Philippe Castagliola
email: philippe.castagliola@univ-nantes.fr

Michael B C Khoo

A Synthetic Scaled

Keywords: Synthetic Xbar chart, Weighted Variance Xbar chart, Scaled Weighted Variance Xbar chart, Johnson distributions Synthetic X chart, Weighted Variance X chart, Scaled, Weighted Variance X chart, Johnson distributions

The synthetic control chart was introduced by [START_REF] Wu | A Synthetic Control Chart for Detecting Small Shifts in the Process Mean[END_REF] as an improvement over the Shewhart X chart for detecting shifts in the mean of a normally distributed process. The synthetic chart for the mean integrates the Shewhart X chart and the conforming run length (CRL) chart. [START_REF] Wu | A Synthetic Control Chart for Detecting Small Shifts in the Process Mean[END_REF] showed that for moderate shifts in the mean, the synthetic chart reduces the out-of-control average run length (ARL) by nearly half while maintaining the same in-control ARL. They also demonstrated that the synthetic chart outperforms the exponentially weighted moving average (EWMA) chart and the joint X-EWMA charts when the mean shift is greater than 0.8σ.

Other works on synthetic charts are as follow : Wu & Spedding (2000a) presented a program in C to design a synthetic chart that minimizes the out-ofcontrol ARL based on an optimization model. Wu & Yeo (2001) and [START_REF] Wu | A Synthetic Control Chart for Detecting Fraction Nonconforming Increases[END_REF] proposed synthetic charts for detecting increases in the fraction nonconforming. [START_REF] Calzada | The Robustness of the Synthetic Control Chart to Non-normality[END_REF] found that the synthetic chart of [START_REF] Wu | A Synthetic Control Chart for Detecting Small Shifts in the Process Mean[END_REF] is reasonably close to the normal theory values for moderate nonnormality or when the sample size n is large. [START_REF] Davis | Evaluating and Improving the Synthetic Control Chart[END_REF] presented a Markov chain model of the synthetic chart suggested by Wu & Spedding (2000a) and used it to evaluate the chart's zero-state and steady-state ARL performances, besides altering the chart to achieve a better ARL performance. [START_REF] Sim | Combined X and CRL Charts for the Gamma Process[END_REF] studied the performance of the synthetic chart based on the gamma and exponential distributions for known and unknown parameters, respectively and concluded that the synthetic chart outperforms the Shewhart X chart with either asymmetric probability limits or 3-sigma control limits. [START_REF] Scariano | A Note on the Lower-sided Synthetic Chart for Exponentials[END_REF] discussed a synthetic chart for exponential data, derived an expression for its ARL and design parameters and showed that the chart outperforms the Shewhart chart for individuals but is inferior to the EWMA and cumulative sum (CUSUM) charts in detecting decreases in the exponential mean. Huang & Chen (2005) suggested a synthetic chart for monitoring process dispersion by combining the sample standard deviation, S chart and the CRL chart. Chen & Huang (2005) combined the sample range, R chart and the CRL chart to form a synthetic chart for process dispersion. [START_REF] Costa | A Synthetic Control Chart for Monitoring the Process Mean and Variance[END_REF] proposed a synthetic chart based on a noncentral chi-square statistic that is superior to the joint X and R chart in detecting shifts in the mean and/or standard deviation. [START_REF] Costa | Synthetic Control Chart for Monitoring the Process Mean and Variance[END_REF] considered a synthetic chart with two-stage sampling to monitor the process mean and variance, and claimed that the chart is more convenient to administer than the joint X and S chart with double sampling, although both charts have similar performances. Similar to the X, EWMA and CUSUM charts, the synthetic chart for the mean proposed by [START_REF] Wu | A Synthetic Control Chart for Detecting Small Shifts in the Process Mean[END_REF] requires the assumption that the distribution of the quality characteristic is normal or approximately normal. But in some situations, it may happen that this condition does not hold (for instance, see [START_REF] Jacobs | Watch Out for Nonnormal Distributions[END_REF]). Experience in the chemical industry shows that there are a number of reasons why a process that is operating in a state of statistical process control, yields non normal skewed distributions. Some of these reasons are:

• measurements or operation in the vicinity of a material's physical limits, e.g. saturation, phase change, boiling point, tensile strength.

• measurements of a characteristic that has zero as a natural limit, e.g. moisture content, impurity content, warpage, bow.

• mathematical relationships between variables, e.g. a variable with an Arrhenius-type exponential dependence on process temperature.

To deal with nonnormal underlying distributions, the approaches that are currently used are (i) transforming the data to attain an approximate normal distribution, (ii) increasing the sample size so that the sample average follows an approximate normal distribution, and (iii) employing heuristic control charts for skewed populations. The existing heuristic charts for skewed populations are the X and R charts based on the weighted variance (WV) method proposed by [START_REF] Bai | X and R Control Charts for Skewed Populations[END_REF], the X chart based on the scaled weighted variance (SWV) method suggested by [START_REF] Castagliola | X Control Chart for Skewed Populations Using a Scaled Weighted Variance Method[END_REF], the X, CUSUM and EWMA charts using the weighted standard deviation method presented by [START_REF] Chang | Control Charts for Positively-skewed Populations with Weighted Standard Deviations[END_REF] and the X and R charts based on the skewness correction method proposed by [START_REF] Chan | Skewness Correction X and R Charts for Skewed Distributions[END_REF].

Some of the other works on control charts for skewed populations are made by (i) [START_REF] Schneider | Control Charts for Skewed and Censored Data[END_REF] who discussed methods to establish control limits when the data are positively skewed and censored from below, (ii) [START_REF] Wu | Asymmetric Control Limits of the X Chart for Skewed Process Distributions[END_REF] who proposed an approach to optimize the control limits of the X chart for skewed populations so that the average number of scrap products is minimized without increasing the Type-I error, (iii) [START_REF] Dou | One-sided Control Charts for the Mean of Positively Skewed Distributions[END_REF] who suggested a procedure to construct a one-sided X chart for positively skewed distributions using the Edgeworth expansion method, (iv) [START_REF] Chen | Economic Design of X Control Charts for Non-normal Data Using Variable Sampling Policy[END_REF] who presented an economic design of X charts for nonnormal data using variable sampling policy, (v) [START_REF] Nichols | A Bootstrap Control Chart for Weibull Percentiles[END_REF] who considered a bootstrap control chart for Weibull percentiles, (vi) [START_REF] Kan | The Individuals Control Charts for Burr Distributed and Weibull Distributed Data[END_REF] who proposed a skewness correction method in setting the asymmetric limits of the individuals charts for Burr and Weibull distributed data, and (vii) [START_REF] Tsai | Skew Normal Distribution and the Design of Control Charts for Averages[END_REF] who developed two control charts and process capability ratios based on the skew normal distribution to monitor the process mean and evaluate the process capability of nonnormal data.

Recently, Khoo, Z. Wu & Atta (2008) proposed a synthetic control chart for monitoring shifts in the process mean of skewed populations using the WV method, where no assumption of the distribution of the underlying process is needed.

This chart was shown to provide vast improvements over all the existing charts for skewed populations, in terms of false alarm and mean shift detection rates for cases with known and unknown parameters.

This paper extends the work of [START_REF] Khoo | A Synthetic Control Chart for Monitoring the Process Mean of Skewed Populations based on the Weighted Variance Method[END_REF] by proposing a synthetic Scaled WV (SWV) control chart for monitoring the mean of skewed populations.

The synthetic SWV-X chart will be shown to outperform the synthetic WV-X chart of [START_REF] Khoo | A Synthetic Control Chart for Monitoring the Process Mean of Skewed Populations based on the Weighted Variance Method[END_REF] for the case with a negative shift in the mean, when the same in-control ARL is considered for the two charts. For this case, the superiority of the synthetic SWV-X chart increases with the level of skewness.

Note that for a positive shift in the mean, the synthetic SWV-X chart is only slightly less effective than the synthetic WV-X chart. Thus, for a process having a skewed population, where past experience indicates that whenever a signal is triggered a negative shift usually occurs, then the synthetic SWV-X chart can be a favourable substitute for the synthetic WV-X chart. The rest of this paper is organized as follows : Section 2 gives a review on the synthetic X, the WV-X and the SWV-X charts. Section 3 presents the proposed synthetic SWV-X chart and details the methodology used for comparing both the synthetic WV-X and synthetic SWV-X charts. Section 4 illustrates the use of the synthetic SWV-X with an example. Section 5 completes the paper with the main conclusions drawn from our study.

2 Literature review The synthetic X chart comprises a X/S sub-chart and a CRL/S sub-chart.

The CRL is defined as the number of inspected units between two consecutive nonconforming units (including the ending nonconforming unit). Figure 1 is an example that shows how the CRL value is determined, assuming that a process starts at t = 0. Here, CRL 1 = 5, CRL 2 = 2 and CRL 3 = 4. The operation of the synthetic X chart is based on the following steps:

Step 1 : Set the lower control limit: L ∈ {1, 2, . . .} of the CRL/S sub-chart and set the constant K > 0 of the X/S sub-chart defined by the following control limits:

LCL X = µ -Kσ (1) U CL X = µ + Kσ (2)
Step 2 : Take a random sample of n observations at each inspection point and compute the sample mean, X.

Step Step 4 : Count the number of X samples between the current and the last nonconforming sample (which includes the current but excludes the last nonconforming sample) as the CRL value of the CRL/S sub-chart.

3 : If LCL X < X < U CL X ,
Step 5 : If the value of CRL ≥ L, the process is declared in-control and the control flow moves back to Step 2. Otherwise, the process is out-ofcontrol and the control flow advances to Step 6.

Step 6 : Signals an out-of-control status to indicate a process mean shift.

Step 7 : Find and remove assignable cause(s). Then move back to Step 2. [START_REF] Wu | A Synthetic Control Chart for Detecting Small Shifts in the Process Mean[END_REF] demonstrated that the Average Run Length (ARL) of the synthetic X control chart corresponding to specific values of K, L, n and where Φ(.) is the standard normal distribution function. In particular, for δ = 0, we have

δ = |µ -µ 1 |/σ (desired magnitude of standardized mean shift) is equal to ARL(δ) = 1 π(1 -(1 -π) L ) (3) with π = Φ(-(K + δ) √ n) + Φ(-(K -δ) √ n) (4
ARL(0) = 1 2Φ(-K √ n)(1 -(1 -2Φ(-K √ n)) L )
Using these equations, [START_REF] Wu | A Synthetic Control Chart for Detecting Small Shifts in the Process Mean[END_REF] suggested optimal combinations of K and L (useful for the Step 1 described above) that minimize the out-ofcontrol ARL for desired magnitudes of the standardized mean shift δ and an in-control ARL (ARL 0 ) of interest.

