Nonparametric Moving Average Control Chart for Process Variability

Nonparametric Moving Average Control Chart for Process Variability

V. Y. Pawar

Department of Statistics PDVP College

Tasgaon, 416 312 INDIA

D. T. Shirke

Department of Statistics Shivaji University Kolhapur 416 004 INDIA

Abstract – In this paper a nonparametric moving average control chart for monitoring process variability is proposed. The chart uses statistic based on the first and third quartiles. The proposed chart is simple to use. The performance of the chart is studied using Average Run Length criterion. The proposed chart performs better than the chart based on quartiles due to Amin et al. (1995) and synthetic chart due to Khilare and Shirke (2012) for various distributions. When underlying distribution is double exponential or Gamma, the Average Run Length performance of the chart is better than that of a well-known S2 chart. An example is provided to illustrate the working of the chart.

Keywords: moving average, nonparametric, sign statistic, average run length.

1. INTRODUCTION

   Control charts are one of the primary tools of statistical process control. Usually control charts to monitor process characteristics are based on some distributional assumptions. Nonparametric control charts do not require the assumption of any specific probability distribution for the underlying process. It is observed that nonparametric charts are simple to use and therefore in the recent years nonparametric charts have attracted researchers attention. Shewhart type nonparametric charts proposed to monitor process location or process variation are based on a suitable nonparametric statistic. It is well-known fact that Shewhart type control charts do not perform well for small shifts in the process parameter. To overcome this demerit, control charts with memory are used. Moving Average control charts are control charts with memory. They are found to be superior to the Shewhart chart in detecting the small process shifts because they use the information contained not only in the current sub group sample but also from previous samples. The present work is focused on nonparametric moving average control charts for monitoring increase in the process variability.

   In quality control applications McGilchrist and Woodyer (1975) proposed a distribution-free cumulative sum technique for monitoring rainfall amounts. Bakir and Reynolds (1979) studied a nonparametric procedure for process control based on within-group ranking. Amin and Searcy (1991) studied the behavior of the EWMA control chart using the

   Wilcoxon signed-rank statistic. Amin et al. (1995) developed nonparametric quality control charts based on the sign statistic. Bakir (2006) developed distribution-free quality control charts based on signed-rank-like statistic. Bakir (2004) proposed a distribution-free Shewhart quality control chart based on signed-ranks. Chakraborti and Eryilmaz (2007) proposed control charts based on signed-rank statistic. Chakraborti and Van de Wiel (2008) proposed Mann- Whiteny statistic based control chart. Human et al. (2010) studied nonparametric Shewhart-type sign control charts based on runs. Ho and Costa (2011) proposed monitoring a wandering mean with an np chart and this chart also works with sign statistic. Pawar and Shirke (2010) proposed a nonparametric Shewhart-Type Synthetic Control Chart for location based on signed-rank statistic.

   Amin et al. (1995) proposed a sign chart for process variation based on quartiles; however, performance of this chart is not up to the mark as compared to S2 chart for normal and some non-normal distributions. Das (2008a) proposed nonparametric control chart for monitoring variability based on the two sample variability test due to Ansari and Bradley (1960). Its in-control performance is better than that of the S2 chart where as the out-of-control performance is not up to the mark. Das and Bhattacharya (2008) proposed a chart based on nonparametric test on variability and shown that it has better in-control performance than the Shewhart S chart. Das (2008b) developed a nonparametric control charts for controlling process variability based on rank test. Khilare and Shirke (2012) developed a nonparametric synthetic chart for process variability based on sign statistic. In this article we propose a moving average control chart for process variability based on sign statistic. Rest of the paper is organized as follows:

   Sign chart and synthetic sign chart for process variability are described briefly in section 2. In section 3 moving average control chart and proposed moving average control chart for process variability using sign statistic are described. The Average Run Length (ARL) performance of the proposed chart and its comparison with the other charts is studied in section 4. Section 5 illustrates the use of the proposed chart with numerical example.

2. SIGN CHART AND SYNTHETIC SIGN CHART FOR MONITORING PROCESS VARIABILITY

   Lehmann (1975) studied a nonparametric statistical method based on ranks. Control charts using test statistic for comparing two variances would require to obtain an initial sample of size m when the process is in-control and then at each sample time i, a sample of size n is taken from the process. These two samples are then pooled to get a pooled sample of size (m+n). The observations in the pooled sample are then ranked from smallest to the largest element and charting statistic based on the ranks of the observations is calculated.

   Another approach given by Bradley (1968) is based on Westenbergs two sample inter-quartile range test. The test is based on pooling two samples S1 and S2 into one sample and then counting the number of observations belonging to the S1 sample that are above the third quartile (Q3) or below the first quartile (Q1), where Q1 and Q3 are based on the pooled sample. Amin et al.(1995) adapted the idea of this test to the one sample case and provided a chart based on quartiles. In the control chart applications Q1 and Q3 would need to be specified by process engineers or more likely estimated from the process data when the process is in-control. Let Xij be the jth observation in the ith sample (i=1, 2, 3, ; j=1, 2, , n). Define,

   The upper control limit d is determined using equation (2) for desired in-control ARL=ARL(0) when the process is in control ( = 0). The synthetic control chart for monitoring process variability is proposed by Khilare and Shirke (2012). The synthetic control chart is described in the following.

