K Mabude

J.-C Malela-Majika

P Castagliola

S C Shongwe

S C Shongwe

Distribution-free mixed GWMA-CUSUM and CUSUM-GWMA Mann-Whitney charts to monitor unknown shifts in the process location

Keywords: Asymptotic control limits, Distribution-free, Exact control limits, Mann-Whitney U statistic, Mixed GWMA-CUSUM scheme, Mixed CUSUM-GWMA scheme, Phase I, Phase II

The Mann-Whitney (MW) test is one of the most important nonparametric tests used in the comparison of the location parameters of two populations. Unlike the t-test, the MW test can be used when the assumption of normality fails to hold. In this paper, the MW U statistic is used to construct two efficient distribution-free monitoring schemes, namely the mixed generally weighted moving average-cumulative sum (GWMA-CUSUM) MW U scheme (denoted as U-MGC) as well as its reversed version, i.e. the CUSUM-GWMA MW U scheme (denoted as U-MCG). The performances of the proposed schemes are investigated using the average run-length (ARL) and average extra quadratic loss (AEQL) values through extensive simulations. The newly proposed charts are found to be superior in small shifts detection than their competing (existing and others that are briefly introduced here) distribution-free Shewhart, EWMA, CUSUM, mixed EWMA-CUSUM, mixed CUSUM-EWMA and GWMA MW U charts in many situations. A real-life example is used to demonstrate the design and implementation of the new schemes.

Introduction

Shewhart control charts (or monitoring schemes) are well-known and appreciated because of their simplicity and quick detection of large shifts in the process parameters. However, they are relatively slow in detecting small and moderate shifts in the process. To solve this problem, alternative monitoring schemes such as the cumulative sum (CUSUM) and the exponentially weighted moving average (EWMA) are recommended (see [START_REF] Page | Continuous inspection schemes[END_REF][START_REF] Roberts | Control chart tests based on geometric moving averages[END_REF]. The CUSUM and EWMA monitoring schemes are efficient in detecting small shifts in the process parameters. Many authors have contributed to the improvement of these monitoring schemes; see for example, [START_REF] Lucas | Combined Shewhart-CUSUM quality control schemes[END_REF], [START_REF] Shamma | Development and evaluation of control charts using double exponentially weighted moving averages[END_REF], [START_REF] Abbas | Mixed exponentially weighted moving averagecumulative sum charts for process monitoring[END_REF] and [START_REF] Zaman | Mixed cumulative sumexponentially weighted moving average control charts: an efficient way of monitoring process location[END_REF]. [START_REF] Lucas | Combined Shewhart-CUSUM quality control schemes[END_REF] proposed a composite Shewhart-CUSUM (CSCUSUM) monitoring scheme that combines the strengths of both Shewhart and CUSUM schemes. To improve the sensitivity of the EWMA scheme toward large shifts, [START_REF] Lucas | Exponentially weighted moving average control schemes[END_REF] designed a

Design of the proposed control charts

The MW 𝐔 statistic

Assume that 𝑋 ={𝑥 𝑖 , i = 1, 2, 3, …, 𝑚} represents the IC Phase I (or reference) sample with unknown continuous cumulative density function (c.d.f.) 𝐹 𝑋 (𝑥) and 𝑌 ={𝑦 𝑡𝑗 , 𝑗 = 1, 2, …, 𝑛; 𝑡 =1, 2, 3, …} represents the Phase II (or test) sample with c.d.f. 𝐹 𝑌 (𝑦). The test samples at time 𝑡 (𝑡 = 1, 2, 3, …) are assumed to be independent and identically distributed (i.i.d.), from each other and, from the reference sample. Let us assume that 𝐹 𝑌 (𝑦) = 𝐹 𝑋 (𝑦 -𝛿), where 𝛿 is the shift in the location. The process is considered as IC if 𝛿 = 0, i.e. if 𝐹 𝑌 (𝑦) = 𝐹 𝑋 (𝑦).

The MW U statistic at time 𝑡 = 1, 2, 3, … (denoted as 𝑈 𝑡 statistic) represents the total number of pairs (𝑥 𝑖 , 𝑦 𝑡𝑗 ) for which the 𝑦 𝑡𝑗 (Phase II sample) are strictly greater than the 𝑥 𝑖 (Phase I sample), i.e. where 𝑡 = 1, 2, 3, … and 𝐼(𝑑 𝑡𝑖𝑗 ) is an indicator function defined as follows:

𝑑 𝑡𝑖𝑗 = 𝑦 𝑡𝑗 -𝑥 𝑖 (2)
and 𝐼(𝑑 𝑡𝑖𝑗 ) = { 1 if 𝑑 𝑡𝑖𝑗 > 0 0 if 𝑑 𝑡𝑖𝑗 ≤ 0 .

(3)

Note that there are 𝑚𝑛 pairs of (𝑥 𝑖 , 𝑦 𝑡𝑗 ) for each Phase II sample; therefore, 0 ≤ 𝑈 𝑡 ≤ 𝑚𝑛. For the two most extreme orderings every 𝑥 𝑖 precedes every 𝑦 𝑡𝑗 (so that 𝑈 𝑡 = 0) and every 𝑦 𝑡𝑗 precedes every 𝑥 𝑖 (so that 𝑈 𝑡 = 𝑚𝑛), respectively. Assuming that no tie is observed, the mean 𝜇 𝑈 𝑡 and variance 𝜎 𝑈 𝑡 2 of the statistic 𝑈 𝑡 are known to be 𝜇 𝑈 𝑡 = 𝑚𝑛 2 and (4)

𝜎 𝑈 𝑡 2 = 𝑚𝑛(𝑚 + 𝑛 + 1) 12 ,
respectively. To keep the notations simple, in the rest of the manuscript 𝜇 𝑈 𝑡 and 𝜎 𝑈 𝑡 2 will be denoted as 𝜇 𝑈 and 𝜎 𝑈 2 , respectively. For more details on the MW monitoring scheme, readers are referred to [START_REF] Chakraborti | A nonparametric control chart based on the Mann-Whitney statistic[END_REF] and [START_REF] Malela-Majika | Distribution-free Phase II Mann-Whitney control charts with runs-rules[END_REF].

The CUSUM MW U scheme

moderate shifts in the process mean. The plotting statistics of this chart consider an equal weighted combination of the current and past observations. In this section, we introduce a similar plotting statistic using the MW U statistic. Thus, the plotting statistics of the CUSUM chart based on the MW U statistic are given by 𝐶 𝑡 + = max [0, (𝑈 𝑡 -𝜇 𝑈 -𝐾 𝐶 ) + 𝐶 𝑡-1 + ] and

(5)

𝐶 𝑡 -= max [0, (𝜇 𝑈 -𝐾 𝐶 -𝑈 𝑡 ) + 𝐶 𝑡-1 -]

for 𝑡 = 1, 2,3, …, where 𝐾 𝐶 = 𝑘 𝐶 𝜎 𝑈 is the reference parameter (or coefficient) of the CUSUM chart. The starting values 𝐶 0 + and 𝐶 0 -are typically initialized to zero. The charting statistics are plotted against the upper control limit 𝐻 𝐶 > 0, with 𝐻 𝐶 = ℎ 𝐶 𝜎 𝑈 , where ℎ 𝐶 > 0 is the control limit coefficient of the CUSUM scheme. The process is considered to be OOC if the charting statistic (i.e. 𝐶 𝑡 + or 𝐶 𝑡 -) falls on or above the control limit 𝐻 𝐶 for any value of 𝑡, that is 𝐶 𝑡 + ≥ 𝐻 𝐶 (or 𝐶 𝑡 -≥ 𝐻 𝐶 ), otherwise, the process is considered to be IC. Note that in this paper, the CUSUM monitoring scheme using the MW U statistic is denoted as U-CUSUM chart.

The GWMA MW U scheme

Following [START_REF] Sheu | The generally weighted moving average control chart for detecting small shifts in the process mean[END_REF]'s idea, the charting statistics of the U-GWMA scheme, denoted as 𝐺 𝑡 , is given by 𝐺 𝑡 = ∑(𝑞 (𝑖-1) 𝛼 -𝑞 𝑖 𝛼 )𝑈 𝑡-𝑖+1 𝑡 𝑖=1 + 𝑞 𝑡 𝛼 𝑈 0 , 𝑡 = 1, 2, 3, …

where 𝑞 ∈ [0, 1) and 𝛼 > 0 are two parameters to be fixed. The starting value of the U statistic at t = 0 is considered to be equal to 𝜇 𝑈 (i.e. 𝑈 0 = 𝜇 𝑈 ). For more details on the GWMA plotting statistic, readers are referred to [START_REF] Sheu | The generally weighted moving average control chart for detecting small shifts in the process mean[END_REF].

In particular, the expected value of ( 6) is given by 𝐸(𝐺 𝑡 ) = 𝜇 𝐺 𝑡 = 𝐸 [∑(𝑞 (𝑖-1) 𝛼 -𝑞 𝑖 𝛼 )𝑈 𝑡-𝑖+1 𝑡 𝑖=1 + 𝑞 𝑡 𝛼 𝜇 𝑈 ] = 𝜇 𝑈 ,

and the variance of ( 6) is then defined by

𝑉𝑎𝑟(𝐺 𝑡 ) = 𝜎 𝐺 𝑡 2 = 𝜎 𝑈 2 𝑄 𝑡 , (8)
where 𝑄 𝑡 = ∑(𝑞 (𝑖-1) 𝛼 -𝑞 𝑖 𝛼 ) 2 𝑡 𝑖=1 .

Therefore, the time varying control limits of the U-GWMA monitoring scheme can be calculated as

𝑈𝐶𝐿 𝐺 𝑡 / 𝐿𝐶𝐿 𝐺 𝑡 = 𝜇 𝐺 𝑡 ± 𝐿𝜎 𝑈 √𝑄 𝑡 (9)
where 𝐿 (𝐿 > 0) is the coefficient representing the distance from the center line to the control limits. This coefficient is used to fix the predefined nominal IC 𝐴𝑅𝐿 (𝐴𝑅𝐿 0 ) value.

