the EARL T for T (assuming δ X = 1) defined as:

EARL T = δT ∈ΩT f δT (δ T )ARL(1, δ T )
• the EARL XT for both of X and T defined as:

EARL XT = δX ∈ΩX δT ∈ΩT f δX (δ X )f δT (δ T )ARL(δ X , δ T )
where Ω X and Ω T are the "range of possible shifts" for δ X and δ T , respectively, f δX (δ X ) and f δT (δ T ) are the p.m.f. (probability mass functions) of the shifts δ X and δ T over Ω X and Ω T , respectively. Since the goal of a TBEA control chart is to detect an increase in the amplitude X and/or a decrease in the time between events T , we suggest to define Ω X = {1.05, 1.1, 1.15, 1.2, 1.25, 1.3} and Ω T = {0.7, 0.75, 0.8, 0.85, 0.9, 0.95}. As the distributions of δ X and δ T are unknown, we assume that f δX (δ X ) and f δT (δ T ) are the p.m.f. of discrete uniform distributions on Ω X and Ω T , respectively.

In the parametric framework, the values EARL X , EARL T and EARL XT depend on the values ARL(δ X , 1), ARL(1, δ T ) and ARL(δ X , δ T ) that can be computed using formulas presented in Rahali et al. [START_REF] Rahali | Evaluation of Shewhart Time-Between-Events-and-Amplitude Control Charts for Several Distributions[END_REF]. Concerning the distribution-free upper-sided EWMA TBEA chart, the same formulas for EARL X , EARL T and EARL XT can be used with the difference that, for each parametric scenario, the values of δ X and δ T have to be transformed into equivalent probabilities p X and p T and the values ARL(δ X , 1), ARL(1, δ T ) and ARL(δ X , δ T ) have to be replaced by ARL(p X , 0.5), ARL(0.5, p T ) and ARL(p X , p T ), that can be computed using [START_REF] Borror | Robustness of the Time Between Events CUSUM[END_REF]. This allows a direct comparison between the parametric methods proposed in Rahali et al. [START_REF] Rahali | Evaluation of Shewhart Time-Between-Events-and-Amplitude Control Charts for Several Distributions[END_REF] and the distribution-free method proposed in this paper.

The results of EARL X , EARL T and EARL XT for the distribution-free upper-sided EWMA TBEA control chart are in Table 3 (see the values in bold) for both scenarios #1 and #2, i.e.

• EARL X = 24.91, EARL T = 45.08 and EARL XT = 10.49 for scenario #1,

• EARL X = 44.30, EARL T = 23.54 and EARL XT = 9.93 for scenario #2.

In Table 3, we also have the value of p X and p T (which depend on the scenario) corresponding to the values δ X ∈ Ω X = {1.05, 1.1, 1.15, 1.2, 1.25, 1.3} and δ T ∈ Ω T = {0.7, 0.75, 0.8, 0.85, 0.9, 0.95}, respectively. For each of these combinations (δ X , δ T ) or (p X , p T ) we have the optimal parameters (λ * , K * ) with the corresponding values for (ARL, SDRL). For example, in scenario #1, the combination (δ X = 1.2, δ T = 0.9) corresponds to (p X = 0.9773, p T = 0.3105) and the optimal parameters for the distribution-free upper-sided EWMA TBEA control chart are (λ * = 0.15, K * = 2.643) with (ARL = 9.78, SDRL = 4.23).

The values of EARL X , EARL T and EARL XT for the 3 parametric Shewhart control charts proposed in Rahali et al. [START_REF] Rahali | Evaluation of Shewhart Time-Between-Events-and-Amplitude Control Charts for Several Distributions[END_REF] (based on statistics Z 1 , Z 2 and Z 3 ) are presented in Table 4 for scenarios #1 and #2. The upper control limits used for each case have also been recorded in this table. A comparison between Tables 3 and4 immediately shows that, no matter the scenario or the statistic considered, the values of EARL X , EARL T and EARL XT for the distribution-free upper-sided EWMA TBEA control chart are always smaller than the ones obtained for the parametric Shewhart control charts proposed in Rahali et al. [START_REF] Rahali | Evaluation of Shewhart Time-Between-Events-and-Amplitude Control Charts for Several Distributions[END_REF], thus showing the advantage of using the proposed distribution-free control chart in situations where the distributions for T and X are unknown. 2017, (see also the bold values in Table 5), confirming that a decrease in the time between fires occurred with a concurrent increase in the amplitude of these fires. Similar conclusions have been obtained using the parametric approaches in Rahali et al. [START_REF] Rahali | Evaluation of Shewhart Time-Between-Events-and-Amplitude Control Charts for Several Distributions[END_REF] and assuming a lognormal distribution for both T i and X i . • It is interesting to investigate the robustness of the Z * i after the generation of new normal random values for S * i . In this case, are the new Z * i totally different from the others? Is it possible to detect the same out-of-control situations? How robust is the "continuousify" method if it is replicated several times? In order to answer these questions, we plotted, in Figure 4, 10 replicated sequences of Z * i corresponding to the same Phase I and II fires example. The D i , T i , X i , ST i , SX i and S i are exactly the same as in Table 5: only the S * i have been randomly generated and the Z * i recomputed. As it can be seen, the 10 trajectories are quite similar and none of them significantly diverges from the others. During the Phase I they are all below the UCL (confirming that the process is actually in-control) and during the Phase II, they all exhibit out-of-control situations more or less at the same moments (the "peaks" occur almost at the same time). Therefore, we can conclude that the "continuousify" method is robust vs. the random generation of the S * i values.

