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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.18 No.3 pp.315-329
DOI : https://doi.org/10.7232/iems.2019.18.3.315

A Comparative Analysis of Bootstrap Multivariate Exponentially-Weighted Moving Average (BMEWMA) Control Limits

O. Ikpotokin*, I. U. Siloko
Department of Mathematics, Faculty of Physical Sciences, Ambrose Alli University, Ekpoma, Nigeria
Department of Mathematics and Computer Science, Edo University Iyamho, Edo State, Nigeria
Corresponding Author, E-mail: ikpotokinosayomore@yahoo.co.uk, osayomoreikpotokin@aauekpoma.edu.ng
January 17, 2019 May 24, 2019 June 1, 2019

ABSTRACT


In the manufacturing and service industries, multivariate statistical quality control charts are mostly used to determine whether a process is performing as intended or if there are some special causes resulting in the variation of an overall statistics. Normally, control charts are obtained under the assumption that the variable under study follows some form of parametric distribution. When this assumption is violated, the performance of such control chart often gives false alarm signals. To address this problem, the Multivariate Exponential Weighted Moving Average (MEWMA) control limits have been proposed in existing literature. This article focuses on reviewing this existing method with a view to developing a novel approach based on the use of bootstrap. Results from a performance study shows that the proposed method enables the setting of control limits that can enhance the easy detection of out of control signals.



초록


    1. INTRODUCTION

    In the industrial sector, simultaneous examination of related quality variables is common phenomenon of multivariate statistical process control (MSPC). Many control charts have been proposed for multivariate data, with the most popular being multivariate Shewhart (Hotelling’s X2 and T2) chart, the multivariate exponentially- weighted moving average (MEWMA) chart, and the multivariate cumulative sum (MCUSUM) chart (Montgomery, 2001;Sullivan and Stoumbos, 2001;Stoumbos and Sullivan, 2002).

    Control charting methodology shared similarities with the classical statistical inference such as hypothesis testing and confidence intervals. Many statistical inference procedures are obtained following some form of parametric distribution such as multivariate normal assumption that is required for many control charts (Gibbons and Chakraborti, 2003;Jarrett and Pan, 2007;Champ and Aparisi, 2007). For instance, the absence of normality assumption affects the robustness of Shewhart Hotelling’s T2 control method and therefore, the control limits obtained under it increases the rate of false alarm similar to type I or type II error, that is, detecting out-of-control signals when the process is in-control (type I error) or not detecting signals when the process is out-of-control (type I error). The presence of type I or type II error will result to sub-standard quality of product and waste of both material and financial resources in any production process. To overcome this problem or difficulty, the distribution-free or nonparametric control methods such as the bootstrap serve as better alternatives. The significance of this study therefore, is geared towards setting bootstrap multivariate exponential weighted moving average (BMEWMA) control limits to avoid the general assumption that observed data follow normal distribution with mean vector (μ) and variance covariance matrix (Σ).

    Multivariate exponentially weighted moving average (MEWMA) established by (Lowry et al., 1992) as an extension of the univariate exponentially weighted moving average (EWMA) is to speedily discover little differences that might be available in a process than the Hoteling’s T2 and MCUSUM methods. Suppose X = (X1, X2, ..., Xd) , is a d-dimensional quality variable obtained from a process of interest, where X N d ( μ 0 , Σ 0 ) , (X is ddimensional vector having mean μ0 as well as variance– covariance matrix Σ0), where both parameters are unknown. But if μ0 and Σ0 can be estimated from a set of K training samples each with size n, and the system was under control at the time these K training set of observations were selected, a MEWMA control chart proposed by (Lowry et al., 1992) is as follows:

    Z i = Λ X i + ( 1 Λ ) Z i 1
    (1)

    where Λ is the diag( λ 1 , λ 2 , ... , λ d = λ ), 0 λ i 1 for i = 1 , 2 , , d . If λ 1 = λ 2 = ... = λ d = λ , the initial value of Z0 is determined as the same value of mean vector under control. The MEWMA method is equal to the T2 - method and is denoted as:

