Exponentially Weighted Moving Average (EWMA) Charts

Transcription

1 Chapter 249 Exponentally Weghted Movng Average (EWMA) Charts Introducton Ths procedure generates exponentally weghted movng average (EWMA) control charts for varables. Charts for the mean and for the varablty can be produced. The format of the control charts s fully customzable. The data for the subgroups can be n a sngle column or n multple columns. Ths procedure permts the defnng of stages. The target value can be entered drectly or estmated from the data, or a sub-set of the data. Sgma may be estmated from the data or a standard sgma value may be entered. Means and ranges/standard devatons can be stored to the spreadsheet. Exponentally Weghted Movng Average Control Charts Smlarly to the CUSUM chart, the EWMA chart s useful n detectng small shfts n the process mean. These charts are used to montor the mean of a process based on samples taen from the process at gven tmes (hours, shfts, days, wees, months, etc.). The measurements of the samples at a gven tme consttute a subgroup. The EWMA chart reles on the specfcaton of a target value and a nown or relable estmate of the standard devaton. For ths reason, the movng average chart s better used after process control has been establshed

2 Other Control Charts for the Mean and Varaton of a Process Common control charts for detectng large process changes, or for establshng ntal control of the process are the X-bar and R charts (or X-bar and s charts). Once a relable estmate of the mean and standard devaton s avalable, the EWMA and CUSUM charts are useful n detectng smaller shfts n the process mean. The EWMA chart may also be used when only a sngle response s avalable at each tme pont. Another opton for sngle responses are the ndvduals and movng range (I-MR) control charts. Control Chart Formulas Suppose we have subgroups, each of sze n. Let x j represent the measurement n the j th sample of the th subgroup. Formulas for the Ponts on the Chart The th subgroup mean s calculated usng xj j= x = 1 n The ponts of the chart are obtaned from the x s by the followng exponental smoothng operaton n ( 1 λ) z 1 z = λ x + The value of z 0 s set to the target mean. The value of λ s specfed by the user. Values for the ponts of the EWMA varaton chart are gven by substtutng x wth the range or standard devaton. Estmatng the EWMA Chart Center Lne (Grand Mean) In the EWMA Charts procedure, the target mean may be nput drectly, or t may be estmated from a seres of subgroups. If t s estmated from the subgroups the formula for the grand average s x n = 1 j= 1 = = 1 n x j. If the subgroups are of equal sze, the above equaton for the grand mean reduces to x = x = 1 x1 + x2 + + x =

3 Estmatng Sgma Sample Ranges Ether the range or the standard devaton of the subgroups may be used to estmate sgma, or a nown (standard) sgma value may be entered drectly. If the standard devaton (sgma) s to be estmated from the ranges, t s estmated as where R R = = 1 ( R) ˆ σ = E µ R d 2 = = σ σ The calculaton of E(R) requres the nowledge of the underlyng dstrbuton of the x j s. Mang the assumpton that the x j s follow the normal dstrbuton wth constant mean and varance, the values for d 2 are derved through the use of numercal ntegraton. It s mportant to note that the normalty assumpton s used and that the accuracy of ths estmate requres that ths assumpton be vald. When n s one, we cannot calculate R snce t requres at least two measurements. The procedure n ths case s to use the ranges of successve pars of observatons. Hence, the range of the frst and second observaton s computed, the range of the second and thrd s computed, and so on. The average of these approxmate ranges s used to estmate σ. R d 2 Estmatng Sgma Sample Standard Devatons If the standard devaton (sgma) s to be estmated from the standard devatons, t s estmated as where s s = = 1 E( s) µ s c 4 = = σ σ ˆ σ = s c 4 The calculaton of E(s) requres the nowledge of the underlyng dstrbuton of the x j s. Mang the assumpton that the x j s follow the normal dstrbuton wth constant mean and varance, the values for c 4 are obtaned from n Γ 2 2 c 4 = n 1 n 1 Γ