Note that when 𝛼 = 1, it can be shown that

𝑉𝑎𝑟(𝐺 𝑡 ) = 𝜎 𝐺 𝑡 2 = (1 -𝑞) 1 -𝑞 2𝑡 1 + 𝑞 𝜎 𝑈 2 . ( 10
)
Therefore, when the process has been running for a long time, that is, when 𝑡 → ∞, the variance of the U-GWMA scheme is given by

𝑉𝑎𝑟(𝐺 𝑡 ) = 𝜎 𝐺 2 = 𝑚𝑛 ( 𝑚 + 𝑛 + 1 12 ) ( 1 -𝑞 1 + 𝑞 ). (11)
In Equation ( 11), when 1 -𝑞 = 𝜆 (i.e., 𝑞 = 1 -𝜆), the U-GWMA scheme is equivalent to the U-EWMA monitoring scheme. The U-GWMA scheme gives a signal at the sampling time t if the U-GWMA scheme plotting statistic, 𝐺 𝑡 , plots outside of the control limits defined in Equation ( 9).

Design of the mixed GWMA-CUSUM MW U monitoring scheme

The U-MGC scheme is constructed by combining the GWMA and CUSUM charts using the MW U statistic. That is, the charting statistic of the U-GWMA scheme in Equation ( 6) is used as input of the charting statistics of the U-CUSUM scheme in Equation (5). Hence, at the sampling time t, the charting statistics of the proposed U-MGC monitoring scheme (denoted as 𝑀𝐺𝐶 𝑈 𝑡 + and 𝑀𝐺𝐶 𝑈 𝑡 -) are then defined by

𝑀𝐺𝐶 𝑈 𝑡 + = max [0, (𝐺 𝑡 -𝜇 𝑈 -𝐾 𝑡 ) + 𝑀𝐺𝐶 𝑈 𝑡-1 + ]
and ( 12)

𝑀𝐺𝐶 𝑈 𝑡 -= max [0, (𝜇 𝑈 -𝐾 𝑡 -𝐺 𝑡 ) + 𝑀𝐺𝐶 𝑈 𝑡-1 -]
where 𝐾 𝑡 is a time-varying reference value. The starting values 𝑀𝐺𝐶 𝑈 0 + and 𝑀𝐺𝐶 𝑈 0 -are typically taken to be equal to 0 and 𝐾 𝑡 is given by

𝐾 𝑡 = 𝑘 (𝑉𝑎𝑟(𝐺 𝑡 )) 1 2 = 𝑘 𝜎 𝑈 √𝑄 𝑡 , ( 13
)
where 𝑘 is the reference parameter (or coefficient) of the GWMA scheme.

When 𝛼 = 1, Equation (13) simplifies to

𝐾 𝑡 = 𝑘𝜎 𝑈 ((1 -𝑞) 1 -𝑞 2𝑡 1 + 𝑞 ) 1 2 . ( 14
)
Therefore, the exact (or time varying) upper control limit (i.e. Case E) is given by

𝐻 𝑡 = ℎ(𝑉𝑎𝑟(𝐺 𝑡 )) 1 2 = ℎ 𝜎 𝑈 √𝑄 𝑡 , (15)
where ℎ > 0 is the control limit coefficient used to fix the predefined nominal IC 𝐴𝑅𝐿 0 .

When 𝛼 = 1, as 𝑡 → ∞, the exact and asymptotic (i.e. Case E and Case A) upper control limits of the proposed U-MGC scheme are given by

𝐻 𝑡 = ℎ𝜎 𝑈 ((1 -𝑞) 1 -𝑞 2𝑡 1 + 𝑞 ) 1 2 (16)
and

𝐻 = ℎ𝜎 𝑈 ( 1 -𝑞 1 + 𝑞 ) 1 2 , ( 17
)
respectively. The charting statistic of the U-MGC scheme are plotted against the upper control limit 𝐻 𝑡 > 0 (or 𝐻 > 0) in Case E (or Case A) and the process is said to be OOC if the charting statistic 𝑀𝐺𝐶 𝑈 𝑡 (i.e. 𝑀𝐺𝐶 𝑈 𝑡 + or 𝑀𝐺𝐶 𝑈 𝑡 -) falls on or above the upper control limit for any value of 𝑡, that is, 𝑀𝐺𝐶 𝑈 𝑡 ≥ 𝐻 𝑡 (or 𝑀𝐺𝐶 𝑈 𝑡 ≥ 𝐻).

Design of the mixed CUSUM-GWMA MW U chart

Unlike the U-MGC chart presented in the previous sub-section, the U-MCG chart is constructed by combining the CUSUM and GWMA charts using the MW U statistic. That is, the charting statistic of the U-CUSUM chart in Equation ( 5) is used as input of the charting statistic of the U-GWMA chart in Equation ( 6). Hence, the two charting statistics of the proposed U-MCG chart are defined by

𝑀𝐶𝐺 𝑈 𝑡 + = ∑(𝑞 (𝑖-1) 𝛼 -𝑞 𝑖 𝛼 ) 𝐶 𝑡-𝑖+1 + 𝑡 𝑖=1 + 𝑞 𝑡 𝛼 𝑀𝐶𝐺 𝑈 0 + , 𝑡 = 1, 2, 3, …
and ( 18)

𝑀𝐶𝐺 𝑈 𝑡 -= ∑(𝑞 (𝑖-1) 𝛼 -𝑞 𝑖 𝛼 ) 𝐶 𝑡-𝑖+1 - 𝑡 𝑖=1 + 𝑞 𝑡 𝛼 𝑀𝐶𝐺 𝑈 0 -, 𝑡 = 1, 2, 3, …

Performance study of the proposed control charts

Performance measures

The performance of a monitoring scheme for specific shifts in the process parameter is usually measured using the average run-length (ARL) values. The ARL is defined as the number of rational subgroups to be plotted before the control chart signals for the first time. In practice, to avoid many false alarms, one must ensure that when the process is IC (i.e. 𝛿 = 0, where 𝛿 is the shift in the location as defined in sub-section 2.1), the IC ARL (denoted as 𝐴𝑅𝐿 0 ) value is set to be equal (or close) to some high desired values such as 370 and 500. When the process is OOC (i.e. 𝛿 ≠ 0), small OOC ARL (𝐴𝑅𝐿 1 ) values reveal that the chart performs better for that specific shifts. When the actual shift delta is unknown and the only available information is an interval [𝛿 𝑚𝑖𝑛 , 𝛿 𝑚𝑎𝑥 ] in which this shift is likely to occur, the literature recommends the use of some overall performance measure (see for example, [START_REF] Ou | A comparison study of effectiveness and robustness of control charts for monitoring process mean[END_REF], [START_REF] Sanusi | Using FIR to improve CUSUM charts for monitoring process dispersion[END_REF] and Malela-Majika and Rapoo (2017)) like, for instance, the average extra quadratic loss (AEQL) defined as

𝐴𝐸𝑄𝐿 = 1 ∆ ∑ 𝛿 2 × 𝐴𝑅𝐿(𝛿), 𝛿 𝑚𝑎𝑥 𝛿 𝑚𝑖𝑛 ( 21
)
where ∆ is the number of increment between 𝛿 𝑚𝑖𝑛 and 𝛿 𝑚𝑎𝑥 . The AEQL metric is recommended when the magnitude of the shift is more important (i.e. the quality practitioner is interested in shifts according to their magnitude). When comparing several schemes or sets of parameters, the one that yields the minimum AEQL value is considered to be the best. In other words, the smaller the AEQL value, the more efficient the chart is in detecting shifts in the process parameter.

Performance analysis

Determination of the optimal design parameters

A short description on how the optimal values are obtained for the control limits coefficients (i.e. h and L) of the U-MGC and U-MCG schemes that are used to fix a specified nominal 𝐴𝑅𝐿 0 value are provided in this section. In this paper, h and L are determined as follows: (i) the values of h (or L) that yield attained 𝐴𝑅𝐿 0 values as close as possible to the nominal 𝐴𝑅𝐿 0 value of 500 are computed using the N(0,1) distribution for Cases A and E; (ii) the OOC ARLs and the corresponding AEQL values are computed separately for Case A and for Case E. The h (or L)

that provides the minimum value of the AEQL is considered to be the optimal design parameter.

This is also checked under other probability distributions to make sure that the attained 𝐴𝑅𝐿 0 remains closer to the nominal value of 500.

IC design and robustness of the proposed MCG and MGC monitoring schemes

The IC robustness is one of the most important keys in the design and implementation of monitoring schemes. If the IC characteristics of a monitoring scheme, such as the IC average and median of the run-length (𝐴𝑅𝐿 0 , 𝑀𝑅𝐿 0 , etc.) are the same (or almost the same) across all continuous distributions, the scheme under consideration is said to be IC robust. To demonstrate the IC robustness of the proposed monitoring schemes, a Monte Carlo simulation was conducted to compute the IC characteristics of the proposed monitoring schemes for some symmetrical and skewed distributions. The following distributions are considered in this paper:

(i) Standard normal distribution, denoted as N(0,1), to study the effect of symmetrical distributions,

(ii) Gamma distribution with the parameter 𝜔 = 3 and 𝛽 = 1, denoted as GAM(3, 1), to study the effect of skewed distributions, (iii) Student's t distribution with degrees of freedom 𝜈 = 3, denoted as t(3), to study the effect of heavy-tailed distributions.