The Weighted Variance and Scaled Weighted Variance X charts

Now, let us consider that the distribution of the quality characteristic X is no longer normal but is some continuous unimodal skew distribution f X (x), where

µ = E(X) is the in-control mean, σ = σ(X)
is the the in-control standarddeviation and θ = P (X ≤ µ) is the in-control probability that X is less than or equal to the mean µ. The Weighted Variance X chart (WV-X chart in short)

was initially proposed by [START_REF] Choobineh | Control-limits of QC Charts for Skewed Distributions Using Weighted-variance[END_REF] who suggested the use of the semivariance approximation of [START_REF] Choobineh | A Simple Approximation for Semivariance[END_REF] in order to provide control limits for the mean in the case of a quality characteristic having a skew distribution. [START_REF] Bai | X and R Control Charts for Skewed Populations[END_REF] provided computations and tables to simplify the implementation of the WV-X chart proposed by [START_REF] Choobineh | Control-limits of QC Charts for Skewed Distributions Using Weighted-variance[END_REF]. The control limits of the WV-X chart are

LCL W V = µ -K W V L σ (5) U CL W V = µ + K W V U σ (6)
where

K W V L and K W V U are equal to K W V L = Φ -1 (1 -α 2 ) 2(1 -θ) n (7) K W V U = Φ -1 (1 -α 2 ) 2θ n ( 8
)
where n is the sample size, Φ -1 (.) the inverse standard normal distribution (distribution of X is symmetrical) then

K W V L = K W V U = Φ -1 (1 -α 2 ) √ n
and the control limits in equation ( 5) and equation ( 6) are reduced to the classical Shewhart X control limits. The Scaled Weighted Variance X chart (SWV-X chart in short) was suggested by [START_REF] Castagliola | X Control Chart for Skewed Populations Using a Scaled Weighted Variance Method[END_REF] as an improvement over the WV-X chart. [START_REF] Castagliola | X Control Chart for Skewed Populations Using a Scaled Weighted Variance Method[END_REF] provided explanations concerning the shortcomings of the WV method and how these shortcomings were addressed using the SWV method. The control limits of the SWV-X chart are as follow (see [START_REF] Castagliola | X Control Chart for Skewed Populations Using a Scaled Weighted Variance Method[END_REF]):

LCL SW V = µ -K SW V L σ (9) U CL SW V = µ + K SW V U σ ( 10
)
where

K SW V L and K SW V U are equal to K SW V L = Φ -1 1 - α 4θ 1 -θ nθ (11)
K SW V U = Φ -1 1 - α 4(1 -θ) θ n(1 -θ) (12)
It is worth to note that the two constants above can only be computed if α 4 < θ < 1 -α 4 and, as for the WV-X chart, if θ = 1 2 then

K SW V L = K SW V U = Φ -1 (1 -α 2 ) √ n
and the control limits in equation ( 9) and equation ( 10) are also reduced to the classical Shewhart X control limits.

3 The Synthetic Weighted Variance and Synthetic Scaled Weighted Variance X charts

The synthetic WV-X chart, suggested by [START_REF] Khoo | A Synthetic Control Chart for Monitoring the Process Mean of Skewed Populations based on the Weighted Variance Method[END_REF], is based on the idea of integrating the WV method of [START_REF] Bai | X and R Control Charts for Skewed Populations[END_REF] with the synthetic X chart of [START_REF] Wu | A Synthetic Control Chart for Detecting Small Shifts in the Process Mean[END_REF]. The operation of the synthetic WV-X chart is similar to that of the synthetic X chart described in section 2.1, except that the control limits in equation ( 1) and equation ( 2) are replaced with the control limits in equation ( 5) and equation ( 6) for the WV-X/S sub chart. [START_REF] Khoo | A Synthetic Control Chart for Monitoring the Process Mean of Skewed Populations based on the Weighted Variance Method[END_REF] compared by simulation the synthetic WV-X chart with different other alternatives (i.e. SC-X chart by [START_REF] Chan | Skewness Correction X and R Charts for Skewed Distributions[END_REF] and the WSD-X, WSD-CUSUM and WSD-EWMA charts by [START_REF] Chang | Control Charts for Positively-skewed Populations with Weighted Standard Deviations[END_REF]) and concluded that the former gives the most favourable results, in terms of false alarms and mean shift detection rates, in both the known and unknown parameter cases, where the results of the synthetic WV-X chart are even better when the skewness of the underlying distribution is larger.

In this paper, we suggest to integrate the SWV method of [START_REF] Castagliola | X Control Chart for Skewed Populations Using a Scaled Weighted Variance Method[END_REF] with the synthetic X chart of [START_REF] Wu | A Synthetic Control Chart for Detecting Small Shifts in the Process Mean[END_REF] by replacing the control limits in equation ( 1) and equation ( 2) with the control limits in equation ( 9) and equation ( 10) for the SWV-X/S sub chart. The resulting chart will be called a synthetic SWV-X chart. The goal of this paper is to evaluate the respective efficiencies of both the synthetic WV-X and synthetic SWV-X charts in terms of the out-of-control ARL. In order to compare these two charts, we have chosen an innovative methodology that does not involve any simulation. This methodology is decribed below:

1. For the sake of simplicity, we assume that µ = 0 and σ = 1.

2. Let β = E(( X-µ σ ) 3 ) and ψ = E(( X-µ σ ) 4 ) -3 be the skewness and kurtosis coefficients, respectively of the quality characteristic X. In our study, we restrict the values of the skewness coefficient β ∈ {0.5, 1 , 1.5, 2, 2.5, 3, 3.5, 4, 4.5} and, for each of these values, we select 7 different values ψ 1 , . . . , ψ 7 for the kurtosis coefficient ψ. As Figure 2 clearly shows, for each selected skewness coefficient β, the 6 smaller kurtosis coefficients ψ 1 , . . . , ψ 6 are uniformly distributed within the curve corresponding to the lower limit for any possible distributions and the curve corresponding to the lognormal distribution while the largest kurtosis coefficient ψ 7 is just above the curve corresponding to the lognormal distribution. This strategy guarantees to cover a large spectrum of distributions, including the gamma, Weibull Table 1. 3. For each combination (β, ψ i ), for i = 1, . . . , 7, in Table 1, we compute the parameters (a i , b i , c i , d i ) of the [START_REF] Johnson | Systems of Frequency Curves Generated by Methods of Translation[END_REF] distribution having µ = 0 for mean, σ = 1 for standard-deviation, β for skewness coefficient and ψ i , i = 1, . . . , 7 for kurtosis coefficient (the estimation algorithm is due to [START_REF] Hill | Fitting Johnson Curves by Moments[END_REF]). Based on Johnson's work, we know that there is an unique set of parameters (a i , b i , c i , d i ) satisfying this condition (the main properties of the Johnson system of distributions are summarized in the Appendix). Let F J (x|a, b, c, d) be the Johnson distribution function of parameters (a, b, c, d). For each β ∈ {0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5}, we

can compute θ i = P (X ≤ µ) = F J (µ|a i , b i , c i , d i ) and θ = 1 7 (θ 1 + • • • + θ 7
) being the average probability that X is less than or equal to the mean µ over the spectrum of considered distributions.

4. If the distribution of the quality characteristic X is known, the distribution of the sample mean X is generally unknown (except for some rare cases) and, therefore, there is no closed-form for it. Nevertheless, it is well known that the mean µ X , the standard-deviation σ X , the skewness coefficient β X and the kurtosis coefficient ψ X of X are related to µ, σ, β and ψ through the following simple formulae:

µ X = µ σ X = σ √ n β X = β √ n ψ X = ψ n
Consequently, if the distribution of the sample mean X is actually unknown, we simply suggest to approximate it with the unique Johnson distribution with parameters (a X , b X , c X , d X ) having µ X for mean, σ X for standard-deviation, β X for skewness coefficient and ψ X for kurtosis coefficient and estimated with the algorithm of [START_REF] Hill | Fitting Johnson Curves by Moments[END_REF]. In order to validate this approach, we conducted a thorough study (not presented here) where we computed for different combinations of (β, ψ), the cumulative distribution function of X either by intensive simulations or by fitting with a Johnson distribution. The result of this study clearly demonstated that the cumulative distribution function of X obtained by fitting a Johnson distribution is extremely close to the real one obtained by simulation, thus providing an easy-to-use and accurate approximation for the cumulative distribution function of X.

5. For a combination (β, ψ i ), the ARL of both the Synthetic WV-X and Synthetic SWV-X charts are computed using equation (3) where π in equation ( 4) is replaced by

π = F J (K L |a X , b X , c X + δ, d X ) + 1 -F J (K U |a X , b X , c X + δ, d X ). Here (K L , K U ) are the constants (K W V L , K W V U ) for the Synthetic WV- X chart or (K SW V L , K SW V U
) for the Synthetic SWV-X chart. Consquently, for values of L, K L , K U and β, we can compute 7 differents values ARL 1 , . . . , ARL 7 corresponding to the 7 kurtosis ψ 1 , . . . , ψ 7 and we can also compute ARL = 1 7 (ARL 1 + • • • + ARL 7 ) as the average ARL over the spectrum of considered distributions.

In Tables 2, 3 and 4 we have computed the values of the constants K L , K U and L, for both the Synthetic WV-X chart and the Synthetic SWV-X chart, for n = 5, β ∈ {0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5} and for δ ∈ {-1.5, -1, -0.7, -0.5, -0.4, -0.3, -0.2, -0.1, 0.1, 0.2, 0.3, 0.4, 0.5, 0.7, 1, 1.5}. The average in-control ARL is ARL 0 = 370.4. For example, if the value of the skewness coefficient is β = 1.5 then the average probability that X is less than or equal to the mean µ is θ = 0.636. If we want to detect a standardized mean shift δ = -0.5 (i.e. a decrease of 0.5σ), we have K L = 0.851, K U = 1.126, L = 7 for the Synthetic WV-X chart and the average out-of-control ARL is ARL = 5.1 while, for the Synthetic SWV-X chart, we have K L = 0.789, K U = 1.252, L = 9 and the average out-of-control ARL is ARL = 3.7. In Tables 2, 3 and 4, the ARL values in bold characters correspond to the lowest out-of-control average ARL's. This clearly demonstrates that when the standardized mean shift δ < 0, the Synthetic SWV-X chart always have smaller average out-of-control ARL than the Synthetic WV-X chart. When the standardized mean shift δ > 0, the previous conclusion is reversed.

