trial lecture hobaek haff

8/11/2019 Trial Lecture Hobaek Haff

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Outline Motivation Univariate charts Multivariate charts Conclusions References

The use of multivariate control charts for qualitycontrol

Trial lecture

Ingrid Hobk Haff

Norsk Regnesentral/Norwegian Computing CenterStatistics for Innovation (sfi)2

[email protected]

September 10th, 2012

Ingrid Hobk Haff The use of multivariate control charts for quality control


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Motivation and main concepts

Univariate control charts

Multivariate control charts

Conclusions



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Motivation and main concepts



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How it all started

In the beginning of the last century, Bell Telephone

produced amplifiers and other equipment that had to be

buried underground. Reducing failures and repairs was

therefore very important. This was restricted to inspection

of finished products to remove defective items.

However, in 1924, Dr. Walter A. Shewhart suggested

monitoring the production quality by means of a graphical

tool, now known as a control chart. The idea was to detect

potential problems in the production process based on

statistical methods.


O li M i i U i i h M l i i h C l i R f


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Quality control

Quality control, also called quality improvement, is animportant field in the manufacturing sector.

It consists in monitoring certain quality characteristics of themanufactured products over time in order to

ensure that the quality of the products is stable identify problems in the production improve the quality of the products.

Statistical quality control is quality control using statistical

methods. Control charts are one of the main tools for statistical quality

control.


O tli M ti ti U i i t h t M lti i t h t C l i R f


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Quality control

A process is said to be in statistical control, or simplyin-control if the probability distribution of the qualitycharacteristics in question is stable over time.

Typically, this means that the mean value and variability ofthese characteristics are more or less constant.

Likewise, the process is out-of-control if the distribution haschanged.

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Control charts

A control chart is a time sequence plot of some measure of

quality, with added control or decision lines. The purpose is to determine whether the process in question

is in-control. The control lines are determined in such a way that

observations outside these limits suggest that the process isout-of-control. If some points fall outside the control limits, the process

should be scrutinised in order to detect the source(s) of thechange.

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Control charts

Although they were originally developed for industrial processes,control charts have been applied within a number of areas,including:

hospital infection control (Sellick, 1993)

prediction of business failures (Theodossiou, 1993)

monitoring the impact of human disturbance of ecologicalsystems (Anderson and Thompson, 2004)

quality management of higher education (Mergen et al., 2000)

corroborating bribery (Charnes and Gitlow, 1995)

improving athletic performance (Clark and Clark, 1997)




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Main concepts

A control chart must contain an upper control limit (UCL)and potentially a lower control limit (LCL).

The construction of control charts consists of two stages.

Phase I is a retrospective data analysis to assess whether theprocess has been in-control in the past.

Phase IIconsists in determining the control limits for futureobservations based on the past observations.

If the Phase I analysis indicates that the process has been

in-control, one may proceed directly to Phase II, and use allthe observed data.

Otherwise, one must try to detect the sources of the change.




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Main concepts

If these sources are identified and can be removed, the

out-of-control observations are discarded and the controllimits are adjusted accordingly in Phase II.

Phase I can be seen as a preprossessing step, whereas Phase IIis the analysis one actually is interested in.

In the remains of the lecture, I will focus on Phase II.

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Outline Moti ation Uni ariate charts Multi ariate charts Conclusions References

Main concepts

The average run length (ARL) is the expected time until apoint falls outside the control limit(s).

By design, the probability of observing a point outside thecontrol limit(s) is very low when the process is in-control.

However, as long as this probability is >0, that will happenfrom time to time.

The expected time until such a false alarm is called thein-control ARL.

The expected time until a true change in the process is

detected is called the parameter-change ARL.

These two ARLs constitute the ARL-properties of the chart.

In practice, it is impossible to distinguish between a false andreal alarm just by looking at the data.




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Main concepts

Ideally, one would like to minimise the parameter-change ARL(true parameter change), while maximising the in-control ARL(false alarm).

Unfortunately, a decrease in the parameter-change ARL will

usually entail a decrease in the in-control ARL.

Likewise, if one attempts to reduce the number of false alarmsby increasing the in-control ARL, the chart will generallybecome less sensitive to changes in the process.

The choice of control limits must therefore be a trade-offbetween these two concerns, and the optimal choice will besituation dependent.




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Univariate control charts




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Control charts for measurement data

Measurement data follow a continuous probability distribution.

They may be either product or process variables.

Let X1,X2, . . ., be the measurements at time = 1, 2, . . ..

These are assumed to be independent follow a normal distribution N(,

2).

The measurements are divided into m subgroups in time, for

instance weekly, and one computes the average xt for eachsubgroup, representing time t.