    T i 2 = Z i ' Z i 1 Z i > h i = 1 , 2 , ....
    (2)

    where ΣZi is the variance-covariance matrix of Zi. The value h is obtained via simulation to attain a desire ARL, and Zi is estimated as:

    Z i = λ 2 λ [ 1 ( 1 λ ) 2 i ]
    (3)

    An estimation of ΣZi as it tends to infinity is expressed as:

    Z i = λ 2 λ
    (4)

    From the MEWMA vector in Equation (3), Zi is extended recursively to obtain:

    Ζ i = λ X i + λ (1- λ ) X i 1 + λ (1- λ ) 2 X i 2 + ... + λ (1- λ ) t-1 X 1 + (1- λ ) i Z 0
    (5)

    When μ0 and Σ0 are unknown, then K incontrol samples of n dimension each being used to approximate the parameters. The control mean vector is estimated by:

    X ¯ ¯ = i = 1 k X i ¯ k and S ¯ = i = 1 k S i k
    (6)

    where

    X i ¯ = h = 1 n X i h n and S i = h = 1 n ( X i h X i ¯ ) ( X i h X i ¯ ) ' n i = 1 , 2 , ... , k respectively .
    (7)

    The study of optimal design of MEWMA charts by means of the average run length and the median run length was carried out by Lee and Khoo (2006). The attributes of MEWMA method as soon as parameters are determined was studied by Champ and Jones-Farmer (2007). MEWMA control method for clinical trials and investigation were developed by Joner et al. (2008). Effect of MEWMA control method adopting some diverse recommended figures of smoothing parameters of λ when the in control parameters are estimated was investigated by Mahmoud and Zahran (2010). A modified MEWMA control scheme for an analytical process data was initiated by Patel and Divecha (2013) with a view to detecting shifts of all kinds in case of highly vector autoregressive model of order one as denoted by VAR (1). A new maximum exponentially weighted moving average control chart for monitoring process mean and dispersion was developed by Haq et al. (2015a). An extended nonparametric exponentially weighted moving average sign control chart was proposed by Lu (2015). New exponentially weighted moving average control charts for monitoring process dispersion was introduced by Haq et al. (2015b). The EWMA method for detecting mean differences of a long-memory process was studied by Rabyk and Schmid (2016). In the existing MEWMA control method, the control limit (h) that gives the desire average run length (ARL0) of say 200 when the process is assumed to be incontrol at the initial stage is difficult or time taking to obtain due to “trial-and-error” nature of selection. That is, the control limit will be adjusted until the desired ARL0 of say 200 is obtained (Prabhu and Runger, 1997;Boone, 2010). To overcome the problem of trial-and-error, the BMEWMA percentile procedure was introduced in order to maximize the advantages of detecting large, moderate and small shift in any given process by adjusting the value of α accordingly.

    The bootstrap control methods are advantageous for its elasticity resulting from not presuming any form of parametric assumption to construct and implement the charting statistic. Apparently, this is very useful in the area of control process in set upstages for the presence of few data. The bootstrap method is also very effective and efficient and less prone to outliers when considering statistic such as p-values, confidence interval, control limits, etc. Efron (1982) introduced the “bootstrap” as a general method by which one available sample size gives rise to many others by method of resampling known as the Monte Carlo algorithm. A bootstrap sample say ( x 1 * , x 2 * ,   x 3   * , , x n * ) is defined to be a random sample of seize n drawn with replacement from the original data set of the same size n say ( x 1 ,   x 2 , x 3 , , x n ) from which the statistic of interest is estimated. In this type of sampling there is the chance of some value appearing once, twice, zero times, hence the name ‘bootstrap resampling’ method. It is a computer based sampling method for assessing measures of accuracy to statistical process control estimates, such as the control limits, p-value based on extreme values of the test statistic, Ikpotokin and Ishiekwene (2017). The objectives of this study therefore are to: (i) introduce a BMEWMA procedure for obtaining control limits as an alternative to the standard practices that observed process data are normally distributed with mean vector (μ) and variance covariance matrix (Σ), (ii) overcome the problem of “trial-and-error” in setting MEWMA control limit (h) that gives the desired in-control ARL0 by introducing the BMEWMA percentile procedure in obtaining control limits for detecting large, moderate and small shift in any given process and (iii) to ascertain comparatively the performance of BMEWMA control limits as the bootstrap replications increase from 1000 to 10000 with the existing method.