   A synthetic control chart for process variability is obtained by combining the sign chart based on quartiles and Conforming Run Length (CRL) chart due to Bourke (1991). The operations of the synthetic control chart are out-lined below:

   1. Determine control limits d and L of sign chart and

      CRL chart respectively.

   2. At time i, take a sample of n units for inspection and calculate Ui.

   3. If Ui < d, a sample is a conforming one and control flow goes back to step (2). Otherwise, a sample is a nonconforming one and control flow continues to the next step.

   4. Count the number of samples between the current and previous nonconforming samples. This number is taken as CRL value for synthetic chart.

   5. If CRL > L, then the process is said to be under control and control flow goes back to the step (2). Otherwise the process is considered as out-of-control and control flow

      1 if

      Xij Q1 or Xij Q3 ,

      continues to next step.

      Uij 0

      1

      if Xij Q1 if Q1 X ij

      or Xij

      Q3.

      Q3

   6. Take action to locate and remove the assignable cause.

The ARL of Synthetic Sign chart for shift in is

The sign statistic corresponding to the ith sample is given

by

() = 1/(1

1 1 1 ) (3)

= , i=1,2, (1)

where 1 =

=1

The random variable =

+

2

follows binomial

+

= 2

=

(1 )

distribution with parameters n and p , where p is the probability of a nonconforming item which is given by p pr[Xij Q1 or Xij Q3 ],

which is function of process parameter(s).

While monitoring process variability, most of the times an increase in the process standard deviation, from its specified value 0 is of interest. In this case a one-sided

and = (1 < 3 > | = 0)

The parameters d and L of the synthetic sign chart are determined using equation (3) for desired in-control ARL=ARL(0) when the process is in control ( = 0). One may refer to Khilare and Shirke (2012) for more details about the synthetic chart.

In the following section we propose a chart based on moving averages to monitor process standard deviation.

control chart is desirable. A one-sided sign chart signals when III. NONPARAMETRIC CONTROL CHART BASED

Ui d,

where d is an upper control limit of the chart to be

ON MOVING AVERAGES

fixed suitably. This is called an upper one-sided sign chart for detecting an increase (or an upward shift) in .

The ARL of a sign chart for shift of magnitude = 1/0 in is given by,

() = 1 ( ), (2)

Let (xi1, xi2 , xin) be a sample (sub group) of size n (>1) of independent observations from the process output at sampling instance i, i = 1, 2, . . .. The probability distribution of the process is assumed to be continuous with process standard deviation . We assume that, when the process is in control, = 0. Let Ui = U(xi1, xi2 , xin) be a suitable

where

=

+

= 2

(1 )

nonparametric statistic for based on the subgroup sample. Charting statistic of Moving Average chart is based on the average of the most recent w (>1) nonparametric statistics

with = (1 < 3 > | = 0 )

defined as

=

1 + 2 + +

+ 1 + + +1

; <

; .

(4)

control and out-of-control ARL values. Computer programs are written in open source software R to study the performance of the control charts under different process

Here w is called as the span of the moving average. If we are interested only in detecting shift in in positive direction, the control chart will have only upper control limit. Let UCL be the upper control limit of the chart. UCL is chosen so as to have the desired in-control ARL. If the interest is to detect shift in negative direction, the control chart will have only lower control limit (LCL). The control chart to detect shifts in both the directions, both LCL and UCL are required by the chart.

A Nonparametric Moving Average chart gives an out of control signal for the first sampling instant i, if

1. if i UCL (when interest is to detect shift in in positive direction).

2. if i LCL (when interest is to detect shift in in negative direction).

3. if i LCL or i UCL (when interest is to detect shift in in both the directions).

In order to obtain LCL/UCL, we need to know the distribution of the charting statistic. Since the sequence of charting statistics, i, i = 1, 2, . is a sequence of dependent variables, it is not easy to obtain exact distribution of i , i =

distributions. We compared the performance of the moving average chart with sign chart, synthetic sign chart and S2 chart for normal, double exponential Cauchy and gamma distributions. We do not consider S2 chart for Cauchy distribution while comparing various charts. The ARL values are calculated under the assumption that the standard deviation of the distributions is one when the process is in- control. As in Bakir (2004), the scale parameter is set to be

= 1/2 for the double exponential distribution to achieve a standard deviation of 1. For the Cauchy distribution, = 0.2605 is chosen to achieve a tail probability of 0.05 above µ

+ 1.645, the same as that of a normal distribution with a mean

µ and a standard deviation of 1. The ARL values for sign chart and synthetic sign chart are exact. For S2 chart under normal distribution, the ARL values are exact while under other distributions they are simulated. The ARL values of moving average sign chart are simulated based on 10,000 runs for samples of sizes n=7 and n=10.