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X-chart

This is the most common univariate control chart.

The purpose of this chart is to detect changes in the mean ofthe quality characteristic.

The chart consists of: the averages xtplotted against time the middle line given by the overall average x the upper control limit UCL=x+LX the lower control limit LCL=xLX,

where X is an estimate of the variance ofXand typically

L= 3. Under the model assumptions, these limits constitute a

confidence interval for the in-control mean, with level 99.73%forL= 3.

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S-chart

This is a chart over the sample standard deviations stof the

subgroups. The purpose of this chart is to detect changes in the variance

of the quality characteristic. The chart consists of:

the sample standard deviations stplotted against time the middle line given by the overall sample standard deviation s the upper control limit UCL= s+LS the lower control limit LCL= sLS,

where S is an estimate of the variance ofSand typicallyL= 3.

This is not a confidence interval for the in-control standarddeviation.

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Strengths and weaknesses

Strengths: Shewhart-charts are easy to make and easy to interpret. They are good at detecting large parameter shifts.

Weaknesses: They do not detect small and medium parameter shifts very

well. They are quite sensitive to the model assumptions, in

particular independence between observations and normality.

Deviations from normality may be amended by atransformation of the original data.

To account for dependence between consecutive observations,one may

filtrate the data with an adequate time series model andconstruct a control chart for the resulting residuals

adjust the control limits.




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Sensitivity to parameter change

The reason why Shewhart-charts have poor detection skills forsmaller shifts in the parameters is probably that they considereach observation (or subgroup) separately, instead of

accumulating information as new observations are made. More specifically, the control limits from Phase I are kept

constant, instead of updating them according to newobservations.

That is precisely what the following types of control charts tryto achieve.




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Cumulative sum (CUSUM) charts

The aim of the cumulative sum (CUSUM) charts, originallydeveloped by Page (1954), is to decrease the

parameter-change ARL for small to medium parameter shiftsrelative to Shewhart-charts, without substantially increasingthe in-control ARL.

That is achieved by updating information by accumulationover past observations.




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CUSUM-charts for the mean

Let

zt= xtx

X

for each subgroup t and

SHt =max{ztk+SHt1 , 0} (1)

SLt =min{zt+k+SLt1 , 0}. (2)

The CUSUM-chart is made by plotting the values SHt and SLtagainst time the upper control limit h forSHt the lower control limit h forSLt.




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CUSUM-charts

The reference value k is chosen as 1/2 the size of the meanshifts one wants to detect in units, typically k= 0.5.

The limit h is chosen to optimise the ARL-properties of thechart, typically h= 4.

CUSUM-charts are optimal for detecting mean shifts of size2k fork


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Exponentially weighted moving average (EWMA) charts

Exponentially weighted moving average (EWMA) charts,originally proposed by Roberts (1959), are another option fordetecting small to medium sized parameter shifts.

These are also based on accumulating information from pastobservations.

Letwt=rxt+ (1r)wt1

for each subgroup t, where 0< r1.

Make the chart by plotting

wtagainst time the upper control limit UCLt=x+LW,t the lower control limit LCLt=xLW,t,

where W,tis an estimate of the standard deviation ofWtunder the assumption of normality.




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EWMA vs CUSUM

The pair of parameters (r, L) is chosen to achieve the desiredARL-properties of the chart. The choice (r, L) = (1, 3) gives

the standard X-chart. For good choices of (r, L), EWMA-charts are comparable to

CUSUM-charts.

The former are easier to interpret than the latter.




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Example

As an illustration, I have constructed X-, CUSUM- andEWMA-charts for a real data set.

The data are measurements of a particular electricalcharacteristic that was involved in the assembly of electronicunits, observed in 7 strips in each of 11 ceramic sheets.

These data were originally analysed by Ott (1949).




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Example

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Control charts for count data

In some cases, the quality measures of interest concern thenumber of defective units or the number of defects on eachinspected unit.

Then, the observed data follow a discrete distribution, that isassumed to be either the binomial or Poisson.

Shewhart-charts are made using normal approximations.

As for measurement data, Shewhart-charts for count datadetect large parameter shifts rather well, provided the normalapproximation is good.

For small parameter shifts, these charts perform very poorly.

CUSUM- and EWMA-charts can also be made for count data.

These have better ARL-properties for smaller parameter shifts.




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Multivariate control charts




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Multivariate quality control

In a research project on ambulatory monitoring, theMinnesota Supercomputing Institute has equipped severalsubjects with instruments that with regular intervals measureand record certain physiological variables that are risk factors

for heart attacks and strokes. These variables are the systolic blood pressure, the diastolic

blood pressure, the heart rate and the overall mean arterialpressure.