An illustrative example

In order to illustrate the use of the Synthetic SWV-X chart, let us consider a 125g yogurt cup filling process where the quality characteristic X is the weight of each yogurt cup. A long term study (Phase I, realized in a local company) based on a large database of yogurt cup weights showed that the distribution of the quality characteristic X is significantly skewed. This study also allowed accurate estimations of the in-control mean µ = 124.9, the in-control standard-deviation σ = 0.76 and the in-control probability θ = 0.679 that X is less than or equal to its mean µ. The quality practitioner in charge of this process decided to take n = 5 yogurt cups every hour. Based on Table 3, he decided to choose the value of θ = 0.682 (which is the closest to the in-control probability θ = 0.679) and to use the constants K L , K U and L optimally designed for detecting a mean shift δ = -0.3, i.e. K L = 0.701, K U = 1.306 and L = 9, yielding the following Synthetic SWV-X control limits LCL = 124.9 -0.701 × 0.76 = 124.37 U CL = 124.9 + 1.306 × 0.76 = 125.89

In Table 5, we recorded 30 samples corresponding to a 30 hours sequence of production (Phase II) from the 101th hour to the 130th hour. In each row we have the values corresponding to n = 5 yogurt cups weighed every hour. The last column is the mean Xi of these n = 5 values. The samples in Table 5 are also plotted in Figure 3 (top). In Figure 3 (bottom), we plotted the mean Xi of the 30 samples with the control limits LCL = 124.37 and U CL = 125.89 of the SWV-X/S sub chart. Concerning the values of Xi , for i = 1, . . . , 100, (corresponding to the starting phase of the process, but not recorded in Table 5), they all verify that LCL < Xi < U CL. As we can see in Figure 3 Thus, we have CRL 3 = 127 -123 = 4 < L = 9 and we can conclude that an out-of-control situation occured corresponding to a downward shift in the process mean (i.e. less yogurt in each of the cups), probably due to a clog in the pipe used for filling the cups.

WV-X chart SWV-X chart β θ δ K L K U L ARL K L K U L ARL 0.5 0.
WV-X chart SWV-X chart β θ δ K L K U L ARL K L K U L ARL 2.0 0.663 -1.5 0.

Conclusions

A synthetic SWV-X chart for skewed populations is suggested in this paper.

The ARL results have shown that the synthetic SWV-X chart gives a more favourable performance than the synthetic WV-X chart when the mean of an underlying process from a skewed population shifts downward or in the negative direction. Consequently, the synthetic SWV-X chart can be a favourable substitute for the synthetic WV-X chart in process monitoring when the mean of a skewed population is likely to shift downward, whenever a change occurs.

  function and α is a desired Type-I error. It is worth to note that if θ =

9

  Page . The values of the skewness β and kurtosis ψ coefficients used in this paper are listed in

Figure 2 :

 2 Figure 2: Selected skewness β and kurtosis ψ i coefficients covering the area corresponding to the gamma, the Weibull and the lognormal distributions.

  Figure 3: (top) 30 samples of size n = 5 corresponding to a 30 hours sequence of production, (bottom) the corresponding Synthetic SWV-X chart

Page 6 of 23 URL: http://mc.manuscriptcentral.com/lssp E-mail: comstat@univmail.cis.mcmaster.ca Communications in Statistics -Simulation and Computation

	F o r
	P
	e Figure 1: Conforming Run Length
	e r
	R
	e
	v i e
	w
	O n l the sample is considered as a conforming sample and the control flow moves back to Step 2. Otherwise, the y
	sample is a nonconforming sample and the control flow advances to
	Step 4.
	5

Table 1 :

 1 Skewness β and kurtosis ψ i coefficients used for the comparison of the synthetic WV-X and synthetic SWV-X charts

	ψ

11 Page 12 of 23 URL: http://mc.manuscriptcentral.com/lssp E-mail: comstat@univmail.cis.mcmaster.ca Communications in Statistics -Simulation and Computation

	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

Table 2 :

 2 Constants K L , K U and L, for both the Synthetic WV-X and Synthetic

	554 -1.5 0.868 0.968	2	1.1	0.843 1.004	2	1.1
	-1.0 0.918 1.023	4	1.8	0.890 1.063	4	1.7
	-0.7 0.955 1.065	7	3.7	0.925 1.108	7	3.3
	-0.5 0.994 1.109 13	8.7	0.960 1.153 13	7.3
	-0.4 1.020 1.137 20	16.0	0.984 1.183 20	12.6
	-0.3 1.043 1.163 30	34.3	1.009 1.215 33	25.0
	-0.2 1.056 1.177 38	89.1	1.037 1.251 60	58.8
	-0.1 0.972 1.083	9	236.8	1.057 1.277 98	164.8
	0.1 1.145 1.277 263	191.5 1.110 1.344 472	261.6
	0.2 1.114 1.242 125	90.9 1.082 1.309 190	127.7
	0.3 1.083 1.208 65	45.6 1.054 1.273 90	62.5
	0.4 1.054 1.175 37	24.7 1.026 1.237 47	32.6
	0.5 1.028 1.146 23	14.3 0.999 1.203 27	18.3
	0.7 0.984 1.097 11	5.9	0.956 1.148 12	7.0
	1.0 0.933 1.040	5	2.3	0.904 1.081	5	2.6
	1.5 0.868 0.968	2	1.1	0.843 1.004	2	1.2
	1.0 0.600 -1.5 0.817 1.001	2	1.1	0.778 1.077	2	1.1
	-1.0 0.863 1.058	4	1.6	0.818 1.140	4	1.5
	-0.7 0.889 1.089	6	2.9	0.839 1.174	6	2.6
	-0.5 0.920 1.128 10	6.5	0.869 1.221 11	5.0
	-0.4 0.936 1.147 13	12.0	0.886 1.248 16	8.1
	-0.3 0.944 1.158 15	27.2	0.905 1.278 25	15.5
	-0.2 0.914 1.120	9	77.9	0.927 1.313 44	37.0
	-0.1 0.817 1.001	2	210.6	0.935 1.327 55	121.9
	0.1 1.081 1.326 177	176.6 1.003 1.432 514	244.9
	0.2 1.052 1.290 100	88.5 0.984 1.403 248	132.8
	0.3 1.024 1.256 60	47.4 0.965 1.373 130	72.4
	0.4 0.998 1.224 38	27.1 0.946 1.343 73	41.1
	0.5 0.974 1.195 25	16.4 0.926 1.312 43	24.4
	0.7 0.931 1.141 12	7.0	0.888 1.252 17	9.8
	1.0 0.877 1.075	5	2.7	0.839 1.174	6	3.4
	1.5 0.817 1.001	2	1.2	0.778 1.077	2	1.3
	1.5 0.636 -1.5 0.770 1.019	2	1.1	0.723 1.127	2	1.1
	-1.0 0.796 1.053	3	1.5	0.755 1.188	4	1.4
	-0.7 0.829 1.097	5	2.5	0.772 1.221	6	2.1
	-0.5 0.851 1.126	7	5.1	0.789 1.252	9	3.7
	-0.4 0.860 1.138	8	9.4	0.803 1.279 13	5.7
	-0.3 0.841 1.113	6	21.9	0.817 1.306 19	10.3
	-0.2 0.796 1.053	3	64.2	0.833 1.335 29	24.5
	-0.1 0.724 0.959	1	183.3	0.819 1.309 20	96.7
	0.1 1.046 1.384 129	180.6 0.913 1.485 328	218.1
	0.2 1.012 1.339 75	94.8 0.897 1.456 193	124.3
	0.3 0.981 1.298 47	52.8 0.881 1.425 116	72.1
	0.4 0.953 1.261 31	30.9 0.865 1.396 74	43.4
	0.5 0.929 1.230 22	18.9 0.850 1.368 48	27.1
	0.7 0.882 1.166 11	8.1	0.823 1.316 22	11.6
	1.0 0.829 1.097	5	3.0	0.784 1.243	8	4.1
	1.5 0.770 1.019	2	1.2	0.723 1.127	2	1.3
	SWV-X charts, for n = 5, β ∈ {0.5, 1, 1.5} and ARL 0 = 370.4.
		14

Table 3 :

 3 Constants K L , K U and L, for both the Synthetic WV-X and Synthetic

	730 1.024	2	1.1	0.672 1.141	2	1.1
	-1.0 0.759 1.065	3	1.4	0.689 1.177	3	1.3
	-0.7 0.781 1.095	4	2.2	0.709 1.222	5	1.8
	-0.5 0.798 1.119	5	4.4	0.723 1.251	7	3.0
	-0.4 0.798 1.119	5	8.0	0.733 1.274	9	4.4
	-0.3 0.759 1.065	3	18.6	0.745 1.300 12	7.7
	-0.2 0.730 1.024	2	53.4	0.751 1.314 14	19.4
	-0.1 0.684 0.959	1	159.4	0.709 1.222	5	86.7
	0.1 1.040 1.459 114	190.6 0.866 1.558 281 213.5
	0.2 0.998 1.400 63	105.7 0.850 1.524 170 126.6
	0.3 0.961 1.348 39	61.2 0.831 1.485 102	76.8
	0.4 0.929 1.303 26	36.8 0.814 1.446 64	47.7
	0.5 0.899 1.261 18	22.8 0.797 1.411 42	30.4
	0.7 0.852 1.195 10	9.6	0.766 1.345 20	13.3
	1.0 0.798 1.119	5	3.4	0.728 1.263	8	4.6
	1.5 0.730 1.024	2	1.2	0.672 1.141	2	1.4
	2.5 0.682 -1.5 0.705 1.033	2	1.1	0.632 1.139	2	1.1
	-1.0 0.738 1.081	3	1.3	0.651 1.185	3	1.2
	-0.7 0.762 1.117	4	2.0	0.664 1.216	4	1.7
	-0.5 0.762 1.117	4	3.9	0.682 1.261	6	2.6
	-0.4 0.738 1.081	3	7.1	0.689 1.278	7	3.8
	-0.3 0.705 1.033	2	16.4	0.701 1.306	9	6.7
	-0.2 0.652 0.956	1	45.3	0.689 1.278	7	18.3
	-0.1 0.652 0.956	1	164.1	0.632 1.139	2	85.2
	0.1 1.071 1.570 144	202.5 0.845 1.647 286 215.2
	0.2 1.010 1.481 64	119.3 0.829 1.609 174 133.7
	0.3 0.953 1.398 33	72.1 0.810 1.563 104	85.1
	0.4 0.913 1.338 21	44.4 0.789 1.515 63	55.0
	0.5 0.875 1.283 14	27.9 0.769 1.468 40	35.9
	0.7 0.823 1.207	8	11.7 0.733 1.383 18	16.0
	1.0 0.762 1.117	4	3.8	0.682 1.261	6	5.3
	1.5 0.652 0.956	1	1.1	0.632 1.139	2	1.3
	3.0 0.697 -1.5 0.687 1.042	2	1.1	0.607 1.146	2	1.1
	-1.0 0.724 1.098	3	1.3	0.627 1.199	3	1.2
	-0.7 0.724 1.098	3	1.9	0.641 1.237	4	1.6
	-0.5 0.724 1.098	3	3.6	0.652 1.266	5	2.4
	-0.4 0.687 1.042	2	6.6	0.661 1.290	6	3.4
	-0.3 0.629 0.955	1	15.0	0.669 1.310	7	6.2
	-0.2 0.629 0.955	1	42.0	0.627 1.199	3	18.4
	-0.1 0.629 0.955	1	229.9	0.574 1.058	1	87.9
	0.1 1.123 1.703 225	213.6 0.841 1.749 326 220.3
	0.2 1.052 1.595 85	134.1 0.826 1.709 200 144.3
	0.3 0.956 1.450 30	84.7 0.804 1.654 114	96.4
	0.4 0.893 1.354 16	53.6 0.779 1.590 64	64.6
	0.5 0.854 1.296 11	34.1 0.755 1.531 39	43.2
	0.7 0.792 1.201	6	14.2 0.711 1.417 16	19.5
	1.0 0.724 1.098	3	4.3	0.661 1.290	6	6.2
	1.5 0.629 0.955	1	1.1	0.574 1.058	1	1.3
	SWV-X charts, for n = 5, β ∈ {2, 2.5, 3} and ARL 0 = 370.4.
		15

