Anan Tang

Philippe Castagliola

Xuelong Hu

Xiaojian Zhou

Optimal one-and two-sided adaptive EWMA scheme for monitoring Poisson count data

Keywords: Adaptive EWMA chart, Poisson Count Data, Markov Chain

The Poisson distribution assumption often arises in several industrial applications for modeling defects or non-conformities. In this work, we investigate the one-and two-sided performance of a new adaptive EWMA (Exponentially Weighted Moving Average)-type chart for monitoring Poisson count data. An appropriate discrete-state Markov chain technique is provided to compute the exact ARL (Average Run Length) properties. Moreover, comparative studies are conducted to demonstrate the higher sensitivity of the proposed chart in the detection of shifts with various magnitudes. Advices on how to select the appropriate chart parameters are provided and an illustrative numerical example is proposed.

INTRODUCTION

Control charts have become a key means of ensuring the quality of processes in the field of SPM (Statistical Process Monitoring). When we deal with different kinds of processes, data can usually be classified as either continuous or discrete. The corresponding control charts are called as variable or attribute charts, respectively.

A variable control chart is commonly used for the monitoring of quality characteristics which are measured on a continuous scale. For example, the Shewhart X and the R/S charts are widely employed to detect shifts in the process mean and variance, respectively. More advanced control charts, like the EWMA and the CUSUM (Cumulative Sum) charts, that combine the information from the beginning up to the current state of the process are expected to be more sensitive to small shifts than Shewhart-type charts. Specifically, Capizzi and Masarotto [START_REF] Capizzi | An adaptive exponentially weighted moving average control chart[END_REF] have proposed an adaptive EWMA (AEWMA) chart in which the smoothing parameter is adjusted based on the difference between the last AEWMA statistic and the new observation. This adaptive strategy has proven to be effective in providing additional protection for small and large shifts simultaneously as well as in compensating the negative effect of inertia, see Woodall and Mahmoud [START_REF] Woodall | The inertial properties of quality control charts[END_REF], Reynolds Jr and Stoumbos [START_REF] Jr | Comparisons of some exponentially weighted moving average control charts for monitoring the process mean and variance[END_REF]. Due to the remarkable adaptability of this AEWMA scheme, it has been investigated by several researchers, see, for instance, Shu [START_REF] Shu | An adaptive exponentially weighted moving average control chart for monitoring process variances[END_REF], Su et al. [START_REF] Su | Adaptive EWMA procedures for monitoring processes subject to linear drifts[END_REF], Aly et al. [START_REF] Aly | A reevaluation of the adaptive exponentially weighted moving average control chart when parameters are estimated[END_REF], Zaman et al. [START_REF] Zaman | An Adaptive EWMA Scheme-Based CUSUM Accumulation Error for Efficient Monitoring of Process Location[END_REF], Haq [START_REF] Haq | A New Adaptive EWMA Control Chart for Monitoring the Process Dispersion[END_REF], Mitra et al. [START_REF] Mitra | An Adaptive Exponentially Weighted Moving Average-Type Control Chart to Monitor the Process Mean[END_REF], Tang et al. [START_REF] Tang | An adaptive exponentially weighted moving average chart for the mean with variable sampling intervals[END_REF][START_REF] Tang | The effect of measurement errors on the adaptive EWMA X chart[END_REF][START_REF] Tang | Optimal design of the adaptive EWMA chart for the mean based on median run length and expected median run length[END_REF].

On the other hand, in several applications, the quality characteristics is not a continuous variable. Such data are widely prevalent in the manufacturing industry, the public health surveillance and the network trafficking, see, for example, Woodall [START_REF] Woodall | The use of control charts in health-care and publichealth surveillance[END_REF], Weiß [START_REF] Weiß | Controlling correlated processes of Poisson counts[END_REF], Yu et al. [START_REF] Yu | Using EWMA control schemes for monitoring wafer quality in negative binomial process[END_REF]. In that case, inspection by attribute is used due to its simple implementation and low testing cost. Considering the EWMA-type scheme for monitoring Poisson processes, Gan [START_REF] Gan | Monitoring Poisson observations using modified exponentially weighted moving average control charts[END_REF] presented a modified EWMA scheme, which rounds the EWMA statistic to a whole integer-value at each step. Furthermore, Borror et al. [START_REF] Borror | Poisson EWMA control charts[END_REF] proposed a new Poisson EWMA (PEWMA) chart having a superior ARL performance than the Shewhart c and the Gan's EWMA charts. For the detection of shifts on a specific direction, Shu et al. [START_REF] Shu | Exponentially weighted moving average control charts for monitoring increases in Poisson rate[END_REF] investigated a one-sided Poisson EWMA chart which focuses on the detection of an increasing rate. An extensive amount of researches has also been performed on Poisson processes with time varying sampling rate, see Ryan and Woodall [START_REF] Ryan | Control charts for Poisson count data with varying sample sizes[END_REF], Zhou et al. [START_REF] Zhou | Likelihood-based EWMA charts for monitoring Poisson count data with time-varying sample sizes[END_REF], Shen et al. [START_REF] Shen | Monitoring Poisson count data with probability control limits when sample sizes are time varying[END_REF]. A detailed state-of-the-art con-cerning control charts for attribute data is included in Woodall [START_REF] Woodall | Control charts based on attribute data: bibliography and review[END_REF], Topalidou and Psarakis [START_REF] Topalidou | Review of multinomial and multiattribute quality control charts[END_REF], Saghir and Lin [START_REF] Saghir | Control charts for dispersed count data: an overview[END_REF]. It has to be noted that, the use of the classical EWMA scheme (with a real-valued smoothing parameter λ) or the classical AEWMA scheme (with a real-valued score function ϕ) for Poisson count data (defined in an integer-valued domain) will lead to statistics that are no longer integer-valued. In this case, the calculation of the ARL properties is not reliable since it is not converging to a stable value based neither on pure simulation techniques nor on a Markov chain method. If the former approach is clearly dependent on the number of simulation cycles generated, the second one, unfortunately, is also heavily affected by the level of discretization of the control limit interval. Weiß [START_REF] Weiß | EWMA monitoring of correlated processes of Poisson counts[END_REF],

Rakitzis et al. [START_REF] Rakitzis | A new memorytype monitoring technique for count data[END_REF] and Castagliola et al. [START_REF] Castagliola | An EWMA-Type Sign Chart with Exact Run Length Properties[END_REF] have shown that, the ARL results of the Markov chain method strongly fluctuate because the discrete nature of this statistic depends on the number of states selected. For this reason, Rakitzis et al. [START_REF] Rakitzis | A new memorytype monitoring technique for count data[END_REF] proposed a discrete EWMA-type chart with exact run length properties, denoted as the CEWMA chart, for monitoring Poisson count data. This CEWMA chart uses two integer-valued parameters (γ X , γ Z ), instead of the classical real smoothing parameter λ, to allocate the weight of past observations. The main motivation of this work is that, there are few adaptive EWMA-type attribute charts for monitoring discrete data (compared to adaptive EWMAtype variable charts). In this paper, we study the one-and two-sided statistical performance of a discrete AEWMA chart for monitoring Poisson count data. An appropriate Markov chain technique is also developed to guarantee exact run length properties for this scheme. The rest of the paper is organized as follows.

In Section 2, the CEWMA chart is reviewed and our proposed CAEWMA chart is introduced. Section 3 introduces an appropriate discrete-state Markov chain methodology and Section 4 presents a detailed procedure for the optimal design of the CAEWMA scheme. Section 5 evaluates the performance of the CEWMA and the CAEWMA charts for detecting a wide range of shifts in the process mean. A numerical example is presented in Section 6 to illustrate the application of this new chart. Concluding remarks are drawn in Section 7.