    2. THE NON-PARAMETRIC BOOTSTRAP MULTIVARIATE EXPONENTIALLYWEIGHTED MOVING AVERAGE (BMEWMA) CONTROL LIMITS

    The non-parametric kernel density estimation (KDE) Hotelling’s T2 control method was proposed by Chou et al. (2001). Hotelling’s T2 control limits obtained from this method is carried out without considering the assumption of normality but with limitations on how to determine some statistics such as kernel functions, smoothing statistic, and mathematical integration to have a full structure of the KDE method. The use of a common structure to construct both the univariate and multivariate control charts conditions was initiated by Polansky (2005). The bootstrap method is adopted to obtain a discrete distribution and numerical integration to obtain control limits. Identifying multivariate out of control variables using the boosting procedure was introduced by Alfaro et al. (2009) with the view of doing boosting with categorization to find out the quality characteristics that is responsible for out of control signals. Chronological change-point tests for linear regression by mean of bootstrapping was introduced by Kirch (2008), Hušková and Kirch (2012). The bootstrap-based design of residual control charts and the non-parametric bootstrap cumulative sum control limits were proposed by Capizzi and Masarotto (2009), Chatterjee and Qiu (2009).

    The percentile bootstrap method as a means of obtaining Hoteling’s T2 control limits assuming that the distribution is not multivariate normal was proposed by Phaladiganon et al. (2011). The idea of bootstrapping method to tackle the matter of quality characteristics that are not correlated as well as the minimum maximum control methods were studied by Adewara and Adekeye (2012). The use of balance bootstrap percentile method to estimate critical value and control limits for autocorrelated processes was introduced by Kalgonda (2013). Definite restricted bootstrap control chart method for performance evaluation was proposed by Gandy and Kvaløy (2013). The block bootstrap control limits for multivariate autocorrelated process was proposed by Kalgonda, (2015) having view around the control procedure based on Zstatistics. Assuming there are d quality variables and each of the quality variable contains n set of samples (xij); (i = 1, 2, …, n; j = 1, 2, …, d) as can be summarized in the matrix below.

    x 1 x 2 x d ( x 11 x 12 x 1 d x 21 x 22 x 2 d x n 1 x n 2 x n d )   d x n

    If the matrix notations of dxn dimensions can be transposed as expressions below:

    x 1 = ( x 11 , x 21 , , x n 1 ) x 2 = ( x 12 , x 22 , , x n 2 ) x d = ( x 1 d , x 2 d , , x n d )

    The introduced procedure for obtaining BMEWMA control limits is as follows:

    1. Combine the sample sizes x 1 , x 2 , , x d of the sets of observation such that: x = ( x 11 , x 21 , , x n 1 ; x 12 , x 22 , , x n 2 ; ; x 1 d , x 2 d , , x n d )

    2. Draw a bootstrap sample of size x * = x 1 * , x 2 * , , x d * with replacement from Step (i) x * = x 11 * , x 21 * , , x n 1 * ; x 12 * , x 22 * , , x n 2 * ; ; x 1 d * , x 2 d * , , x n d *

    3. Repeat Step (ii) a large number of times to obtain bootstrap replications as: x * = x 11 * ( i ) , x 21 * ( i ) , , x n 1 * ( i ) ; x 12 * ( i ) , x 22 * ( i ) , , x n 2 * ( i ) ; ; x 1 d * ( i ) , x 2 d * ( i ) , , x n d * ( i ) where ( i * = 1 , 2 , , B ) , and B = 1000, 2000, 3000, …,10000

