9.3
Estimating σ
• It is not recommended to estimate the process variability σ 2 by the sample variance
Pi
2
j=1 (xj − x)
2
s =
i−1
• s2 gives an estimate of the process variability that is too large because it takes into account
the inherent process variability, measurement error, as well as long-term variability due to
assignable causes (such as shifts in the mean).
• The detection of variability due to assignable causes is the goal of quality control procedures.
Thus, incorporating these additional sources of variability when estimating the process variability defeats the purpose of the cusum procedure.
• Therefore, an estimate of variability that contains inherent process variability and measurement error (but excludes assignable cause variability) is desired.
• The mean squared successive difference (MSSD) is a good estimator of this variability:
M SSD =
where there are i samples to be included in the cusum.
• The estimate of the process standard deviation is σ
b=
√
M SSD.
• If preliminary data exist and little change in process variability is expected, then calculate σ
b
from the preliminary data and use it to monitor the process.
• If preliminary data is not available, then σ
b can be updated as successive samples are taken.
175
Piston Ring Diameter Cusum Example (σ unknown):
• We will now make cusum plots and generate the tabular cusum for the 25 sample means
assuming σ is unknown.
• We assume the in-control mean µ0 = 74 and the estimated σ
b = M SSD = .00982998.
• The following table contains the tabular cusum values with resets after each signal.
CUSUM with Reset after Signal (sigma unknown)
sample
xbar n
cusum_l
hsigma cusum_h flag
1 74.0102 5
0 0.017584
0.008002
2 74.0006 5
0 0.017584
0.006404
3 74.0080 5
0 0.017584
0.012206
4 74.0030 5
0 0.017584
0.013008
5 74.0034 5
0 0.017584
0.014210
6 73.9956 5 .002201945 0.017584
0.007612
7 74.0000 5 .000003890 0.017584
0.005414
8 73.9968 5 .001005836 0.017584
0.000016
9 74.0042 5
0 0.017584
0.002018
10 73.9980 5
0 0.017584
0.000000
11 73.9942 5 .003601945 0.017584
0.000000
12 74.0014 5 .000003890 0.017584
0.000000
13 73.9984 5
0 0.017584
0.000000
14 73.9902 5 .007601945 0.017584
0.000000
15 74.0060 5
0 0.017584
0.003802
16 73.9966 5 .001201945 0.017584
0.000000
17 74.0008 5
0 0.017584
0.000000
18 74.0074 5
0 0.017584
0.005202
19 73.9982 5
0 0.017584
0.001204
20 74.0092 5
0 0.017584
0.008206
21 73.9998 5
0 0.017584
0.005808
22 74.0016 5
0 0.017584
0.005210
23 74.0024 5
0 0.017584
0.005412
24 74.0052 5
0 0.017584
0.008414
25 73.9982 5
0 0.017584
0.004416
176
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
177
15
16
17
18
19
20
21
22
23
24
25
73.998200
74.005200
74.002400
74.001600
73.999800
74.009200
73.998200
74.007400
74.000800
73.996600
74.006000
73.990200
73.998400
74.001400
73.994200
73.998000
74.004200
73.996800
74.000000
73.995600
74.003400
74.003000
74.008000
74.000600
74.010200
Subgroup
Sample Subgroup
sample
Size
Mean
0.01617714
0.00870057
0.01192896
0.00743640
0.00816701
0.00798123
0.00846759
0.00698570
0.01056882
0.00779744
0.00731437
0.01530359
0.01045466
0.00421900
0.00286356
0.00628490
0.00554076
0.01225561
0.00552268
0.00870632
0.01221884
0.00908295
0.01474788
0.00750333
0.01477159
Subgroup
Std Dev
0.0118 0.02940000
0.0096 0.03120000
0.0074 0.02600000
0.0052 0.02360000
0.0030 0.02200000
0.0008 0.02220000
-0.0014 0.01300000
-0.0036 0.01480000
-0.0058 0.00740000
-0.0080 0.00660000
-0.0102 0.01000000
-0.0124 0.00400000
-0.0146 0.01380000
-0.0168 0.01540000
-0.0190 0.01400000
-0.0212 0.01980000
-0.0234 0.02180000
-0.0256 0.01760000
-0.0277 0.02080000
-0.0299 0.02080000
-0.0321 0.02520000
-0.0343 0.02180000
-0.0365 0.01880000
-0.0387 0.01080000
0.0470
0.0492
0.0514
0.0536
0.0558
0.0580
0.0602
0.0624
0.0646
0.0668
0.0690
0.0712
0.0734
0.0756
0.0778
0.0800
0.0822
0.0844
0.0865
0.0887
0.0909
0.0931
0.0953
0.0975
0.0997
V-Mask V-Mask
Upper
Limit
Cusum
Limit Exceeded
-0.0409 0.01020000
V-Mask
Lower
Limit
Cumulative Sum Chart Summary for diameter
The CUSUM Procedure
CUSUM for Piston-Ring Diameters (sigma unknown)
• We will now make a cusum plot for the 25 sample means (n = 5) assuming a process in-control
mean µ0 = 74 with an estimated process standard deviation σ
b = M SSD = .00982998.