THE CAEWMA CONTROL CHART

Suppose that {X t , t = 1, 2, . . .} are the number of non-conformities in a unit which are assumed to follow a Poisson distribution with mean θ. By definition, the p.m.f. (probability mass function) of X t is equal to

f p (x|θ) = e -θ θ x x! , x = 0, 1, 2, . . . . (1)
The process is said to be in-control if θ = θ 0 and it is out-of-control if θ = θ 1 = θ 0 . When θ 1 > θ 0 , it corresponds to an upward shift in the process mean. Of course, we need to detect this kind of shift as soon as possible and make adjustments to return the system to its normal state. On the other hand, a downward shift (0 < θ 1 < θ 0 ) reflects the decrease in the number of non-conformities.

Modern SPM methods also emphasize the monitoring of this kind of shift as soon as possible since any beneficial preventive interventions and strategies to the process may be worth of interest.

The classical continuous EWMA chart is defined as

Z t = (1 -λ)Z t-1 + λX t ,
where λ ∈ (0, 1] is a real-valued smoothing parameter. However, note that, when applying this EWMA chart to monitor discrete count data, the ARL properties, obtained based on the Markov Chain approach presented in Brook and Evans [START_REF] Brook | An approach to the probability distribution of CUSUM run length[END_REF], are known to not strictly monotonically converge with the increase in the number of Markov chain states. The same fact has also been emphasized in Weiß [START_REF] Weiß | EWMA monitoring of correlated processes of Poisson counts[END_REF].

At this point, Rakitzis et al. [START_REF] Rakitzis | A new memorytype monitoring technique for count data[END_REF] proposed a CEWMA monitoring technique for count data, in which not only the observations but also the smoothing constants (γ X , γ Z ) and the EWMA statistic Z t are all integers. The CEWMA statistics are defined as follows:

Z t = γ X X t + γ Z Z t-1 + R t-1 γ X + γ Z , (2)
R t = γ X X t + γ Z Z t-1 + R t-1 -(γ X + γ Z )Z t , (3)
where γ X and γ Z are two positive integer-valued parameters, . . . denotes a rounded down integer (e.g. 1.5 = 1). By definition, Z t corresponds to the quotient of the Euclidean division and R t ∈ {0, 1, . . . , γ X + γ Z -1} is the remainder of this division. Please note that these R 1 , R 2 , . . . values are only recorded but will not be used for monitoring. The monitored statistics are only the values Z 1 , Z 2 , . . ., and ( 2) and ( 3) can be rewritten as

(γ X + γ Z )Z t + R t = γ X X t + γ Z Z t-1 + R t-1 . (4)
This CEWMA chart is optimally designed to detect a specific shift magnitude by adjusting the relative value of γ X and γ Z . More weight can be given to past observations when the ratio γ X /(γ X + γ Z ) gets smaller, thus allowing a better sensitivity to the detection of small shifts. Otherwise, large values of γ X /(γ X + γ Z ) must be used to quickly detect large shifts. However, it is quite difficult to pre-determine the exact magnitude of shifts occurring in practice. A natural option is to find a potential tendency presented in the data and then to dynamically adjust the weight coefficient according to the predicted difference.

Therefore, we suggest the following adaptive EWMA (denoted as CAEWMA) scheme proposed in Tang et al. [START_REF] Tang | A new nonparametric adaptive EWMA control chart with exact run length properties[END_REF] for count data:

Z t = ϕ(X t -Z t-1 ) + (γ X + γ Z )Z t-1 + R t-1 γ X + γ Z , (5)
R t = ϕ(X t -Z t-1 ) + (γ X + γ Z )Z t-1 + R t-1 -(γ X + γ Z )Z t , (6)
where 5) and ( 6) can be rewritten as

ϕ(e t ) = ϕ(X t -Z t-1 ) is a score function. If we define C t def = (γ X +γ Z )Z t + R t then (
(γ X + γ Z )Z t + R t = ϕ(e t ) + (γ X + γ Z )Z t-1 + R t-1 , C t C t-1 (7)
where the initial values for Z t and R t are fixed as Z 0 = z 0 and R 0 = r 0 . So, at time t, when the X t , Z t-1 , R t-1 values as well as the score function ϕ(e t ) are fixed, we can uniquely get the values of Z t and R t . It goes without saying that the error terms e t between X t and Z t-1 have to be also integers, so a new score function is defined as

ϕ(e) =          e(γ X + γ Z ) + γ Z k if e < -k eγ X if |e| ≤ k e(γ X + γ Z ) -γ Z k if e > k , (8)
where k is also a positive integer-valued parameter. As mentioned in Capizzi and

Masarotto [START_REF] Capizzi | An adaptive exponentially weighted moving average control chart[END_REF], when we construct the score function, the following conditions need to be met: i) ϕ(e) must be monotone increasing;

ii) if k → ∞ and we have ϕ(e) = eγ X and ( 7) becomes (γ

X + γ Z )Z t + R t = γ X X t + γ Z Z t-1 + R t-1
, which agrees with the CEWMA scheme in Rakitzis et al. [START_REF] Rakitzis | A new memorytype monitoring technique for count data[END_REF];

iii) if k = 0, we have ϕ(e) = e(γ X + γ Z ) and ( 7) reduces to a Shewhart-type scheme.

The process is said to be out-of-control whenever

Z t < h L or Z t > h U ,
= (γ X + γ Z )z t-1 + r t-1 . As z t-1 ∈ {h L , h L + 1, . . . , h U } and r t-1 ∈ {0, 1, . . . , γ X + γ Z -1}, the minimum and the maximum values of c t-1 are c min = (γ X + γ Z )h L , (9)
c max = (γ X + γ Z )(h U + 1) -1, (10)
     (γ X + γ Z )(x t -ct-1 γ X +γ Z ) + kγ Z + c t-1 ≥ c min if e t < -k, γ X (x t -ct-1 γ X +γ Z ) + c t-1 ≥ c min if |e| ≤ k, (11)
and

     (γ X + γ Z )(x t -ct-1 γ X +γ Z ) -kγ Z + c t-1 ≤ c max if e t > k, γ X (x t -ct-1 γ X +γ Z ) + c t-1 ≤ c max if |e| ≤ k, (12)
from which we obtain the minimum x min and the maximum x max values as

x min = max 0, c min -c t-1 -kγ Z γ X + γ Z + c t-1 γ X + γ Z , c min -c t-1 γ X + c t-1 γ X + γ Z , (13)
x max = min c max -c t-1 + kγ Z γ X + γ Z + c t-1 γ X + γ Z , c max -c t-1 γ X + c t-1 γ X + γ Z , (14)
where . . . denotes a rounded up integer. Now, the transition probabilities q i,j of going from one state to another can be calculated using the following steps:

Given n, h L , h U , γ X , γ Z , k c min ← (γ X + γ Z )h L c max ← (γ X + γ Z )(h U + 1) -1 m ← c max -c min For i = 0, 1, . . . , m c t-1 ← i + c min z t-1 ← ct-1 γ X +γ Z x min ← max 0, cmin-ct-1-kγ Z γ X +γ Z + ct-1 γ X +γ Z , cmin-ct-1 γ X + ct-1 γ X +γ Z x max ← min cmax-ct-1+kγ Z γ X +γ Z + ct-1 γ X +γ Z , cmax-ct-1 γ X + ct-1 γ X +γ Z For x t = x min , x min + 1, . . . , x max e t ← x t -z t-1 If e t < -k ϕ(e t ) ← (γ X + γ Z )e t + kγ Z Else if e t > k ϕ(e t ) ← (γ X + γ Z )e t -kγ Z Else -k ≤ e t ≤ k ϕ(e t ) ← γ X e t
End If

z t ← ϕ(et)+ct-1 γ X +γ Z r t ← ϕ(e t ) + c t-1 -(γ X + γ Z )z t j ← (γ X + γ Z )z t + r t -c min q i,j ← f p (x t |θ)
End For

End For

Let Q be the (m + 1, m + 1) transition matrix containing all of the probabilities q i,j , and let s be the initial probability vector represented by a (m + 1, 1)

vector s = (s 0 , s 1 , . . . , s m ), where s contains the probabilities that the Markov chain starts in a given state. We will set 1 in the entry corresponding to (γ X + γ Z )Z 0 + r 0 and zeros in the remaining ones. For example, If z 0 = θ 0 and r 0 = 0, the entry s (γ X +γ Z ) θ0 +r0 = 1 and s i = 0 for i = (γ X + γ Z ) θ 0 + r 0 .