    4. Estimate the bootstrap replication mean vector ( x ¯ * ) , bootstrap replication variance and covariance matrix (S*) of the bootstrap sample variables in Step (iii) such that:

      x ¯ * = 1 B i = 1 B ( x 1 i * x 2 i * x d i * ) = ( x ¯ 1 * x ¯ 2 * x ¯ d * ) and S * = ( s 11 * s 12 * s 1 d * s 21 * s 22 * s 2 d * s B 1 * s B 2 * s B d * )

      where s 11 * = 1 B 1 ( x 1 i * 2 ( x 1 i * ) 2 B ) ,   s 22 * = 1 B 1 ( x 2 i * 2 ( x 2 i * ) 2 B ) , , s B d * = 1 B 1 ( x k i * 2 ( x k i * ) 2 B ) ; ( B = d )

      is the variance of the bootstrap samples replicated

      s 12 * = 1 B 1 ( x 1 i * x 2 i * x 1 i * x 2 i * B ) , s 13 * = 1 B 1 ( x 1 i * x 3 i * x 1 i * x 3 i * B ) s 23 * = 1 B 1 ( x 2 i * x 3 i * x 2 i * x 3 i * B ) . s B d * = 1 B 1 ( x B i * x d i * x B i * x d i * B )

      Bd is the covariance matrix of the bootstrap sample replicated.

    5. Obtain BMEWMA ( T i 2 * ) statistics from the dataset in Step (iv) as

      T i 2 * = Z i * ' Z i * 1 Z i *
      (8)

      where

      Z i * = λ x 1 * + ( 1 λ ) Z i 1 *
      (9)

      Z i * = λ 2 λ [ 1 ( 1 λ ) 2 i ] * λ = 0.1 , 0.2 , , 0.9 , 1.0
      (10)

      For instance, i = 1; Equation (10) becomes

      Z 1 * = λ 2 λ [ 1 ( 1 λ ) 2 ( 1 ) ] *
      (11)

      meaning a scalar multiplying variance covariance matrix (Σ*), and Equation (9) becomes

      Z 1 * = λ x 1 * + ( 1 λ ) Z 1 1 * = λ ( x ¯ 11 * , x ¯ 12 * , , x ¯ 1 p * ) + ( 1 λ ) Z 0 * Z 0 * = 0 ,

      and

      Z 1 * = λ ( x ¯ 1 * , x ¯ 2 * , , x ¯ p * )
      (12)

      Substitute the values of Z 1 * & z 1 * in Equations (12) and (11) into Z 1 * ' Z 1 * 1 Z 1 * in Equation (8) to obtain T 1 2 * as a scalar.

      For i=2; Equation (10) becomes

      Z 2 * = λ 2 λ [ 1 ( 1 λ ) 2 ( 2 ) ] *
      (13)

      meaning a scalar multiplying variance covariance matrix (Σ*), and Equation (9) becomes

      Z 2 * = λ x 2 * + ( 1 λ ) Z 2 1 * = λ ( x ¯ 21 * , x ¯ 22 * , , x ¯ 2 p * ) + ( 1 λ ) Z 1 *
      (14)

      Substitute the values of Z 2 * & Z 2 * in Equations (14) and (13) into Z 2 * ' Z 2 * 1 Z 2 * in Equation (8) to obtain T 2 2 * as a scalar.

      For i = B; Equation (10) becomes

      Z B * = λ 2 λ [ 1 ( 1 λ ) 2 ( B ) ] *
      (15)

      meaning a scalar multiplying variance covariance matrix (Σ*), and Equation (9) becomes

      Z B * = λ x B * + ( 1 λ ) Z B 1 * = λ ( x ¯ 21 * , x ¯ 22 * , , x ¯ 2 p * ) + ( 1 λ ) Z B 1 *
      (16)

      Substitute the values of Z B * & Z B * in Equations (16) and (15) into Z B * ' Z B * 1 Z B * in Equation (8) to obtain T B 2 * as a scalar.