CUSUM for Piston-Ring Diameters (sigma unknown)
The CUSUM Procedure
5
5
5
5
5
5
5
5
5
5
5
5
14
15
16
17
18
19
20
21
22
23
24
25
5
9
5
5
8
13
5
7
5
5
6
12
5
5
5
5
4
11
5
3
5
5
2
10
5
1
178
73.998200
74.005200
74.002400
74.001600
73.999800
74.009200
73.998200
74.007400
74.000800
73.996600
74.006000
73.990200
73.998400
74.001400
73.994200
73.998000
74.004200
73.996800
74.000000
73.995600
74.003400
74.003000
74.008000
74.000600
74.010200
0.01617714
0.00870057
0.01192896
0.00743640
0.00816701
0.00798123
0.00846759
0.00698570
0.01056882
0.00779744
0.00731437
0.01530359
0.01045466
0.00421900
0.00286356
0.00628490
0.00554076
0.01225561
0.00552268
0.00870632
0.01221884
0.00908295
0.01474788
0.00750333
0.01477159
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00120195
0.00000000
0.00760195
0.00000000
0.00000390
0.00360195
0.00000000
0.00000000
0.00100585
0.00000390
0.00220195
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
Decision
Decision Interval
Cusum Interval Exceeded
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
73.998200
74.005200
74.002400
74.001600
73.999800
74.009200
73.998200
74.007400
74.000800
73.996600
74.006000
73.990200
73.998400
74.001400
73.994200
73.998000
74.004200
73.996800
74.000000
73.995600
74.003400
74.003000
74.008000
74.000600
74.010200
Subgroup
Sample Subgroup
sample
Size
Mean
0.01617714
0.00870057
0.01192896
0.00743640
0.00816701
0.00798123
0.00846759
0.00698570
0.01056882
0.00779744
0.00731437
0.01530359
0.01045466
0.00421900
0.00286356
0.00628490
0.00554076
0.01225561
0.00552268
0.00870632
0.01221884
0.00908295
0.01474788
0.00750333
0.01477159
Subgroup
Std Dev
0.00441560
0.00841365
0.00541170
0.00520975
0.00580780
0.00820585
0.00120390
0.00520195
0.00000000
0.00000000
0.00380195
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00201755
0.00001560
0.00541365
0.00761170
0.01420975
0.01300780
0.01220585
0.00640390
0.00800195
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
Decision
Decision Interval
Cusum Interval Exceeded
Cumulative Sum Chart Summary for diameter
Cumulative Sum Chart Summary for diameter
Subgroup
Std Dev
The CUSUM Procedure
The CUSUM Procedure
Subgroup
Sample Subgroup
sample
Size
Mean
UPPER ONE-SIDED CUSUM (sigma unknown)
LOWER ONE-SIDED CUSUM (sigma unknown)
• We will now generate the upper and lower tabular cusums for the 25 sample means
followed by one-sided cusum plots.