The mean value ARL = E(RL), as well as the standard deviation SDRL = σ(RL) can be easily obtained as

ARL = s (I -Q) -1 1, (15)
SDRL = 2s (I -Q) -2 Q1 -ARL 2 + ARL. ( 16
)
where I is the (m + 1, m + 1) identity matrix and 1 is a (m + 1, 1) vector of 1's.

DESIGN OF THE CAEWMA SCHEME

When there is an objective of detecting a shift in a particular direction (either upward or downward), one-sided schemes could be straightforwardly adapted from the two-sided situation. For the upper-sided chart, h L = 0 is set, and it results in c min = 0 and c max = m. Analogously, for the lower-sided case, an upper restriction h U = A will be picked to prevent the statistic Z t from shifting to infinity.

Next, we provide a procedure for the design of the proposed CAEWMA scheme, in which the statistical performances are evaluated in terms of the ARL. Usually, the same ARL 0 value is taken when the process is in-control, and when the process is out-of-control, the smaller the out-of-control ARL 1 value, the better the performance of control charts. Due to the discreteness of the monitoring statistic Z t , ARL(h L , h U , γ X , γ Z , k, θ 0 ) = ARL 0 cannot be exactly attained. Therefore, to satisfy ARL(h L , h U , γ X , γ Z , k, θ 0 ) ≈ ARL 0 , only the combinations of (h L , h U , γ X , γ Z , k, θ 0 ) with a ARL value close to the design one, i.e. ARL(h L ,h U ,γ X ,γ Z ,k,θ0)-ARL0 ARL0 ≤ ζ, will be considered, where ζ is a pre-determined threshold.

• For the upper-sided CAEWMA chart, we suggest to find parameters (h U , γ X , γ Z , k)

giving an optimal performance for detecting a range of shifts:

Min EARL(h U , γ X , γ Z , k, Ω), (17)
s.t. ARL(h U , γ X , γ Z , k, θ 0 ) ≈ ARL 0 , (18)
where

EARL = Ω f (θ 1 )ARL(h U , γ X , γ Z , k, θ 1 )dθ 1 , (19)
and Ω is assumed to be a uniform distribution for the mean shift U [θ 1min , θ 1max ] with f (θ 1 ) = 1 θ1 max -θ1 min .

• For the two-sided CAEWMA chart, one important problem that must be taken into account is that the ARL 0 value is not always larger than the corresponding ARL 1 values. Inspired by Su et al. [START_REF] Su | Adaptive EWMA procedures for monitoring processes subject to linear drifts[END_REF], a two-steps searching procedure is recommended:

1. For a desired ARL 0 value, find the optimal combination of (γ X , γ Z ) for the CEWMA chart, which gives a minimum ARL 1 value at θ 1min .

Some of these combinations could be found in Rakitzis et al. [START_REF] Rakitzis | A new memorytype monitoring technique for count data[END_REF].

2. Based on the same (γ X , γ Z ) value obtained in Step 2, for the CAEW-MA chart, find an optimal k value having the minimum ARL 1 value at θ 1max subject to ARL(h L , h U , γ X , γ Z , k, θ 0 ) ≈ ARL 0 .

PERFORMANCE COMPARISONS

Since it is quite difficult to predict the magnitude and direction of shifts occurring in practice, Han and Tsung [START_REF] Han | A reference-free cuscore chart for dynamic mean change detection and a unified framework for charting performance comparison[END_REF] recommended using the RMI (Relative Mean Index) to evaluate the global performance of control charts, which is defined as follows:

RMI(I) = 1 N N i=1 ARL(I, θ (i) 1 ) -ARL * (θ (i) 1 ) ARL * (θ (i) 1 ) , (20)
where ARL(I, δ (i) ) is the ARL value for a particular chart I for the shift θ 1 (i), and among these comparative charts I = 1, 2, . . . , ARL * (θ

(i)
1 ) denotes the the 10 minimum ARL value when detecting θ (i)

1 . Here, a discrete uniform distribution for the shift θ 1 is assumed, for example, when θ 0 = 8, consider upward shifts θ 1 ∈ {9, 10, . . . , 20}, θ (i) 1 denotes the i th shift, i.e. θ We have compared the CAEWMA chart with the CEWMA chart proposed in Rakitzis et al. [START_REF] Rakitzis | A new memorytype monitoring technique for count data[END_REF]. From Tables 1 to 6, we can draw the following conclusions:

• It can be seen that, when designing the CEWMA charts, a small 'smoothing ratio' γ X /(γ X + γ Z ) value is effective for detecting small shifts, and a large one is effective for larger shifts. Therefore, for a single CEWMA control chart, it is difficult to balance the overall performance for different shift sizes, at the same time. Taking the upper-sided case as an example, in Table 1, if the CEWMA chart, with (γ X , γ Z ) = (1, 83), can be efficient in the detection of small shifts (e.g. θ 1 = 9), it is unfortunately insensitive to moderate or large ones (e.g. θ 1 > 12).

• For a specified in-control ARL 0 = 1000, Table 1 and 2 compare the performance for upper-sided shifts in the Poisson mean θ 0 ∈ {8, 12}.

Note that, according to Rakitzis et al. [START_REF] Rakitzis | A new memorytype monitoring technique for count data[END_REF], the optimal combination (γ X , γ Z ) = (1, 83) of the CEWMA chart could help in detecting small shifts θ 1 ∈ {9, 10} (when θ 0 = 8) faster. However, based on the results shown in Table 1, the CAEWMA chart always has the minimum ARL 1 for θ 1 ∈ {9, 10, 11, 12}. The same case can also be observed in Table 2, when θ 0 = 12, the CAEWMA chart has the minimum ARL 1 = 2.1 value for θ 1 = 24 compared with the optimal ARL 1 = 2.2 obtained with the CEWMA chart.

• For the lower-sided case in Table 3, when θ 0 = 16, note that the CEWMA chart with (γ X , γ Z ) = [START_REF] Borror | Poisson EWMA control charts[END_REF][START_REF] Woodall | The inertial properties of quality control charts[END_REF] is the only optimally recommended combination in Rakitzis et al. [START_REF] Rakitzis | A new memorytype monitoring technique for count data[END_REF] for all shift sizes. However, the comparison reveals that the CAEWMA chart has a higher detection probability for decreasing shifts θ 1 ∈ {15, 14, 13, 12} and θ 1 ∈ {5, 4, 3} while, for shifts θ 1 ∈ {10, 9, 8, 7}, the differences between these two schemes are negligible.

The advantage of the CAEWMA chart in detecting different shift sizes is also fully verified in terms of the RMI performance. For example, when θ 0 = 20 in Table 4, the CAEWMA chart gives the minimum RMI = 0.206 value compared with all CEWMA charts.

• For the two-sided case in Tables 5 and6, it is interesting to note when similar values of γ X and γ Z are selected, the CAEWMA chart guards against small and large shifts more effectively than the CEWMA chart.