    6. For B = 10000, repeat the processes in Step (v) 10000 times by changing the values of Z B & Z B appropriately to obtain T 1 2 * , T 2 2 * , , T B 2 * .

    7. Set the upper control limit such that in each of the bootstrap statistic 2* 2* 2* (T1 , T2 ,…, TB ) is arranged from the lowest to highest figure, determine the position of B(1−α )th value such that:

      B M E W M A C L = # { ( T 1 2 * , T 2 2 * , , T B 2 * ) B ( 1 α ) }
      (17)

    8. From the control limit established in Step (vii), determine those variables that are under control process from those that are out of control process. That is, if any variable is greater than BMEWMACL, pronounce that particular variable as out of control

    3. APPLICATION TO NUMERICAL EXAMPLE

    The data used in this study is on Detergent Production Processing Company adapted from Oyeyemi (2011). Four quality variables (   X 1 , X 2 , X 3 a n d X 4 ) representing active detergent, moisture content, bulk density and ph level respectively, under which thirty five samples were recorded at different time periods as summarized in Appendix I. Adopting BMEWMA procedure which was translated to visual basic code (see Appendix II), bootstrap samples are replicated 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, and 10000 times from the original data set. Their bootstrap multivariate exponential weighted moving average control limits (BMEWMACL) are computed such that the false alarm rate is fixed to values of α = 0.01, 0.025, 0.05, 0.1, 0.2, 0.25 and smoothening parameters λ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 as shown in Tables 1-10 and Figures 110 respectively.Tables 2, 3, 4, 5, 6, 7, 8, 9, Fig. 2, 3, 4, 5, 6, 7, 8, 9

    3.1 Discussion and Interpretation of Results

    Tables 110 summarise the control limits obtained for BMEWMA at chosen values of α = 0.01, 0.025, 0.05, 0.1, 0.2, 0.25 and smoothening parameters λ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. An evaluation was made for the bootstrap samples replicated 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, and 10000 times. A critical look at Tables 1-10 shows that as the value of α increases, the control limits obtained for each of the smoothening parameter (λ) decreases as reflected in Figures 110. From the figures, a curve slopping from left to right indicates the ability of the BMEWMA to detect large, moderate and small out-of-control signals in any given process. Comparative analyses of control limits within the smoothing parameters (λ) indicate no difference as observed in the figures (smoothening curves). This further corroborates the fact that standard error does not influence the control limits as bootstrap replications increases. Rather, an increase in the sizes of bootstrap replications from 1000 to 10000 as shown in Tables 410 and Figures 410 indicates consistency and reliability in the BMEWMA method.

    In Table 11, the column denoted by h shows the results of MEWMA control limit (h) obtained from Prabhu and Runger (1997) as the quality characteristics p = 4, when it is assumed that there is no shift in the process, and the desired in-control average run length (ARL0) is 200 while columns denoted by B = 1000 to 10000 represent the control limits obtained from the proposed BMEWMA replicated from the given data in appendix I. A common feature in both methods is that the quality characteristics is equal to four (p = 4) for a selected smoothing parameter (λ). However, the false alarm rate value (α) for the BMEWMA was assumed to be 0.002, which is approximately zero (0). A comparative analysis of results of control limits from Table 11 shows that the two methods are not quite different from each other in their performance.

    4. CONCLUSION

    This study has critically looked at the BMWEMA approach of setting control limits whether or not the essential distribution is identified, and the underlying assumption of multivariate normality is fulfilled. By means of an experimental illustration, the BMWEMA results obtained in this study at different alpha (α) levels and smoothening parameters (λ) have the ability to detect small, moderate and large shifts in any production process. In general, control limits decrease with the increase in alpha (α) levels as shown in Tables (1) to (10) and Figures (1) to (10), for the fact that it is reasonably easier to detect a larger shift than a smaller shift. In conclusion, the BMWEMA control limits attained in this study will help quality administrators to make decisions that will enhance their production, especially in being able to identify both small and large shifts in production process.