UPPER ONE-SIDED CUSUM (sigma unknown)
The CUSUM Procedure
LOWER ONE-SIDED CUSUM (sigma unknown)
The CUSUM Procedure
179
SAS Cusum Code for Piston-Ring Diameter Data
DM ’LOG; CLEAR; OUT; CLEAR;’;
ODS GRAPHICS ON;
* ODS PRINTER PDF file=’C:\COURSES\ST528\SAS\CUSUM2.PDF’;
OPTIONS NODATE NONUMBER;
DATA piston;
DO sample=1 TO 25;
DO item=1 TO 5;
INPUT diameter @@;
diameter = diameter+70; OUTPUT;
END; END;
LINES;
4.030 4.002 4.019 3.992 4.008
3.995 3.992 4.001 4.011 4.004
3.988 4.024 4.021 4.005 4.002
4.002 3.996 3.993 4.015 4.009
3.992 4.007 4.015 3.989 4.014
4.009 3.994 3.997 3.985 3.993
3.995 4.006 3.994 4.000 4.005
3.985 4.003 3.993 4.015 3.988
4.008 3.995 4.009 4.005 4.004
3.998 4.000 3.990 4.007 3.995
3.994 3.998 3.994 3.995 3.990
4.004 4.000 4.007 4.000 3.996
3.983 4.002 3.998 3.997 4.012
4.006 3.967 3.994 4.000 3.984
4.012 4.014 3.998 3.999 4.007
4.000 3.984 4.005 3.998 3.996
3.994 4.012 3.986 4.005 4.007
4.006 4.010 4.018 4.003 4.000
3.984 4.002 4.003 4.005 3.997
4.000 4.010 4.013 4.020 4.003
3.988 4.001 4.009 4.005 3.996
4.004 3.999 3.990 4.006 4.009
4.010 3.989 3.990 4.009 4.014
4.015 4.008 3.993 4.000 4.010
3.982 3.984 3.995 4.017 4.013
;
SYMBOL1 v=dot width=3;
PROC CUSUM DATA=piston;
XCHART diameter*sample=’1’
/ MU0=74 SMETHOD=noweight H=4.0 K=0.5 DELTA=1.0
DATAUNITS HAXIS = 1 TO 25
TABLESUMMARY TABLEOUT OUTTABLE = qsum ;
INSET ARL0 ARLDELTA H K SHIFT / POS = n;
LABEL diameter=’Diameter Cusum’
sample = ’Piston Ring Sample’;
TITLE ’CUSUM for Piston-Ring Diameters (sigma unknown)’;
PROC CUSUM DATA=piston;
XCHART diameter*sample=’1’
/ MU0=74 SMETHOD=noweight H=4.0 K=0.5 DELTA=1.0
DATAUNITS HAXIS=1 TO 25
SCHEME=onesided TABLESUMMARY TABLEOUT;
INSET ARL0 ARLDELTA H K SHIFT / POS = n;
LABEL diameter=’Diameter Cusum’
sample = ’Piston Ring Sample’;
TITLE ’UPPER ONE-SIDED CUSUM (sigma unknown)’;
180
PROC CUSUM DATA=piston;
XCHART diameter*sample=’1’
/ MU0=74 SMETHOD=noweight H=4.0 K=0.5 DELTA=-1.0
DATAUNITS HAXIS=1 TO 25
SCHEME=onesided TABLESUMMARY TABLEOUT;
INSET ARL0 H K SHIFT / POS = ne;
LABEL diameter=’Diameter Cusum’
sample = ’Piston Ring Sample’;
TITLE ’LOWER ONE-SIDED CUSUM (sigma unknown)’;
*** The following code will make a table with resetting ***;
***
after an out-of-control signal is detected
***;
DATA qsum; SET qsum;
h=4;
k=.5;
sigma=.00983;
aim=74;
** enter values **;
xbar=_subx_; n=_subn_;
hsigma=h*sigma/SQRT(_subn_);
ksigma=k*sigma/SQRT(_subn_);
RETAIN cusum_l 0 cusum_h 0;
IF (-hsigma < cusum_l < hsigma)
THEN DO;
cusum_l = cusum_l + (aim - ksigma) - xbar;
IF cusum_l < 0 then cusum_l=0;
END;
IF (-hsigma < cusum_h < hsigma)
THEN DO;
cusum_h = cusum_h + xbar - (aim + ksigma);
IF cusum_h < 0 then cusum_h=0;
END;
IF MAX(cusum_l,cusum_h) ge hsigma
THEN DO;
IF (cusum_l ge hsigma) THEN DO;
flag=’lower’; OUTPUT; END;
IF (cusum_h ge hsigma) THEN DO;
flag=’upper’; OUTPUT; END;
cusum_l=0; cusum_h=0;
END;
ELSE OUTPUT;
PROC PRINT DATA=qsum;
ID sample;
VAR xbar n cusum_l hsigma cusum_h flag;
TITLE ’CUSUM with Reset after Signal (sigma unknown)’;
RUN;
181
9.4
Designing the Cusum Control Chart for x or x
• σ = the process standard deviation.