For example, in Table 5, when compared with the CEWMA chart with γ X = 5 and γ Z = 37, the CAEWMA chart with γ X = 5 and γ Z = 38 provides a slightly worse performance for detecting increasing shifts θ 1 ∈ {22, 24, 26}, but better performance for detecting decreasing shifts θ 1 ∈ {18, 16, 14, 12} and increasing shifts θ 1 ∈ {30, 35, 40}. In Table 6, when γ X = 1 and γ Z = 15, there is no loss in the efficiency for the ARL 1 in detecting decreasing shifts θ 1 ∈ {18, 16, . . . , 6}, however, the CAEWMA chart performs better in the detection of increasing shifts θ 1 ∈ {22, 24, . . . , 40} compared to the CEWMA chart.

To sum up, it is shown that the CAEWMA chart outperforms the CEWMA chart in terms of the out-of-control ARL 1 and RMI to efficiently detect both small and large shifts. As the practitioner has usually no clear idea about the actual magnitude of potential shifts, we recommend the use of the CAEWMA chart for practical purposes.

(Please Insert Tables 1 to 6 Here)

AN ILLUSTRATIVE EXAMPLE

The implementation of the proposed CAEWMA chart is illustrated with an example from the traffic flow monitoring. Based on historical data, the number X t of vehicles passing every minute in a particular section of the road follows a Poisson distribution with in-control mean θ 0 = 12. We actually have two data sets recording the number of vehicles passing in a certain section of the road • in the morning period 8 : 00 -9 : 00 (Table 7),

• in the afternoon period 17 : 00 -18 : 00 (Table 8).

For comparison, two upper-sided CEWMA charts are also implemented for the same date sets. The desired ARL 0 = 1000 value is fixed, and based on Rakitzis et al. [START_REF] Rakitzis | A new memorytype monitoring technique for count data[END_REF], the optimal combinations (h U , γ X , γ Z ) = [START_REF] Ryan | Control charts for Poisson count data with varying sample sizes[END_REF][START_REF] Aly | Optimal design of the adaptive exponentially weighted moving average control chart over a range of mean shifts[END_REF][START_REF] Su | Adaptive EWMA procedures for monitoring processes subject to linear drifts[END_REF] and [START_REF] Shen | Monitoring Poisson count data with probability control limits when sample sizes are time varying[END_REF][START_REF] Aly | A reevaluation of the adaptive exponentially weighted moving average control chart when parameters are estimated[END_REF][START_REF] Capizzi | An adaptive exponentially weighted moving average control chart[END_REF] of the CEWMA charts (denoted as CEWMA-1 and CEWMA-2, respectively) are considered here. Concerning the CAEWMA chart, the unique optimal combination is obtained to be (h U , γ X , γ Z , k) = [START_REF] Saghir | Control charts for dispersed count data: an overview[END_REF][START_REF] Borror | Poisson EWMA control charts[END_REF][START_REF] Ryu | Optimal design of a CUSUM chart for a mean shift of unknown size[END_REF][START_REF] Jr | Comparisons of some exponentially weighted moving average control charts for monitoring the process mean and variance[END_REF].

All results are presented in Figures 1 and2. For the morning case (Table 7), the CAEWMA and CEWMA-1 charts give an out-of-control signal at the 38 th observation. It indicates that the traffic congestion caused by the morning peak occurred around 8 : 38. While, no point exceeds the control limit in the CEWMA-2 chart, so no alarm is issued. For the afternoon case (Table 8), both the CAEWMA and CEWMA-2 charts find anomalies at the 23 th observation and the CEWMA-1 chart signals at the 26 th observation. It indicates that the traffic congestion caused by the afternoon peak occurred around 17 : 23. In short, the CAEWMA chart is not only as effective as the CEWMA-1 chart when the shift size is small, but it is also as effective as the CEWMA-2 chart for large shifts.

(Please Insert Figures 1 and2 Here)

CONCLUSIONS

In this work, we have extended the AEWMA scheme to the monitoring of Poisson count data. An appropriate Markov chain technique has been developed to guarantee exact ARL results for discrete statistics, without approximation.

Particularly, to enhance its detection capability for asymmetric distribution, an in-depth study for the one-and two-sided cases have been performed. The one-sided CAEWMA scheme is recommended when shifts toward a particular side are definitely more significant than toward the opposite one. The two-sided CAEWMA scheme should be used when shifts toward either side are equally important. The comparative results support the fact that the CAEWMA chart is able to provide a more balanced detection property against shifts of different magnitudes than the CEWMA chart in both one-and two-sided conditions.

Therefore, this scheme is worth to be considered when the exact size of the shift cannot be pre-determined.

  and the total number of shifts in this example is N = 12. Obviously, the smaller the value of RMI(I), the better the performance of the proposed chart.

Figure 1 :Figure 2 :

 12 Figure1: Control charts applied to the morning period 8 : 00 -9 : 00

  respectively. So, at time t -1, all the discrete states are c t-1 ∈ {c min , c min + 1, . . . , c max }, and the total number of states is m + 1 = c max -c min + 1. When the state c t-1 moves to the state c t = (γ X + γ Z )z t + r t , the score ϕ(e t ) needs to satisfy c t = ϕ(e t ) + c t-1 ∈ {c min , c min + 1, . . . , c max }. Since ϕ(e t ) is monotone increasing in e t , the allowable values of x t ≥ 0 must simultaneously satisfy

Table 1 :

 1 Comparisons of the CEWMA and CAEWMA charts, upper-sided case, ARL 0 = 1000

	and θ 0 = 8
	θ		CEWMA		CAEWMA
	8	995.1 987.8 1011.6	1008.8
	9	83.7 125.7 154.5	75.8
	10	42.7	32.0	40.8	21.4
	11	28.7	13.4	16.3	11.1
	12	21.7	7.7	8.7	7.4
	13	17.4	5.2	5.6	5.6
	14	14.6	4.0	4.1	4.5
	15	12.6	3.2	3.2	3.7
	16	11.1	2.7	2.7	3.2
	17	9.9	2.4	2.3	2.8
	18	8.9	2.1	2.0	2.5
	19	8.2	1.9	1.8	2.2
	20	7.5	1.8	1.7	1.9
	21	7.0	1.6	1.5	1.8
	RMI	2.542 0.129 0.207	0.119
	h L	0	0	0	0
	h U	8	12	13	10
	γ X	1	4	6	8
	γ Y	83	7	7	43
	k				13

Table 2 :

 2 Comparisons of the CEWMA and CAEWMA charts, upper-sided case, ARL 0 = 1000

	and θ 0 = 12
	θ		CEWMA		CAEWMA
	12	1016.0 1032.0 1008.3	1009.3
	13	74.5	92.0	151.5	135.0
	14	28.1	27.7	41.2	35.8
	15	17.0	14.7	17.6	16.3
	16	12.2	9.9	10.0	9.9
	17	9.6	7.4	6.8	7.0
	18	7.9	6.0	5.1	5.4
	19	6.7	5.1	4.1	4.4
	20	5.9	4.4	3.5	3.7
	21	5.2	3.9	3.0	3.1
	22	4.7	3.5	2.7	2.7
	23	4.3	3.2	2.4	2.4
	24	4.0	2.9	2.2	2.1
	25	3.7	2.7	2.0	1.9
	RMI	0.523	0.210	0.141	0.113
	h L	0	0	0	0
	h U	13	14	16	15
	γ X	1	1	2	3
	γ Y	19	8	5	14
	k				12

Table 3 :

 3 Comparisons of the CEWMA and CAEWMA charts, lower-sided case, ARL 0 = 1000

	and θ 0 = 16
	θ		CEWMA		CAEWMA
	16	879.0 1015.0 934.8	1010.2
	15	131.4 100.2	66.6	68.6
	14	35.3	27.6	21.6	21.8
	13	15.1	13.4	12.1	12.1
	12	8.7	8.6	8.3	8.3
	11	6.0	6.3	6.4	6.3
	10	4.5	5.0	5.2	5.1
	9	3.7	4.1	4.4	4.3
	8	3.1	3.5	3.8	3.7
	7	2.7	3.1	3.4	3.2
	6	2.4	2.8	3.1	2.8
	5	2.2	2.5	2.8	2.4
	4	2.0	2.2	2.6	2.0
	3	2.0	2.1	2.3	1.5
	RMI	0.172 0.175 0.177	0.079
	h L	13	14	15	15
	h U	30	30	30	30
	γ X	3	3	3	5
	γ Y	13	28	68	114
	k				12