Conclusions

In this paper we have investigated a distribution-free EWMA control chart based on sign statistics for monitoring time between events and amplitude data. Implementing a distribution-free control chart allows the problem of estimating the in-control distributions of time between events and amplitudes to be overcome. Only a couple of selected quantiles from the in-control distributions need to be estimated to start the implementation of the EWMA TBEA control chart. Using the proposed distribution-free EWMA TBEA control chart allows to efficiently monitor TBEA data with an out-of-control performance significantly better than any parametric Shewhart TBEA control charts currently available in literature. Being the EWMA TBEA control chart based on a discrete sample statistic, we have defined a technique, called "continuousify", which allows to compute the ARL values for the EWMA scheme by using Markov chains in a reliable and replicable way.

Future research in the same area will consider the case of possible ties in the data, i.e. situations where ST i = 0 or SX i = 0 (like it happens in the Illustrative example). In this case, the distribution of S i and S * i are no longer the same and this new situation is worth to be investigated. Furthermore, this technique can be extended to any kind of discrete distribution like the Poisson or the binomial distribution and zero-inflated versions of these distributions, for instance. It could also be adapted to work with the multivariate version of the EWMA for multivariate discrete data like multivariate Poisson data.

D i T i X i ST i SX i S i S * i Z * i D i T i X i ST i SX i S i S * i Z * i 9 9

Figure 2 .

 2 Figure 2. p.d.f. of S * i when the process is in-control for σ ∈ {0.1, 0.125, 0.15, 0.2}

  3 and p X = 0.6, i.e. (λ * = 0.045, K * = 2.387), ARL = 30.79 and SDRL = 18.25. • As long as σ ∈ {0.1, 0.125, 0.15, 0.2}, the optimal design parameters (λ * , K * ) and the out-of-control ARL and SDRL values are almost the same. For instance, if p T = 0.3 and p X = 0.6, then the optimal parameters are (λ * = 0.045, K * = 2.387) (irrespective of the value of σ) and the out-of-control ARL and SDRL values are (ARL = 30.55, SDRL = 18.04), (ARL = 30.79, SDRL = 18.25), (ARL = 31.08, SDRL = 18.51) and (ARL = 31.81, SDRL = 19.15) when

Figure 4 .

 4 Figure 4. 10 new trajectories for the Z * i based on randomly regenerated S * i values corresponding to the data set in Table 5

Table 2 .

 2 Optimal values for (λ * , K * ) with the corresponding out-of-control values of (ARL, SDRL) for p T ∈ {0.1, 0.2, . . . , 0.4}, p X ∈ {0.5, 0.6, . . . , 0.9} and σ ∈