– In the formulas, we will assume the rational (sample) subgroup size is n = 1.
√
– When n > 1, replace σ with σ/ n in all formulas.
• , the specified magnitude of the smallest shift in the process mean to be detected.
•
K is called the reference or slack value.
– Generally, we want to choose k relative to the size of the shift we want to detect. This
will determine K. It is suggested that K = ∆2 (or equivalently, k = 2δ ) because this
choice of K has been shown to work well in practice.
– The choice of k = 2δ yields a cusum control chart that comes close to minimizing the
ARL for detecting a shift of size ∆ = δσ.
•
H is called the decision interval.
– Montgomery suggests using h = 4 or, equivalently, H = 4σ. DuPont suggests using
h = 5 or, equivalently, H = 5σ.
• Because the ARL table uses h and k (the standardized values of H and K) and a specified σ,
or an estimate σ
b is required in order to convert from standardized units k and h to the units
of the original data K = kσ and H = hσ (or, K = kb
σ and H = hb
σ.
• There are two common methods for assigning cusum parameter values:
METHOD 1:
Given k, δ, and the in-control ARL.
– Using k and a specified in-control ARL, a reasonable value of h can be found from the ARL
table or use the cusumarl function in SAS. Then, K = kσ and H = hσ.
– Given this value of h, the researcher should verify that the out-of-control ARL corresponding
to a shift of ∆ = δσ is sufficiently small.
– Adjustments to h and k can be made to find an acceptable balance between the in-control
and out-of-control ARL’s.
– If σ has to be estimated, replace σ with σ
b in the formulas.
METHOD 2:
Given the in-control ARL0 and the out-of-control ARL1 for specified δ.
– A second way to choose h and k is to enter the ARL table with desired values for the in-control
ARL0 and for the out-of-control ARL1 for a specified value of δ, say δ0 .
– The columns contain ARL values for various values of δ. The column labeled δ = 0 represents
the in-control ARL0 .
– By following the δ = 0 column and the δ = δ0 column down until an acceptable pair of
in-control and out-of-control ARL’s is found, the values of h and k can be found by tracing
that row over to the first two columns labeled h and k (or use the cusumarl function in SAS).
182
9.4.1
Estimation of the ARL
• Several techniques have been developed to calculate/estimate the ARL for a cusum.
– A commonly-adopted approach based on Markov chains was developed by Brooks and
Evans (Biometrika (1972) 59, 3, 539-549.)
– Hawkins (J. of Quality Technology (1992), 24, 1, 37-43) provides an estimating equation
that yields accurate ARL values estimates.
– SAS uses the integral equation method described by Goel and Wu Technometrics (1971),
13, 221-230) and Lucas and Crosier (Technometrics (1982), 24, 199-205).
Find h such that ARL0 ≈ 300 given k = .5.
Find h such that ARL0 ≈ 200 given k = .25.
Example 1 for Method 1:
Example 2 for Method 1:
EXAMPLE 1 METHOD 1
k
h
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
k
h
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
4.50
4.51
4.52
4.53
4.54
4.55
4.56
4.57
4.58
4.59
4.60
arl2
167.684
185.868
205.976
228.210
252.793
279.973<-310.023<-343.243
379.968
420.566
465.444
arl2
279.973
282.844
285.744
288.674
291.633
294.621
297.640
300.689<-303.769
306.880
310.023
EXAMPLE 2 METHOD 1
arl1
k
335.367
371.736
411.952
456.420
505.587
559.947
620.046
686.487
759.937
841.133
930.889
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
arl1
559.947
565.689
571.489
577.348
583.265
589.243
595.280
601.379
607.539
613.761
620.046
h
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
k
h
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
6.80
6.81