Table 4 :

 4 Comparisons of the CEWMA and CAEWMA charts, lower-sided case, ARL 0 = 1000

	and θ 0 = 20
	θ		CEWMA		CAEWMA
	20	993.9 1207.2 1005.1	1002.8
	19	285.4 123.1	78.2	79.1
	18	95.1	34.3	26.0	26.1
	17	38.0	16.7	14.6	14.6
	16	18.1	10.7	10.0	10.0
	15	10.2	7.8	7.7	7.7
	14	6.6	6.1	6.2	6.2
	13	4.7	5.1	5.2	5.2
	12	3.6	4.4	4.5	4.5
	11	2.9	3.8	4.0	4.0
	10	2.5	3.4	3.6	3.5
	9	2.2	3.1	3.3	3.2
	8	1.9	2.9	3.1	2.9
	7	1.7	2.6	2.9	2.6
	RMI	0.625 0.274	0.233	0.206
	h L	14	18	19	19
	h U	30	30	30	30
	γ X	3	3	2	1
	γ Y	4	35	55	27
	k				15

Table 5 :

 5 Comparisons of the CEWMA and CAEWMA charts, two-sided case, ARL 0 = 1000

	and θ 0 = 20
	θ		CEWMA		CAEWMA
	6	2.4	2.0	1.9	2.4
	8	2.9	2.2	2.1	2.9
	10	3.6	2.8	2.6	3.6
	12	4.7	3.9	3.6	4.6
	14	6.9	6.6	5.9	6.8
	16	13.5	16.9	14.0	13.2
	18	62.4	100.8	70.9	58.2
	20	1017.2 1000.3 982.8	1000.0
	22	50.2	110.4 335.9	53.2
	24	14.7	21.1	42.2	15.2
	26	8.3	8.8	13.3	8.4
	28	5.8	5.4	7.0	5.8
	30	4.5	3.9	4.7	4.3
	35	3.0	2.4	2.7	2.5
	40	2.3	1.8	2.0	1.9
	RMI	0.168	0.220 0.656	0.131
	h L	17	15	15	17
	h U	23	26	27	23
	γ X	5	7	3	5
	γ Y	37	17	7	38
	k				17

Table 6 :

 6 Comparisons of the CEWMA and CAEWMA charts, two-sided case, ARL 0 = 1500 and θ 0 = 20

	θ		CEWMA		CAEWMA
	6	3.0	2.1	2.2	3.0
	8	3.5	2.4	2.7	3.5
	10	4.2	3.0	3.4	4.2
	12	5.4	4.1	4.7	5.4
	14	7.7	6.3	8.1	7.7
	16	13.5	13.7	22.4	13.5
	18	47.0	70.7	191.8	47.0
	20	1520.4 1495.6 1517.1	1504.0
	22	50.1	142.9	85.0	50.0
	24	17.2	23.2	17.9	17.2
	26	10.4	10.0	8.3	10.3
	28	7.5	6.3	5.3	7.4
	30	5.9	4.6	3.9	5.7
	35	3.9	2.9	2.5	3.5
	40	3.6	2.2	1.9	2.2
	RMI	0.319	0.258	0.373	0.249
	h L	18	16	15	18
	h U	22	25	25	22
	γ X	1	6	7	1
	γ Y	15	25	23	15
	k				21
			23

Table 7 :

 7 Data sets for the morning period 8 : 00 -9 : 00

	t	X t	t	X t	t	X t
	1	13 21	2	41 10
	2	15 22 11 42 19
	3	15 23 10 43 16
	4	12 24 10 44 14
	5	12 25 20 45 14
	6	9	26 22 46	8
	7	16 27 10 47 19
	8	15 28 12 48 14
	9	11 29 19 49 13
	10	8	30 15 50 15
	11 10 31	9	51 13
	12 17 32 14 52 11
	13	6	33 15 53 15
	14 12 34 10 54 15
	15 20 35 22 55 16
	16 17 36 17 56 15
	17 14 37 16 57 15
	18 11 38 17 58 12
	19	8	39 15 59 12
	20 14 40 10 60 11
			24

Table 8 :

 8 Data sets for the afternoon period 17 : 00 -18 : 00

	t	X t	t	X t	t	X t
	1	17 21 14 41 19
	2	7	22 19 42 20
	3	10 23 26 43 20
	4	10 24 18 44 26
	5	10 25 21 45 20
	6	12 26 28 46 22
	7	16 27 31 47 23
	8	10 28 20 48 21
	9	16 29 16 49 24
	10	6	30 18 50 25
	11 15 31 12 51 16
	12	5	32 20 52 25
	13 14 33 21 53 21
	14 13 34 11 54 16
	15 13 35 24 55 20
	16 16 36 16 56 23
	17	8	37 25 57 22
	18 14 38 17 58 18
	19 13 39 19 59 23


1

Shakespeare and the Origins of European Culture Wars

Jean-Christophe Mayer, CNRS and Université Paul-Valéry Montpellier 3

As both national and European politics have come under increasing criticism in the aftermath of the major 2008 financial crisis, which continues to affect most European economies, politicians have been tempted to divert their peoples' attention by focusing less on practical policy building and more on culture wars. Thus, issues such as sexual freedom, ethnic diversity, migrancy, or individuals' relationship to the state have come to the fore and are dividing Europe, as they become subjects of bitter wrangles, not only between politicians of various nations, but also between Europeans themselves.

The idea of culture wars is not new. In the late 1970s political expert Ronald Inglehart argued that in western societies, what he called "postmaterialist" values were becoming more important than traditional "materialist values" (such as the state's role in a market economy) [START_REF] Inglehart | The Silent Revolution[END_REF]. In other words, and according to Inglehart, as differences between major parties were less marked, societies tended to be structured by cultural feuds and oppositions.

More recently, political scholar and columnist Michael Behrent pointed out that public issues in Europe at the moment were shaped and influenced by culture war notions. 1 Sociologists, such as Irene Taviss Thomson, remarked, however, that "there is, of course, an intuitive appeal -a surface plausibility -to the culture war idea", but that cultural wars were more a means of diverting people's attention from unresolved economic and political problems (Thomson, 2012: 12). The cultural war idea has in fact been used in public debates although no serious study has proved its actual sociological reality.

Be that as it may, cultural wars have affected and are affecting every corner of society including literary studies. As the world's most popular playwright, Shakespeare and his works have been the site of much cultural -and sometimes bitter -argument. 2 Shakespeare and literature in general may seem far remote from European cultural and political issues, but in fact, Shakespeare, like other authors with strong societal auras, can be seen as particularly useful cultural tools. As Douglas Lanier noted, Shakespeare is now "a resource for doing certain kinds of cultural work" (Lanier, 2002: 14).

This was not always the case. In what follows, I shall argue that -from a historical point of view -Shakespeare became engulfed in cultural wars in the eighteenth century, precisely at a time when the public sphere was expanding greatly. As we shall see, like other cultural figures, Shakespeare was used to express various agendas and as a means of broaching political and even European issues. By focusing on the beginning of Shakespearean culture wars in the eighteenth century between the two super-powers of the time (France and England), I hope to raise our awareness of how cultural forms, and literature in particular, can structure public and diplomatic discourse and be appropriated, manipulated, and become instruments in a covert and at times overt race for political hegemony.

So, let us first concentrate on where the story began: the first half of the eighteenth century, when the question of cultural and political dominance between European nations, and more specifically between England and France, really affirmed itself in the field of literature.