4 Estmatng Sgma Weghted Approach When the sample sze s varable across subgroups, a weghted approach s recommended for estmatng sgma (Montgomery, 2013): ˆ σ = s = = 1 ( n = 1 1) s n 2 1/ 2 EWMA Chart Lmts The lower and upper control lmts for the EWMA chart are calculated usng the formulas 2 [ 1 ( λ) ] ˆ σ λ LCL = µ 0 m 1 n 2 λ 2 [ 1 ( λ) ] ˆ σ λ UCL = µ 0 + m 1 n 2 λ where m s a multpler (often set to 2.7 or 3), and λ s the exponental smoothng constant. Note that the values of lmts change wth each successve subgroup, but tend to level off at around the tenth subgroup. Values for the lmts of the EWMA varaton chart are gven by substtutng 0 µ wth the range or standard devaton center lne value (such as R-bar or s-bar)

5 Data Structure In ths procedure, the data may be n ether of two formats. The frst data structure opton s to have the data n several columns, wth one subgroup per row. Example dataset S1 S2 S3 S4 S The second data structure opton uses one column for the response data, and ether a subgroup sze or a second column defnng the subgroups. Alternatve example dataset Response Subgroup In the alternatve example dataset, the Subgroup column s not needed f every subgroup s of sze 5 and the user specfes 5 as the subgroup sze. If there are mssng values, the Subgroup column should be used, or the structure of the frst example dataset

6 Procedure Optons Ths secton descrbes the optons avalable n ths procedure. To fnd out more about usng a procedure, go to the Procedures chapter. Varables Tab Ths panel specfes the varables that wll be used n the analyss. Input Type Specfy whether the data s n a sngle response column or n multple columns wth one subgroup per row. Response Column and Subgroup Column or Subgroup Sze Response Subgroup Multple Columns wth One Subgroup Per Row X1 X2 X Varables Response Column Response Varable Specfy the column wth the data values. The data values are separated nto subgroups below usng the Subgroup Specfcaton optons. Subgroup Specfcaton Specfy whether subgroups are defned by a Subgroup ID varable, or by a subgroup sze. If the subgroup sze s 3, then subgroups are formed by gong down the response column n groups of 3. The frst subgroup would be 5, 6, 4; the second would be 3, 7, 6; and so on

7 Subgroup ID Varable Specfy the column contanng the subgroup dentfers. Response ID Varable A new subgroup s created for each change n the Subgroup ID Varable, gong down. Subgroup Sze Specfy the number of ndvduals n each subgroup. Response If the subgroup sze s 3, then subgroups are formed by gong down the response column n groups of 3. The frst subgroup would be 5, 6, 4; the second would be 3, 7, 6; and so on. Varables Multple Columns Data Varables Specfy the columns contanng the sample responses. Each row represents a subgroup. X1 X2 X

8 If only one varable s specfed, NCSS automatcally generates an ndvduals chart wth a movng-range of sze 2. Stages Number of Stages Specfy whether the analyses and charts are to be produced based on a sngle set of subgroups, or a seres of sets of subgroups. Typcally a sngle stage s used unless you wsh to have separate estmaton for varous segments of the subgroups. When usng multple stages, the software requres that there be at least one subgroup n every stage specfed. Stage Specfcaton Specfy whether the varous stages wll be defned usng a varable (column) wth a unque value for each stage, or by enterng a range of rows for each stage. Stage Varable X1 X2 X3 Stage Enter a range for each stage 1-50, , Ths would produce three stages. The frst stage would be made up of rows 1 to 50, the second stage would be rows 51 to 100, and the thrd stage would be rows 101 to 150. Stage Varable Specfy the varable (column) that contans the dentfers for each stage. X1 X2 X3 Stage Varable A new stage s created for each change n the Stage Varable, gong down