6.82
6.83
6.84
6.85
6.86
6.87
6.88
6.89
6.90
183
arl2
125.402
132.572
140.120
148.064
156.427
165.228
174.491
184.240
194.498<-205.292<-216.650
228.601
241.175
254.404
268.321
282.962
298.363
314.565
331.608
349.534
368.390
arl1
250.804
265.144
280.239
296.129
312.854
330.457
348.983
368.479
388.996
410.585
433.301
457.202
482.350
508.807
536.641
565.923
596.727
629.130
663.215
699.068
736.780
arl2
194.498
195.553
196.613
197.679
198.750
199.826<-200.908
201.996
203.089
204.188
205.292
arl1
388.996
391.106
393.226
395.357
397.500
399.653
401.817
403.992
406.179
408.376
410.585
SAS Code for Examples 1 and 2 for Method 1:
DM ’LOG; CLEAR; OUT; CLEAR;’;
ODS LISTING;
OPTIONS LS=72 PS=54 NODATE NONUMBER;
*** DESIGNING A CUSUM: METHOD 1, EXAMPLE 1;
DATA csmarl;
k = .5;
* DO h = 4 TO 5 BY .1;
DO h = 4.5 TO 4.6 BY .01;
arl2 = CUSUMARL(’twosided’,0,h,k);
arl1 = CUSUMARL(’onesided’,0,h,k); OUTPUT;
END;
PROC PRINT DATA=csmarl; ID k h;
RUN;
*** DESIGNING A CUSUM: METHOD 1, EXAMPLE 2;
DATA csmarl;
k = .25;
* DO h = 6 TO 8 BY .1;
DO h = 6.8 TO 6.9 BY .01;
arl2 = CUSUMARL(’twosided’,0,h,k);
arl1 = CUSUMARL(’onesided’,0,h,k); OUTPUT;
END;
PROC PRINT DATA=csmarl; ID k h;
RUN;
Example for Method 2:
Find h and k such that ARL0 ≈ 150 and ARL1 ≈ 9 for a 1σ shift (δ = 1).
METHOD 2 EXAMPLE
Obs
k
h
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
0.25
0.25
0.25
0.25
0.25
0.30
0.30
0.30
0.30
0.70
0.70
0.75
0.75
0.80
0.80
0.85
0.85
6.30
6.31
6.32
6.33
6.34
5.62
5.63
5.64
5.65
2.92
2.93
2.74
2.75
2.58
2.59
2.43
2.44
arl0
arl1
148.064
148.882
149.703
150.528
151.358
148.498
149.459
150.427
151.401
149.190
151.337<-148.937
151.233
149.179
151.636
148.069
150.663
9.12655
9.13988
9.15321
9.16655
9.17988
8.73056
8.74485
8.75913
8.77342
8.56659
8.59707<-8.77763
8.81201
9.03707
9.07601
9.31376
9.35796
arl2
4.24303
4.24874
4.25445
4.26016
4.26587
3.95485
3.96074
3.96663
3.97252
2.95506
2.96279 GOOD
2.90746
2.91555
2.87143
2.87990
2.83984
2.84870
*** DESIGNING A CUSUM: METHOD 2 EXAMPLE;
DATA csmarl;
DO k = .25 TO 1 BY .05;
DO h = 2 TO 8 BY .01;
arl0 = CUSUMARL(’twosided’,0,h,k);
arl1 = CUSUMARL(’twosided’,1,h,k);
arl2 = CUSUMARL(’twosided’,2,h,k); *** 2 sigma shift for comparison;
IF (148 le arl0 le 152) and (8.5 le arl1 le 9.5) THEN OUTPUT;
END; END;
PROC PRINT DATA=csmarl;
RUN;
184
185
9.5
Fast Initial Response (FIR) for Cusum Charts
• The situation may arise where the researcher is concerned that the process may be in the
out-of-control state at start-up or when the process is restarted after an adjustment has been
made to the process.
• The standard cusum procedure may be slow in detecting a shift in the process that is present
immediately upon start-up or restart.
• Lucas and Crosier (1982) suggested a modification to the standard cusum procedure that
would decrease the response time required to detect an out-of-control signal that is present
immediately upon start-up or restart.
• In the computational tabular form, a standard cusum has both the high and low cusum
initially set to 0 (C0− = C0+ = 0).
• The suggested modification is to initially set both the high and low cusum to some specified
positive, nonzero constant c (C0− = C0+ = c) between 0 and decision interval H. This is called
the headstart value for a fast initial response (FIR) cusum.
• Lucas and Crosier suggested setting c =
H
.
2
• The design of the FIR cusum is similar to the design of the standard cusum. The two major
differences between the two designs are:
1. Rather than using the ARL table for a standard cusum design, one must use a table of
ARL’s for the FIR cusum design.
2. Rather than using C0− = C0+ = 0 in the formulae of the computational form, one should
use C0− = C0+ = H/2.