François-Marie Arouet, better known as Voltaire, was a French eighteenth-century man of letters, philosopher and also, to some extent, cultural ambassador of neo-classical values. In exile in England for almost three years from 1726 to 1728, Voltaire went to the theatre at a time when England was gaining ground politically and internationally, but when Shakespeare was almost an unknown entity in France and on the continent. Voltaire, who was at times very critical of the political system in his own country saw England's constitutional monarchy as more progressive than France's absolutist system, but his views of the arts and of Shakespeare in particular were more mitigated. Voltaire admired Shakespeare for being "natural and sublime", naturalness being a quality arguably lacking in French theatre of the period, but there was much in Shakespeare that disagreed with the neo-classical aesthetics which were dominating so much of Europe at the time, under the aegis of France. In his Lettres philosophiques, composed about 1729 and first printed in English in 1733 under the title Letters Concerning the English Nation, he wrote: Shakespeare's brilliant monstrosities please a thousand times more than today's elegance. The poetic genius of the English still seems more like a bushy tree planted by nature, branching out at random, growing unevenly and strongly; it dies if you try to alter its nature and prune its branches into the topiary Marly was a castle built under the reign of Louis XIV, whose gardens were famous for being pristine. While the description is a touch condescending, the horticultural metaphor also underlines in passing the potential for growth of the arts in England and perhaps already their potential for growing wildly and for invading other gardens and well-kept neo-classical territories such as France. During the first half of the eighteenth century, Voltaire continued nevertheless to see Britain as more advanced than France, politically speaking, and confessed his admiration for English philosophy and science to his friend, the British merchant and later diplomat Sir Everard Fawkener in the dedication of his play Zaïre in 1736:

You have to submit yourselves to the rules of our theatre, as we have to embrace your philosophy. We have made as good investigations of the human heart as you have in physics. The art of giving pleasure seems to belong to the French, while yours appears to be the art of thinking.

[Vous devez vous soumettre au rêgles de notre théâtre, comme nous devons embrasser votre philosophie. Nous avons fait d'aussi bonnes experiences sur le coeur humain que vous sur la physique.

L'art de plaire semble l'art des Français, et l'art de penser paraît le vôtre] . (Voltaire, 1877: vol. 2, 554) Voltaire's attitude gradually changed during the second half of the eighteenth century, as both the cultural and political terrain shifted. The Seven Years' War, which was in effect a world war involving several European nations from 1756 to 1763, but which also set Great Britain against the Bourbon dynasty (France and Spain) over trade and colonial dominion, no doubt precipitated these changes. The cultural balance was also shifting -Shakespeare's fame began to grow in Europe as Britain sought to establish its cultural and political authority and the playwright was appropriated increasingly to serve English nationalist agendas.

On the cultural terrain, a few significant salvos were fired, as a couple of articles translated from the English and comparing Shakespeare to Corneille and Otway to Racine appeared respectively in October and November 1760 in the French Journal encyclopédique.

Both articles underlined the English authors' superiority. Not long after, in December 1760, Voltaire shared his displeasure in a letter to Marie de Vichy de Chamrond. Interestingly, the letter simultaneously refers to the loss of the city of Pondicherry on the Indian subcontinent (one of France's colonial outposts besieged by the English in 1760) and to the claim of Shakespeare's alleged superiority:

(… ) and, for that matter, I'm angry at the English. Not only is it my belief that they've taken our Pondicherry, but they've just printed that their Shakespeare is far superior to Corneille. Voltaire also aired his views publicly in 1761 in his "Appeal to all nations of Europe regarding the judgement of an English writer" ("Appel à toutes les nations de l'Europe des jugements d'un écrivain anglais"). In this work, Voltaire pointed out that Shakespeare, unlike Racine, for instance, was hardly known outside Britain and called upon all nations "from Saint Petersburg to Naples" to decide whether he was right and -implicitly -to support French cultural supremacy. Voltaire ceased to be diplomatic as soon as he perceived that literature, and Shakespeare in particular, was employed for nationalistic reasons by the British. This may explain why he had chosen to treat warfare and literature separately until then, but now employed a military vocabulary as a form of resistance to what he considered as attempts on behalf of the British to establish their cultural as well as military dominance. Of course, losing battles in the two main theatres of the Seven Years' War, North America and India, was no mere detail and while Voltaire could be intellectually dismissive about these losses, they would nonetheless lead ultimately to Britain's linguistic and cultural dominance in those parts of the world. In other words, French cultural dominance was on its way out.

More than a decade later, with the war of American independence serving as a backdrop this time, the cultural battle around Shakespeare continued to rage between the British and the French. In 1776, the first complete translation into French of Shakespeare's works by Pierre Le Tourneur was published. The twenty volumes, in which Le Tourneur praised Shakespeare with an enthusiasm that was also self-serving, were sold by subscription. Voltaire was horrified to discover that King Louis XVI was at the top of the list of subscribers, as well as other persons from all over Europe. The writer and philosopher Denis Diderot had also ordered six copies, which, for Voltaire, was the equivalent of high treason. What upset Voltaire particularly was that he himself had been partly responsible for this situation and had let the enemy inside the walls through his early-mitigated praise of Shakespeare at a time when hardly anyone had heard of him. Voltaire's words were blunt as he wrote again to the Count of Argental in 1776:

It was I who was the first to speak of this Shakespeare at an earlier time; it was I who was the first to show the French people some pearls that I found in his huge heap of dung. D'Alembert himself had fully embraced Voltaire's project and delivering his friend's speech to the Académie was like accomplishing a warlike mission. In a letter written a few days before the speech was aired, D'Alembert hoped that French men of letters would accomplish a better mission on the terrain of cultural warfare than French generals and soldiers did on the battlefield. He also had vowed to punish all traitors:

At last, my dear master, the battle has begun and the signal has been given. Either Shakespear or Racine will be left standing; we have to show these sad and insolent English that our men of letters can fight them better that our soldiers and our generals. Unfortunately, there are quite a few deserters and false brothers among those men of letters. But the deserters will be caught and hanged; what annoys me is that the fat of these hanged men will be good for nothing; for they are quite dry and lean. Adieu, my Clearly, Shakespeare's reputation was at the centre of a war of words, but also of deeds. While, in the past, Voltaire had had a measure of admiration for some aspects of Shakespeare's works, as well as for the English constitutional system, he was now forced to fight against what he no doubt considered as a form of "regressive nationalism" (Prince, 2012: 282), which mobilized Shakespeare as an instrument in a war of propaganda.

The British had in fact also been using warlike language to defend Shakespeare against Voltaire's attacks for quite a while. In his review of Samuel Johnson's edition of Shakespeare in 1765, William Guthrie accused Johnson of pandering to French taste too much and of judging Shakespeare by "the rules of the French academy", whereas, according to Guthrie:

[Shakespeare] proceeds by storm. He knows nothing of regular approaches to the fort of the human heart. He effects his breach by the weight of his metal, and makes his lodgement though the enemy's artillery is thundering round him from every battery of criticism, learning, and even probability (apud Rhodes, 2004: 220) Shakespeare had been used in England as a counter-establishment writer in the first half of the eighteenth century. Indeed, in the words of Michael Dobson, "Shakespeare became national poet in the 1730s as an Opposition playwright rather than an Establishment one" (Dobson, 1992: 136). Shakespeare's defenders in those days were part of the Patriots, an anti-Walpole faction within the Whig party, which often used Shakespeare criticism and quotations to criticise the government, particularly in The Craftsman, a newspaper that was an important Patriot mouthpiece. Yet, in the second half of the eighteenth century, Shakespeare was appropriated by the agents of a more conservative British nationalism, to which progressive men like Voltaire could react violently.