Table 4 :

 4 Constants K L , K U and L, for both the Synthetic WV-X and Synthetic

	Page 17 of 23
	SWV-X charts, for n = 5, β ∈ {3.5, 4, 4.5} and ARL 0 = 370.4.
	16

URL: http://mc.manuscriptcentral.com/lssp E-mail: comstat@univmail.cis.mcmaster.

ca Communications in Statistics -Simulation and Computation

				Page 18 of 23
	i	X	Xi	CRL
	F o r P e 101 125.3 124.9 124.4 124.6 125.2 124.88 102 124.8 124.8 126.4 124.6 126.4 125.40 103 124.7 124.4 124.7 125.2 128.6 125.52 104 125.5 124.6 124.8 124.4 124.3 124.72 105 124.7 124.3 124.6 125.0 124.4 124.60 106 124.5 124.8 124.7 124.4 124.9 124.66 106 124.4 125.1 125.6 124.5 124.8 124.88 107 124.4 125.3 124.3 125.0 124.3 124.66 108 124.5 126.8 124.4 125.9 124.8 125.28 110 124.4 125.4 124.4 125.0 129.2 125.68 111 126.2 124.3 125.4 124.9 124.5 125.06 112 124.4 124.2 124.3 124.4 124.3 124.32 112 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • e r R e v i 113 124.6 124.6 124.8 124.4 124.5 124.58 • • • 114 124.4 124.5 124.7 124.6 126.1 124.86 • • • 115 125.3 124.8 124.4 124.7 124.5 124.74 • • • 116 124.2 124.9 125.4 124.1 124.6 124.64 • • • 117 125.0 124.5 124.1 124.4 125.0 124.60 • • • 118 124.5 124.7 124.6 124.6 124.7 124.62 • • • 119 125.3 124.2 125.2 124.6 124.3 124.72 • • • 120 125.6 124.3 124.4 124.7 124.9 124.78 • • • 121 124.9 124.6 125.0 124.3 124.2 124.60 • • • 122 124.9 124.3 124.5 124.1 124.6 124.48 • • • 123 124.2 124.0 124.5 124.6 124.3 124.32 11 e w 124 125.4 125.5 124.5 124.5 124.2 124.82 • • • 125 124.1 124.1 126.1 124.3 124.2 124.56 • • • 126 124.1 125.9 124.1 124.7 125.3 124.82 • • • 127 124.1 124.3 124.1 124.2 124.5 124.24 4
	O 128 124.2 124.2 124.4 126.3 124.5 124.72 129 124.2 124.7 124.6 124.3 125.1 124.58 130 126.5 124.1 125.3 124.3 124.4 124.92 n • • • • • • • • • l Table 5: 30 samples of size n = 5 corresponding to a 30 hours sequence of
	production			y
		17

URL: http://mc.manuscriptcentral.com/lssp E-mail: comstat@univmail.cis.mcmaster.

ca Communications in Statistics -Simulation and Computation

  the value of Xi for i = 101, . . . , 111 also verify that LCL < Xi < U CL. Then the 112th sample ( X112 = 124.32) is below LCL = 124.37 of the SWV-X/S sub chart. This implies that CRL 1 = 112 > L = 9 and we can conclude that, up to this point, the process seems to be perfectly in-control. From the 113th sample to the 122nd sample, the values of Xi verify that LCL < Xi < U CL.The 123rd sample ( X123 = 124.32) is again below LCL = 124.37. This implies that CRL 2 = 123 -112 = 11 > L = 9 and we can conclude that up to this point, the process is still in-control. Then samples 124, 125 and 126 show that LCL < Xi < U CL, but sample 127 ( X127 = 124.24) is below LCL = 124.37.

	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

Communications in Statistics -Simulation and Computation

Appendix

Let us focus on transformations of form Z = a + bg(Y ) of the random variable Y , where a and b > 0 are two parameters, where g is a monotone increasing function, and where Z is a N (0, 1) random variable. It is very easy to show that the cumulative distribution function of the random variable Y is:

If c and d > 0 are two additional parameters such that Y = X-c d , then we can


Raphaël Soulard
email: soulard@sic.univ-poitiers.fr

Philippe Carré
email: carre@sic.univ-poitiers.fr

QUATERNIONIC WAVELETS FOR IMAGE CODING

The Quaternionic Wavelet Transform is a recent improvement of standard wavelets that has promising theoretical properties. This new transform has proved its superiority over standard wavelets in texture analysis, so we propose here to apply it in a wavelet based image coding process. The main point is the interpretation and coding of the QWT phase, which is not dealt with in the literature. At equal bitrates, our algorithm performs better visual quality than standard wavelet based method.

INTRODUCTION

It has been well known since the early 90s that wavelet representations are strikingly well suited for image coding (see JPEG-2000). This transform separates the information so that one can code progressively the global image structure and then the details with a few coefficients, carrying out scalable bitstreams at high compression rates.

In 2001, the importance of the Fourier phase for signal representation led to an enhancement of the standard wavelet transform (DWT) : the Complex Wavelet Transform (CWT) [START_REF] Selesnick | The dual-tree complex wavelet transform[END_REF], whose coefficients have a shift invariant magnitude and a complex phase, giving them innovating properties. This improvement was furthered in 2004 with the Quaternionic Wavelet Transform (QWT) [START_REF] Chan | Coherent multiscale image processing using dual-tree quaternion wavelets[END_REF]. Based on fundamentals brought by T. Bülow in 1999 [START_REF] Bülow | Hypercomplex spectral signal representation for the processing and analysis of images[END_REF], this representation -specifically defined for 2D signals -provides a coherent description of local structures through a shift-invariant magnitude, analogous to a standard DWT analysis, and a 3-angle 2D phase, carrying geometric information.

Our previous work has shown superiority of the QWT over DWT in a texture analysis context [START_REF] Soulard | Quaternionic wavelets for texture classification[END_REF]. We expect an improvement of wavelet based image coding, thanks to the structural analysis brought by the QWT phase. The QWT is overcomplete and its redundancy is 4:1 so it may be thought unadapted to compression. However this redundancy sorts out the information better than DWT, so even if we have more coefficients, many of them will be discarded or hardly quantized so we get in fine a better coding than with DWT. In particular, the magnitude should contain less significant coefficients to code, and the phase should be hardly quantized without loss of visual quality.

Given the promising theoretical properties of this new transform, we aim at studying its potential for a famous application of wavelets. Hence we propose to study the QWT in comparison with standard wavelets, in a compression context, without emphasis on state of the art techniques.

A necessary first point in image coding is quantization. Present wavelet based coding methods (EZW, SPIHT, EBCOT, TCE, SPECK . . . ) that are today the best alternative use quantized coefficients. The QWT magnitude can intuitively be processed like a standard DWT but the main point is the phase quantization -far from straightforward. With a first QWT quantization algorithm this work gives an application not did yet to our knowledge and furthers the practical use of QWT coefficients.

After a presentation of the transform we verify that a part of the information originally coded in the magnitude has been moved into the phase; through a study of the magnitude quantization that compares DWT with QWT. Then the interpretation of the QWT phase is discussed, and we propose a quantization algorithm that is compared with DWT in terms of image quality.

THE QWT

The Quaternionic Wavelet Transform (QWT) is an orthogonal 2D filterbank analysis for grayscale images. It provides a quaternionic scale space analysis, based on fundamental work by Bülow [START_REF] Bülow | Hypercomplex spectral signal representation for the processing and analysis of images[END_REF]. Bülow showed that complex algebra C is only optimal for handling 1D signals and that 2D signals are best described by embedding signal processing tools in the more general quaternion algebra H.

Whereas DWT coefficients are real QWT is quaternion valued i.e. 4-vectors made of one magnitude and a 3-angle phase. Thus the information is better separated to describe more explicitly the image content.

In 2004 the Rice University from Houston proposes to use their dual-tree algorithm to carry out a QWT with perfect reconstruction filterbanks [START_REF] Chan | Coherent multiscale image processing using dual-tree quaternion wavelets[END_REF] (that we use in this work). At the same time Bayro proposes a quaternionic Gabor pyramid [START_REF] Bayro-Corrochano | The theory and use of the quaternion wavelet transform[END_REF].

Evolution of DWT : QWT

A standard wavelet transform (DWT) provides a scale-space analysis of an image; yielding a matrix in which each coefficient is related to a 'subband' (localization in the 2D Fourier domain) and to a position in the image. A 'subband' means both an oscillation scale (i.e. a 1D frequency band) and a spatial orientation (i.e. rather vertical, horizontal or diagonal).

The QWT is an improvement of the DWT providing a richer scale-space analysis for 2-D signals. Contrary to DWT it is nearshift invariant and provides a magnitude-phase local analysis of images. It is based on the 2D generalization of both the Fourier transform and the analytic signal defined in [START_REF] Bülow | Hypercomplex spectral signal representation for the processing and analysis of images[END_REF] in the quaternion algebra H -more adapted than C to describe 2D signals. So in the one hand the QWT can be viewed like a local '2D Quaternionic Fourier Transform' (QFT) and in the other hand its subbands are '2D Quaternionic Analytic Signals' associated with bandpass filtered versions of the original signal.