Or, instead of using tables, you can also just utilize the ‘headstart’ FIR option in the cusumarl
function in SAS.
• If the process is in the in-control state initially, the cusums Ci+ and Ci− will go to zero in a
small number of samples.
• If the process is in the out-of-control state initially, then one of the cusums will exceed H after
a small number of samples.
• Once the FIR cusum scheme has been set up, one can proceed with quality control testing in
the same manner as with the standard cusum scheme.
• The benefit of the FIR cusum procedure is that the decrease in the out-of-control ARL1 is
relatively large, while the decrease in the in-control ARL0 is relatively small.
• This implies that an out-of-control signal can be detected in fewer runs with the FIR than
without it.
• To adjust for the slight decrease in the in-control ARL0 , H can be adjusted to give the desired
in-control ARL0 .
• To use a FIR cusum to generate the tabular cusum in PROC CUSUM in SAS, insert as one
of the cusum options HEADSTART=nn where nn = H/2 (or any other headstart value).
186
179
187
GENERATING FIR CUSUM ARLs
h
k delta fir
arl1
arl2
h
k delta fir
arl1
arl2
4
4
4
4
4
4
4
4
4
4
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.0
0.0
0.5
0.5
1.0
1.0
1.5
1.5
2.0
2.0
0
2
0
2
0
2
0
2
0
2
77.08
66.57
13.29
8.99
6.06
3.70
3.91
2.35
2.93
1.77
38.54
28.03
13.20
8.68
6.06
3.69
3.91
2.35
2.93
1.77
6
6
6
6
6
6
6
6
6
6
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.0
0.0
0.5
0.5
1.0
1.0
1.5
1.5
2.0
2.0
0
3
0
3
0
3
0
3
0
3
4
4
4
4
4
4
4
4
4
4
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.0
0.0
0.5
0.5
1.0
1.0
1.5
1.5
2.0
2.0
0
2
0
2
0
2
0
2
0
2
335.37
316.38
26.68
20.25
8.38
5.29
4.75
2.86
3.34
2.01
167.68
148.70
26.63
20.06
8.38
5.29
4.75
2.86
3.34
2.01
8
8
8
8
8
8
8
8
8
8
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.0
0.0
0.5
0.5
1.0
1.0
1.5
1.5
2.0
2.0
0
4
0
4
0
4
0
4
0
4
6
6
6
6
6
6
6
6
6
6
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.0
0.0
0.5
0.5
1.0
1.0
1.5
1.5
2.0
2.0
0
3
0
3
0
3
0
3
0
3
250.80
225.44
20.90
13.48
8.73
5.05
5.51
3.17
4.07
2.37
125.40
100.04
20.89
13.38
8.73
5.05
5.51
3.17
4.07
2.37
8
8
8
8
8
8
8
8
8
8
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.0
0.0
0.5
0.5
1.0
1.0
1.5
1.5
2.0
2.0
0 18967.17 9483.58
4 18783.72 9300.14
0
84.00
84.00
4
63.25
63.24
0
16.37
16.37
4
9.42
9.42
0
8.75
8.75
4
4.87
4.87
0
6.01
6.01
4
3.37
3.37
DM ’LOG; CLEAR; OUT; CLEAR;’;
ODS LISTING;
OPTIONS LS=72 PS=54 NODATE NONUMBER;
*** DESIGNING A CUSUM WITH A FIR;
DATA firarl;
DO h = 4 TO 8 BY 2;
DO k = .25 TO .5 BY .25;
DO delta = 0 to 2 by .5;
DO fir = 0 to h/2 by h/2;
arl1 = CUSUMARL(’onesided’,delta,h,k,fir);
arl2 = CUSUMARL(’twosided’,delta,h,k,fir);
END; END; END; END;
PROC PRINT DATA=firarl;
ID h k delta fir;
TITLE ’GENERATING FIR CUSUM ARLs’;
RUN;
188
OUTPUT;
2553.11 1276.55
2490.87 1214.32
51.34
51.34
38.76
38.71
12.37
12.37
7.38
7.38
6.75
6.75
3.87
3.87
4.68
4.68
2.70
2.70
736.78
684.30
28.76
17.86
11.39
6.39
7.11
3.97
5.21
2.94
368.39
315.91
28.76
17.83
11.39
6.39
7.11
3.97
5.21
2.94