The aim is to detect changes in the mean and variance of one

or several of these variables as quickly as possible.

It should be taken into account that these are highlycorrelated.




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Why multivariate charts?

The quality of most manufacturing processes depends onseveral, possibly related characteristics, rather than just one.

In such cases, quality control requires the simultaneousmonitoring of all these characteristics.

Constructing separate charts for the characteristics is not

recommended for several reasons:1. If the characteristics are dependent, one risks both

not detecting when the process is out-of-control falsely detecting the process as out-of-control when in fact

it is not.

2. Even when the characteristics are independent, the numberof false alarms becomes much larger.

3. If the number of characteristics is high, it is cumbersome,if not impossible, to monitor all the individual charts.




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Illustration in the bivariate case




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Multivariate control chart

The aim is to find a scalar statistic that summarises the necessary

information from all the quality characteristics construct a control chart based on this statistic.

The challenge is to find such a statistic that has the power detect parameter changes in the joint

characteristic distribution for which it is possible to compute adequate control limits.




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Multivariate control charts for measurement data

Once more, let us start with measurement data.

Let X1,X2, . . ., be the measurements at time = 1, 2, . . .,with X = (Xt1, . . . ,Xp)

T, of the pquality characteristics

of interest. These are assumed to

be serially independent follow a multinormal distribution Np(,).

The measurements are divided into m subgroups of size n,and one computes the averagext for each subgroup t.



2


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T2-chart

This is the classic multivariate control chart, and is themultivariate analogue of the X-chart.

The purpose of this chart is to detect changes in one orseveral of the components of mean vector.

This chart is based on Hotellings T2

-statistic

T2t =n(xtx)TS1(xtx),

wherex is the overall average vector and S is the sample

covariance matrix. Under the model assumptions, this statistic follows a

Hotellings T2-distribution, which is a scaledFischer-distribution.



T 2 h


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T2-chart

The chart consists of: the T2ts plotted against time

the upper control limitUCL= p(m+1)(n1)mnmp+1F

1p,mnmp+1(1),

where Fa,b() is the cumulative distribution function of theFischer distribution with parameters (a, b).

This is a confidence region for the in-control mean vector.

The confidence level is chosen to obtain the desired

ARL-properties of the chart.



|S|1/2 h


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|S|1/2-chart

This is the multivariate analogue of the S-chart.

The purpose of this chart is to detect changes in one orseveral of the pvariances or in one or several of thep(p1)/2 correlations.

One computes the sample covariance matrix St for eachsubgroup t.

The chart consists of: |St|

1/2 plotted against time the middle line given by b3|S|

1/2

the upper control limit UCL= b3|S|1/2 +L

b1b23|S|

1/2

the lower control limit LCL= b3|S|1/2 Lb1b23|S|1/2,

where S is the overall sample covariance matrix,b1 = (n1)

pp

j=1(nj),

b3 = (2/(np))p/2(n/2)/((np)/2) and typically L= 3.

This does not define a confidence region for the in-control

covarariance matrix.Ingrid Hobk Haff The use of multivariate control charts for quality control


S h d k


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Strengths and weaknesses

Multivariate Shewhart-charts have the same strengths andweaknesses as the univariate equivalents:

they detect large parameter shifts well,

but are not effective for more subtle parameter changes.



M lti i t CUSUM (MCUSUM) h t


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Multivariate CUSUM (MCUSUM) charts

Several multivariate CUSUM (MCUSUM) charts have beenproposed for faster detection of parameter changes.

One of the most promising for mean shifts is the following,suggested by Crosier (1988).

Let

SH,t=

0, Ctk

(xtx + SH,t1)

1 kCt

, Ct>k

Ct= ((x

tx

+S

H,t

1)

TS1

(x

tx

+S

H,t

1))

1/2

andyt= (S

TH,tS

1SH,t)1/2.



MCUSUM h t


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MCUSUM-charts

To construct the chart, plot the yts against time the upper control limit h.

The reference value kdetermines the size of mean shifts forwhich the chart is optimal, and is typically chosen to be 0.5.

The limit h is chosen to optimise the ARL-properties of thechart. The standard is h= 4.



Multivariate EWMA (MEWMA) charts


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Multivariate EWMA (MEWMA) charts

The first multivariate EWMA (MEWMA) chart was suggestedby Lowry et al. (1992).

Letzt=R(xtx) + (I R)zt1,

where I is the ppidentity matrix and R is a ppdiagonalmatrix with diagonal entries 0


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MEWMA-charts

An MEWMA-chart for detecting changes in the mean vectoris then constructed by plotting

the w2ts against time the upper control limit UCL= L.

The parameters r1, . . . , rp and Lare chosen to achieve thedesired ARL-properties of the chart.