The above distributions have been transformed to have a unit variance and a mean equal to 0 for a fair comparison of the U-MGC and U-MCG schemes under different probability distributions. Tables 1 and2 display the actual 𝐴𝑅𝐿 0 values and design parameters of the proposed U-MCG and U-MGC schemes when (m, n) = (100, 5) for a nominal 𝐴𝑅𝐿 0 of 500. For instance, when 𝛼 = 1, 𝑞 = 0.5 and 𝑘 = 0.1, then it is found that the design parameter ℎ = 44.589 is such that the U-MGC scheme in Case A yields 𝐴𝑅𝐿 0 values of 502.68, 509.23 and 503.47 under the N(0,1), GAM(3,1) and t(3) distributions, respectively. It is important to note that when 𝛼 = 1

(with 𝑞 = 1 -𝜆), then the GWMA scheme reduces to the EWMA scheme; thus, this implies that the results in Table 1 correspond to the mixed EWMA-CUSUM and mixed CUSUM-EWMA MW U (denoted as U-MEC and U-MCE) schemes. Stated differently, when 𝛼 = 1

(with 𝑞 = 1 -𝜆), then the U-MGC scheme reduces to the U-MEC scheme and similarly, the U-MCG scheme reduces to the U-MCE scheme. The results in Tables 1 and2 show that for both Cases A and E, the 𝐴𝑅𝐿 0 values of the U-MGC and U-MCG schemes are close enough to the nominal value of 500 across all the distributions considered in this study. These findings confirm that the U-MGC and U-MCG schemes are IC robust.

Moreover, the findings in Tables 1 and2 as well as in Figures 1 and2 can be summarized as follows:

• Regardless of Case A or E, the control limit coefficient (i.e. ℎ) is an increasing function of 𝑞; i.e. the larger (smaller) the value of 𝑞, the wider (narrower) the control limit of the U-MGC scheme (see Figure 1(a)).

• For small reference constants (i.e. 𝑘 values), the ℎ values increases rapidly for both Cases E and A. However, for large 𝑘 values, ℎ increases at a small rate for both Cases E and A. Moreover, the smaller (larger) the magnitude of 𝑘, the wider (narrower) the control limit of the U-MGC scheme.

• The Case A's h values of the U-MGC scheme are slightly different from the Case E's h values (see Figure 1(a)).

• For the U-MCG scheme, in Case A, the control limit constant (i.e. 𝐿) is an increasing function of 𝑞. However, in Case E, 𝐿 is a decreasing function of 𝑞 at a small rate (see • For both Case A and Case U, regardless of the value of 𝛼, the smaller (larger) the magnitude of 𝑘, the wider (narrower) the control limit (see Figure 2). Moreover, when 𝑘 is kept fixed the larger (smaller) the value of 𝑞, the wider (narrower) the control limit of the U-MGC scheme (see Figures 2(a)-(d)).

• A thorough examination of Figure 2 shows that for very large values of 𝛼, the control limits converge toward a unique value regardless of the value 𝑞. (q,k)=(0.7,0.1) (q, k) =(0.1, 0.5) (q, k)=(0.5, 0.5) (q,k)=(0.7,0.5) (q, k) =(0.1, 0.1) (q, k)=(0.5, 0.1) Alpha L 2.5 1.5 0.8 0.1 50 40 30 20 10 0 Variable (q,k)=(0.7,0.1) (q, k) =(0.1, 0.5) (q, k)=(0.5, 0.5) (q,k)=(0.7,0.5) (q, k) =(0.1, 0.1) (q, k)=(0.5, 0.1) (c) Case E U-MGC scheme (d) Case E U-MCG scheme Figure 2. Optimal parameters of the proposed monitoring schemes when (m, n) = (100, 5), 𝛼 ∈ {0.1, 0.8, 1.5, 2.5}, 𝑘 ∈ {0.1, 0.5} and (m, n) = (100, 5) for a nominal 𝐴𝑅𝐿 0 of 500

OOC Performance of the U-MCG and U-MGC monitoring schemes

This section investigates the OOC performance of the proposed U-MCG and U-MGC schemes for specific shifts as well as for an overall performance. The performance of the new schemes is first investigated when 𝛿 𝑚𝑖𝑛 = 0, 𝛿 𝑚𝑎𝑥 = 2.5, (𝑞, 𝛼, 𝑘) = (0.1, 0.1, 0.1), 𝑚 ∈ {50, 100, 400} and 𝑛 ∈ {5, 10} for a nominal 𝐴𝑅𝐿 0 value of 500; and secondly, when 𝛿 𝑚𝑖𝑛 = 0, 𝛿 𝑚𝑎𝑥 = 2.5, (𝑚, 𝑛) = (100, 5) and 𝑘 ∈ {0.1, 0.5} for different values of 𝛼 and 𝑞 under the N(0,1), GAM(3,1) and t(3) distributions. Variable (q,k)=(0.7,0.1) (q, k) =(0.1, 0.5) (q, k)=(0.5, 0.5) (q,k)=(0.7,0.5) (q, k) =(0.1, 0.1) (q, k)=(0.5, 0.1) Alpha L 2.5 1.5 0.8 0.1 50 40 30 20 10 0 Variable (q,k)=(0.7,0.1) (q, k) =(0.1, 0.5) (q, k)=(0.5, 0.5) (q,k)=(0.7,0.5) (q, k) =(0.1, 0.1) (q, k)=(0.5, 0.1) Table 3: Case E 𝐴𝑅𝐿 and 𝐴𝐸𝑄𝐿 values of the U-MGC and U-MCG schemes along with their design parameters for different Phase I and Phase II sample sizes when (𝑞, 𝛼, 𝑘) = (0.1, 0.1, 0.1) for a nominal 𝐴𝑅𝐿 0 value of 500 The findings in Tables 4 to 6 can be summarized as follows:

(i) When 𝛼 = 1 (see Tables 4 and5),

• for small reference constants, in Case A, the sensitivity of the U-MGC scheme increases as 𝑞 increases regardless of the nature of the underlying process distribution. However, the sensitivity of the U-MCG scheme is almost the same regardless of the value of 𝑞 except for very small shifts (0 < 𝛿 ≤ 0.25) where the sensitivity of the U-MCG scheme decreases in the intervals 0.1 ≤ 𝑞 ≤ 0.3 and 0.5

≤ 𝑞 ≤ 0.7,
• in Case E, when 𝑘 is small, the sensitivity of the U-MGC scheme decreases as 𝑞 increases regardless of the nature of the underlying process distribution,

• in Case A, when 𝑘 is large, the sensitivity of the U-MGC scheme decreases as 𝑞 increases for small shifts in the location parameter, and it performs the worst for large values of 𝑞. However, for moderate and large shifts in the location parameter, the sensitivity of the U-MGC scheme increases as 𝑞 increases. For the U-MCG scheme, under the 𝑁(0,1) distribution, for small shifts, when 0 < 𝑞 ≤ 0.5, the sensitivity of the U-MCG scheme decrease as 𝑞 increases. When 0.5 < 𝑞 < 1, the sensitivity of the U-MCG scheme increases as 𝑞 increases. For moderate and large shifts in the location parameter, the sensitivity of the U-MCG scheme is almost the same for all 𝑞 values. Under non-normal distribution, the U-MCG scheme performs the worst for small values of 𝑞. Its performance tends to be the same as 𝑞 increases.

For moderate and small shifts, the sensitivity of the U-MCG scheme is almost the same for all 𝑞 values, and

• in Case E, when 𝑘 is large, regardless of the nature of the underlying process distribution, the sensitivity of the U-MGC scheme decreases as 𝑞 increases except for very small shifts where the performance of the U-MGC scheme increases as 𝑞 increases. From small-to-large shifts, the sensitivity of the U-MCG scheme is almost the same for all values of 𝑞 regardless of the nature of the distribution, except for very small shift under skewed distribution, where the sensitivity of the U-MCG scheme increases as 𝑞 increases.

(ii) When 𝛼 ≠ 1 (see Table 6),

• for small values of 𝑞 (i.e. 0 < 𝑞 < 0.5), the sensitivity of both the U-MGC and U-MCG schemes increases as 𝛼 increases regardless of the nature of the distribution,

• for 0.5 ≤ 𝑞 < 0.7, regardless of the nature of the distribution, the sensitivity of the U-MGC scheme increases in the interval 0 < 𝛼 ≤ 1.5 and remains almost the same for 𝛼 > 1.5. However, the sensitivity of the U-MCG scheme increases when 0 < 𝛼 ≤ 1 and remains almost the same when 𝛼 > 1, and

• in Case E, for large values of 𝑞 (i.e. 0.7 ≤ 𝑞 < 1), regardless of the nature of the distribution, the sensitivity of the U-MGC scheme decreases in the interval 0 < 𝛼 < 1 and increases when 𝛼 > 1. For small and moderate shifts, the sensitivity of the U-MCG scheme remains almost the same under skewed and heavy-tailed distributions when 0 < 𝛼 < 1. Under symmetrical distributions, the sensitivity of the U-MCG scheme decreases when 0 < 𝛼 < 1. For large shifts and 𝛼 > 1, under symmetrical and heavy-tailed distributions, the sensitivity of the U-MCG scheme is almost the same.

Note that the Case A of the above discussion yields the same findings. Therefore, to preserve writing space, the results are not displayed in this study. • the overall performance of the proposed U-MGC and U-MCG schemes deteriorates as 𝑞 increases except for the 0.7 ≤ 𝑞 < 1 where the sensitivity of the U-MCG scheme for 𝑘 = 0.5 increases,

• the U-MGC and U-MCG schemes perform better for large values of the reference constants (i.e. k) regardless of the underlying process distribution, and

• the U-MCG scheme is superior to the U-MGC scheme for all values of 𝑞 regardless of the underlying process distribution.

(ii) In Case A (see Figures 3(c)-(d)),

• regardless of the nature of the distribution, the sensitivity of the U-MCG scheme deteriorates as 𝑞 increases when 𝑘 = 0.1; whereas, the sensitivity of the U-MGC scheme for a reference constant of 0.1 increases as 𝑞 increases. However, when 𝑘 = 0.5, the sensitivity of the U-MGC scheme increases in the interval 0 < 𝑞 ≤ 0.6 and decreases for 0.6 < 𝑞 < 1; whereas, the sensitivity of the U-MCG scheme deteriorates in the interval 0 < 𝑞 ≤ 0.6 and increases for 0.6 < 𝑞 < 1.