	{0.1, 0.125, 0.15, 0.2}		σ = 0.1
				p X
	p T	0.5	0.6	0.7	0.8	0.9
	0.5	(-,-)
		(370.40,-)
	0.4	(0.010,1.773)	(0.025,2.174)
		(105.66,74.04) (50.77,32.32)
	0.3	(0.025,2.174)	(0.045,2.387) (0.070,2.515)
		(51.54,32.55)	(30.55,18.04) (20.50,11.38)
	0.2	(0.040,2.348)	(0.070,2.515) (0.100,2.591) (0.145,2.639)
		(31.30,17.51)	(20.74,11.40)	(14.85,7.67)	(11.19,5.55)
	0.1	(0.060,2.474)	(0.090,2.571) (0.135,2.634) (0.180,2.645) (0.240,2.627)
		(21.40,10.76)	(15.16,7.37)	(11.32,5.40)	(8.76,3.84)	(6.99,2.74)
				σ = 0.125
				p X
	p T	0.5	0.6	0.7	0.8	0.9
	0.5	(-,-)
		(370.40,-)
	0.4	(0.010,1.774)	(0.025,2.174)
		(106.19,74.55) (51.11,32.63)
	0.3	(0.025,2.174)	(0.045,2.387) (0.070,2.515)
		(51.88,32.87)	(30.79,18.25) (20.68,11.53)
	0.2	(0.040,2.348)	(0.065,2.496) (0.100,2.592) (0.140,2.638)
		(31.53,17.72)	(20.91,11.27)	(14.99,7.80)	(11.32,5.57)
	0.1	(0.060,2.474)	(0.090,2.572) (0.135,2.634) (0.175,2.648) (0.225,2.639)
		(21.57,10.92)	(15.30,7.50)	(11.44,5.49)	(8.88,3.89)	(7.10,2.75)
				σ = 0.15
				p X
	p T	0.5	0.6	0.7	0.8	0.9
	0.5	(-,-)
		(370.40,-)
	0.4	(0.010,1.775)	(0.025,2.175)
		(106.83,75.16) (51.53,33.01)
	0.3	(0.025,2.175)	(0.045,2.387) (0.070,2.515)
		(52.30,33.26)	(31.08,18.51) (20.90,11.71)
	0.2	(0.040,2.348)	(0.065,2.496) (0.095,2.584) (0.135,2.636)
		(31.82,17.97)	(21.13,11.45)	(15.17,7.81)	(11.47,5.61)
	0.1	(0.055,2.449)	(0.090,2.573) (0.130,2.632) (0.170,2.651) (0.215,2.646)
		(21.79,10.76)	(15.47,7.64)	(11.59,5.53)	(9.02,3.96)	(7.23,2.80)
				σ = 0.2
				p X
	p T	0.5	0.6	0.7	0.8	0.9
	0.5	(-,-)
		(370.40,-)
	0.4	(0.010,1.777)	(0.025,2.176)
		(108.43,76.68) (52.57,33.96)
	0.3	(0.020,2.085)	(0.045,2.387) (0.065,2.496)
		(53.33,32.51)	(31.81,19.15) (21.44,11.90)
	0.2	(0.040,2.348)	(0.065,2.496) (0.090,2.574) (0.125,2.630)
		(32.53,18.61)	(21.66,11.92)	(15.60,8.01)	(11.84,5.74)
	0.1	(0.055,2.449)	(0.085,2.562) (0.120,2.624) (0.155,2.652) (0.195,2.658)
		(22.31,11.21)	(15.89,7.83)	(11.95,5.64)	(9.34,4.06)	(7.53,2.91)

Table 4 .

 4 Values EARL X , EARL T and EARL XT of the 3 parametric Shewhart control charts proposed in Rahali et al.[START_REF] Rahali | Evaluation of Shewhart Time-Between-Events-and-Amplitude Control Charts for Several Distributions[END_REF] for scenarios #1 and #2

			Scenario #1			Scenario #2
		Z 1	Z 2	Z 3	Z 1	Z 2	Z 3
	UCL	0.5550	1.9692	2.9115	0.5470	1.6742	2.6171
	EARL X	52.4749 93.1968 134.3478	85.9319 121.0851 110.8255
	EARL T	55.3093 44.5776	41.6612	69.5180	57.4231	57.5436
	EARL XT	10.9375 14.8817	18.9865	18.8108	20.8753	20.5259

Table 5 .

 5 Phase I and II values of D i , T i , X i , ST i , SX i , S i , S * i and Z * i for the forest fires example.

	Phase 1

  Figure 3. Time T i in days between fires (top), amplitudes X i as the burned surface in ha (middle) and the distribution-free EWMA TBEA chart with statistic Z * i (bottom) corresponding to the data set in Table 5