    <Appendix II> Multivariate Bootstrap Control System

    Splash Screen Form Code

    Private Sub tmrDisplay_Timer()

    Unload Me

    mdiMain.Show

    End Sub

    Main Form Code

    Private Sub mnuAboutSoftware_Click()

    frmAbout.Show

    Private Sub mnuBootstrap_Click()

    frmBootstrap.Show

    End Sub

    Private Sub mnuBootstrapModifiedMEWMA_Click()

    frmModifiedMEWMABootstrap.Show

    End Sub

    Private Sub mnuBootstrapRealMEWMA_Click()

    frmMEWMABootstrap.Show

    End Sub

    Private Sub mnuMEWMABootstrapClearAll_Click()

    On Error Resume Next

    If formMEWMABootstrap = True Then DataEnv.ClearMEWMABootstrap

    If formModifiedMEWMABootstrap = True Then DataEnv.ClearModifiedMEWMABootstrap

    DataEnv.rsMEWMABootstrap.Requery

    DataEnv.rsModifiedMEWMABootstrap.Requery

    End Sub

    Private Sub mnuMEWMAClearAll_Click()

    On Error Resume Next

    If formRealMEWMA = True Then DataEnv.ClearRealMEWMA

    If formModifiedMEWMA = True Then DataEnv.ClearModifiedMEWMA

    DataEnv.rsModifiedMEWMA.Requery

    DataEnv.rsRealMEWMA.Requery

    End Sub

    Private Sub mnuModifiedMEWMA_Click()

    frmModifiedMEWMA.Show

    End Sub

    Private Sub mnuRealMEWMA_Click()

    frmRealMEWMA.Show

    End Sub

    Private Sub mnuSaveChanges_Click()

    On Error Resume Next

    With DataEnv

     .rsF_T.UpdateBatch adAffectAll

     .rsBoostrap.UpdateBatch adAffectAll

     .rsRealMEWMA.UpdateBatch adAffectAll

     .rsModifiedMEWMA.UpdateBatch adAffectAll

     .rsMEWMABootstrap.UpdateBatch adAffectAll

    End With

    End Sub

    Bootstrap MEWMA Form Code

    Dim nCount, Max_n, Max_b, iCount

    Dim genNum As Integer

    Dim mSum As Single, vSum As Single, Variance As Single, Mean As Single

    Dim covSum As Single, Covariance As Single, preZi As Single

    Dim MatrixLine(10, 10) As Single

    Dim MatrixInverse(10, 10) As Variant

    Option Explicit

    Private Sub cmdSOLVE_Click()

    On Error Resume Next

     Call ComputeMean

     Call ComputeVarCov

     Call Build_Matrix

     Call Calculate_Inverse

     Call Calculate_Transpose

     Call Type_Result

     DataEnv.rsMEWMABootstrap.UpdateBatch adAffectAll

    End Sub

    Private Sub DataGrid_Click()

    txtTotalRecord.Text = DataGrid.ApproxCount

    txtCurrentRecord.Text = DataGrid.Row + 1

    End Sub

    Private Sub DataGrid_KeyDown(KeyCode As Integer, Shift As Integer)

    txtTotalRecord.Text = DataGrid.ApproxCount

    txtCurrentRecord.Text = DataGrid.Row + 1

    End Sub

    Private Sub DataGrid_MouseDown(Button As Integer, Shift As Integer, X As Single, Y As Single)

    If Button = vbRightButton Then

     PopupMenu mnuGridMenu

    End If

    End Sub

    Private Sub Form_Load()

    Call getInfo

    formMEWMABootstrap = True

    mdiMain.mnuBootstrapRealMEWMA.Checked = True

    'Give numbers 1-10 to combo1

    For n = 1 To 10

     Combo1.AddItem n

    Next n

    'Default = 4 (Matrix dimensions 4X4)