• under the normal distribution, the U-MGC scheme with a reference constant of 0.5 is superior, except for large values of 𝑞 for which the U-MGC scheme with small values of 𝑘 outperforms all other designs followed by the U-MCG scheme with 𝑘 = 0.5. However, under non-normal distributions, the U-MGC scheme with a reference constant of 0.5 is superior except for large values of 𝑞 for which the U-MGC scheme with small values of 𝑘 outperforms all other designs followed by the U-MCG scheme with 𝑘 = 0.5.

• under both normal and non-normal distributions, the U-MCG scheme with 𝑘 = 0.1 is inferior compared to other designs. Variable (q,k)=(0.7,0.5) (q,k)=(0.1,0.5) (q,k)=(0.5,0.5) Alpha AEQL 2.5 1.5 0.8 0.1 Variable (q,k)=(0.7,0.1) (q,k)=(0.1,0.1) (q,k)=(0.5,01) Variable (q,k)=(0.7,0.5) (q,k)=(0.1,0.5) (q,k)=(0.5,0.5)

Alpha

Alpha AEQL 2.5 1.5 0.8 0.1 Variable (q,k)=(0.7,0.1) (q,k)=(0.1,0.1) (q,k)=(0.5,01) Alpha AEQL 2.5 1.5 0.8 0.1 30 25 20 15 10 Variable (q,k)=(0.7,0.5) (q,k)=(0.1,0.5) (q,k)=(0.5,0.5) (c) Under t(3) distribution (k = 0.1) (d) Under t(3) distribution (k = 0.5)
Panel 2: U-MCG scheme • Under the 𝑁(0,1) distribution and small reference coefficients, the U-MGC scheme performs better for small values of 𝑞 for all possible values of 𝛼. Moreover, the U-MGC scheme designed using 𝑘 = 0.5 is superior to the one with 𝑘 = 0.7 when 0 < 𝛼 ≤ 1.25; however, the converse is observed when 𝛼 > 1.25. For large reference coefficients, the U-MGC scheme with large values of 𝑞 performs better, except when 0.68 < 𝛼 ≤ 1.15, as the U-MGC scheme with 𝑘 = 0.1 performs better in that scenario.

Under non-normal distributions, for small reference coefficients, the U-MGC scheme performs better for small values of 𝑞, for all possible values of 𝛼. However, for large reference coefficients, the U-MGC scheme with large values of 𝑞 performs better except when 0.59 < 𝛼 ≤ 1.31, as the U-MGC scheme with 𝑘 = 0.1 performs better in this scenario (see Figure 4-Panel 1).

• Under both normal and non-normal distributions, for small reference coefficients, the U-MCG scheme performs better for small values of 𝑞, for all possible values of 𝛼. For large reference coefficients, the U-MCG scheme performs better for large values of 𝑞, for all possible values of 𝛼. However, the U-MCG scheme designed using 𝑘 = 0.5 is Variable (q,k)=(0.7,0.5) (q,k)=(0.1,0.5) (q,k)=(0.5,0.5) superior to the one with 𝑘 = 0.1 when 0 < 𝛼 ≤ 0.75; whereas, the converse is observed when 𝛼 > 0.75 (see Figure 4-Panel 2).

Performance comparison

In this section, the proposed U-MCG and U-MGC schemes are compared to a number of MW U-type monitoring schemes: (i) Shewhart (denoted MW U), (ii) EWMA (denoted U-EWMA), (iii) CUSUM (denoted U-CUSUM), (iv) mixed EWMA-CUSUM (denoted U-MEC), (v) mixed CUSUM-EWMA (denoted U-MCE) and, (vi) GWMA (denoted U-GWMA). The MW U scheme was first discussed in [START_REF] Chakraborti | A nonparametric control chart based on the Mann-Whitney statistic[END_REF], the U-CUSUM and U-EWMA schemes are equivalent to the existing CUSUM and EWMA WRS schemes discussed in [START_REF] Li | Nonparametric CUSUM and EWMA control charts for detecting mean shifts[END_REF], the U-GWMA scheme is introduced in Section 2.3 here and the U-MEC and U-MCE schemes are introduced in Table 1 here. The comparison is conducted when (𝑚,𝑛) = (100,5), 𝑘 = 0.5, (q, α, 𝜆) = (0.9, 1.5, 0.1) and 𝛿 𝑚𝑎𝑥 = 2.5 for a nominal 𝐴𝑅𝐿 0 value of 500 under the 𝑁(0,1), 𝑡(3) and 𝐺𝐴𝑀(3,1) distributions. Tables 7 to 9 present the IC and OOC performances of the competing schemes for specific shifts as well as their overall performances and their corresponding design parameters. The latter are determined such that the actual 𝐴𝑅𝐿 0 value is closer or equal to the nominal 𝐴𝑅𝐿 0 value. The schemes that perform better are shaded in grey. When two columns are shaded, the schemes under consideration are similar (or almost similar) in performance. To study the overall performance for small, moderate and large shifts, the 𝐴𝐸𝑄𝐿 values were calculated using Equation ( 21) where 𝐴𝐸𝑄𝐿 1 , 𝐴𝐸𝑄𝐿 2 and 𝐴𝐸𝑄𝐿 3 correspond to small, moderate and large shifts with 𝛿 𝑚𝑎𝑥 = 0.75, 1.5 and 2.5, respectively; whereas, 𝐴𝐸𝑄𝐿 4 and 𝐴𝐸𝑄𝐿 5 correspond to small-to-moderate (i.e., 0 < 𝛿 ≤ 1.5) and small-tolarge shifts (i.e., 0 < 𝛿 ≤ 2.5), respectively.

Table 7: Case E performance comparison of eight monitoring schemes under the N(0, 1) distribution when (m, n) = (100, 5), (q, α, 𝜆) = (0.9, 1.5, 0.1) for a nominal 𝐴𝑅𝐿 0 value of 500 𝐴𝐸𝑄𝐿 2 is the AEQL for moderate shifts only (0.75 < 𝛿 ≤ 1.5) 𝐴𝐸𝑄𝐿 3 is the AEQL for large shifts only (1.5 < 𝛿 ≤ 2.5) 𝐴𝐸𝑄𝐿 4 is the AEQL for small-to-moderate shifts (0 < 𝛿 ≤ 1.5) 𝐴𝐸𝑄𝐿 5 is the AEQL for small-to-large shifts (0 < 𝛿 ≤ 2.5)

Table 8: Case E performance comparison of eight monitoring schemes under the t(3) distribution when (m, n) = (100, 5), (q, α, λ) = (0.9, 1.5, 0.1) for a nominal 𝐴𝑅𝐿 0 value of 500 Tables 9: Case E performance comparison of eight monitoring schemes under the 𝐺𝐴𝑀(3,1) distribution when (m, n) = (100, 5), (q, α, 𝜆) = (0.9, 1.5, 0.1) for a nominal 𝐴𝑅𝐿 0 value of 500 From Table 7, it can be seen that under the 𝑁(0,1) distribution, the U-MCG scheme performs better for small shifts in the location parameter; whereas, the U-CUSUM, U-EWMA, U-MEC and U-MCE have the worst performances for small shifts (see the 𝐴𝐸𝑄𝐿 1 values). However, the U-GWMA and MW U schemes perform better than all competing schemes considered in this study under moderate and large shifts (see the 𝐴𝐸𝑄𝐿 2 and 𝐴𝐸𝑄𝐿 3 values). Moreover, for small-to-moderate shifts, the U-MCG scheme outperform all competing schemes (see the 𝐴𝐸𝑄𝐿 4 values); while the U-GWMA scheme performs better for small-to-large shifts (see the 𝐴𝐸𝑄𝐿 5 values). From Tables 8 and9, it can be observed that under the 𝑡(3) and 𝐺𝐴𝑀(3,1)

distributions, the U-MCG scheme outperforms the competing schemes under small shifts (see the 𝐴𝐸𝑄𝐿 1 values). Under moderate shifts, the U-GWMA scheme is superior compared to all other competing schemes (see the 𝐴𝐸𝑄𝐿 2 values). For large shifts, the Shewhart MW U scheme performs better (see the 𝐴𝐸𝑄𝐿 3 values). In addition, under the 𝑡(3) distribution, for small-tomoderate shifts in the location parameter, the U-MCG scheme is superior to all competing schemes (see the 𝐴𝐸𝑄𝐿 4 values); while the U-GWMA scheme performs better for small-tolarge shifts (see the 𝐴𝐸𝑄𝐿 5 values). Under the 𝐺𝐴𝑀(3,1) distribution, for both small-tomoderate and small-to-large shifts in the location parameter, the U-MCG scheme is superior to all competing schemes considered in this study (see the 𝐴𝐸𝑄𝐿 4 and 𝐴𝐸𝑄𝐿 5 values). In the latter comparison, the U-CUSUM, U-EWMA, U-MEC and U-MCE schemes do not have a range of shift values where they yield a better performance for the considered AEQL situations.

It can also be seen that the proposed U-MCG scheme with (q, α) = (0.9, 1.5) has very interesting run-length characteristics regardless of the size of the shift. However, for these parameters, the U-MGC schemes performs relatively worst especially for moderate and large shifts. This poor performance of the U-MGC scheme is due to the large value of 𝑞. In some other situations, the U-MGC scheme presents very interesting run-length characteristics (see for instance, Tables 4 to 6).