9 Stage Ranges Enter a range for each stage usng a dash. Separate each stage wth a comma. Example: 1-50, , Ths would produce three stages. The frst stage would be made up of rows 1 to 50, the second stage would be rows 51 to 100, and the thrd stage would be rows 101 to 150. Specfy Rows n Calculatons Specfcaton Method Select whch method wll be used to specfy the rows of the data to be used to form subgroups. All Rows All rows n the response column(s) wll be used. Enter Frst Row and Last Row Specfy the frst row and the last row of the data for use n calculatons. Frst N Rows (Enter N) The data begnnng at Row 1 and endng at Row N wll be used n calculatons. Last N Rows (Enter N) Subgroups wll be formed from the last N rows of the dataset. Keep Rows Varable Specfy a varable and a value n that varable column that wll be used to determne whch rows are used to form the subgroups. Remove Rows Varable Specfy a varable and a value n that varable column that wll be used to determne whch rows wll not be used to form the subgroups. Frst Row Specfy the begnnng row to be used for the frst subgroup. Last Row Specfy the last row to be used for the last subgroup. N Enter the number of rows to be used n formng subgroups. Keep Rows Varable Ths varable (column) s used to specfy whch rows of the data wll be used to form the subgroups for the calculatons. Keep Rows Value Ths value determnes whch rows of the Keep Rows Varable wll be used n the calculaton porton of the analyss

10 Remove Rows Varable Ths varable (column) s used to specfy whch rows of the data wll not be used to form the subgroups for the calculatons. Remove Rows Value Ths value determnes whch rows of the Remove Rows Varable wll not be used n the calculaton porton of the analyss. Specfy Rows n Charts Specfcaton Method Select whch method wll be used to specfy the rows of the data to be used to form subgroups for the charts. All Rows All rows n the response column(s) wll be used. Enter Frst Row and Last Row Specfy the frst row and the last row of the data for use n the plots. Frst N Rows (Enter N) The data begnnng at Row 1 and endng at Row N wll be used n the plots. Last N Rows (Enter N) Subgroups wll be formed from the last N rows of the dataset. Keep Rows Varable Specfy a varable and a value n that varable column that wll be used to determne whch rows are used to form the subgroups. Remove Rows Varable Specfy a varable and a value n that varable column that wll be used to determne whch rows wll not be used to form the subgroups. Frst Row Specfy the begnnng row to be used for the frst subgroup. Last Row Specfy the last row to be used for the last subgroup. N Enter the number of rows to be used n formng subgroups. Keep Rows Varable Ths varable (column) s used to specfy whch rows of the data wll be used to form the subgroups for the plots. Keep Rows Value Ths value determnes whch rows of the Keep Rows Varable wll be used n the plots. Remove Rows Varable Ths varable (column) s used to specfy whch rows of the data wll not be used to form the subgroups for the plots

11 Remove Rows Value Ths value determnes whch rows of the Remove Rows Varable wll not be used n the plots. Labels (Optonal) Subgroup Label Varable Specfy a varable (column) that contans the desred x axs (subgroup) labels for plots. If left blan, the plot wll use the subgroup number. The format of the labels s controlled on the x axs tab of the plot format wndow. Pont Label Varable Specfy a varable (column) that contans the desred ndvdual pont labels for plots. If left blan, no pont labels are shown. The format of the labels s controlled on the man chart tab of the plot format wndow. Target & Sgma Tab The optons on the Target & Sgma tab are used to specfy the target value and sgma. Target Value Optons Target Value Specfcaton Specfy whether the target value s estmated from the data, or whether t wll be specfed drectly. From Rows n Calculatons Data Estmate the target value from the subgroups specfed for calculatons. Enter Target Value(s) Specfy the target value drectly. If multple stages are used, separate the target value for each stage by spaces. Use a Varable wth Target Value(s) Specfy a column contanng the target value n row 1. If multple stages are used, a target value should be entered n a separate cell for each stage, begnnng wth row 1 for the frst stage. Target Value(s) Enter the target value to be used. If multple stages are used, separate the target values for each stage by spaces. Target Value(s) Varable Specfy a column contanng the target value n row 1. If multple stages are used, a target value should be entered n a separate cell for each stage, begnnng wth row 1 for the frst stage. Lmt Multplers Prmary Lmt Multpler Ths opton specfes the multpler of sgma for the prmary control lmts. For the well-nown 3-sgma lmts, the multpler s set to 3. Specfcaton Lmts Lower Lmt Enter an optonal lower specfcaton lmt for dsplay on the chart. These lmts are not the control lmts

