8/13/2019 CUSUM PDF

1/9

Introduction

A major disadvantage of a Shewart control chart it is relatively insensitive to small process

shifts, say on the order of 1.5 or less. One alternative to the Shewart control chart is thecumulative sum(cusum) control chart which may be used when small process shifts are of

interests.

Basic Principles : The Cusum Control Chart for Monitoring the Process Mean

Consider the data in table 1.1. The first 20 of these observations were drawn at random

from a normal distribution with mean and standard deviation . These observations

have been plotted on a Shewhart control chart in Fig. 9.1. The center line three sigma control

limits on these charts are

Table 1.1 Data for the Cusum Example

8/13/2019 CUSUM PDF

2/9

Fig. 1.1 A Shwart control chart for table 1.1

Note that all 20 observations plot in control. The last 10 observations in column (a) of

Table 1.1 were drawn from a normal distribution with mean and standard deviation

consequently, we can think of these last 10 observations as having benn drawn from the

process when it is out of control-that is, after the process has experienced a shift in the mean of

1. These last 10 observations are also plotted on the control chart in Fid .1. None of these

points plots outside the control limits, so we have no strong evidence that the process is out of

control. Note that there is an indication of a shift in process level for the last 10 points, because

all but one above the center line. However, if we rely on the traditional signal of an out-of-

control process, one or more points beyond the three-sigma control limit, then the Shewhart

control has failed to detect the shift.

The reason for this failure, of course, is the relatively small magnitude of the shift. The

Shewart chart for averages is very effective if the magnitude of the shift is 1.5to 2 or larger.

For smaller shifts, it is not as effective. The cumulative sum( cusum) control chart is a good

alternative when small shifts are important.

The cusum directly incorporates all the information in the sequence of the sample values by

plotting the cumulative sums of the deviations of the sample values from a target value. For

8/13/2019 CUSUM PDF

3/9

example, suppose that the sample of size are collected, and xjis the average of the

sample. Then if is the target for the process mean, the cumulative sum control chart is

formed by plotting the quantity

x

(1.1)

Against the sample number i. is called the cumulative sum up to and including the ith

sample. Because they combine information from several samples, cumulative sum charts are

more effective than Shewhart charts for detecting small process shifts. Furthermore, they are

particularly effective with samples of size This makes the cumulative sum control chart a

good candidate for use in the chemical and process industries where rational subgroups are

frequently of size 1, and in discrete parts manufacturing with automatic measurement of each

part and on-line process monitoring directly at the work center.

We note that if the process remains in control at the target value the cumulativesum defined in equation (1.1) is a random walk with mean zero. However, If the mean shifts

upward to some value say, then an upward or positive drift will develop in the

cumulative sum . Conversely, if the mean shifts downward to some then a

downward or negative drift in will develop. Therefore, if a significant trend develops in the

plotted points either upward or downward, we should consider this as evidence that the

process mean has shifted, and a search for some assignable cause should be performed.

8/13/2019 CUSUM PDF

4/9

Fig 1.2 Plot of the cumulative sum from column ( c ) of table 1.1

This theory can be easily demonstrated by using the data in column ( a ) of table 1.1

again. To apply the cusum in equation (1) to these observations, we would take x =(Since

our sample size is and the value Therefore, the cusum becomes Column (b) of

table 1.1 contains the differences

, and the cumulative sums are computed in column ( c

). The Starting value for the cusum, is taken to be zero. Figure 1.2 plots the cusum from

column (c) of table 1.1. Note that for the first 20 observations where the cusum tends

to drift slowly, in this case maintaining values near zero.

However, in the last 10 observations, where the mean has shifted to , a strong upward

trend develops.

8/13/2019 CUSUM PDF

5/9

Of course, the cusum plot in Fig. 1.2 is not a control chart because it lacks statistical

control limits. There are ways to represent cusums, the tabular(or algorithmic) cusum,and the

V-mask form of the cusum.Of the two representations, the tabular cusum is preferable. We

now present the construction and use of the tabular cusum.

Sometimes we think of as a target value for the quality characteristic x. This

viewpoint is often taken in the chemical and process industries when the objective is to control

x(viscosity, say) to a particular target value( such as 2000 centistokes at 100 ). If the process

drifts or shifts off this target value, the cusum will signal, and an adjustment is made to some

manipulatable variable (such as the catalyst feed rate) to bring the process back on target.

Also, in some cases a signal from a cusum indicates the presence of an assignable cause

that must be investigated just as in the Shewhart chart case.

The tabular cusum works by accumulating derivations from that are above target

with one statistic and accumulating derivations from that are below target with another

statistic . The statistic . The statistics and are called one-sided upper and lower

cusums,respectively. They are computed as follows:

The Tabular Cusum

(1.2)

(1.3)

Where the starting values are

In equations (1.2) and (1.3) , Kis usually called the reference value ( or the allowance,

or the slack value), and it is often chosen about halfway between the target and the out-of-

control value of the mean that we are interested in detecting quickly.

Thus, if the shifts is expressed in standard deviation units as ||

) then K is one-half the magnitude of the shift or

||

Note that

accumulate deviations from the target value of that are greater than

K, with both quantities reset to zero on becoming negative. If either or

exceed the

decision interval H, the process is considered to be out of control.

8/13/2019 CUSUM PDF

6/9

We have briefly mentioned how to choose K,but how does one choose H?Actually, the

proper selection of these two parameters is quite important, as it has substantial impact on the

performance of the cusum. A reasonable value of H is five times the process standard

deviation.

Recall that the target value is , the subgroup is , the process standard

deviation is , and suppose that the magnitude of the shift we are interested in detecting is

. Therefore, the out-of-control value of the process mean is

We will use a tabular cusum with

(because the shift size is 1.0and ) and

Table 1.2. The Tabular Cusum for table 1.1

(because the recommended value of the decision interval is ).

Table 1.2 presents tabular cusum scheme. To illustrate the calculations, consider period 1. The

equations for

and

are

And

8/13/2019 CUSUM PDF

7/9

Since and . Now , so since

And

For period 2, we would use

And

Since

And

Panels (a) and (b) of table 1.2 summarize the remaining calculations. The quantities

and in table 1.2 indicate the number of consecutive periods that the cusums or

have

been nonzero.

The cusum calculations in table 1.2 show that the upper side cusum are period 29 is

. Since this is the first period at which we would have to conclude that

the process is out of control at that point. The tabular cusum also indicates when the shift

probably occurred. The Counter recored the number of consecutive periods since the

upper-side cusum rose above the value of zero. Since at period 29, was last in

control at 29-7=22, so the shift likely occurred between 22 and 23.

8/13/2019 CUSUM PDF

8/9

Semi-Finals Report

In Partial Fulfilment of the course

CHETE-1 ADVANCE STATISTICs

Submitted to:

Engr. Melba Mendoza

Faculty, Chemical Engineering Department

Xavier University-Ateneo De Cagayan

College of Engineering

Corrales, Cagayan de Oro City

Submitted by:

Joeferlo A. Valcarcel

BS CHE-5

October 3, 2013

8/13/2019 CUSUM PDF

9/9