Illustrative example

In this section, the implementation and application of the proposed U-MGC and U-MCG monitoring schemes is illustrated using the data on the inside diameters of piston rings manufactured by a forging process from Montgomery (2005, page 223). The first set of data For the U-MCG scheme, when 𝑘 = 0.1, (𝑞, 𝛼) = (0.5, 1.5), (0.1, 1.5) and (0.5, 0.5), we have 𝐿 = 18.606, 19.076 and 33.642 with 𝐴𝑅𝐿 0 values equal to 500.78, 498.38 and 499.46, respectively. However, when 𝑘 = 0.5, we have 𝐿 = 4.829, 5.743 and 7.646 with 𝐴𝑅𝐿 0 values of 503.01, 499.44 and 500.57, respectively. The plot of the U-MCG scheme plotting statistics for 𝑘 = 0.1 and 0.5 are shown in Figure 6. It is observed that when 𝑘 = 0.1, (𝑞, 𝛼) = (0.5, 1.5) and (0.1, 1.5), the proposed U-MGC scheme gives a signal on the twelfth and fourteenth Phase II subgroups, respectively (see Figures 6(a)-(b)). When 𝑘 = 0.1 and (𝑞, 𝛼) = (0.5, 0.5), the proposed U-MCG scheme does not give a signal in Phase II (see Figure 6(c)). However, when 𝑘 = 0.5, (𝑞, 𝛼) = (0.5, 1.5), (0.1, 1.5) and (0.5, 0.5), the U-MCG scheme gives a signal on the fifth, third and second Phase II subgroups, respectively (see Figure 6(d)-(f)). Therefore, in these conditions, the U-MGC scheme is more sensitive compared to the U-MCG scheme since it gives a signal sooner. Variable (q,alpha)=(0.5,0.5) H(0.5,1.5) H(0.1,1.5) H(0.5,0.5) (q,alpha)=(0.5,1.5) (q,alpha)=(0.1,1.5) (q,alpha)=(0.5,0.5) H(0.5,1.5) H(0.1,1.5) H(0.5,0.5) (q,alpha)=(0.5,1.5) (q,alpha)=(0.1,1.5)

Sample number/ sampling number

Figure 1

 1 Figure 1(b)). Note that in Case A, for small 𝑘 values, 𝐿 increases at a high rate (see Figure 1(b)).

  (a) U-MGC scheme (b) U-MCG scheme Figure 1. Optimal parameters of the proposed monitoring schemes when 𝛼 = 1, 𝑘 ∈ {0.1, 0.5} and (m, n) = (100, 5) for a nominal 𝐴𝑅𝐿 0 of 500 (a) Case A U-MGC scheme (b) Case A U-MCG scheme

  SchemeMGC MCG m 50 100 400 50 100 400 Distribution Shift n = 5 n = 10 n = 5 n = 10 n = 5 n = 10 n = 5 n = 10 n = 5 n = 10 n = 5 n = 10

  terms of the 𝐴𝑅𝐿 and 𝐴𝐸𝑄𝐿 values when (𝑞, 𝛼, 𝑘) = (0.1, 0.1, 0.1), 𝑚 ∈ {50, 100, 400} and 𝑛 ∈ {5, 10} for a nominal 𝐴𝑅𝐿 0 value of 500. For 𝑚 ≤ 50, in terms of the 𝐴𝐸𝑄𝐿 values, it can be observed that the overall sensitivity of the U-MGC and U-MCG schemes increases as the Phase I and/or Phase II sample size(s) increase(s). In terms of the 𝐴𝑅𝐿 values, the U-MGC and U-MCG schemes perform better under heavy-tailed distributions regardless of the size of the shift in the location parameter. However, under small and moderate shifts, they perform better under symmetrical distributions compared to the skewed ones. Note though, they are similarly sensitive under symmetrical and skewed distributions for large shifts. When 𝑚 > 50, in terms of the 𝐴𝑅𝐿 and 𝐴𝐸𝑄𝐿 values, the sensitivity of the U-MGC and U-MCG schemes increase as the Phase I and/or Phase II sample size(s) increase(s). They both perform better under nonnormal distributions. Moreover, when the Phase I is kept fixed, the larger (smaller) the Phase II sample size, the larger (smaller) the design parameter. However, when the Phase II sample size is kept fixed, the larger (smaller) the Phase I sample size, the smaller (larger) the design parameter. Note that similar findings are also observed for the Case A performance of the U-MGC and U-MCG schemes. To preserve writing space, the rest of this paper will focus on the investigation of the performance of the U-MGC and U-MCG scheme when 𝑚 = 100 and 𝑛 = 5.

  Figure 3 displays the overall performance of the proposed schemes when 𝛿 𝑚𝑎𝑥 = 2.5, (𝑚, 𝑛) = (100, 5), 𝑘 ∈ {0.1, 0.5} and 𝛼 = 1 for different values of 𝑞 under normal and non-normal distributions. From Figure 3, the following is observed in terms the overall performance (i.e. AEQL values):

  Under t(3) distribution (k = 0.1) (d) Under t(3) distribution (k = 0.5) Panel 1: U-MGC scheme (a) Under N(0,1) distribution (k = 0.1) (b) Under N(0,1) distribution (k = 0.5)

Figure 4 .

 4 Figure 4. AEQL values of the U-MGC and U-MCG schemes when 𝛼 = {0.1, 0.8, 1.5, 2.5}, 𝑘 ∈ {0.1, 0.5} and (m, n) = (100, 5) for a nominal 𝐴𝑅𝐿 0 of 500 under the normal and non-normal distribution

  contains hundred and twenty five Phase I observations (m = 125) collected when the process was considered to be IC. These data are used as the Phase I data for which a goodness of fit test for normality is not rejected. The second set of data contains fifteen test samples each of size n = 5 which are considered to be the Phase II data. The proposed monitoring schemes are designed in Case E when 𝑘 ∈ {0.1, 0.5}, 𝛼 ∈ {0.5, 1.5} and 𝑞 ∈ {0.1, 0.5} for a nominal 𝐴𝑅𝐿 0 value of 500. When 𝑘 = 0.1, (𝑞, 𝛼) = (0.5, 1.5), (0.1, 1.5) and (0.5, 0.5), we have ℎ = 38.862, 22.362 and 47.764 which are the control limit coefficients of the U-MGC scheme with 𝐴𝑅𝐿 0 values of 499.58, 500.52 and 501.57, respectively. However, when 𝑘 = 0.5, we have ℎ = 11.423, 6.224 and 12.569 so that the corresponding 𝐴𝑅𝐿 0 values are 499.58, 500.52 and 501.57, respectively. The plot of the U-MGC schemes' plotting statistics for 𝑘 = 0.1 and 0.5 are shown in Figure5. It is observed that when 𝑘 = 0.1, (𝑞, 𝛼) = (0.5, 1.5), (0.1, 1.5) and (0.5, 0.5), the proposed U-MGC scheme gives a signal on the seventh, sixth and ninth Phase II subgroups, respectively (see Figure5(a)). However, when 𝑘 = 0.5, (𝑞, 𝛼) = (0.5, 1.5), (0.1, 1.5) and (0.5, 0.5), the proposed U-MGC scheme gives a signal on the third, second and fourth Phase II subgroups, respectively (see Figure5(b)).

  (a) MGC scheme with 𝑘 = 0.1 (b) MGC scheme with 𝑘 = 0.5 Figure 5. Implementation of the U-MGC scheme for the Montgomery (2005)'s piston ring data

Figure 6 .

 6 Implementation of the U-MCG scheme for the Montgomery (2005)'s piston ring data

  The control limits of the proposed U-MCG chart are𝑈𝐶𝐿/𝐿𝐶𝐿 = 𝜇 𝐶 𝑡 ± 𝐿 • 𝜎 𝐶 𝑡 √𝑄 𝑡 , (20)where 𝐿 is the width coefficient and 𝜇 𝐶 𝑡 and 𝜎 𝐶 𝑡 are the time-varying mean and variance of the charting statistics of the U-CUSUM scheme in the IC situation up to specific time 𝑡 and when 𝑡 → ∞, they both become constant.The U-MCG chart is constructed by plotting the charting statistics 𝑀𝐶𝐺 𝑈 𝑡 + and 𝑀𝐶𝐺 𝑈 𝑡 -against the sample number or sampling time 𝑡. The process is said to be OOC if the charting statistic 𝑀𝐶𝐺 𝑈 𝑡 (i.e. 𝑀𝐶𝐺 𝑈 𝑡 + or 𝑀𝐶𝐺 𝑈 𝑡 -) falls beyond the control limits defined in Equation (20), that is, for any value of 𝑡, 𝑀𝐶𝐺 𝑈 𝑡 ≥ 𝑈𝐶𝐿 (or 𝑀𝐶𝐺 𝑈 𝑡 ≤ 𝐿𝐶𝐿).

	where the initial (or starting) values of 𝑀𝐶𝐺 𝑈 𝑡 + and 𝑀𝐶𝐺 𝑈 𝑡 -charting statistics are equal to 𝜇 𝑈 as
	defined in Equation (7), respectively; that is, 𝑀𝐶𝐺 𝑈 0 + = 𝑀𝐶𝐺 𝑈 0 -= 𝜇 𝐶 𝑡 . The mean and variance
	of the 𝑀𝐶𝐺 𝑈 𝑡 + and 𝑀𝐶𝐺 𝑈 𝑡 -charting statistics are given by
	𝐸(𝑀𝐶𝐺 𝑈 𝑡 + ) = 𝐸(𝑀𝐶𝐺 𝑈 𝑡 -) = 𝜇 𝐶 𝑡
	and	(19)
	𝑉𝑎𝑟(𝑀𝐶𝐺 𝑈 𝑡 + ) = 𝑉𝑎𝑟(𝑀𝐶𝐺 𝑈 𝑡 -) = 𝑄 𝑡 𝜎 𝐶 𝑡 2 .

