Process or Product Monitoring and Control_14 doc

The moving mean plot shows a large number of control points.. The moving range plot shows a large number of control points.. The mean control chart shows a large number of control poi

Trang 1

8 Generate a mean control chart

using lot-to-lot variation.

1 The moving mean plot shows

a large number of control points.

2 The moving range plot shows

3 The mean control chart shows

4 The sd control chart shows

no out-of-control points.

5 The mean control chart shows

6 The sd control chart shows

is 36% of the total.

8 The mean control chart shows one point that is on the boundary of being out of control.

Trang 2

6 Process or Product Monitoring and Control

6.6 Case Studies in Process Monitoring

6.6.2 Aerosol Particle Size

Trang 3

6.6.2 Aerosol Particle Size

6.6.2.1 Background and Data

Data Source The source of the data for this case study is Antuan Negiz who

analyzed these data while he was a post-doc in the NIST StatisticalEngineering Division from the Illinois Institute of Technology

transferred to the hot gas stream leaving behind dried small-sizedparticles

The response variable is particle size, which is collected equidistant intime There are a variety of associated variables that may affect theinjection process itself and hence the size and quality of the depositedparticles For this case study, we restrict our analysis to the responsevariable

Applications Such deposition process operations have many applications from

powdered laundry detergents at one extreme to ceramic molding at animportant other extreme In ceramic molding, the distribution andhomogeneity of the particle sizes are particularly important becauseafter the molds are baked and cured, the properties of the final moldedceramic product is strongly affected by the intermediate uniformity ofthe base ceramic particles, which in turn is directly reflective of thequality of the initial atomization process in the aerosol injection device

Trang 4

to determine a prediction model of particle size as a function oftime such a model is frequently autoregressive in nature Such ahigh-quality prediction equation would be essential as a first step indeveloping a predictor-corrective recursive feedback mechanism whichwould serve as the core in developing and implementing real-timedynamic corrective algorithms The net effect of such algorthms is, ofcourse, a particle size distribution that is much less variable, muchmore stable in nature, and of much higher quality All of this results infinal ceramic mold products that are more uniform and predictableacross a wide range of important performance characteristics

For the purposes of this case study, we restrict the analysis todetermining an appropriate Box-Jenkins model of the particle size

Case study

114.63150114.63150116.09940116.34400116.09940116.34400116.83331116.34400116.83331117.32260117.07800117.32260117.32260117.81200117.56730118.30130117.81200118.30130117.81200118.30130118.30130118.54590118.30130117.07800116.09940

Trang 5

118.30130118.79060118.05661118.30130118.54590118.30130118.54590118.05661118.30130118.54590118.30130118.30130118.30130118.30130118.05661118.30130117.81200118.30130117.32260117.32260117.56730117.81200117.56730117.81200117.81200117.32260116.34400116.58870116.83331116.58870116.83331116.83331117.32260116.34400116.09940115.61010115.61010115.61010115.36539115.12080115.61010115.85471115.36539115.36539115.36539115.12080

Trang 6

114.87611114.87611115.12080114.87611114.87611114.63150114.63150114.14220114.38680114.14220114.63150114.87611114.38680114.87611114.63150114.14220114.14220113.89750114.14220113.89750113.65289113.65289113.40820113.40820112.91890113.40820112.91890113.40820113.89750113.40820113.65289113.89750113.65289113.65289113.89750113.65289113.16360114.14220114.38680113.65289113.89750113.89750113.40820113.65289113.89750113.65289

Trang 7

113.65289114.14220114.38680114.63150115.61010115.12080114.63150114.38680113.65289113.40820113.40820113.16360113.16360113.16360113.16360113.16360112.42960113.40820113.40820113.16360113.16360113.16360113.16360111.20631112.67420112.91890112.67420112.91890113.16360112.91890112.67420112.91890112.67420112.91890113.16360112.67420112.67420112.91890113.16360112.67420112.91890111.20631113.40820112.91890112.67420113.16360

Trang 8

113.65289113.40820114.14220114.87611114.87611116.09940116.34400116.58870116.09940116.34400116.83331117.07800117.07800116.58870116.83331116.58870116.34400116.83331116.83331117.07800116.58870116.58870117.32260116.83331118.79060116.83331117.07800116.58870116.83331116.34400116.58870116.34400116.34400116.34400116.09940116.09940116.34400115.85471115.85471115.85471115.61010115.61010115.61010115.36539115.12080115.61010

Trang 9

115.85471115.12080115.12080114.87611114.87611114.38680114.14220114.14220114.38680114.14220114.38680114.38680114.38680114.38680114.38680114.14220113.89750114.14220113.65289113.16360112.91890112.67420112.42960112.42960112.42960112.18491112.18491112.42960112.18491112.42960111.69560112.42960112.42960111.69560111.94030112.18491112.18491112.18491111.94030111.69560111.94030111.94030112.42960112.18491112.18491111.94030

Trang 10

112.18491112.18491111.20631111.69560111.69560111.69560111.94030111.94030112.18491111.69560112.18491111.94030111.69560112.18491110.96170111.69560111.20631111.20631111.45100110.22771109.98310110.22771110.71700110.22771111.20631111.45100111.69560112.18491112.18491112.18491112.42960112.67420112.18491112.42960112.18491112.91890112.18491112.42960111.20631112.42960112.42960112.42960112.42960113.16360112.18491112.91890