				0.5								Phase 1						Phase 2
				40								Phase 1					Phase 2
				0.4
				35
																			UCL
				30 0.3
			Z * i	25
				20 0.2 T i
				3.68 15 0.1		1	-1	-1.0 -0.917 0.000		258	1	1.00	-1	-1	0.0	-0.078	0.000
	26	17		1.99 10		1	-1	-1.0 -0.802 0.000		260	2	3.70	-1	-1	0.0	0.119	0.008
	60	34		6.00			1		1	0.0	-0.081 0.000		262	2	3.17	-1	-1	0.0	-0.063	0.003
	67 70 72 86	7 3 2 14	1.19 135.80 5 0 Oct 14.37 8.10 Oct -1 1 0 1 0 16	-1 1 1 1 Nov Nov 16	-1.0 -0.901 0.000 0.5 0.552 0.039 Dec Jan Feb Mar 1.0 1.113 0.114 0.0 -0.104 0.099 Dec Jan Feb Mar 16 17 17 17	265 268 Apr 269 271 Apr 17	3 3 May 1 2 May 17	18.40 1.00 Jun 2.22 19.09 Jun 17	Jul 17	0 0 -1 -1 Jul	1 -1 -1 1 Aug Aug 17	0.5 -0.5 -0.145 0.333 Sep 0.0 0.208 1.0 1.001 Sep 17	0.026 0.014 0.096 0.028
	88	2		32.31	16	-1	16	1	1.0 16	0.892 17	0.154 17 17	272 17	1 17	2.00 17	-1 17	-1 17	0.0 17	0.027	0.091
	94	6		3.07			1	-1	-1.0 -1.056 0.069		274	2	34.28	-1	1	1.0	1.086	0.161
	95	1		10.03			-1		1	1.0	0.867	0.125		276	2	3.00	-1	-1	0.0	0.070	0.154
	96	1		7.93 4000		-1		1	1.0	1.033	0.189 Phase 1		277	1	6.63	-1	1	1.0 Phase 2	0.955	0.210
	97	1		1.50			-1	-1	0.0	0.409	0.204		278	1	4.47	-1	-1	0.0	-0.097	0.189
	103	6		23.30 3500		1		1	0.0	-0.116 0.182		285	7	8.24	1	1	0.0	0.160	0.187
	106	3		3.73			0	-1	-0.5 -0.708 0.120		286	1	769.45	-1	1	1.0	1.024	0.246
	109	3		4.73 3000		0	-1	-0.5 -0.677 0.064		287	1	4.37	-1	-1	0.0	-0.144	0.218
	111	2		3.19			-1	-1	0.0	0.179	0.072		288	1	90.70	-1	1	1.0	0.961	0.270
	113	2		6.25 2500		-1		1	1.0	1.032	0.139		289	1	11.49	-1	1	1.0	1.044	0.324
	114	1		3.60			-1	-1	0.0	-0.155 0.118		295	6	3590.78	1	1	0.0	0.033	0.304
	115 118	1 3		6.12 1.50 2000 X i		-1 0	1 -1	1.0 -0.5 -0.740 0.123 1.112 0.188		296 297	1 1	1427.92 255.96	-1 -1	1 1	1.0 1.0	0.949 1.054	0.349 0.399
	122 134 12 4		1.33 1.42 1500		1 1	-1 -1	-1.0 -1.009 0.044 -1.0 -1.037 0.000		298 302	1 4	1.00 13.88	-1 1	-1 1	0.0 0.0	-0.051 -0.074	0.367 0.336
	137 140	3 3		5.75 3.47 1000		0 0	1 -1	0.5 -0.5 -0.507 0.005 0.629 0.044		303 305	1 2	138.28 8.90	-1 -1	1 1	1.0 1.0	1.117 1.153	0.391 0.444
	142 143	2 1		13.31 26.31 500		-1 -1		1 1	1.0 1.0	1.217 1.041	0.090 0.157		308 312	3 4	1.50 34.63	0 1	-1 1	-0.5 -0.342 0.0 -0.217 0.347 0.389
	144	1		18.54			-1		1	1.0	0.923	0.210		313	1	82.56	-1	1	1.0	0.811	0.379
	146 147 150	2 1 3		66.17 9.90 4.22	0	-1 -1 Oct 0 16	1 1 Nov -1 16	1.0 1.0 Dec -0.5 -0.534 0.260 1.124 0.274 0.916 0.319 Jan Feb 16 17 17	Mar 17	314 315 Apr 319 17	1 1 May 4 17	2.00 162.08 Jun 3.26 17	-1 -1 Jul 1 17	-1 1 Aug -1 17	0.0 1.0 Sep -1.0 -1.056 -0.019 1.071 17	0.351 0.402 0.300
	157	7		34.28			1		1	0.0	-0.110 0.234		321	2	285.91	-1	1	1.0	0.729	0.330
	161 162	4 1		2.23 1.84 0.5			1 -1	-1 -1	-1.0 -1.102 0.140 0.0 0.152 0.141 Phase 1		322 325	1 3	2.00 11.57	-1 0	-1 1 Phase 2 0.0 0.5	-0.283 0.347	0.287 0.291
	163	1		2.88			-1	-1	0.0	-0.018 0.130		334	9	34.70	1	1	0.0	0.068	0.275
	164	1		21.46			-1		1	1.0	1.087	0.197		335	1	431.00	-1	1	1.0	1.150	0.337
	165	1		4.46 0.4			-1	-1	0.0	-0.001 0.183		336	1	10.89	-1	1	1.0	1.003	0.383
	166 167	1 1		58.27 8.84			-1 -1		1 1	1.0 1.0	1.034 0.863	0.243 0.286		340 346	4 6	1.00 1.50	1 1	-1 -1	-1.0 -1.004 -1.0 -0.921 UCL	0.286 0.202
	180 13		1.03 0.3			1	-1	-1.0 -0.905 0.203		347	1	1.17	-1	-1	0.0	0.100	0.195
	187	7		16.57			1		1	0.0	0.156	0.199		349	2	1.27	-1	-1	0.0	0.129	0.190
	201 14	4.96 i Z *			1	-1	-1.0 -1.084 0.110		350	1	26.25	-1	1	1.0	1.098	0.254
	202 205	1 3		1.37 23.39 0.2			-1 0	-1 1	0.0 0.5	-0.087 0.096 0.498 0.124		353 354	3 1	11.66 3.03	0 -1	1 -1	0.5 0.0	0.332 0.127	0.259 0.250
	225 20		1.70			1	-1	-1.0 -1.032 0.043		355	1	12.00	-1	1	1.0	1.130	0.311
	247 22		5.30			1		0	-0.5 -0.727 0.000		356	1	1.00	-1	-1	0.0	-0.206	0.275
	248	1		15.64 0.1			-1		1	1.0	1.161	0.081
	257	9		5.14			1	-1	-1.0 -0.921 0.011
				0	Oct			Nov		Dec	Jan	Feb	Mar		Apr	May	Jun	Jul	Aug	Sep
					16			16		16	17		17	17		17	17	17	17	17	17