Table 1 :

 1 Attained 𝐴𝑅𝐿 0 values and optimal parameters of the U-MGC and U-MCG monitoring schemes for a nominal 𝐴𝑅𝐿 0 of 500 when 𝑘 ∈ {0.1, 0.5}, 𝛼 = 1, 𝑞 ∈

				{0.1,0.2,…,0.8} and (m, n) = (100, 5) under different distributions
	Case	Chart	𝑘	Distribution	0.1	0.2	0.3	0.4	𝒒	0.5	0.6	0.7	0.8
				N(0,1)	501.55	500.72	499.73	500.30		502.68	502.41	498.77	499.59
	A Case	U-MGC U-MCG	0.1 0.5 0.1 0.5	GAM(3,1) t(3) 𝒉 N(0,1) GAM(3,1) t(3) 𝒉 N(0,1) GAM(3,1) t(3) 𝑳 N(0,1) GAM(3,1) t(3)	511.05 502.30 22.868 500.56 498.77 509.34 6.199 500.52 494.22 493.05 21.384 499.02 494.29 509.07	509.61 493.47 26.719 499.18 507.06 496.44 7.332 501.59 490.27 500.24 23.634 502.20 496.54 509.83	513.42 501.25 31.367 500.24 513.54 504.00 8.733 500.05 489.26 499.64 26.431 500.61 498.20 509.82	509.46 510.11 36.684 505.73 501.47 498.99 10.524 502.11 512.65 510.12 29.362 500.10 500.73 512.67		509.23 503.47 44.589 500.55 508.26 503.21 12.922 501.76 498.30 491.20 32.582 503.41 501.81 503.25	487.33 499.40 55.194 499.81 500.06 502.01 16.394 498.01 509.01 495.92 37.162 501.37 510.03 500.15	513.24 507.82 69.497 501.76 510.11 504.20 21.672 509.48 494.35 508.22 44.313 501.35 509.79 498.29	501.54 512.64 94.289 506.84 509.55 501.49 31.454 499.05 514.81 497.24 54.313 504.53 503.53 496.42
				L	5.740	6.234	6.784	7.384		8.088	8.898	9.8283	11.029
				N(0,1)	502.19	500.96	505.90	501.85		500.26	500.71	503.10	501.86
	E Case	U-MGC	0.1 0.5	GAM(3,1) t(3) 𝒉 N(0,1) GAM(3,1) t(3) 𝒉 N(0,1)	521.47 493.23 22.909 500.48 491.32 501.77 6.203 500.06	510.49 507.81 26.931 500.99 500.05 490.00 7.298 498.81	502.54 501.29 31.673 499.19 495.97 516.01 8.755 499.58	492.97 493.81 36.869 498.95 489.44 483.76 10.505 510.67		506.48 500.12 44.362 508.45 491.09 520.18 12.981 504.65	504.49 497.08 55.340 500.54 498.23 506.92 16.491 509.79	501.02 496.28 70.240 502.21 509.91 493.82 21.631 501.92	505.33 506.18 95.241 504.73 509.90 496.20 31.592 502.10
		U-MCG	0.1 0.5	GAM(3,1) t(3) 𝑳 N(0,1) GAM(3,1) t(3)	504.71 504.39 19.484 500.59 511.02 509.51	507.03 521.43 19.478 499.76 500.25 496.24	508.08 498.09 19.315 500.94 493.41 516.14	493.52 498.91 19.163 502.84 509.73 494.61		510.50 506.11 19.092 505.76 501.30 514.44	509.50 517.94 18.931 504.57 495.31 489.48	506.76 495.47 18.420 505.96 500.42 512.07	511.01 518.89 18.001 504.44 510.62 509.29
				L	5.210	5.060	4.952	4.831		4.651	4.418	4.161	3.6933

Table 2 :

 2 Attained 𝐴𝑅𝐿 0 values and optimal parameters of the U-MGC and U-MCG monitoring schemes for a nominal 𝐴𝑅𝐿 0 of 500 when 𝑘 ∈{0.1, 0.5}, 𝑞 ∈ {0.1,0.5,0.7}, 𝛼 ∈ {0.1, 0.8, 1.5, 2.5} and (m, n) = (100, 5) under different distributions

				𝒒		0.1				0.5				0.7
	Case	Chart	𝑘	𝜶	0.1	0.8	1.5	2.5	0.1	0.8	1.5	2.5	0.1	0.8	1.5	2.5
				N(0,1)	500.45	499.54	502.48	501.39	500.78	498.51	500.76	502.91	501.00	500.43	500.11	506.38
	A Case	U-MGC U-MCG	0.1 0.5 0.1 0.5	GAM(3,1) t(3) 𝒉 N(0,1) GAM(3,1) t(3) 𝒉 N(0,1) GAM(3,1) t(3) L N(0,1) GAM(3,1) t(3)	511.33 504.01 21.493 502.04 510.93 501.33 5.632 501.54 506.23 500.85 20.886 501.68 503.26 505.54	509.36 501.44 22.718 502.04 504.34 501.96 6.226 502.05 500.01 503.88 21.462 499.98 502.33 500.16	505.53 499.29 22.849 499.81 500.43 503.46 6.229 500.16 504.45 501.49 21.496 500.16 502.05 504.11	505.19 503.32 22.826 498.78 502.44 505.46 6.223 501.16 503.66 502.23 21.539 498.33 505.18 500.46	503.20 504.18 28.256 504.04 503.18 500.23 6.616 499.51 500.95 502.51 26.836 503.56 501.2 502.42	507.59 502.44 46.794 501.99 504.27 500.33 13.429 502.18 510.06 504.05 34.036 501.29 499.26 500.19	504.16 501.24 40.084 502.22 511.35 501.67 11.648 497.88 501.39 501.77 30.386 500.1 501.2 503.34	510.82 501.55 34.337 500.92 503.41 500.04 9.787 498.11 505.18 502.22 27.726 506.04 501.46 500.79	502.04 505.39 31.343 501.9 501.74 503.11 7.029 501.51 500.46 509.34 32.936 500.52 507.29 501.65	498.32 504.21 76.963 500.1 502.31 501.64 24.045 500.31 508.22 501.56 47.053 502.03 512.07 502	499.77 503.01 54.601 500.6 504.06 500.46 16.378 501.44 502.26 500.49 37.374 498.84 503.15 500.39	504.60 501.11 37.676 501.55 503.12 501.66 10.848 500.25 501.05 503.21 29.289 503.06 497.46 502.69
				L	5.534	5.721	5.736	5.742	8.127	8.079	7.709	7.186	12.897	9.857	9.031	7.476
				N(0,1)	499.42	498.90	501.51	507.35	495.90	503.80	495.92	502.03	501.75	501.48	506.48	508.28
	E Case	U-MGC	0.1 0.5	GAM(3,1) t(3) 𝒉 N(0,1) GAM(3,1) t(3) 𝒉 N(0,1)	500.95 509.03 21.349 500.51 496.60 506.77 5.626 509.35	507.08 501.34 22.728 501.47 503.44 494.06 6.206 503.24	500.23 502.19 22.890 499.87 491.19 501.45 6.213 509.75	512.73 499.65 23.032 494.86 500.57 511.04 6.216 502.37	510.13 492.73 28.252 500.16 508.17 501.24 6.642 498.94	504.79 501.07 47.060 502.77 500.08 507.10 13.367 499.06	495.41 500.55 39.871 498.22 494.64 501.14 11.515 500.74	500.99 504.42 34.674 498.40 501.38 499.13 9.780 503.89	514.41 492.89 31.203 503.54 499.14 510.41 7.103 496.12	509.26 494.73 78.034 515.71 498.28 499.20 23.826 511.32	512.70 512.12 54.604 498.07 501.17 506.56 16.503 506.48	497.24 496.48 37.501 508.03 493.94 496.05 10.755 508.28
		U-MCG	0.1 0.5	GAM(3,1) t(3) L N(0,1) GAM(3,1) t(3)	503.76 516.42 20.876 502.12 501.55 499.32	498.30 493.75 21.359 499.77 497.07 505.01	494.24 499.96 21.532 500.43 500.49 502.20	510.24 490.53 21.542 499.78 506.43 510.19	491.01 496.31 26.856 497.22 498.61 498.64	493.45 491.46 34.040 500.21 503.97 499.72	501.94 498.72 30.397 501.63 500.92 505.37	504.24 509.28 27.754 500.33 499.81 503.32	504.72 511.44 32.952 509.09 518.19 494.66	503.68 497.77 47.067 503.70 503.85 504.11	514.10 499.03 37.389 500.35 499.01 512.92	498.79 493.93 29.298 490.07 493.61 505.08
				L	5.589	5.803	5.832	5.846	8.207	8.122	7.949	7.287	13.112	9.974	9.152	7.606

Table 3

 3

displays the Case E performance of the U-MGC and U-MCG monitoring schemes in

Table 4 :

 4 Case A OOC 𝐴𝑅𝐿 values of the U-MGC and U-MCG monitoring schemes for a nominal 𝐴𝑅𝐿 0 value of 500 when 𝑘 ∈ {0.1, 0.5}, 𝛼 = 1, 𝑞 ∈ {0.1,0.3, 0.5, 0.7} and (m, n) = (100, 5)