12 Upper Lmt Enter an optonal upper specfcaton lmt for dsplay on the chart. These lmts are not the control lmts. Spec Value Enter an optonal specfcaton value for dsplay on the chart. Sgma Estmaton Optons Range or SD Estmaton Specfy whether the estmaton of sgma wll be based on the average range or the average standard devaton. Range The average range wll be used to estmate sgma. SD The average standard devaton wll be used to estmate sgma. Sgma Specfcaton Specfy the method by whch Sgma wll be estmated for use n the charts. From Data R-bar or s-bar Estmate Estmate sgma based on the average of the ranges or standard devatons (whchever s specfed under Range or SD Estmaton). Only the subgroups specfed for use n calculatons wll be used. From Data Weghted Approach Estmate Ths method estmates s-bar usng a specal formula that s recommended when the subgroup sze vares across subgroups. Ths opton should only be used when the Range or SD Estmaton s set to SD. Only the subgroups specfed for use n calculatons wll be used. Enter Standard Sgma Value(s) In ths case the sgma value s entered drectly. The correspondng R-bar or s-bar value s bac-calculated from ths value. If multple stages are used, separate the sgma values for each stage by spaces. Use a Varable wth Standard Sgma Value(s) Specfy a column contanng the standard sgma value n row 1. The correspondng R-bar or s-bar value s bac-calculated from ths value. If multple stages are used, a sgma value should be entered n a separate cell for each stage, begnnng wth row 1 for the frst stage. Sgma Value(s) Enter the value to be used for the standard sgma. If multple stages are used, separate the sgma values for each stage by spaces. Sgma Varable Specfy a column contanng the standard sgma value n row 1. The correspondng R-bar or s-bar value for varaton plots s bac-calculated from ths value. If multple stages are used, a sgma value should be entered n a separate cell for each stage, begnnng wth row 1 for the frst stage

13 Reports Tab The followng optons control the format of the reports. Specfy Reports Target Value and Sgma Summary Secton Ths report gves the numerc values of the target value and sgma, as well as the sgma estmaton. Report Optons Precson Specfy the precson of numbers n the report. A sngle-precson number wll show seven-place accuracy, whle a double-precson number wll show thrteen-place accuracy. Note that the reports are formatted for sngle precson. If you select double precson, some numbers may run nto others. Also note that all calculatons are performed n double precson regardless of whch opton you select here. Ths s for reportng purposes only. Varable Names Ths opton lets you select whether to dsplay varable names, varable labels, or both. Page Ttle Ths opton specfes a ttle to appear at the top of each page. Plot Subttle Ths opton specfes a subttle to appear at the top of each plot. EWMA Charts Tab Ths panel sets the optons used to defne the appearance of the two EWMA charts. Select Plots EWMA Chart, EWMA Varablty Chart These charts are controlled by three form objects: 1. A checbox to ndcate whether the chart s dsplayed. 2. A format button used to call up the plot format wndow (see Qualty Control Chart Format Optons below for more formattng detals). 3. A second checbox used to ndcate whether the chart can be edted durng the run. EWMA Parameter Ths specfes the value of the smoothng parameter (λ) n the EWMA charts. The range for ths parameter s 0 to 1 (not ncludng 0). Typcally, a value between 0.05 and 0.25 wors well n practce. Popular choces nclude 0.05, 0.10, and

14 Storage Tab The optons on ths panel control the automatc storage of the means on the current dataset. Storage Columns Store Means or Ranges/SDs n Column You can automatcally store the means, ranges, or standard devatons of each subgroup nto the column specfed here. Whether the range or standard devaton s stored depends on the choce of Range or SD Estmaton on the Target & Sgma tab. Warnng: Any data already n ths column s replaced. Be careful not to specfy columns that contan mportant data. Qualty Control Chart Format Wndow Optons Ths secton descrbes a few of the specfc optons avalable on the frst tab of the control chart format wndow, whch s dsplayed when a qualty control chart format button s pressed. Common optons, such as axes, labels, legends, and ttles are documented n the Graphcs Components chapter