Trang 11

112.91890112.67420112.42960112.42960112.42960112.91890113.16360112.67420113.16360112.91890112.42960112.67420112.91890112.18491112.91890113.16360112.91890112.91890112.91890112.67420112.42960112.42960113.16360112.91890112.67420113.16360112.91890113.16360112.91890112.67420112.91890112.67420112.91890112.91890112.91890113.16360112.91890112.91890112.18491112.42960112.42960112.18491112.91890112.67420112.42960112.42960

Trang 12

112.18491112.42960112.67420112.42960112.42960112.18491112.67420112.42960112.42960112.67420112.42960112.42960112.42960112.67420112.91890113.40820113.40820113.40820112.91890112.67420112.67420112.91890113.65289113.89750114.38680114.87611114.87611115.12080115.61010115.36539115.61010115.85471116.09940116.83331116.34400116.58870116.58870116.34400116.83331116.83331116.83331117.32260116.83331117.32260117.56730117.32260

Trang 13

117.07800117.32260117.81200117.81200117.81200118.54590118.05661118.05661117.56730117.32260117.81200118.30130118.05661118.54590118.05661118.30130118.05661118.30130118.30130118.30130118.05661117.81200117.32260118.30130118.30130117.81200117.07800118.05661117.81200117.56730117.32260117.32260117.81200117.32260117.81200117.07800117.32260116.83331117.07800116.83331116.83331117.07800115.12080116.58870116.58870116.34400

Trang 14

115.85471116.34400116.34400115.85471116.58870116.34400115.61010115.85471115.61010115.85471115.12080115.61010115.61010115.85471115.61010115.36539114.87611114.87611114.63150114.87611115.12080114.63150114.87611115.12080114.63150114.38680114.38680114.87611114.63150114.63150114.63150114.63150114.63150114.14220113.65289113.65289113.89750113.65289113.40820113.40820113.89750113.89750113.89750113.65289113.65289113.89750

Trang 15

113.40820113.40820113.65289113.89750113.89750114.14220113.65289113.40820113.40820113.65289113.40820114.14220113.89750114.14220113.65289113.65289113.65289113.89750113.16360113.16360113.89750113.65289113.16360113.65289113.40820112.91890113.16360113.16360113.40820113.40820113.65289113.16360113.40820113.16360113.16360112.91890112.91890112.91890113.65289113.65289113.16360112.91890112.67420113.16360112.91890112.67420

Trang 16

112.91890112.91890112.91890111.20631112.91890113.16360112.42960112.67420113.16360112.42960112.67420112.91890112.67420111.20631112.42960112.67420112.42960113.16360112.91890112.67420112.91890112.42960112.67420112.18491112.91890112.42960112.18491

Trang 17

Non-stationarity can often be removed by differencing the data orfitting some type of trend curve We would then attempt to fit aBox-Jenkins model to the differenced data or to the residuals afterfitting a trend curve

Although Box-Jenkins models can estimate seasonal components, theanalyst needs to specify the seasonal period (for example, 12 formonthly data) Seasonal components are common for economic timeseries They are less common for engineering and scientific data

Run Sequence

Plot

Trang 18

of the Run

Sequence Plot

We can make the following conclusions from the run sequence plot

The data show strong and positive autocorrelation

Trang 19

The next step is to examine the sample autocorrelations of thedifferenced data.

Autocorrelation

Plot of the

Differenced

Data

Trang 20

The autocorrelation plot of the differenced data with a 95%

confidence band shows that only the autocorrelation at lag 1 issignificant The autocorrelation plot together with run sequence ofthe differenced data suggest that the differenced data are stationary.Based on the autocorrelation plot, an MA(1) model is suggested forthe differenced data

To examine other possible models, we produce the partialautocorrelation plot of the differenced data

Trang 21

Note that whatever method is used for model identification, modeldiagnostics should be performed on the selected model.

Trang 22

# NONLINEAR LEAST SQUARES ESTIMATION FOR THE PARAMETERS OF # # AN ARIMA MODEL USING BACKFORECASTS # #############################################################

SUMMARY OF INITIAL CONDITIONS -

MODEL SPECIFICATION

FACTOR (P D Q) S

1 2 1 0 1

DEFAULT SCALING USED FOR ALL PARAMETERS.

##STEP SIZE FOR

######PARAMETER

##APPROXIMATING #################PARAMETER DESCRIPTION STARTING VALUES

#####DERIVATIVE INDEX #########TYPE ##ORDER ##FIXED ##########(PAR)

##########(STP)

1 AR (FACTOR 1) 1 NO 0.10000000E+00 0.77167549E-06

2 AR (FACTOR 1) 2 NO 0.10000000E+00 0.77168311E-06

3 MU ### NO 0.00000000E+00 0.80630875E-06

NUMBER OF OBSERVATIONS (N) 559 MAXIMUM NUMBER OF ITERATIONS ALLOWED (MIT) 500

MAXIMUM NUMBER OF MODEL SUBROUTINE CALLS ALLOWED 1000

CONVERGENCE CRITERION FOR TEST BASED ON THE FORECASTED RELATIVE CHANGE IN RESIDUAL SUM OF SQUARES (STOPSS)

Định dạng
Số trang	22
Dung lượng	1,11 MB