In his correspondence with D'Alembert, Voltaire expressed his disappointment at seeing the American Revolution apparently failing. However, the former encouraged him to carry on the fight against bardolatry in France, because, as D'Alembert put it, "since philosophy and reason have been conquered in New York, they must at least prevail in their own small domain" (apud Prince, 2012: 288). That, in Voltaire's mind, Shakespeare's rise to prominence was allied with British imperialism, and its concurrent desire to crush American liberties, is made extremely evident in his letter of October 1776 to French statesman Jacques Necker:

You are a great man, Sir, yourself, but I will never let Shakespeare become a fearful figure for France, one for whom Corneille and Racine could be burnt at the stake. I tend to be on the same side as those we call the American insurgents -I do not wish to be a slave to the English.

[Grand homme vous même, Monsieur; mais je ne consentirai jamais que Shakespear en soit un si redoutable pour la France, et qu'on lui immole Corneille et Racine. Je suis assez comme ceux qu'on appelle les insurgens d'Amérique, je ne veux point être l'esclave des Anglais] (Besterman, 1967 : 215)

Conclusion

As we know, Voltaire was wrong about the fate of the American Revolution, but his nightmare of British cultural dominance through Shakespeare turned out to be true in some regards. While early eighteenth-century critics had sought excuses for what could be considered as wild extravagances in the works of Shakespeare, when compared to French neoclassical norms in particular, the various conflicts, which set the British nation against its neighbours and particularly France changed the way the national corpus of literature came to be regarded by the end of the century. Several of Shakespearean plays, where the theme of international relations was prominent, and which lent themselves well to topical interpretations, were of course popular: Henry V, Coriolanus or Cymbeline especially, served such purposes (Prince, 2012: 277).

By the end of the eighteenth century, Shakespeare was on a safer textual ground and was being exported to other lands and to the confines of the British colonial empire. This was partly the Shakespeare that Voltaire disliked so much -one whose works, especially after the French Revolution -ceased to be regarded as a disordered garden, but became synonymous with "notions of order, self-restraint and authority" (Prince, 2012: 291) and were in fact set against Republican disorder. Thus, Edmund Burke would use Shakespeare to try to "impose order on the chaos of the French Revolution" (ibidem). Yet Voltaire's pessimism was, of course, largely blind to the fact that Shakespeare would be repeatedly transformed, appropriated by other countries and could again become an instrument of cultural and political negotiation between nations other than the British and French. As the British had liberated themselves from the yoke of French neoclassicism, they themselves had to resist the rise of German Romanticism in the early nineteenth century, the German Romantics famously seeing Shakespeare as theirs: "ganz unser" (completely ours), as August Wilhelm Schlegel called him) (Paulin, 2012: 323), thus opening the way for further national appropriations of Shakespeare worldwide. Shakespeare's works, like other important art forms, continue to be at the heart of culture wars today. That art forms are exploited in this way poses an important problem for any society. Crucial art will always be appropriated, and this is a normal process -it is its manipulation by political or market forces that threatens societies. Indeed, a society or a group of nations such as the European union suffers from those who claim that culture wars exist and that they are tied to that other great fantasy: the clash of civilisations. If this were true, as sociologist Irene Taviss Thomson points out again, A society experiencing a culture war would face grave difficulties. It would lack common standards and assumptions, and as a result, the ability to make public policy decisions would be severely compromised. Indeed, a society without such common ground could barely function. (Thomson, 2010: 12) Fortunately, not everyone buys into the fantasy of the superiority of certain values in the current so-called culture war, in which famous European artists can be manipulated in order to stand for alleged decent values. Even a quick look at Shakespearean academic criticism or at current theatrical productions would be enough to dispel these illusions. However appealing and politically convenient the idea of culture wars in Europe might be, it relies on a misconception touching the notion of culture itself. Since the end of the twentieth century, the concept of culture has come under scrutiny in academic circles. How could culture wars be a social reality, when social reality itself is devoid of concrete structures, coherence and stability? Those of us who study Shakespearean adaptation, for instance, know full well that culture is more akin to a "toolkit", or a "repertoire of skills and styles", with which artists create mediation and pastiches (Thomson, 2010, 13). It is my hope that this brief exploration of the origins of and reasons for the exploitation of Shakespeare's works for nationalistic and ideological reasons has gone some way towards throwing light on these issues.

Works Cited

Besterman, Theodore (ed.) (1967), Voltaire on Shakespeare, Studies on Voltaire and the Eighteenth Century, liv, Geneva, Institut et Musée Voltaire.

  gardens of Marly. (Leigh (ed.), 2007: 74) [Les monstres brillans de Shakespear plaisent mille fois plus que la sagesse moderne. Le génie poëtique des Anglais ressemble jusqu'à présent à un arbre touffu planté par la nature, jetant au hazard mille rameaux & croissant inégalement & avec force; il meurt, si vous voulez forcer sa nature & le tailler en arbre des jardins de Marly. (Lanson (ed.), 1917: 87-88)]

  [(…) D'ailleurs je suis fâché contre les Anglais. Non seulement ils m'ont pris Pondicheri à ce que je crois, mais ils viennent d'imprimer que leur Shakespear est infiniment supérieur à Corneille] (Besterman (ed.), 1967: 62)

3

 3 As the Seven Years' War was still not over, Voltaire began working on an edition of Corneille in 1762. That same year, Henry Home, Lord Kames, brought out his Elements of Criticism, in which he wrote rather disparaging words on Corneille and Racineeven ridiculing passages in some of their work -and sang the praises of Shakespeare. Voltaire reviewed Kames's book in the Gazette Littéraire in April 1764 in a tone that was part angry, part ironical, as Voltaire obliquely wondered how a Scottish judge like Kames who wrote on literature as well as gardening could pretend to become an arbiter of taste (Besterman (ed.), 1967: 88). That same year, in a letter to the Count and Countess of Argental, Voltaire talked about his review of Kames's Elements, and made the following extraordinary statement: As long as the British have been content to take our vessels and seize Canada and Pondicherry, I have been content to maintain a noble silence. But now that they push barbarity to the point of finding Racine and Corneille ridiculous, I have to take up arms. [Tant que les Anglais se sont contentés de prendre nos vaisseaux et de s'emparer du Canada et de Pondicheri, j'ai gardé un noble silence. Mais à présent qu'ils poussent la barbarie jusqu'à trouver Racine et Corneille ridicules, je dois prendre les armes] (Voltaire, 1953-65, liv, 42).

  [C'est moi qui autrefois parlai le premier de ce Shakespear; c'est moi qui le premier montrai au Français quelques perles que j'avais trouvées dans son énorme fumier] (Besterman (ed.), 1967: 175) Voltaire was exaggerating his distaste for Shakespeare, of course. What annoyed him most was the wave of Anglomania that was threatening to submerge France at a time when the British seemed still in a position to crush the hopes of the American revolutionaries whom Voltaire supported. To counter what he perceived as an assault also on French culture and values, he asked his friend Jean Le Rond d'Alembert, who was secretary of the Académie française, to read out a letter of protest. The letter, in which Voltaire underlined Shakespeare's shortcomings and criticized Kames's disrespectful treatment of Racine in his Elements of Criticism, was read out on 25 August 1776 at the Académie in the presence of the British ambassador and Elizabeth Montagu, who had specifically attacked Voltaire in her Essay on the Writings and Genius of Shakespeare (1769). Revealingly, Voltaire's correspondence in those months is full of military vocabulary, as he saw himself waging war and conducting battles under "General" D'Alembert, as he calls him in one of his letters (Besterman (ed.), 1967, pp. 182-3).

  dear and illustrious friend. As I mount the charge on Sunday, I shall cry 'Long live Saint Denis and Voltaire, and death to George Shakespear!' [Enfin, mon cher maître, voilà la bataille engagée et le signal donné. Il faut que Shakespear ou Racine demeurent sur la place; il faut faire voir à ces tristes & insolens Anglois, que nos gens de lettres savent mieux se battre contre eux que nos soldats & nos généraux. Malheureusement il y a parmi ces gens de lettres bien des déserteurs et des faux frères. Mais les déserteurs seront pris & pendus; ce qui me fâche, c'est que la graisse de ces pendus ne sera bonne à rien; car ils sont bien secs et bien maigres. Adieu, mon cher et illustre ami. Je crierai dimanche en allant à la charge, Vive s t Denis Voltaire & meure George Shakespear!] 4

  Shakespeare entered the sphere of respected printed literature first through his folios and in the ensuing series of eighteenth-century editions. Despite their still controversial nature and the multiple wrangles between editors, eighteenth-century textual studies made great strides thanks to Shakespeare and to the dual enterprise of establishing his text and developing reliable philological tools -Samuel Johnson's mutually dependent projects of a Dictionary (1755) and of an edition of Shakespeare's works (1765) being good examples.


