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In light of rapid development of customer requirements, control procedures of quality concept use multivariate analysis. This is because of recent advances in information technology and in recording. The charting procedures are based on Mahalanobis distance but their performance needs normality and a type-I error rate choice. The DD-diagram is an alternative scheme that uses data depth to avoid these conditions rarely met in practice. For a given error-free sample, the performance of DD-diagram and that of multivariate EWMA control procedures are compared through a real example on individual observations taken from a multivariate quality process.

Multivariate control schemes are valuable when several process features are being observed to identify instability within the manufacturing quality process that exhibits substantial cross-correlations. These control schemes use relationships between quality characteristics to generate powerful control procedures that are sensitive to shifts in position and/or in dispersion.

Multivariate Shewhart control chart was first introduced in 1947 and it is known as Hotelling’s

In this paper, an application of both MEWMA control chart and DD-diagram is conducted using individual observations taken off a real case of quality process from the industry. The data of the samples are collected during two different times of the production process. The reference sample measures are drawn from a production process during which the process is considered in control. However, the empirical sample measures are drawn later in the frame work of a quality control routine. The MEWMA control chart and the DD-diagram are given in Sections 2 and 3, respectively. In Section 4 these monitoring techniques are applied. The empirical analysis is given in Section 5 and after that we dedicate Section 6 to draw some conclusions.

As the number of process variables grows the traditional multivariate control charts such as the

Let

characteristics

is a vector-valued output at time

As indicated by [

where

where

The statistic

and the covariance matrix

Replacing

such that

Let F be a probability distribution in

Therefore, if the quality of the

According to [

the Mahalanobis depth at x with respect to F is defined to be

The sample version of Equation (10) is obtained by replacing

Henceforth, D or

Given a notion of data depth, one can compute the depths of all quality measures

Given the definitions (10) or (11), the sample becomes

When the depth-equivalence class contains more than one point measure,

On this basis and using data depth, the Equations (12) and (13) fix out a centre or a multivariate median. Moreover, [

A data depth plots is a graphical comparison between two multivariate distributions based on data depth. So in addition to the reference sample

Since

In general, the distributions are rarely known so instead we use an empirical version of the DD-diagram. If F and G are unknown distributions for the samples X and Y, then the DD-diagram is obtained when plotting

if Equation (11) is used to compute the data depth.

If

In most cases, the departure from the diagonal line usually takes the form of pulling down from the point

In analogous manner to the classical multivariate control procedures, [

It is conversely proportional to the sum of the depth of the reference sample centre,

In this section individual observations data are collected from a production process during which the process is considered in control. These observations are then used to estimate the parameters of the underlying distribution F of the considered process. Then, another series of individual observations are drawn when the distribution of the process has drifted to a distribution G. Both series of observations are used to construct and to argue the performance of the monitoring schemes the MEWMA chart and the DD-diagram. Based on a previous work of [

Processing the observed data begins with the start-up stage that consists of estimating the parameters of F, constructing the control limit of MEWMA chart and determining the centre of F as reference sample.

According to Equation (5), the vector of sample mean is

and to Equation (6), the sample covariance matrix is

To construct a multivariate EWMA control chart, [

To determine the centre, data depths of all observations of the reference sample are calculated using Equation (11). As recorded in the work of [

In order to detect graphically any point that is not satisfying the limiting variation interval, the

as the least acceptable data depth value and below which the corresponding point is considered out of control i.e. at least one of the p-characteristics exceeds its limiting variation interval.

It is clear that in the reference sample, the point of order 45 is characterized by the maximum data depth in either case before and after centering with respect to the computed vector-valued centre

The second stage consists of using both control schemes to evaluate the stability of the observed production process when an empirical sample is drawn. At this phase, the parameters of the reference sample F obtained in

the start-up stage are used to monitor any taken empirical sample in the future. Specifically, after drawing the empirical sample G given in [

Hereafter, the vector-valued centre

The first line of _{2}) exceeds its specified measure. The other observations (2, 3, 27, 29, 33, 39, 43, 50 and 52) are considered out of control because the humidity rate (X_{3}) is lower than its minimum value.