Definition of the Transform

The Quaternionic 2D Analytic Signal

A quaternion is a generalization of a complex number, related to 3 imaginary units i, j, k, written q = a + bi + c j + dk, or q = |q|e iϕ e jθ e kψ in its polar form. It is thus defined by one modulus, and three angles that we call phase.

The (quaternionic) analytic signal associated with a 2D function is defined by means of its partial (H 1 , H 2 ) and total (H T ) Hilbert transforms (HT) :

f A (x, y) = f (x, y) + iH 1 f (x, y) + jH 2 f (x, y) + kH T f (x, y)

Quaternionic Wavelets

The mother wavelet is a quaternionic 2D analytic filter, and yields coefficients that are 'analytic'. Thus, it inherits the 'local magnitude' and 'local phase' concepts from the 1D analytic signal, very useful in signal analysis.

Note that the usual interpretation of the magnitude remains analogous to 1D, as it indicates the relative 'presence' of a feature,

[-π; π], θ ∈ [-π 2 ; π 2 ], ψ ∈ [-π 4 ; π 4
]. The 3 terms of phase are represented in color, the hue corresponding to the angle (cyan for 0, red for ±π). Darker zones in phase correspond to negligible magnitude (making phase absurd).

whereas the local phase is now represented by 3 angles that make a complete description of this 2D feature.

From a practical point of view, if the mother wavelet is separable i.e. ψ(x, y) = ψ h (x)ψ h (y), the 2D HT's are equivalent to 1D HT's along rows and/or columns. Then considering the 1D Hilbert pair of wavelets (ψ h , ψ g = H ψ h ) and scaling functions (φ h , φ g = H φ h ), the analytic 2D wavelets are written in terms of separable products.

ψ D = ψ h (x)ψ h (y)+iψ g (x)ψ h (y)+ jψ h (x)ψ g (y)+kψ g (x)ψ g (y) ψ V = φ h (x)ψ h (y)+iφ g (x)ψ h (y)+ jφ h (x)ψ g (y)+kφ g (x)ψ g (y) ψ H = ψ h (x)φ h (y)+iψ g (x)φ h (y)+ jψ h (x)φ g (y)+kψ g (x)φ g (y) φ = φ h (x)φ h (y)+iφ g (x)φ h (y)+ jφ h (x)φ g (y)+kφ g (x)φ g (y)
This means the decomposition is heavily dependent on the position of the image with respect to x and y axis (rotation-variance), and the wavelet is not isotropic, but the advantage is an easy computation with separable filterbanks.

Each subband of the QWT can be seen as the analytic signal associated with a narrowband 1 part of the image. The QWT magnitude |q| is shift-invariant and represents features at any space position in each frequency subband. The 3 phase angles (ϕ, θ , ψ) describe the 'structure' of those features. We discuss below the interpretation of these phases.

Filterbank Implementation

The QWT uses the Dual-Tree algorithm [START_REF] Selesnick | The dual-tree complex wavelet transform[END_REF], a filterbank implementation that uses a Hilbert pair as a complex 1D wavelet, bringing shift invariance and analytic coefficients with little redundancy.

Two complementary 1D filter sets lead to four 2D filterbanks -one pixel shifted each other -providing the near-shift invariance for a redundancy of only 4:1. Originally combined by Kingsbury to compute two directional complex analytic wavelets, the 4 outputs of the Dual-Tree here constitute one 4-valued quaternionic wavelet analysis, embedding the structural information into a local phase concept, rather than an oriented separation. As the Dual-Tree makes an approximation, the QWT coefficients are approximately analytic, so the extraction of 2-D local amplitude and phase, as well as their interpretation, are actually approximate. The Fig. 1 shows an example of a QWT decomposition.

MAGNITUDE CODING

As a preliminary and to be convinced that QWT magnitude and phase carry complementary information; we first observed the effect of magnitude quantization with both transforms. The process is to code QWT (resp. DWT) magnitude by classic uniform quantization with a fixed step, while keeping exact the phase information 1 The 1D analytic signal provides a time analysis considering the entire frequency spectrum. So in practice, the extracted local (instantaneous) characteristics are only meaningful when the signal itself is narrowband.

(resp. the sign). This first experiment cannot be used in a coding scheme, but it is a way to verify that the information is better separated in QWT coefficients. As the QWT phase contains some rich information about local structures that cannot be carried by the DWT sign; we should obtain better results with QWT.

Experimental process

We describe here the procedure we used to produce reconstructions, which stands for every one showed in this paper :

• Process DWT and/or QWT; The DWT uses biorthogonal CDF 9/7 filters, and the QWT is defined in [START_REF] Chan | Coherent multiscale image processing using dual-tree quaternion wavelets[END_REF]. • Apply the quantization method to the DWT and/or the 4 outputs of QWT, followed by the reconstruction of approximated values. • Process reverse DWT and/or QWT.

Because an image coding experiment is strongly dependent on the image chosen for the test, we use several images (photos) from the base "LIVE" [START_REF] Sheikh | Live image quality assessment database[END_REF] in their 8 bit grayscale version. For practical convenience, images were cropped to 512×512.

The quality of the reconstructed image is measured by a classical Peak Signal to Noise Ratio (PSNR). Our quantization algorithm is evaluated with rate-distortion curves by calculating the average number of bits needed to code a coefficient -in number of bits per pixel (bpp). The original coding of our grayscale images is 8 bpp.

Note that our bitrates are higher than those of a whole coder, as quantization is only one step of image coding. For example, the literature commonly consider 'low bitrates' around 0.1 bpp, for complete compression schemes that take into account many dependencies between the coefficients, and use entropy coding. But in our context a 'high bitrate' corresponds to the number of bits needed to quantize wavelet coefficients and have perfect reconstruction, which is around 15 bits in practice. So we consider in this paper 'low bitrates' under 6 bpp.

Distorted reconstructions

We evaluate the impact of the quantization step size on the reconstruction, by calculating the PSNR. Table 1 lists some PSNR's obtained by 5 bits and 8 bits magnitude quantization. With all tested images the DWT is never significantly superior to QWT -sometimes equivalent. The image sailing1 is slightly better reconstructed by DWT because an important part of the image is quite textural (sea surface). The QWT is clearly adapted to code geometric structures and seems less efficient for describing textures. Some experiments we made with textural images confirmed this; it is part of our future work.

Mostly the QWT rate-distortion curve is over the DWT curve. We can see Fig. 2 that the PSNR of the QWT reconstruction is always more than 2 dB better than DWT for monarch image. That means that the QWT phase compensates for the loss of information due to magnitude quantization. The example of reconstruction with 3 bits magnitude quantization shows the obvious superiority of QWT, that retrieves the shape of the contours far better than DWT (See original image Fig. 1). Moreover, as the quaternionic wavelets are non-oscillating, it reduces considerably the well known oscillations that usually occur after a non linear wavelet domain processing.

Conclusion About Magnitude

The QWT generally allows harder magnitude quantization and the reconstructions have a smoother aspect with fewer artifacts than DWT. That confirms that the QWT phase contains far more information than the DWT sign; which is a positive result. So if we are able to quantize this phase so as to allocate a number of bits comparable to this of the DWT sign; we can achieve a superior image representation than DWT. In the sequel we study the QWT phase in order to quantize it efficiently.

THE QWT PHASE

For now, the literature is quite poor about the QWT and the major difficulty with the use of this transform is the interpretation of the phase.

Use of QWT phase

In his thesis [START_REF] Bülow | Hypercomplex spectral signal representation for the processing and analysis of images[END_REF], Bülow shows the importance of phase in image analysis, defines a quaternionic Fourier transform (QFT), a quaternionic 2D phase and 2D quaternionic analytic Gabor filters.

In a Gabor based texture segmentation, the filtered images are 2D analytic and form a scale-space analysis of the image from which Bülow extracts magnitudes and local phases at each point to characterize the texture.

First, due to the QFT shift theorem the two first terms of phase ϕ and θ describe small shifts of the coded structure, around the quaternionic coefficient position. This information is analogous to the classical instantaneous 1D phase that codes an impulse shift.

Note that in 1D, that shift information is equivalent to the structure information. A phase of 0 or π just means an "impulse" (positive or negative) and a phase around ± π 2 describes a "step" (rising or falling) -being in fact the edge of a shifted impulse. In 2D that shift is not sufficient to describe every structure; in particular "i2D" structures (e.g. corners, T-junctions) that are more complex than lines or edges.

The third term ψ completes the structure analysis and is seen as a texture feature. Bülow found a near-linear relation between ψ and a "λ " parameter in a superposition of two plane waves defined :

f λ (x,y) = (1-λ) cos(ω 1 x + ω 2 y) + λ cos(ω 1 x -ω 2 y)
We found three recent references [START_REF] Chan | Coherent multiscale image processing using dual-tree quaternion wavelets[END_REF][START_REF] Bayro-Corrochano | The theory and use of the quaternion wavelet transform[END_REF][START_REF] Zhou | Quaternion wavelet phase based stereo matching for uncalibrated images[END_REF] where ϕ and θ are used in disparity estimation. As the QWT performs local QFT's the shift theorem approximately stands for QWT so ϕ and θ code quite simply a shift of the structure.

In another application of [START_REF] Chan | Coherent multiscale image processing using dual-tree quaternion wavelets[END_REF] ("wedgelet" representation), ϕ and θ are used for wedges position and ψ is used for their orientation.

Distribution of Phase

From our compression point of view it is interesting to observe the statistic distribution of the QWT phase. So we combined our LIVE base with the Brodatz Texture album [START_REF] Brodatz | Textures : A Photographic Album for Artists and Designers[END_REF] in order to represent a great variety of images, and the data was cumulated over all images to have more general statistics.

The histograms Fig. 3 are processed for different scales in each subband for ϕ, θ and ψ. As we know that phases of low coefficients have very little meaning and are numerically unstable these cases were ignored in the processing of histograms; in order to make them more meaningful. Coefficients which magnitude is less than 2% of the maximum amplitude are not counted (Empirical threshold keeping 26% of all the QWT coefficients). If we do not use such a threshold the distributions are much more "noisy" i.e. a uniform density is added to all curves.

Note that the distributions of the phase components are strongly dependent on the subband in which it is observed. A first simple explanation is about the behavior of ϕ and θ in horizontal and vertical subbands. In those subbands the coded structures are aligned with x-axis or y-axis. And we know that ϕ and θ can be seen as a 2D space shift. We must remark that a horizontal structure can hardly exhibit a horizontal local shift because it is equivalent to the same structure -same remark for vertical -so only one of the two first terms is significant for horizontal and vertical structures. A second explanation is about ψ. We also know that ψ is around ± π 4 when the structure is diagonal, and around 0 else. Then the horizontal and vertical subbands contain structures that are never diagonal, so the ψ phase is always around 0.