Gerd Behrmann
email: behrmann@cs.aau.dk

Agnès Cougnard
email: acougnar@cs.aau.dk

Alexandre David
email: adavid@cs.aau.dk

Emmanuel Fleury
email: fleury@labri.fr

Kim G Larsen

Didier Lime
email: didier.lime@irccyn.ec-nantes.fr

Uppaal-Tiga: Timed Games for Everyone

In 2005 we proposed the first efficient on-the-fly algorithm for solving games based on timed game automata with respect to reachability and safety properties. Since then we have completely re-implemented the first prototype and made dramatic improvements both in terms of performance (several orders of magnitude) and the input language (complete support of all the language features of Uppaal). In addition, the tool supports the new feature of strategy generation with different compactness levels. In this paper we present this new version of Uppaal-Tiga.

Introduction

Uppaal-Tiga1 is a tool for solving games based x > 1;u1

x < 1 u2 x := 0

x ≥ 2;c2

x < 1 u3 c3

x ≤ 1;c4 Fig. 1. Timed game automaton example.

on timed game automata. It is implementing the on-the-fly algorithm presented in [CDF + 05]. The new version is faster by several orders of magnitude (more than 1000 times faster on large examples) and consumes much less memory (100 times less on large examples) than the first prototype presented at CONCUR'05. These dramatic improvements come from a different internal architecture of the tool and improvements on the DBM library supporting subtractions, partitions, federations, intersections, specific operations for timed games (e.g. predt), and the essential operation of merging several difference bound matrices (DBMs) into one.

In addition, the full input language of Uppaal-4.0[BDH + 06] 2 is now supported, which gives the user access to C-like syntax to declare functions, custom types, and other control statements (loops etc). The input language is given by a network of timed game automata [START_REF] Maler | On the Synthesis of Discrete Controllers for Timed Systems[END_REF] (TGA). Let us consider the example of the TGA A shown in Fig. 1 consisting of a timed automaton with one clock x and two types of edges: controllable (c i ) and uncontrollable (u i ). The reachability game consists in finding a strategy for a controller, i.e. when to take the controllable transitions that will guarantee that the system, regardless of when and if the opponent chooses to take uncontrollable transitions, will eventually end up in the location Goal. Winning (or losing) conditions of the game are specified by TCTL formulas in the tool. In this example, we play a reachability game and the corresponding formula control: A<> A.Goal is satisfied. The tool gives the strategy (obtained with option -t0) shown in Fig. 2. The tool supports both reachability and safety games. If there is no winning strategy, the tool will give a counter strategy for the opponent (environment) to make the controller lose.

The new features of Uppaal-Tiga since version 0.7 are:

-Better performance.

-Definition of winning/losing conditions as TCTL formulas.

-Support for both reachability and safety games.

-Support for the full input language of Uppaal-4.0 (except for priorities that are incompatible). -Output of strategies.

-Different compactness levels for the strategies (can output the strategy as a BDD/CDD). -Ability to play against Uppaal-Tiga with the command line verifier.

Strategy Synthesis

A strategy is a function f : L × R C ≥0 → Act c ∪ {λ} that constantly gives information as to what the controller should do during the course of the game. In a given situation, the strategy could suggest the controller to either "do a particular controllable action" or "do nothing at this point in time (λ)". A strategy is said to be a winning strategy if the controller supervised by the strategy always win the game whatever actions are chosen by the environment. If there is no such winning strategy, there exists a counter-strategy to either make the controller lose (reach a state marked as losing) or just prevent it to win.

Query Extensions

Winning conditions are specified through extensions of the Uppaal query language. Given a timed game automaton A, a set of goal states (win) and/or a set of bad states (lose), both defined by classical Uppaal state formulas, four types of winning conditions can be issued. For all of them, the game is to find a controllable strategy f such that A supervised by f ensures that:

control: A<> win (must reach win) control: A[ not(lose) U win ] (must reach win and must avoid lose) control: A[ not(lose) W win ] (should reach win and must avoid lose) control: A[] not(lose) (must avoid lose) By default, Uppaal-Tiga first check whether there is or not a winning strategy for the timed game given a winning condition. Note that it is always better to start asking for existence because the process of strategy extraction is quite demanding.

Strategy Extraction

The command line tool verifytga in the Uppaal-Tiga package can output a strategy for each given winning condition when used with the -t0 option. If no such strategy exists, a counter-strategy for the environment is given. Two additional options are available to control compactness: -c0 makes sure that states with a given discrete part are unique and -c1 prints the strategy as a BDD/CDD representation.

Game Simulation

The -p option of verifytga is the interactive mode where users can play against the strategy synthesized by Uppaal-Tiga. Users may choose transitions to fire or to delay when it is their turns to play. For example, choices can be: t2 (take transition t 2 ) or w4.3 (wait for 4.3 time units).

Experiments And Conclusion

We compare the new verifier against the old prototype (old) against the current new version (new). The example is the production cell of [START_REF] Lewerentz | Production Cell: A Comparative Study in Formal Specification and Verification[END_REF][START_REF] Melcher | Controller Synthesis for the "Production Cell" Case Study[END_REF]. Time constraints are defined to make the model controllable (c) or uncontrollable (u) with different numbers of plates (c3, c6 . . . ). cannot handle the models with 12 plates, which is shown by "-". The new version is obviously far better.

The new version of Uppaal-Tiga is becoming mature and is fully operational with the GUI of Uppaal and the only missing feature of the tool is the ability to play strategies in the simulator, which we are currently working on. An interesting remark that we cannot develop in this short abstract is that back-propagating only winning or losing information is more efficient than back-propagating both as first thought in [CDF + 05].

  State: ( A.l1 ) While you are in (A.x<1), wait. When you are in (A.x==1), take transition A.l1->A.l2 { x <= 1, tau, 1 } State: ( A.l2 ) While you are in (1<=A.x && A.x<2) || (A.x<1), wait. When you are in (2<=A.x), take transition A.l2->A.Goal { x >= 2, tau, 1 } State: ( A.l3 ) When you are in (A.x<=1), take transition A.l3->A.l4 { 1, tau, 1 } State: ( A.l4 ) When you are in (A.x==1), take transition A.l4->A.l2 { x <= 1, tau, 1 } While you are in (A.x<1), wait.

Fig. 2 .

 2 Fig. 2. Synthesized strategy of the example of Fig. 1.

Table 1 .

 1 Table 1 shows time consumption is given in seconds (s) and memory usage in megabytes (M). The old version Experimental results on the production cell.

	Model	c3	c6	c12	u3	u6	u12
	Old	0.1s 1M 12s 63M -	-	0.2s 3M 235s 273M -	-
	New 0.05s 3.5M 0.05s 3.5M 0.14s 55M 0.02s 3.5M 0.04s 3.5M 0.12s 55M

http://www.cs.aau.dk/∼adavid/tiga/

See http://www.uppaal.com.