For centered measures, the DD-diagram in the right subplot of

The MEWMA control chart, for

Obs. # | |||||
---|---|---|---|---|---|

2 | 0.991 | 7.956 | 0.113 | 105 | 454.60 |

3 | 0.982 | 7.900 | 0.114 | 101 | 454.30 |

19 | 0.996 | 8.000 | 0.117 | 101 | 456.13 |

27 | 0.987 | 7.850 | 0.114 | 102 | 453.00 |

29 | 0.999 | 7.950 | 0.114 | 107 | 452.10 |

33 | 0.991 | 7.951 | 0.113 | 109 | 454.21 |

39 | 0.993 | 7.950 | 0.115 | 111 | 454.00 |

43 | 0.996 | 7.940 | 0.114 | 109 | 453.88 |

50 | 0.982 | 7.959 | 0.111 | 106 | 454.12 |

52 | 0.987 | 7.940 | 0.113 | 107 | 455.00 |

54 | 0.987 | 8.010 | 0.115 | 106 | 450.10 |

Obs. # | |||||
---|---|---|---|---|---|

3 | 0.982 | 7.900 | 0.114 | 101 | 454.30 |

12 | 0.992 | 8.000 | 0.114 | 107 | 454.60 |

13 | 0.994 | 7.940 | 0.116 | 108 | 453.90 |

19 | 0.996 | 8.000 | 0.117 | 101 | 456.10 |

29 | 0.999 | 7.950 | 0.114 | 107 | 452.10 |

33 | 0.991 | 7.951 | 0.113 | 109 | 454.20 |

34 | 0.994 | 7.980 | 0.116 | 110 | 454.10 |

39 | 0.993 | 7.950 | 0.115 | 111 | 454.00 |

43 | 0.996 | 7.940 | 0.114 | 109 | 453.90 |

50 | 0.982 | 7.959 | 0.111 | 106 | 454.10 |

51 | 0.991 | 7.973 | 0.115 | 106 | 453.60 |

52 | 0.987 | 7.940 | 0.113 | 107 | 455.00 |

53 | 0.992 | 7.958 | 0.116 | 106 | 454.10 |

57 | 0.997 | 7.980 | 0.115 | 106 | 454.70 |

Comparing the realized values for the out-of-control observations with respect to their specification intervals indicated in the first line of

The investigation of

The DD-diagram is a graphical comparison that exhibits location shifts and/or scale increase when moving from the distribution F of the reference sample to the distribution G of the empirical one. And to use this dia- gram, we do not need any requirement about the nature of the observed multivariate quality process distribution. Although, this procedure looks like a non parametric method, DD-diagram does not require large samples. It suffices to have a size of the samples that goes beyond 30 to ensure a reasonable performance. So, whenever this size goes bigger the DD-diagram improves in performance.

The above application allows us to say that DD-diagram performs as better as multivariate EWMA control chart because its use does not depend on normality as for the case of MEWMA control chart. In the above application, DD-diagram detects 11 points indicating that their components exceed their specified limits whereas the MEWMA control chart gives 14 points corresponding to a smoothing parameter

When a multivariate quality process changes its distribution from F to G and if the location shift is eliminated i.e. centering the measures with respect to the centre or the deepest point of F, DD-diagram makes it possible to distinguish between the out-of-control observations that were drifted because of location shifts and scale increase respectively and those that were drifted under the effect of variations in dispersion only. This fact is not feasible when using MEWMA control chart.

In general consider the test of a null hypothesis asserting stability of a production process versus an alter- native one that concerns the existence of shifts in location and/or in scale, then the empirical sample has higher dispersion than that of the reference one. This is deduced from the fact that the resulting clouds, of centered measures or not, are located under the limiting variation level line at

MekkiHajlaoui, (2015) On the Charting Procedures: MEWMA Chart and DD-Diagram. Open Journal of Statistics,05,373-381. doi: 10.4236/ojs.2015.55039