The main point of the histograms analysis is that there are a great variety of cases within QWT coefficients, obviously leading to an adaptive quantization that we propose now.

PHASE QUANTIZATION

Systematic Results

Experimentally, we observe that a uniform quantization of each term of the QWT phase gives a monotonic relation between the quantization step and the PSNR. This holds for any term separately and also for simultaneous quantization of the 3 terms. But knowing that there are many different cases of phase these global results are far from being enough so we now present how to exploit this variety.

Adaptive Quantization Ideas

First, it is straightforward that small coefficients do not need their phase to be coded. Depending on the chosen magnitude quantification a QWT may have many zeroes so this point is important.

Considering only first scale -which represents 3/4 of the data -we can assume that a precise description of the local shift (ϕ, θ ) is useless because the resolution of the subband is just twice lower than image resolution. The impact of a wrongly coded shift is very low in this case so we can quantize those phases very roughly too. More generally, it may be intuitive to quantize the phase with a smaller step when scale increases.

Our Proposed Phase Quantization

Based on our QWT phase analysis we propose the following phase quantization with arbitrary values.

Zero Coefficients

For zero coefficients we do not code the phase so the bit allocation is just that for the magnitude. If the magnitude quantization is hard then there are many zeroes; otherwise we use an experimental threshold (0.04% of the max) that guarantees perfect reconstruction when phase is not coded under it.

That reaches a total of 8 bits to code 3 phases in scale 1 knowing that many coefficients are negligible at this scale; so we have a very light code here.

For other scales, the quantization step is adaptive :

• Horizontal subband : the couple (ϕ, ψ) is coded on 4 bits and θ is quantized more precisely, on "1 + scale" bits • Vertical subband : the couple (θ , ψ) is coded on 3 bits and ϕ is quantized more precisely, on "2 + scale" bits. • Diagonal subband : the couple (ϕ, θ ) is coded on 5 bits and ψ is quantized more precisely, on "scale" bits.

Quantization centroids are fitted at multiples of π 4 . We now present the performance of our coding algorithm based on the ideas presented above. It quantizes uniformly the magnitude with the number of bits as a parameter and an adaptive phase quantization is performed with respect to the description above.

MAIN RESULTS

To compare with standard wavelets we force the DWT and the QWT processes to allocate the same number of bits for a same image. More precisely we first choose a fixed magnitude bitrate to code the QWT while calculating the bitrate needed for phase coding to get the total exact bitrate. After that we first quantize DWT magnitude with a similar bitrate. By counting the numerous small DWT coefficients that do not need their sign to be coded the actual bitrate is processed. Then the DWT magnitude quantization step is adjusted until the DWT and QWT bitrates are similar (convergence). 4: Final coding results with zooms.

Result Analysis

Results on the LIVE base are generally good especially at 'lower bitrates' (< 6 bpp, see 3.1). The Fig. 5 shows rate-distortion curves for two images and validates our algorithm with the objective quality measure "PSNR". The reconstructions Fig. 4 show the superiority of QWT. The reason is that the QWT phase needs a very low number of bits. So the advantage of the magnitude presented in section 3 is not lost, thanks to a coding of the phase as light as the DWT sign. Our QWT coding preserves better contour shapes and has no oscillations; this is a great advantage over DWT. Nevertheless, recall that the PSNR quality measure may be inefficient in some cases as it does not take into account the human visual system. That is the reason of the seeming superiority of DWT for 'higher bitrates' (> 6 bpp, see 3.1) whereas the reconstructions show a rather equivalent visual quality. See zoomed reconstructions at 7 bpp Fig. 4 : there is a difference but the quality is actually subjective. In fact, the distortion brought by the QWT is smooth and invisible but still present and numerically influential on PSNR. Moreover, our implementation has some inherent invisible phase distortion that does not get more accurate with the bitrate parameter. At high bitrate, this little incompressible phase distortion is detected by the PSNR, while DWT keeps on improving the quality.

A last experimental point is to validate the algorithm. Generally, for a fixed magnitude coding, the image reconstructed with the exact phase is visually the same than this with the coded phase. That means our phase coding keeps all important information.

So in spite of the rate-distortion curves we can state that the QWT coding process outperforms the standard wavelets.

CONCLUSION

We proposed an innovating wavelet based coding algorithm using the new Quaternionic Wavelet Transform. This first step in applying QWT for image coding turns out to confirm its superiority over standard wavelets. The coded images has visually more acceptable distortion at lower bitrates with smooth degradations, preservation of contour shape, and no oscillations; and the quality is equivalent at higher bitrates.

Here are some ways of improvement. By studying analytical expressions of QWT magnitude and phase pdf's -starting from assumptions about cartesian terms that are classical wavelet transforms -one may optimize quantization and so enhance reconstruction. Moreover the well known dependencies of standard wavelets coefficients across scales are even stronger with the QWT redundancy and may be used to improve compression rate. The final step is to integrate this quantization method in a whole coding scheme to see if the algorithm is well suited to entropy coding.

The study of monogenic wavelets -a theoretic improvement of the QWT more complicated to implement -is part of our prospects in image coding.

Figure 1 :

 1 Figure 1: The quaternionic wavelet transform of image monarch. From left to right : Original image, Magnitude (intensity inverted for visual convenience), ϕ ∈ [-π; π], θ ∈ [-π 2 ; π 2 ], ψ ∈ [-π 4 ; π 4 ]. The 3 terms of phase are represented in color, the hue corresponding to the angle (cyan for 0, red for ±π). Darker zones in phase correspond to negligible magnitude (making phase absurd).

  of image 'monarch' (PSNR)

Figure 2 :

 2 Figure 2: Magnitude quantization.

Figure 3 :

 3 Figure 3: Histograms of QWT phase. The curves are arranged the way the QWT subbands are in Fig. 1.

  of image 'sailing2' (PSNR)

Figure 5 :

 5 Figure 5: Rate-distortion curves from our final QWT coder, for images monarch and sailing2.

Table 1 :

 1 PSNR's with magnitude quantization.

		PSNR (dB)
		5 bits	8 bits
	Image name	DWT QWT DWT QWT
	building2	17.0 18.8 26.0 31.2
	cemetry	20.2 22.5 29.5 34.9
	monarch	24.0 27.6 35.4 40.3
	paintedhouse 22.5 24.6 32.7 37.3
	parrots	26.4 29.3 36.0 39.4
	plane	22.4 22.8 30.1 31.0
	sailing1	22.8 22.6 31.1 30.5
	sailing2	25.3 28.8 35.6 39.9

ACKNOWLEDGEMENTS

This work is supported by the ANR project VERSO -CAIMAN. We also thank the anonymous reviewers for their valuable remarks and suggestions.

High Frequencies

For coefficients of first scale :

), and ψ is set in {-π 8 ; π 8 } (1 bit).


Per Bakke Nicole Le Moual

Ewa Orlowski

Dick Heederik

H Kromhout

S M Kennedy

Bert Rijcken

Francine Kauffmann

P Bakke

Performance of population specific job exposure matrices (JEMs): European collaborative analyses on

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

Results-Significant associations between occupational exposure estimated by the population-specific JEM and lung function were found in the French and the rural Dutch surveys, whereas no significant relationship was found using self-reported exposure. In populations with few subjects in most jobs, exposure cannot be estimated with sufficient precision by a population-specific JEM, which may explain the lack of relationships in the Norwegian and the Dutch (urban area) surveys.

Conclusion-Population-specific JEM, easy to construct and at little cost, seem to perform better than crude self-reported exposures, in populations with a sufficient number of subjects per job.

Key words: Job exposure matrix -Occupational exposure -Lung function Associations between occupational exposures and Chronic Obstructive Lung Disease (COPD) have been investigated mostly in occupational populations [START_REF] Becklake | Occupational Exposure : Evidence for causal association with chronic obstructive pulmonary disease[END_REF] [START_REF] Heederik | Contribution of occupational exposures to the occurrence of chronic nonspecific lung disease[END_REF].

In community-based studies, selection bias due to the healthy worker effect is a less important issue than in occupational cohorts, but the validity of exposure assessment methods in such studies is a matter of debate [START_REF] Goldberg | Measurement of occupational exposure and prevention : principal approaches to research[END_REF]. Self-reported information and more recently Job Exposure Matrices (JEM) [START_REF] Moual | Occupational exposures estimated by means of job exposure matrices in relation to lung function in the PAARC survey[END_REF] [START_REF] Post | Occupational exposures estimated by a population specific job exposure matrix and 25 year incidence rate of chronic nonspecific lung disease (CNSLD) : the Zutphen Study[END_REF] have been used to estimate occupational risk factors in community-based studies on COPD.

Population-specific job exposure matrices, which use the subjects with the same job as 'experts' for that job, have also been proposed [START_REF] Kromhout | Performance of two general Job-Exposure Matrices in a study of Lung Cancer Morbidity in the Zutphen Cohort[END_REF]. Building a population-specific JEM is easy as it results from simple computer calculations on self-reported exposures. A population-specific JEM estimates exposure taking into account the percentage of self-reported exposure per job. Using population specific JEM might lead to less misclassification in exposure than using self-reported exposure.

However, the heterogeneity of exposure in a given job is not taken into account by the JEM [START_REF] Goldberg | Measurement of occupational exposure and prevention : principal approaches to research[END_REF]. Self-reported and population-specific JEM assessment methods have never been compared formally although results using both methods have been published once [START_REF] Sunyer | Pulmonary ventilatory defects and occupational exposures in a population-based study in spain[END_REF].

Three general population surveys in France [START_REF] Coopératif | Pollution atmosphérique et affections respiratoires chroniques ou à répétition. I Méthodes et sujets[END_REF], The Netherlands [START_REF] Van Der Lende | Epidemiology of chronic non-specific lung disease (chronic bronchitis)[END_REF], and Norway [START_REF] Bakke | Prevalence of obstructive lung disease in a general population : relation to occupational title and exposure to some airborne agents[END_REF] provided data on an individual basis on occupational airborne exposure and lung function. In the French [START_REF] Krzyzanowski | The relation of respiratory symptoms and ventilatory function to moderate occupational exposure in a general population. Results from the French PAARC study among 16000 adults[END_REF] and the Norwegian [START_REF] Bakke | Prevalence of obstructive lung disease in a general population : relation to occupational title and exposure to some airborne agents[END_REF] surveys, associations between occupational exposures and lung diseases have already been studied, whereas in the Dutch survey [START_REF] Van Der Lende | Epidemiology of chronic non-specific lung disease (chronic bronchitis)[END_REF] no analyses on occupational exposure have been performed yet. The aim of the present paper is to evaluate how occupational exposures estimated by population-specific job exposure matrices perform compared to self-reported exposures, in their relationships to lung function.