Illustrative example

We consider the same illustrative example as in Rahali et al. [START_REF] Rahali | Evaluation of Shewhart Time-Between-Events-and-Amplitude Control Charts for Several Distributions[END_REF], which is based on a real data set concerning the time T i (in days) between fires in forests of the french region "Provence -Alpes -Côte D'Azur" and their amplitudes X i (burned surface in ha = 10000m 2 , where only surfaces larger than 1ha have been included). This data set reports a total of 92 fires that have been divided into two subsets:

• 47 fires, from October 2016 to approximately mid-June 2017. This subset, corresponding to the "low season" for fires, is used here as a Phase I data set. • 45 fires, from approximately mid-June 2017 to the end of September 2017. This subset, corresponding to the "high season" for fires, is used here as a Phase II data set.

The dates D i (from October 1st 2016, in days), the times between fires T i as well as their amplitudes X i have been recorded in Table 5. The values of T i and X i are also plotted in Figure 3 (top and middle): it is evident that the Phase II values of T i (X i ) are shorter(larger) than those observed during Phase I.

In order to compute the upper control-limit UCL of the distribution-free EWMA TBEA chart, the following values have been fixed: p T = 0.3, p X = 0.7, σ = 0.125 and ARL 0 = 370.4. Using the results in Table 2 we have λ * = 0.07 and K * = 2.515 and we get UCL = 2.515 × 0.07 × (0.125 2 + 0.5) 2 -0.07 = 0.344.

From the Phase I data set, the following in-control median values have been estimated θ T0 = 3 and θ X0 = 5.3. These values are used to compute the values ST i , SX i , S i and S * i in Table 5. As it can be noticed, for some dates we have T i = θ T0 = 3: for these values, we have T iθ T0 = 0. This is not supposed to happen as T i is supposed to be a continuous random variable but, due to the measurement scale (days), this situation actually may occur. In this case, we have decided to keep the corresponding values and to assign ST i = 0 (instead of -1 or +1). For this reason, some values of S i = s = ±0.5 and the corresponding values for S * i are obtained by randomly generating a Nor(s, σ) random variable, as it is already the case for values s ∈ {-1, 0, +1}. For instance, in Table 5, when D i = 70 we have S i = 0.5 and the corresponding value for S * i has been randomly generated from a Nor(0.5, 0.125) distribution (S * i = 0.552). When D i = 140 we have S i = -0.5 and he corresponding value for S * i has been randomly generated from a Nor(-0.5, 0.125) distribution (S * i = -0.507).

The values Z * i have been computed using (2) for both Phase I and II data sets. They are recorded in Table 5 and plotted in Figure 3 (bottom) along with the distributionfree EWMA TBEA upper control limit UCL = 0.344:

• For the Phase I data set, all the values Z * i are below the upper control limit UCL = 0.344; therefore, this data set is in-control and the estimated median values θ T0 = 3 and θ X0 = 5.3 can be used for the Phase II monitoring.

• For the Phase II data set, the distribution-free EWMA TBEA detects several out-of-control situations during the period mid-June 2017 -end of September