					MGC chart			MCG chart
	𝑘	Distribution	𝒒 𝜹	0.1	0.3	0.5	0.7	0.1	0.3	0.5	0.7
			0.25	80.53	79.32	78.07	77.41	79.18	81.82	77.96	84.67
			0.50	24.02	21.73	18.44	14.94	22.71	23.09	23.15	23.97
		N(0,1)	1.00 1.50	12.67 8.76	9.81 7.14	8.44 6.51	7.72 4.79	11.26 8.18	11.51 8.52	11.00 8.84	12.87 9.89
			2.00	7.04	6.07	4.93	4.07	6.95	7.26	7.61	8.63
			2.50	6.30	6.01	5.22	3.43	6.23	6.79	7.04	8.04
			0.25	37.19	33.58	30.40	30.22	38.34	38.52	38.78	38.74
			0.50	16.21	13.71	12.13	9.17	15.91	16.16	16.34	17.38
	0.1	t(3)	1.00 1.50	8.88 7.41	8.00 6.17	7.12 5.31	5.21 4.34	8.93 7.18	9.27 7.51	9.61 7.87	10.63 8.89
			2.00	6.82	5.89	4.65	3.82	6.47	6.91	7.20	8.20
			2.50	6.28	5.43	4.61	3.29	6.14	6.52	7.03	8.03
			0.25	65.66	64.76	63.99	64.76	66.30	76.04	67.25	63.34
			0.50	19.74	17.52	14.67	12.94	19.80	20.08	20.09	21.00
		GAM(3,1)	1.00 1.50	10.22 7.99	9.02 6.73	7.04 5.49	6.16 4.73	10.31 7.82	10.67 8.18	10.91 8.51	11.95 9.55
			2.00	6.73	5.99	4.88	3.91	6.89	7.15	7.51	8.52
			2.50	6.25	5.47	4.79	4.00	6.23	6.86	7.03	8.03
			0.25	99.52	114.52	136.11	204.14	84.23	88.42	93.50	84.90
			0.50	13.24	12.85	18.23	37.42	11.79	11.97	12.21	12.08
		N(0,1)	1.00 1.50	5.18 2.97	4.20 2.97	3.87 2.34	4.23 1.80	4.39 3.07	4.47 3.20	4.66 3.27	4.57 3.12
			2.00	2.42	2.56	2.10	1.21	2.50	2.70	2.78	2.47
			2.50	2.15	2.22	1.65	1.09	2.13	2.25	2.36	2.10
			0.25	35.87	45.33	60.81	128.79	35.33	31.26	32.42	31.72
			0.50	7.54	6.95	7.04	13.05	6.80	6.84	7.02	7.21
	0.5	t(3)	1.00 1.50	4.00 2.69	3.04 2.49	2.67 2.00	2.32 1.42	3.37 2.60	3.46 2.80	3.59 2.90	3.47 2.58
			2.00	2.32	2.20	1.84	1.19	2.23	2.35	2.45	2.21
			2.50	2.14	2.10	1.29	1.11	2.11	2.15	2.18	2.09
			0.25	97.05	137.00	178.90	335.23	102.81	93.09	94.59	92.02
			0.50	10.15	13.35	17.14	43.77	9.71	9.81	10.19	10.01
		GAM(3,1)	1.00 1.50	4.06 3.03	3.78 2.69	3.03 2.31	3.99 2.00	3.90 2.98	4.06 3.06	4.18 3.12	4.12 2.99
			2.00	2.19	2.14	2.16	1 .31	2.49	2. 76	2.87	2.41
			2.50	2.07	2.03	1.93	1.20	2.22	2.23	2.39	2.04

Table 5 :

 5 Case E OOC 𝐴𝑅𝐿 values of the U-MGC and U-MCG monitoring schemes for a nominal 𝐴𝑅𝐿 0 value of 500 when 𝑘 ∈ {0.1, 0.5}, 𝛼 = 1, 𝑞 ∈ {0.1,0.3, 0.5, 0.7} and (m, n) = (100, 5)

					MGC scheme			MCG scheme
	𝑘	Distribution	𝒒 𝜹	0.1	0.3	0.5	0.7	0.1	0.3	0.5	0.7
			0.25	83.54	86.06	90.90	93.78	82.98	83.73	83.95	80.65
			0.50	23.87	26.78	29.26	34.32	22.80	22.89	23.40	23.88
		N(0,1)	1.00 1.50	11.90 8.71	13.44 9.94	15.21 11.34	18.51 14.10	11.29 8.24	11.51 8.48	11.95 8.97	12.86 9.89
			2.00	7.35	8.45	9.76	12.27	7.00	7.23	7.73	8.64
			2.50	6.87	7.90	9.05	11.34	7.00	6.75	7.08	8.05
			0.25	40.69	44.84	47.12	53.06	37.64	38.88	39.72	39.25
			0.50	16.92	18.82	21.04	25.08	16.00	16.07	16.60	17.30
	0.1	t(3)	1.00 1.50	9.49 7.61	10.80 8.74	12.04 10.09	15.19 12.63	9.00 7.24	9.22 7.49	9.72 7.97	10.66 8.90
			2.00	7.00	8.03	9.23	11.65	6.53	6.88	7.27	8.20
			2.50	6.61	7.65	8.98	11.18	6.18	6.49	7.05	8.03
			0.25	70.33	81.09	82.49	85.97	68.42	69.53	61.94	66.77
			0.50	20.85	23.56	25.85	30.54	19.92	20.18	20.50	21.19
		GAM(3,1)	1.00 1.50	10.95 8.35	12.48 9.54	14.10 10.93	17.22 13.63	10.39 7.89	10.60 8.17	11.08 8.63	11.98 9.56
			2.00	7.23	8.32	9.63	12.15	6.93	7.14	7.64	8.53
			2.50	6.92	7.93	9.05	11.39	6.30	6.82	7.06	8.04
			0.25	86.81	82.53	81.64	80.29	87.61	88.89	89.64	89.86
			0.50	11.80	12.21	13.40	15.68	11.71	11.61	12.14	12.07
		N(0,1)	1.00 1.50	4.56 3.22	5.22 3.80	6.28 4.72	8.20 6.39	4.35 3.08	4.47 3.19	4.60 3.26	4.61 3.13
			2.00	2.80	3.16	4.09	5.59	2.54	2.68	2.78	2.51
			2.50	2.37	3.01	4.08	5.07	2.13	2.24	2.34	2.12
			0.25	31.13	30.42	27.09	28.84	32.24	31.42	32.70	31.28
			0.50	6.90	7.65	8.97	11.08	6.82	6.88	7.08	7.18
	0.5	t(3)	1.00 1.50	3.54 2.89	4.16 3.31	5.11 4.21	6.82 5.76	3.38 2.62	3.48 2.79	3.59 2.88	3.51 2.63
			2.00	2.44	3.06	4.02	5.23	2.22	2.33	2.41	2.24
			2.50	2.20	3.01	4.00	5.06	2.11	2.15	2.18	2.11
			0.25	95.04	72.90	71.58	60.21	111.58	99.79	86.32	87.85
			0.50	9.89	10.10	11.36	13.56	9.82	9.69	9.87	10.03
		GAM(3,1)	1.00 1.50	4.11 3.08	4.78 3.66	5.76 4.47	7.64 5.17	3.92 2.99	4.04 3.05	4.18 3.11	4.13 3.02
			2.00	2.88	3.08	4.02	5.52	2.51	2. 74	2.00	2.49
			2.50	2.42	3.00	4.00	5.04	2.06	2.21	2.00	2.06

Table 6 :

 6 Case E OOC 𝐴𝑅𝐿 values of the U-MGC and U-MCG monitoring schemes for a nominal 𝐴𝑅𝐿 0 value of 500 when 𝑘 = 0.1, 𝑞 𝜖 {0.1, 0.5, 0.7}, 𝛼 𝜖 {0.1, 0.8, 1.5, 2.5} and (m, n) = (100, 5)

					MGC scheme			MCG scheme
	Distribution	𝒒	𝜶 𝜹	0.1	0.8	1.5	2.5	0.1	0.8	1.5	2.5
			0.25	84.66	80.04	87.48	83.77	83.68	92.47	92.13	88.61
			0.50	22.32	16.63	14.63	12.60	20.66	13.56	12.05	12.00
		0.1	1.00 1.50	10.06 7.03	7.42 5.21	4.56 4.31	4.55 3.94	9.45 6.54	6.34 4.73	4.35 3.44	4.37 3.21
			2.00	5.74	3.80	3.51	3.43	5.68	3.17	3.03	2.84
			2.50	5.41	3.39	3.32	3.30	5.34	3.01	2.73	2.54
			0.25	81.37	77.29	75.95	80.27	78.99	87.54	88.22	92.00
			0.50	15.31	13.46	12.80	12.46	12.82	11.79	11.85	12.06
	N(0,1)	0.5	1.00 1.50	7.29 5.47	6.52 4.92	5.81 4.32	4.45 4.03	6.36 4.95	4.576 4.22	4.64 3.37	4.61 3.34
			2.00	4.94	4.17	3.88	3.39	4.23	3.49	3.00	3.02
			2.50	4.49	4.00	3.41	3.05	4.04	3.11	2.80	3.00
			0.25	90.08	105.05	97.33	90.12	84.13	78.27	78.83	80.10
			0.50	18.86	26.25	21.66	18.12	14.26	14.75	13.35	13.36
		0.7	1.00 1.50	8.65 7.17	10.49 9.42	8.99 7.43	7.98 10.81	6.94 5.18	8.04 7.43	6.63 5.16	6.51 5.86
			2.00	6.55	8.44	6.70	9.26	4.82	6.14	4.98	4.58
			2.50	5.87	6.89	5.50	8.54	3.69	5.45	4.12	4.44
			0.25	37.48	30.64	29.08	29.87	36.52	33.10	32.56	31.14
			0.50	14.97	9.12	7.64	7.19	14.11	7.66	7.01	6.80
		0.1	1.00 1.50	7.61 6.02	5.51 4.92	4.37 4.14	4.36 3.99	7.11 5.62	4.27 3.52	3.98 3.13	3.53 2.99
			2.00	5.38	3.54	3.39	3.11	5.20	3.11	2.97	2.74
			2.50	5.30	3.18	3.10	3.02	5.11	3.00	2.49	2.40
			0.25	29.40	26.89	29.87	27.91	25.19	30.22	34.08	32.65
			0.50	7.42	9.12	8.39	8.02	5.70	6.93	7.06	7.01
	t(3)	0.5	1.00 1.50	3.79 3.11	5.30 4.32	4.70 3.97	4.36 3.52	3.13 2.75	3.44 2.84	3.68 3.07	3.64 3.08
			2.00	3.00	4.05	3.54	3.17	2.54	2.66	2.89	3.01
			2.50	2.92	4.01	3.23	3.06	2.34	2.42	2.62	3.00
			0.25	45.32	55.93	51.71	47.14	39.93	41.49	39.93	39.00
			0.50	10.89	16.87	12.77	10.11	7.99	8.18	6.59	6.53
		0.7	1.00 1.50	6.68 5.86	10.06 9.82	7.97 6.24	7.23 6.06	5.81 4.81	6.23 5.40	5.49 5.21	5.06 4.82
			2.00	4.21	8.80	5.67	5.34	4.34	4.70	4.51	4.19
			2.50	4.01	7.27	5.22	4.81	3.57	3.78	3.68	3.53
			0.25	86.02	82.09	83.01	80.72	91.05	96.36	90.01	94.69
			0.50	19.06	10.50	9.84	9.93	18.72	9.86	10.00	9.94
		0.1	1.00 1.50	9.08 6.74	6.11 5.01	4.70 4.33	4.52 4.12	8.78 5.84	4.10 3.76	4.17 3.21	3.97 3.03
			2.00	5.63	3.60	3.47	3.15	5.23	3.15	3.00	2.81
			2.50	5.41	3.26	3.18	3.08	5.01	3.04	2.57	2.46
			0.25	77.23	67.08	68.94	78.44	105.63	82.19	89.70	99.84
			0.50	10.05	11.68	10.97	10.63	8.31	9.91	10.00	9.95
	GAM(3,1)	0.5	1.00 1.50	4.45 3.24	6.02 4.73	5.34 4.15	4.98 3.93	3.29 2.89	4.10 3.11	4.24 3.17	4.20 3.15
			2.00	3.00	4.09	3.92	3.28	2.52	2.73	3.00	3.00
			2.50	3.00	4.04	3.47	3.07	2.40	2.57	2.94	3.00
			0.25	75.30	82.57	82.04	75.94	72.79	76.25	68.42	68.38
			0.50	15.50	16.33	17.98	14.88	11.59	11.87	10.42	10.30
		0.7	1.00 1.50	8.49 7.77	10.24 9.89	8.81 7.98	7.90 6.41	7.01 6.81	7.12 6.46	6.75 7.33	6.40 5.49
			2.00	6.41	8.31	6.58	5.16	5.17	5.05	5.94	4.69
			2.50	5.90	7.54	5.02	4.60	4.30	4.42	4.29	3.72