Material and methods

In the French Cooperative PAARC (Pollution Atmosphérique et Affections

Respiratoires Chroniques) study [START_REF] Coopératif | Pollution atmosphérique et affections respiratoires chroniques ou à répétition. I Méthodes et sujets[END_REF], performed in 1975, 20310 men and women, aged 25-59 years, residing in seven French cities (Bordeaux, Lille, Lyons, Mantes-la-Jolie, Marseilles, Rouen, Toulouse), were surveyed at home. The primary aim of the survey was to investigate a possible effect of air pollution on respiratory symptoms and lung function. Households 'headed' by manual workers were excluded to reduce the effect of occupational hazards. Therefore, subjects in the French survey were less occupationally exposed than the general population. In the French questionnaire, derived from the British Medical Research Council/European Coal and Steel Community questionnaire, subjects were classified as non smokers, exsmokers (stopped for at least one month), or smokers (light smokers (< 10 g/day), moderate smokers (10 to 19 g/day) and heavy smokers ( ≥ 20 g/day)), based on grams of tobacco smoked as cigarettes, cigars or pipe per day.

In the Dutch survey [START_REF] Van Der Lende | Epidemiology of chronic non-specific lung disease (chronic bronchitis)[END_REF], 3477 men and 3256 women, aged 15-64 years, were recruited over three years (1965,1967,1969) (petroleum products, solvents, detergents, pigments, plastics, paints/lacquers, insecticides /pesticides)). The occupational titles were coded using the Nordic 3 digit classification [START_REF]Occupational classification. List of occupations with NYK numbers[END_REF].

A job exposure matrix is a table in which each cell contains for each job an estimation of exposure for a given hazard. Two population-specific JEM were constructed using computer calculations for each survey and in the French survey for each gender. The first step was to calculate in each job the proportion of subjects exposed to dusts, gases and fumes using the self-reported dichotomous variable. A JEM with two classes of exposures is then built by considering as exposed any job for which more than x percent of the subjects who practise this job reported themselves exposed. The choice of the cutpoint x is arbitrary and may be chosen to obtain an overall fixed percentage of exposed and modified to increase the specificity or the sensitivity of the measure. By the same method, JEM with several classes of exposure may be defined. A matrix which classified exposure in three categories (P10-50 with the two cutpoints : p < 10%, p ≥ 50%), as described by Post et al [START_REF] Post | Occupational exposures estimated by a population specific job exposure matrix and 25 year incidence rate of chronic nonspecific lung disease (CNSLD) : the Zutphen Study[END_REF] was constructed.

The probability of exposure in an occupation was considered high, moderate or low, when ≥ 50%, 10 to 49% or < 10% of the men (women) working in this occupation reported themselves occupationally exposed, respectively. Another JEM was built in order to have the same statistical power as using self-reported exposure. The construction of that two-class JEM was done in such a way that the prevalence of exposure using the Pp JEM would be the same as the prevalence in the self-reported data (p). For example, in the French survey 4605 men declared themselves as nonexposed. The closest figure of non exposed using the Pp JEM (4674) was obtained by considering as non exposed the jobs in which less than 33% (optimal cutpoint) of workers reported exposure (table 2). In that case, the prevalence of exposure using the Pp JEM (24.8%) was as close as possible to the prevalence based on the selfreported exposure (25.9% (table 4)). For the other populations, a job was classified exposed using the Pp JEM when the percentage of subjects, who declared themselves exposed in a job, was equal or greater to the following cutpoints : 30% in women from the French survey, 50% and 29% in urban and rural Dutch residents, respectively and 50% in the Norwegian survey. The different classifications of exposure for a few typical jobs is illustrated in table 3.

ANALYSED POPULATION

For the current study, analyses were restricted to subjects aged 25-64 years old who performed good forced expiratory volume in one second (FEV 1 ) tracings and whose last occupation was encountered for at least two men (women) in each of the three surveys. Analyses in women were done only in the French survey. In the Dutch and the Norwegian surveys too few women reported exposures to use JEM in optimal conditions (prevalence of exposure about 10% or more) [6] [15]. In the French survey, to compare present results with results previously described [START_REF] Moual | Occupational exposures estimated by means of job exposure matrices in relation to lung function in the PAARC survey[END_REF],

analyses were also performed for subjects whose occupation was encountered at least 10 times, i.e. 5046 men and 5000 women.

STATISTICAL ANALYSIS

Standardised FEV 1 score (m ± SD = 0 ± 1) was obtained for each gender and city, after adjustment of FEV 1 on age, height and smoking habits (expressed by dummy variables, four in France (ex-smokers, light smokers, moderate smokers, heavy smokers), three in The Netherlands (ex-smokers, cigar/pipe smokers, cigarette smokers) and two in Norway (ex-smokers, smokers)), in multiple regression analyses. The relationship between exposure and FEV 1 was tested with a Student's t test for dichotomous variables and with a test for trend for P10-50 JEM [START_REF] Snedecor | Statistical methods[END_REF]. Analyses were also done using FEV 1 score adjusted on age and height for each gender and city and regressing this score on occupational hazards, adjusting for smoking habits. As all analyses gave the same results, reported ones are thus for scores adjusted a priori on smoking. The comparison of the self-reported exposure and the Pp JEM (both exposure assessments classified in two categories) was performed using Cohen's Kappa statistic [START_REF] Fermanian | Mesure de l'accord entre deux juges : cas qualitatif[END_REF].

In order to assess the influence of the number of subjects per job on the precision of the exposure estimates by the population-specific JEM, a resampling procedure was used. Two series of draws were obtained by sampling 100 times, without replacement, in men from the largest survey (French PAARC survey). Each sample included 900 men in the first series and 3200 men in the second series.

Restricting analyses to men whose occupations were encountered at least two times resulted in respectively about 850 (a similar number of subjects as in the urban and rural Dutch residents) and 3170 men in each sample.

Results

In the men from the three populations as well as the women from the French survey, the mean age was about 42 years old (table 4). Fewer men were smokers in the Norwegian survey (non smokers included by design), compared to men in the French or the Dutch surveys. In contrast, in women (French survey) there were about 70% never smokers and only 23% smokers. In the French survey (manual workers households excluded by design), men reported themselves to be less exposed than men from the Dutch or the Norwegian surveys did. French women reported themselves to be less exposed than men did. The distribution of the number of subjects holding the same job differed across the surveys. In the Norwegian survey and the urban Dutch surveys the maximum number of men per job was low compared to the French and the rural Dutch surveys.

OCCUPATIONAL EXPOSURE

Concordance of self-reported exposure and Pp JEM was low in the French survey and the rural Dutch residents and moderate in the other cases. Cohen's kappa values were 0.31, 0.30, 0.58, 0.34, 0.71, in women, men from the French survey, men from urban and rural Dutch residences and men from the Norwegian survey, respectively.

In the French survey, FEV 1 was lower among exposed subjects according to the self-reported exposure than among non exposed subjects, but the difference was not significant in men and of borderline significance in women. In contrast, using both the Pp JEM and the P10-50 JEM significantly lower FEV1 values were observed in subjects classified as exposed (table 5). Furthermore, the results show that the higher was the proportion of people exposed in the group the lower was the mean FEV 1 score. Results from the analyses in the French survey restricted to occupation encountered at least 10 times, showed consistent results (not shown).

In the rural Dutch residents, a significantly lower FEV 1 was observed in men exposed according to the Pp JEM whereas consistent, but non significant, associations were found using self-reported exposure and the P10-50 JEM (table 5). In the urban Dutch residents, no significant associations were found according to the three estimates, with a non significant trend to higher FEV 1 in exposed versus non exposed when using self-reported exposure.

In the Norwegian survey, lower FEV 1 values were observed in men who declared themselves exposed compared to non exposed, but the difference was not significant (table 5). The same trend was observed using the Pp JEM and the P10-50 JEM with however a weaker association than when using self-reported exposure.

No interaction between occupational exposures estimated by the three methods and smoking habits was found in any of these analyses.

PRECISION OF OCCUPATIONAL EXPOSURES ESTIMATED BY POPULATION-SPECIFIC JEM

In the French survey, in both men and women, as in the rural Dutch residents, a large proportion of subjects practised a job encountered at least 50 times (table 4).

In contrast, in the Norwegian survey and the urban Dutch residents there were possibly too few subjects per job to permit a good estimate of exposure.

A resampling procedure using the largest French data set was performed to test the hypothesis that the exposure estimate would be more precise when jobs were reported by a large number of individuals. The resampling was performed with varying sample size to obtain a varying proportion of jobs with few subjects, which in a given population depends on the sample size. Results from the two series of resamplings showed proportions of self-reported exposure similar to the prevalence observed in men from the whole data set (table 4, table 6). In the first series of draws (n around 900 in each sample like in the Dutch populations), about 40% of the men had a job seldom encountered (less than 10 times), whereas in the second series (with n four times larger), around 45% of the men had a job encountered at least 50 times in each sample (table 6). In both series, more significant relationships, between exposure and FEV 1 , were found using the population-specific JEM than using selfreported exposure with more often significant associations in the second series than in the first one. No reverse associations were observed in the second series, and only one significant reverse association between FEV 1 and exposure out of 100 was found in the first series, using both self-reported exposure and the P10-50 JEM.

Discussion

Associations between occupational exposure estimated by the populationspecific JEM and lower lung function were found, whereas no significant relationships were found using self-reported exposure in the French survey, in both men and women, and in the rural Dutch area. No significant relationship was found in the Norwegian survey and in the urban Dutch resident whatever the method used. Results suggest that population-specific JEM perform better than self-reported exposures, when there are enough jobs with a large number of subjects, a hypothesis supported by results obtained by resampling procedures.

ASSOCIATIONS OF OCCUPATIONAL EXPOSURE WITH LUNG FUNCTION

Significant associations between exposure estimated by the population-specific JEM and lung function found in three out of the five populations studied are in agreement with findings previously reported in other community and workforce based surveys [START_REF] Becklake | Occupational Exposure : Evidence for causal association with chronic obstructive pulmonary disease[END_REF] [2]. In the Norwegian survey, our results are not inconsistent with associations previously reported on a larger sample between self-reported occupational exposure to specific hazards and spirometric airflow limitation for subjects over 50 [START_REF] Bakke | Prevalence of obstructive lung disease in a general population : relation to occupational title and exposure to some airborne agents[END_REF]. Although the association between self-reported occupational exposure and FEV 1 was not statistically significant, the magnitude was comparable to the magnitude found in men from the French survey using the Pp JEM. In the urban Dutch area no trend was found whatever the method used. In the three surveys, the analyses were done on the last occupation. In the Norwegian survey, the OR between occupational exposure and obstructive lung disease was higher using exposure in the present job than in the job held longest [START_REF] Bakke | Prevalence of obstructive lung disease in a general population : relation to occupational title and exposure to some airborne agents[END_REF].