SAS code for FIR example using piston ring data
DM ’LOG; CLEAR; OUT; CLEAR;’;
* ODS PRINTER PDF file=’C:\COURSES\ST528\SAS\fir1.pdf’;
ODS LISTING;
OPTIONS LS=78 PS=500 NONUMBER NODATE;
DATA piston;
DO sample=1 TO 25;
DO item=1 TO 5; n= item + 5*(sample-1);
INPUT diameter @@;
diameter = diameter+70; OUTPUT;
END; END;
LINES;
4.030 4.002 4.019 3.992 4.008
3.995 3.992
3.988 4.024 4.021 4.005 4.002
4.002 3.996
3.992 4.007 4.015 3.989 4.014
4.009 3.994
3.995 4.006 3.994 4.000 4.005
3.985 4.003
4.008 3.995 4.009 4.005 4.004
3.998 4.000
3.994 3.998 3.994 3.995 3.990
4.004 4.000
3.983 4.002 3.998 3.997 4.012
4.006 3.967
4.012 4.014 3.998 3.999 4.007
4.000 3.984
3.994 4.012 3.986 4.005 4.007
4.006 4.010
3.984 4.002 4.003 4.005 3.997
4.000 4.010
3.988 4.001 4.009 4.005 3.996
4.004 3.999
4.010 3.989 3.990 4.009 4.014
4.015 4.008
3.982 3.984 3.995 4.017 4.013
;
SYMBOL VALUE=dot WIDTH=3;
4.001
3.993
3.997
3.993
3.990
4.007
3.994
4.005
4.018
4.013
3.990
3.993
4.011
4.015
3.985
4.015
4.007
4.000
4.000
3.998
4.003
4.020
4.006
4.000
PROC CUSUM DATA=piston;
XCHART diameter*sample=’1’
/ mu0=74 sigma0=.005 h=4.0 k=0.5 delta=1.0
dataunits headstart = 2
outtable = qsum tablesummary tablecomp
haxis=0 TO 25 vaxis=0 TO .035 by .005;
INSET
ARL0 ARLDELTA H K SHIFT / POS = se;
LABEL
diameter=’Diameter Cusum’
sample = ’Piston Ring Sample’;
TITLE ’FIR CUSUM for Piston-Ring Diameters’;
DATA qsum; set qsum;
h=4;
k=.5;
sigma=.005;
aim=74; ** enter values **;
xbar=_subx_; n=_subn_;
hsigma=h*sigma/sqrt(_subn_);
ksigma=k*sigma/sqrt(_subn_);
retain cusum_l 0 cusum_h 0;
189
4.004
4.009
3.993
3.988
3.995
3.996
3.984
3.996
4.000
4.003
4.009
4.010
fir = 2*sigma/sqrt(_subn_);
if sample=1 then do;
cusum_l = fir; cusum_h = fir; END;
if (-hsigma < cusum_l < hsigma)
then do;
cusum_l = cusum_l + (aim - ksigma) - xbar;
if cusum_l < 0 then cusum_l=0;
END;
if (-hsigma < cusum_h < hsigma)
then do;
cusum_h = cusum_h + xbar - (aim + ksigma);
if cusum_h < 0 then cusum_h=0;
END;
if max(cusum_l,cusum_h) ge hsigma
then
if (cusum_l ge hsigma) then do;
flag=’lower’;
if (cusum_h ge hsigma) then do;
flag=’upper’;
cusum_l=fir; cusum_h=fir;
else output;
do;
output; END;
output; END;
END;
PROC PRINT DATA=qsum;
ID sample; var xbar n fir cusum_l hsigma cusum_h flag;
TITLE ’FIR CUSUM with Reset after Signal’;
PROC CUSUM DATA=piston;
XCHART diameter*sample=’1’
/ mu0=74 sigma0=.005 h=4.0 k=0.5 delta=1.0
dataunits headstart=2
scheme=onesided tablesummary tablecomp
haxis=1 TO 25;
INSET
ARL0 ARLDELTA H K SHIFT / POS = n;
LABEL
diameter=’Diameter Cusum’
sample = ’Piston Ring Sample’;
TITLE ’Upper One-Sided FIR CUSUM ’;
PROC CUSUM DATA=piston;
XCHART diameter*sample=’1’
/ mu0=74 sigma0=.005 h=4.0 k=0.5 delta=-1.0
dataunits headstart=2
scheme=onesided tablesummary tablecomp
haxis=1 TO 25;
INSET
H K SHIFT / POS = nw;
LABEL
diameter=’Diameter Cusum’
sample = ’Piston Ring Sample’;
TITLE ’Lower One-Sided FIR CUSUM ’;
run;
190
FIR CUSUM with Reset after Signal
sample
xbar
n
fir
cusum_l
hsigma
cusum_h
flag
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
74.0102
74.0006
74.0080
74.0030
74.0034
73.9956
74.0000
73.9968