    Combo1.Text = 4

    End Sub

    Private Sub Form_Resize()

    On Error Resume Next

    'resize datagrid

    If Me.Width - DataGrid.Width > 0 Then

     DataGrid.Width = Me.Width - 9000

     DataGrid.Height = Me.Height - 2000

    End If

    End Sub

    Private Sub Form_Unload(Cancel As Integer)

    formMEWMABootstrap = False

    mdiMain.mnuBootstrapRealMEWMA.Checked = False

    End Sub

    Global Variable Declaration Module

    'Define Global variables

    'Matrix dimensions are set to Max 10x10 for the interface needs, but can be increased here to whatever

    Global Const MAX_DIM = 10

    Global System_DIM As Integer 'Current Matrix [A] dimensions

    Global Matrix_A(1 To MAX_DIM, 1 To MAX_DIM)

    Global Matrix_MEWMA(1 To MAX_DIM, 1 To MAX_DIM)

    Global Operations_Matrix(1 To MAX_DIM, 1 To 2 * MAX_DIM) 'Matrix where the calculations are done

    Global Inverse_Matrix(1 To MAX_DIM, 1 To MAX_DIM) 'Matrix with the Inverse of [A]

    Global Solution_Problem As Boolean 'Determines whether the inverse was found or not

    Global Matrix_Mult(1 To MAX_DIM, 1 To MAX_DIM) 'Matrix with the product [A]*[A-1]=[I] (must be always equal to Singular matrix [I])

    Global temporary_1 As Variant, elem1 As Variant, multiplier_1 As Variant

    Global n As Integer, line_2 As Integer, k As Integer, m As Integer, L As Integer

    Global Vector_Multiple(10)

    Global result_1$, line_1$

    Global formFDistr As Boolean

    Global formBoostrap As Boolean

    Global formRealMEWMA As Boolean

    Global formModifiedMEWMA As Boolean

    Global formMEWMABootstrap As Boolean

    Global formModifiedMEWMABootstrap As Boolean

    Global MEWMA_Vector(10)

    Global strResult As String

    Global T_2 As Variant

    Global Diff_Mean(10), Diff_Mean_Transpose(10)

    Figure

    IEMS-18-3-315_F1.gif

    BMEWMACL when B = 1000.

    IEMS-18-3-315_F2.gif

    BMEWMACL when B = 2000.

    IEMS-18-3-315_F3.gif

    BMEWMACL when B = 3000.

    IEMS-18-3-315_F4.gif

    BMEWMACL when B = 4000.

    IEMS-18-3-315_F5.gif

    BMEWMACL when B = 5000.

    IEMS-18-3-315_F6.gif

    BMEWMACL when B = 6000.

    IEMS-18-3-315_F7.gif

    BMEWMACL when B = 7000.

    IEMS-18-3-315_F8.gif

    BMEWMACL when B = 8000.

    IEMS-18-3-315_F9.gif

    BMEWMACL when B = 9000.

    IEMS-18-3-315_F10.gif

    BMEWMACL when B = 10000.

    Table

    Data on Detergent Production Processing, Oyeyemi (2011)

    BMEWMACL for bootstrap samples replicated 1000 times

    BMEWMACL for bootstrap samples replicated 2000 times

    BMEWMACL for bootstrap samples replicated 3000 times

    BMEWMACL for bootstrap samples replicated 4000 times

    BMEWMACL for bootstrap samples replicated 5000 times

    BMEWMACL for bootstrap samples replicated 6000 times.

    BMEWMACL for bootstrap samples replicated 7000 times

    BMEWMACL for bootstrap samples replicated 8000 times

    BMEWMACL for bootstrap samples replicated 9000 times

    BMEWMACL for bootstrap samples replicated 10000 times

    MEWMA control limit (h) from Prabhu and Runger (1997) and BMEWMACL

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