Another possibility is that there is no effect of occupational exposure on FEV 1 in urban Dutch and Norwegian populations. One could hypothesize that in the Norwegian survey, if self-reported exposure was differentially biased and lead to a false-positive effect the Pp JEM corrected this and correctly indicated no effect of occupational exposure on FEV 1 in this population. Furthermore the differences in questions on exposure in the Norwegian survey compared to the other surveys may also explain differences in the results. However, associations previously reported in the Norwegian survey [START_REF] Bakke | Prevalence of obstructive lung disease in a general population : relation to occupational title and exposure to some airborne agents[END_REF] are not in favour of these hypotheses. The lack of relationship using the population-specific JEM could be explained by the imprecise assessment of exposure due to the low number of subjects per job in the Norwegian survey as in the urban Dutch residents.

Residual confounding for smoking when using JEM, related to job and therefore to social class, instead of individual exposure could be theoretically possible. However, associations of FEV 1 with exposure assessed by external JEM was unchanged by considering different adjustment for smoking (and considering pack years) in a previous analysis in the French survey [START_REF] Moual | Occupational exposures estimated by means of job exposure matrices in relation to lung function in the PAARC survey[END_REF]. In the present analysis, a priori and a posteriori adjustment for smoking led to similar results.

PERFORMANCE OF POPULATION-SPECIFIC JEM COMPARED TO SELF-REPORTED

The study of the performance of a JEM comprises three main elements as described by Bouyer et al [START_REF] Bouyer | Studying the performance of a job exposure matrix[END_REF]: (i) the ability of the JEM to evaluate accurately the exposure itself, (ii) its statistical performance in terms of bias and power, (iii) its ability to detect known associations between risk factors and disease. In our study, only the third point can be directly studied to estimate the performance of the population-specific JEM although the two other aspects may be addressed indirectly.

The pattern of relationships between the probability of exposure and a decrease in An argument in favour of the accurate exposure assessment of the populationspecific JEM is that similar results has been found using both an ad hoc JEM built by experts and a population-specific JEM, in a population-based study on COPD [START_REF] Sunyer | Pulmonary ventilatory defects and occupational exposures in a population-based study in spain[END_REF].

More significant associations between specific exposures and incidence of lung cancer were found using a population-specific JEM than using an external JEM [START_REF] Pannett | A job-exposure matrix for use in population based studies in England and Wales[END_REF] and a theoretical calculation showed that population-specific JEM performed better than the external JEM when the prevalence of exposure was above 10 percent [START_REF] Kromhout | Performance of two general Job-Exposure Matrices in a study of Lung Cancer Morbidity in the Zutphen Cohort[END_REF]. In the French survey, using both the Pp and P10-50 JEM, associations found with lung function are in agreement with findings previously reported [START_REF] Moual | Occupational exposures estimated by means of job exposure matrices in relation to lung function in the PAARC survey[END_REF] using a British [START_REF] Pannett | A job-exposure matrix for use in population based studies in England and Wales[END_REF],

an Italian [START_REF] Macaluso | Job exposure matrices : experience in Italy. In : Job exposure matrices ED Acheson[END_REF] and a JEM built by experts for the survey [START_REF] Moual | Occupational exposures estimated by means of job exposure matrices in relation to lung function in the PAARC survey[END_REF]. In men, both the magnitude of the exposure-FEV 1 relationship and the number of subjects in the highest category of exposure were similar whatever the JEM used (British, Italian, internal or P10-50 JEM), whereas in women these two parameters fluctuated according to the JEM [START_REF] Moual | Occupational exposures estimated by means of job exposure matrices in relation to lung function in the PAARC survey[END_REF]. A further advantage of using the population-specific JEM is the absence of additional error due to job recoding. Whereas the use of an external JEM required translation of occupation codes into the coding system used by that JEM, a population-specific JEM, by its mode of construction, can be applied whatever the classification used to code occupation.

Decrease of classification bias (both non differential and differential) may be obtained by using population-specific JEM. As all threshold methods, populationspecific JEM increase errors of classification and does not take into account all the available information (in our case, the proportion of subjects exposed in a job) [START_REF] Bouyer | Comparison of three methods of estimating odds ratios from a job exposure matrix in occupational case-control studies[END_REF].

However, our results suggest that the error due to misclassification of exposure by subjects may be larger than the error due to the heterogeneity of exposures among subjects with the same job. Significant associations were found using the populationspecific JEM when no association were observed using self-reported exposure.

Group-based occupational exposure assessment strategies have been shown to be very effective and to yield essentially unbiased estimates of exposure response relationships whereas individual-based estimates of exposure might lead to precise but substantially attenuated relations [START_REF] Tielemans | Individual-based and group-based occupational exposure assessment: some equations to evaluate different strategies[END_REF]. Differential bias was diminished compared to self-reported method, as using population-specific JEM exposure was less dependent on health status. The discussion of the performance in term of power cannot be assessed here.

With the same power, associations were significant using Pp JEM whereas they were not significant using self-reported exposure, in the French survey and the rural Dutch area. Usually it is difficult to compare results using different exposure assessment methods because the statistical power is not the same [18][23]. Siematycki et al [START_REF] Siemiatycki | Costs and statistical power associated with five methods of collecting occupation exposure information for populationspecific case-control studies[END_REF] described a method of exposure assessment based on expert evaluation, in which they choose a cutpoint for maximizing the power. Bouyer et al [START_REF] Bouyer | Studying the performance of a job exposure matrix[END_REF] emphasized that having comparable prevalence of exposure in two exposure assessment methods should help to compare the methods. By the mode of construction of the Pp JEM, the prevalence of exposure and therefore the statistical power were the same for the self-reported exposure and the Pp JEM. Although in our study the cutpoint was chosen to optimize the comparison of the two methods, this is not necessarily the best for optimizing the specificity or the sensitivity of the estimate. In future studies using population-specific JEM, there is no a priori need to have the same prevalence as the self-reported one observed in the population.

Our results show that a good estimation of exposure using a populationspecific JEM, required enough jobs with a large number of subjects. By doing resamplings, relationships between exposure and FEV 1 were more often significant using population-specific JEM than self-reported exposure and more frequently significant in large draws (better estimates of exposure) than in small ones.

Therefore, resampling findings are consistent with results found in the three surveys. On the null hypothesis "no relationship between occupational exposure and FEV 1 " the percent of significant relationships observed should be different from 5%, because the samples are not independent, but the exact proportion is difficult to estimate.

Population-specific JEM have been used previously [START_REF] Post | Occupational exposures estimated by a population specific job exposure matrix and 25 year incidence rate of chronic nonspecific lung disease (CNSLD) : the Zutphen Study[END_REF] [6] [START_REF] Sunyer | Pulmonary ventilatory defects and occupational exposures in a population-based study in spain[END_REF] but no formal comparison with self-reported exposure was performed. Our results suggest that the assessment by both methods may be different and lead to different estimates of associations with health.

In conclusion, our results show that a large number of subjects with the same job is required to make a population-specific JEM sufficiently precise. However, the choice of optimal cutpoints depends on the population studied. Population-specific JEM are easy to construct and their applications are not limited by the classification of jobs used in the studies. Furthermore, population-specific JEM perform better than self-reported method, when conditions for their use are fulfilled (large populations or populations with similar jobs). * In the French and the Dutch surveys, subjects were considered exposed if they answered positively to one or two questions on exposure to dusts, gases or chemical fumes. In the Norwegian survey, subjects were considered exposed if they answered positively to at least one of the following hazards : asbestos, quartz, wood dust, aluminium dust, welding, soldering, metal compounds in dust or gas form (chromium, nickel, platinum), or petroleum products, solvents, detergents, pigments, plastics, paints/lacquers, insecticides /pesticides.

FEV 1

 1 found in the French survey and evidence of known associations in two different populations (French survey and rural Dutch residents) are arguments in favour of the validity of the population-specific JEM. The lowest kappa values between self-reported exposure and the Pp JEM were observed in the groups in which the population-specific JEM performed better than self-reported exposure.

  ............................................................................................................................................................................ ............................................................................................................................................................................... .............................................................................................................................................................................. ........................................................................................................................................................................... .............................................................................................................................................................................. ...........................................................................................................................................................................

Table 1

 1 Methods in the three general populations

	France	The Netherlands	Norway
	1975	1965-1969	1985-1988
	7 cities	Urban / Rural

Table 2

 2 Construction of the dichotomous JEM based on the self-reported exposure -Pp JEM

	Percent of subjects who declared	Number of subjects	Cumulative	Cumulative
	themselves exposed in a job		number of subjects	Percent
	0	188	188
	3	30	218
	4	49	267
	5	229	496

  Example in men from the French survey Self-reported exposure : 4605 men non exposed, i.e. 74.1%; 1612 men exposed, i.e. 25.9%. Pp JEM : 4674 men non exposed, i.e. 75.2%; 1543 men exposed, i.e. 24.8%. * Cutpoints for other groups analysed : 30% in women from the French survey; 50% and 29% in urban and rural Dutch residents, respectively; and 50% in the Norwegian survey.

				...
	80	10	6164	99.1
	82	17	6181	99.4
	83	18	6199	99.7
	100	18	6217	100.0

Table 3

 3 Exposure assessments in four jobs in men from the French survey The cutpoint for exposure assessment was 33% in men from the French survey.

	Jobs	Number of	% of self-reported	P10-50 JEM	Pp JEM *
		subjects	exposure in the job
	Physicians	147	9%	low	non exposed
	Office workers	465	19%	moderate	non exposed
	Stock clerks	144	35%	moderate	exposed
	Bakers -Pastrycooks	51	67%	high	exposed
	*

Table 4

 4 Description of subjects in the three general populations

	France		The Netherlands (men)	Norway (men)
	Women	Men	Urban	Rural
	(5571)	(6217)	(854)	(780)	(395)

Acknowledgments

We were grateful to R Vermeulen for the coding of the jobs and to J Lellouch, H M Boezen and J P Schouten for their comments on this paper.

Supported by ECOJEM (European

Collaborative analyses on Occupational risk factors for COPD using Job Exposure Matrices). ECOJEM is a subproject of the European Concerted Action "Epidemiological surveys on chronic obstructive pulmonary diseases (COPD) in different European countries : Prevalence rates and relationship to host and environmental risk factors" coordinated by C. Giuntini -(grant BMH1-CT92-0849).