15 [Xbar] / [Range] Chart Tab Symbols Secton You can modfy the attrbutes of the symbols usng the optons n ths secton. A wde varety of szes, shapes, and colors are avalable for the symbols. The symbols for n-control and out-ofcontrol ponts are specfed ndependently. There are addtonal optons to label out-of-control ponts. The label for ponts outsde the prmary control lmts s the subgroup number. The label for ponts that are out-of-control based on the runs test s the number of the frst runs test that s sgnaled by ths pont. The user may also specfy a column of pont labels on the procedure varables tab, to be used to label all or some of the ponts of the chart. The raw data may also be shown, based on customzable raw data symbols. Lnes Secton You can specfy the format of the varous lnes usng the optons n ths secton. Note that when shadng s desred, the fll wll be to the bottom for sngle lnes (such as the mean lne), and between the lnes for pars of lnes (such as prmary lmts). Lnes for the zones, secondary lmts, and specfcaton lmts are also specfed here

16 Ttles, Legend, Numerc Axs, Group Axs, Grd Lnes, and Bacground Tabs Detals on settng the optons n these tabs are gven n the Graphcs Components chapter. The legend does not show by default, but can easly be ncluded by gong t1o the Legend tab and clcng the Show Legend checbox. Example 1 EWMA Chart Ths secton presents an example of how to produce an EWMA chart. The data represent 50 subgroups of sze 5. The data used are n the QC dataset. We wll analyze the varables D1 through D5 of ths dataset. You may follow along here by mang the approprate entres or load the completed template Example 1 by clcng on Open Example Template from the Fle menu of the EWMA Charts wndow. 1 Open the QC dataset. From the Fle menu of the NCSS Data wndow, select Open Example Data. Clc on the fle QC.NCSS. Clc Open. 2 Open the EWMA Charts wndow. Usng the Analyss or Graphcs menu or the Procedure Navgator, fnd and select the EWMA Charts procedure. On the menus, select Fle, then New Template. Ths wll fll the procedure wth the default template. 3 Specfy the varables. On the EWMA Charts wndow, select the Varables tab. Double-clc n the Data Varables text box. Ths wll brng up the varable selecton wndow. Select D1 through D5 from the lst of varables and then clc O. D1-D5 wll appear n the Data Varables box. 4 Specfy the target value and sgma. On the Movng Average Charts wndow, select the Target & Sgma tab. Set Target Value Specfcaton to Enter Target Value(s). Set the Target Value to 67. Set Sgma Specfcaton to Enter Sgma Value(s). Set the Sgma Value to 8. 5 Run the procedure. From the Run menu, select Run Procedure. Alternatvely, just clc the green Run button. Target Value Secton Target Value Secton for Subgroups 1 to 50 Number of Subgroups 50 Target Value Specfcaton Value User-Specfed Target Value 67 Ths secton dsplays the target value that s used n the EWMA chart. Target Value Specfcaton Ths s the method by whch the target value s obtaned

17 Target Value Ths s value of the target value used to create the movng average chart. Sgma Specfcaton Secton Sgma Specfcaton Secton for Subgroups 1 to 50 Sgma User-Specfed Specfcaton Sgma Value User-Specfed 8 Ths secton shows the specfcaton of the standard devaton used n the EWMA chart. Sgma Specfcaton Ths s the method by whch the sgma value s obtaned. User-Specfed Sgma Value Ths s value of the sgma entered by the user, and s the value used to create the movng average chart. EWMA Chart Ths EWMA plot shows the progress of the weghted movng averages across the 50 subgroups. There does not appear to be an ndcaton of a change n the process mean

18 Example 2 EWMA Chart wth Addtonal Formattng Ths example uses the same setup as Example 1, except that a varety of mprovements are made n the plot format. These mprovements are made by clcng the EWMA Chart format button on the EWMA Chart tab. You can load the completed template Example 2 by clcng on Open Example Template from the Fle menu of the EWMA Charts wndow. EWMA Chart As shown here, a varety of enhancements can be made to the formattng of the control charts to mae the chart as easy to read as possble