Ifr1 =. . .= rp=r, all the pquality characteristics are giventhe same weight. For r= 1, this chart is equivalent to a

T2-chart.



Comparison


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Comparison

As in the univariate case, the Shewhart-charts are good atdetecting large shifts in the parameters.

On the other hand, MCUSUM- and MEWMA-charts aresuperior for smaller changes in the parameters.

They all rely on the assumption of multinormally distributedand serially independent observations.

If these assumptions are not fulfilled, transformations, timeseries models and adjustment of the control limits can be

considered, but this is much more difficult in the multivariatesetting.



Example


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Example

As an illustration, I have constructed T2-, MCUSUM- andMEWMA-charts for a real bivariate data set.

The data consist of two different types of overtime hours for

the Madison, Wisconsin, police department. The first type is legal appearances and the second is

extraordinary events.

Each subgroup represents approximately half a year.

These data were analysed by Johnson and Wichern (1998).



Example


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Example

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Alternative multivariate control charts


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Alternative multivariate control charts

There are several alternatives to the classic Shewhart-, MCUSUM-and MEWMA-charts, among those:

Bayesian control charts (Wang, 2012) control charts based on neural networks (Psarakis, 2011)

nonparametric control charts (Boone and Chakraborti, 2011).



Multivariate charts for count data


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Multivariate charts for count data

Multivariate charts for count data, also called multiattributecontrol charts (MACCs), have been much less studied thancharts for measurement data.

Patel (1973) suggested multivariate extensions ofShewhart-charts for count data based on HotellingsT2-statistic, using the normal approximation.

MCUSUM- and MEWMA-charts for count data have also

been proposed (Yu et al., 2003; Somerville et al., 2002).



Locating the sources of out-of-control signals


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Locating the sources of out of control signals

One of the major challenges when using multivariate controlcharts is to identify the sources of an out-of-control signal.

Since the joint quality characteristic distribution has beensummarised by a single statistic, there is no way of knowingwhich set of variables is responsible.

Several approaches have been suggested, including the construction of individual Bonferroni confidence intervals

(Alt, 1984) analysis of the corresponding principal components (Lowry

et al., 1992) partitioning the T2-statistic into independent components

(Mason et al., 1994).

However, this is still an open problem.



Possible extensions


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Possible extensions

Recall that most multivariate control charts are built on theassumption of multivariate normality.

This implies that

all individual characteristics have the same type of marginaldistribution, namely normal

the dependence between each pair of characteristics is fullydescribed by the corresponding correlation.

One possible extension is to replace the multivariate normal

distribution with another multivariate distribution.



Multivariate control charts based on copulae


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Copulae are tools for constructing multivariate distributions. They can join univariate margins of (potentially) different

types.

They may also account for non-linear dependencies betweenthe quality characteristics.

Fatahi et al. (2012) have proposed a copula-based bivariatecontrol chart for monitoring rare events, i.e. count data.

In the bivariate case, there are many different copula modelsto choose between.

The selection is much more limited in higher dimensions.

Pair-copula constructions (PCCs), that I have studied in mythesis, may be an alternative in higher dimensions.





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p

In order to build a multivariate control chart based on a PCCor another type of copula, one must select an appropriatescalar statistic T(X1, . . . ,Xp).

Once a statistic is chosen, control limits can be computed bysimulating from the estimated model.

The main challenge is to find a statistic that represents the joint characteristic distribution well is able to detect changes in this distribution tolerably fast.




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Conclusions



Summing up


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g p

Control charts are one of the most important tools for qualitycontrol.

They are widely used, also for non-manufacturingapplications, for instance in public health.

In practice, the quality of most processes depends on several,possibly related quality characteristics, rather than just one.

This requires multivariate control charts.

Two of the main challenges related to the use of multivariatecontrol charts are:

finding an adequate scalar statistic that summarises the jointcharacteristic distribution

locating the sources of out-of-control signals.



Other issues


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There are many important issues concerning the use of controlcharts, that I have not mentioned.

These include the impact of measurement errors choosing the sample/subgroup size process capability (six-sigma) robust estimation of the parameters.



Control charts in the future


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With the development of technology for acquiring data andthe increasing computing power, multivariate control chartsare likely to be even more relevant in the future.

Most of the proposed methods so far are built on theassumption of multinormally distributed data.

Natural extensions of control chart methods include the use ofother, more flexible multivariate distributions, for instance

built on copulae or even pair-copula constructions.



Alt, F. (1984). Multivarate quality control. In Kotz, S., Johnson, N., and Read, C.,editors The Encyclopedia of Statistical Sciences John Wiley New York


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Johnson, R. and Wichern, D. (1998). Applied Multivariate Statistical Analysis.

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