74.0042
73.9980
73.9942
74.0014
73.9984
73.9902
74.0060
73.9966
74.0008
74.0074
73.9982
74.0092
73.9998
74.0016
74.0024
74.0052
73.9982
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
.004472136
0.000000
0.002754
0.000000
0.000354
0.000000
0.003282
0.002164
0.004246
0.000000
0.000882
0.005564
0.003046
0.003528
0.012210
0.000000
0.006754
0.004836
0.000000
0.000682
0.000000
0.003554
0.000836
0.000000
0.000000
0.005154
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
.008944272
0.013554
0.003954
0.010836
0.006354
0.008636
0.003118
0.002000
0.000000
0.003082
0.000000
0.000000
0.000282
0.000000
0.000000
0.009354
0.000000
0.000000
0.006282
0.003364
0.011446
0.003154
0.003636
0.004918
0.009000
0.001554
upper
FIR CUSUM for Piston-Ring Diameters
The CUSUM Procedure
191
upper
lower
upper
upper
upper
Upper One-Sided FIR CUSUM
The CUSUM Procedure
Lower One-Sided FIR CUSUM
The CUSUM Procedure
192
193
5
5
5
5
5
5
5
5
5
5
5
5
14
15
16
17
18
19
20
21
22
23
24
25
5
9
5
5
8
13
5
7
5
5
6
12
5
5
5
5
4
11
5
3
5
5
2
10
5
1
73.998200
74.005200
74.002400
74.001600
73.999800
74.009200
73.998200
74.007400
74.000800
73.996600
74.006000
73.990200
73.998400
74.001400
73.994200
73.998000
74.004200
73.996800
74.000000
73.995600
74.003400
74.003000
74.008000
74.000600
74.010200
0.01617714
0.00870057
0.01192896
0.00743640
0.00816701
0.00798123
0.00846759
0.00698570
0.01056882
0.00779744
0.00731437
0.01530359
0.01045466
0.00421900
0.00286356
0.00628490
0.00554076
0.01225561
0.00552268
0.00870632
0.01221884
0.00908295
0.01474788
0.00750333
0.01477159
0.00068197
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00068197
0.00000000
0.00545573
0.00737376
0.00509180
0.01220983
0.00352786
0.00304590
0.00556393
0.00088197
0.00000000
0.00424590
0.00216393
0.00328197
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
Decision
Cusum Interval
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
73.998200
74.005200
74.002400
74.001600
73.999800
74.009200
73.998200
74.007400
74.000800
73.996600
74.006000
73.990200
73.998400
74.001400
73.994200
73.998000
74.004200
73.996800
74.000000
73.995600
74.003400
74.003000
74.008000
74.000600
74.010200
Subgroup
Sample Subgroup
sample
Size
Mean
0.01617714
0.00870057
0.01192896
0.00743640
0.00816701
0.00798123
0.00846759
0.00698570
0.01056882
0.00779744
0.00731437
0.01530359
0.01045466
0.00421900
0.00286356
0.00628490
0.00554076
0.01225561
0.00552268
0.00870632
0.01221884
0.00908295
0.01474788
0.00750333
0.01477159
Subgroup
Std Dev
0.01310163
0.01601966
0.01193769
0.01065573
0.01017376
0.01149180
0.00340983
0.00632786
0.00004590
0.00036393
0.00488197
0.00000000
0.00373769
0.00645573
0.00617376
0.01309180
0.01620983
0.01312786
0.01744590
0.01856393
0.02408197
0.02180000
0.01991803
0.01303607
0.01355410
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
0.0089
Decision
Cusum Interval
Cumulative Sum Chart Summary for diameter
Cumulative Sum Chart Summary for diameter
Subgroup
Std Dev
The CUSUM Procedure
The CUSUM Procedure
Subgroup
Sample Subgroup
sample
Size
Mean
Upper One-Sided FIR CUSUM
Lower One-Sided FIR CUSUM