Sample number/ sampling number

	MGC+
	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

Conclusion and recommendations

In an effort to efficiently monitor small location shifts in a two-sample distribution-free scenario, in this paper, the mixed GWMA-CUSUM and CUSUM-GWMA MW U monitoring schemes (denoted as U-MGC and U-MCG schemes) are proposed. The newly proposed schemes are revealed to be more efficient than the numerous MW-type schemes in monitoring small shifts in the process location. Thus, practitioners are recommended to use the proposed U-MCG monitoring scheme to efficiently detect small shifts. Note though, the U-GWMA scheme discussed herein is recommended for moderate shifts and both the Shewhart MW U and U-GWMA schemes are recommended when monitoring large shifts in the process location.

In terms of the overall performance, the U-MCG scheme is recommended for monitoring small-to-moderate shifts and the U-GWMA scheme is preferred for monitoring small-to-large shifts. Since the proposed U-MGC scheme is relatively insensitive to large shifts, the design of an improved version is needed. Therefore, in future we intend to investigate the performance of the combined Shewhart-MGC MW U schemes as well as the Shewhart-MCG MW U schemes in order to further improve the sensitivity of the proposed schemes in monitoring moderate to large shifts in the location parameter.


How large can be a braid plain? A worldwide scale comparison Quelle est la largeur d'une rivière en tresses ? Ue comparaison à l'échelle globale

Keywords: Active channel width, braid plain, braiding rivers, global scale, hydromorphological functioning

Les rivières en tresses sont des rivières à chenaux multiples caractérisées par de larges bandes actives. Elles se trouvent surtout dans les régions de montagne ou de piémont à travers le monde, là où il existe une connexion active en amont avec les sources sédimentaires. Mais est-ce que les rivières en tresses fonctionnent de la même manière à travers le monde ? L'hypothèse est qu'il existe, à l'échelle globale, une gamme de conditions déterminant le patron de tressage et les largeurs des bandes actives en fonction de la taille du bassin versant et de ses caractéristiques hydroclimatiques, géologiques ou orographiques.

Cette étude est basée sur l'analyse de la relation entre la largeur de la bande active et certaines caractéristiques des bassins versants amont (c'est-à-dire la largeur de la bande active normalisée) de 257 tronçons de rivière en tresses à travers le monde, caractérisés à l'aide d'images satellites disponibles entre septembre 2019 et mai 2020. Nous avons ensuite comparé la gamme de valeurs de largeur de bande active normalisée à certains caractères des bassins (par ex., les conditions climatiques, hydrologiques et de relief). Nos résultats montrent que ces facteurs contrôlent la taille des tresses et qu'il existe ainsi des conditions plus ou moins favorables pour que des systèmes en tresses se développent sur terre.

BRAID P LAIN RIVERS AND THEIR CHARACTERIS ATION

Braiding rivers are multi-thread, highly shifting rivers characterized by a wide area of unvegetated gravel corridor, the active braid plain. These rivers often occupy the piedmont valleys of high mountains and are considered hotspots of biodiversity. Because of their dynamics, braiding rivers are expected to be amongst the most sensitive river systems to global environmental changes, including climate change. These rivers underwent significant changes in their intensity and extent as consequence of human disturbances across the globe, notably in France, Italy and New Zealand, amongst others [START_REF] Piégay | Braided River Management: from Assessment of River Behaviour to Improved Sustainable Development[END_REF].

The intensity of braiding pattern is commonly identified and measured by indices based on bar or channel dimensions or number, which strongly depend on the hydrological conditions when river is observed. Another indicator independent of hydrological conditions at the date of observation is the normalised active channel width (W*), that is the active channel width (water and bare sediment areas, in m) rated by the upstream catchment area (km 0.4 ), and whose size is linked to the connectivity to sediment sources, in terms of amount and transport capacity [START_REF] Piégay | Census and typology of braided rivers in the French Alps[END_REF]. W* is characterized by a high variability. In the French Alps, minimum values are around 10m/km 0.4 , below which a network of braided channels is no more observed. Despite some regional references exist (e.g. French Alps, Piedmont region in Italy, new Caledonia (2,3)), it does not yet exist a global overview of braiding river intensity.

S TUDY AREA AND METHODS

We selected 257 sites displaying a multi-channel braiding-like pattern from visual inspection of Google Earth images of piedmont and mountain areas located in different geographic zones across the world (Figure 1): Albania (n=17), Azerbaïdjan (n=14), Chile (n=17), China (n=44), France (n=53), Georgia (n=14), India (n=37), New Zealand (n=32), United States and Canada (30). For the scope of the work we analysed the relationship between the active channel width and the upstream catchment area measured and calculated the normalised active channel width W* as proxy of braiding intensity at reach scale (where a reach is 3 to 5 times the active channel width). We then explained this relationship and the braiding intensity by analysing a series of controlling variables that characterise transport capacity, mainly flood regime (mean annual precipitation and seasonality) and slope (relief ratio and elongation ratio). All GIS measures and processing steps were realized on ArcMap 10.7 and QGis 3.4.9. All analysis were performed on R version 3.6.2.

PRELIMINARY RES ULTS AND DIS CUS S ION

The largest active channel are observed in India and New Zealand with mean value of 722.9 and 628.5 m, respectively, whereas narrowest reaches are observed in France with mean values lower than 100 m. For all reaches taken together, the relationship between the active channel width and the upstream catchment area is statistically significant (adjusted r-squared = 0.39; p-value <0.0001), with great variability between geographic zones (Figure 1). Notably the relationship is not significant for reaches in New Zealand, suggesting differences between sub-zones. The existence of three subzones for New Zealand is clarified after analysing the residuals to the global model: one sub-zone includes reaches with extremely large active channel widths and highest residuals (>0.5; NZL_G) located in upstream areas close to glaciers in the southern island; a second sub-zone includes reaches with residuals between -0.5 and 0.5 located in piedmont areas in the southern island (NZL_M); a third zone includes reaches with lowest residuals (mainly around zero or negative) located in floodplain areas of the northern and southern islands.

The W* is then calculated for each zone and sub-zone, considering a coefficient 0.36 from a quantile regression model (Q=0.5). Highest W* are observed in India, New Zealand (mainly NZL_G and NZL_M) and USA-Canada, whereas lowest W* are observed in France, Chile, China and Georgia.

Between these two groups, reaches in Albania and Azerbaijan and floodplain reaches in New Zealand display intermediate W* values. The difference between zones and sub-zones is statistically significant in most cases (p-value <0.05), notably between the highest, the lowest and the intermediate W* values mentioned above, but not within these three groups. Interesting and significant (p<0.0001) differences are observed for instance between China and India, which are located northern and southern respectively of the Himalayan chain, suggesting different conditions in terms of sediment production or transport that cause extremely large and potentially active braiding rivers in the Indian side (Figure 1). To explain the observed patterns we related the residuals from the quantile regression, calculated as median values per zone and sub-zone, to potential controlling drivers at catchment scale, namely the precipitation regime over the period 1970-2000, and the relief ratio and elongation ratio. The inter and infra-zones differences in braiding intensity (W*) can be mainly explained in terms of amount of precipitation (p<0.005), notably between China and Indian braiding reaches, but also in terms of precipitation seasonality, such as between France and Albania, or both, such as between sub-zones in New Zealand.

These findings suggest that a range of conditions can determine at global scale the existence of a given planform channel pattern, and as consequence that the expected behaviours (in terms of morphodynamics) might differ, according to the geographic areas related to specific controlling drivers. This has important consequences for braiding river management, for instance in terms of flood risk management or in terms of habitat conservation, braiding rivers being potential sensitive hot-spots at global scale in terms of biodiversity. Further analyses on the temporal trends in W* values from satellite imagery will clarify the braiding river behaviours and potential human pressures affecting them at global scale.

  Selected catchments are characterised by different climatic conditions and hydrological regime, different catchment conditions and geological context and different human pressures.The data included in this study are: (i) satellite images available between September 2019 and May 2020 from QGis & ArcMap from DigitalGlobe, GeoEye, Earthstar Geographics, CNE/Airbus imageries; //www.hydrosheds.org/); (iii) elevation data from SRTM DEM at 30 m resolution (https://earthexplorer.usgs.gov/); (iv) data on climate and precipitations for the years 1970-2020 at 30 seconds resolution from WorldClim2.1 (https://worldclim.org/data/bioclim.html).

Figure 1 .

 1 Figure 1. Study area (USC, USA and Canada; FRA, France, ALB, Albania; CHN, China; IND, India; CHL, Chile, GEO, Georgia; AZE, Azerbaijan; NZL, New Zealand) and variability of the relationship between active channel width (m) and the upstream catchment area (km 0.4 ).