Under normal distribution, the ARL of S2 chart for shift in is

() = 1/(S2 ) , (5)

where S2 = (2 1 | = )

1, 2, . In such cases simulation technique can be used to

1

2 0

obtain control limits and the ARL values. Further, since the distribution of i is discrete, many times, it will not be possible to get LCL/UCL that gives in-control ARL exactly equal to the desired value, say ARL(0). In such a case we choose that LCL/UCL for which in-control ARL is close to the desired ARL(0).

To construct a nonparametric moving average control chart for process variability based on nonparametric sign statistic, we assume that, when the process is in control, the process standard deviation = 0. For nonparametric moving average control chart for variability, our charting statistic i defined in equation (4) is based on nonparametric sign statistic Ui defined in equation (1). The one-sided moving average chart signals when i UCL.

IV. PERFORMANCE STUDY OF THE MOVING AVERAGE CHART

The main task of a control chart is to detect change in the process as early as possible and to give an out-of-control signal. A control chart is more efficient if ARL of the chart is large when the process is in control and small when the process is out of control.

Since the exact distribution of the charting statistic i is not known, the simulation technique is used to compute in-

The upper control limit UCL is determined using equation

1. for desired in-control ARL=ARL(0) when the process is in control ( = 0).

   The performance of different charts is compared by equating their in-control ARLs. For that we computed the adjusted ARL of chart B to match with in-control ARL of the chart A, using following formula

   [ ] = [ ] × [ 0 ] .

   [ 0 ]

   In Table 1, we provide in-control ARL values for various values of Upper Control Limits (d), and span w, when n=7. These values remain same for all the continuous distributions as long as the process standard deviation is 1. It is clear that in-control ARL increases as d or w increases.

   Table 2-Table 5 provide ARL comparison of four charts namely, Moving Average (MA), Synthetic sign, Sign and S2 Charts for n=7. We consider three values of w; the span of MA chart. Table 6- Table 9 provide ARL comparison of the same charts when n=10.

   Table 1: In control ARL values of Moving Average Chart under normal distribution when n=7.

   w=2		w=3		w=4
   d	ARL(0)	d	ARL(0)	d	ARL(0)
   4	40.64	3.00	30.35	3.0	70.03
   5	160.63	3.67	96.86	3.5	209.5
   6	1128.84	4.33	319.68	4.0	65.48

   Table 2: ARL Performance of M. A., Synthetic, Sign and S2 Chart under normal distribution

   (n=7, ARL(0) = 128).

   1 0	MA Chart			Synthetic Chart	Sign Chart	S2 Chart
   	w=2 d=5	w=3 d=4.33	w=4 d=3.5	d=5 L=2	d=7	UCL= 2.906
   1.0	128.00	128.00	128.00	128.00	128.00	128.00
   1.1	58.72	47.49	47.50	58.72	74.92	39.39
   1.2	33.18	22.83	21.95	31.94	48.67	16.77
   1.3	19.99	12.69	12.40	19.68	34.15	8.93
   1.4	13.54	8.32	8.13	13.29	25.40	5.57
   1.5	9.91	5.61	5.75	9.64	19.76	3.89
   1.6	7.50	4.19	4.31	7.38	15.93	2.95
   1.7	6.09	3.28	3.45	5.90	13.22	2.38
   1.8	5.01	2.61	2.87	4.88	11.23	2.02
   1.9	4.26	2.21	2.44	4.16	9.72	1.77
   2.0	3.70	1.91	2.14	3.61	8.55	1.59

   Table 3: ARL Performance of M. A., Synthetic, Sign and S2 Chart under double exponential distribution (n=7, ARL(0) = 128 ).

   1 0	MA Chart			Synthetic Chart	Sign Chart	S2 Chart
   	w=2 d=5	w=3 d=4.33	w=4 d=3.5	d=5 L=2	d=7	UCL= 4.517
   1.0	128.00	128.00	128.00	128.00	128.00	128.00
   1.1	67.93	56.50	56.04	67.25	82.35	61.10
   1.2	39.93	29.72	29.21	39.84	57.02	33.05
   1.3	26.18	17.97	17.31	25.88	41.78	20.54
   1.4	18.30	11.72	11.33	18.04	32.00	13.30
   1.5	13.62	8.04	8.06	13.29	25.40	9.49
   1.6	10.66	5.96	6.16	10.25	20.75	7.22
   1.7	8.27	4.78	4.74	8.21	17.36	5.54
   1.8	6.90	3.78	4.01	6.76	14.81	4.47
   1.9	5.96	3.16	3.34	5.71	12.85	3.78
   2.0	4.98	2.62	2.93	4.93	11.31	3.30

   Table 4: ARL Performance of M. A., Synthetic, Sign and S2 Chart under gamma distribution (n=7, ARL(0) = 128 ).

   1 0	MA Chart			Synthetic Chart	Sign Chart	S2 Chart
   	w=2 d=5	w=3 d=4.33	w=4 d=3.5	d=5 L=2	d=7	UCL= 3.869
   1.0	128.00	128.00	128.00	128.00	128.00	128.00
   1.1	59.17	47.59	46.09	57.89	74.19	49.88
   1.2	31.73	21.99	21.24	30.98	47.61	24.71
   1.3	19.44	12.07	11.66	18.77	32.98	14.49
   1.4	12.76	7.67	7.59	12.49	24.21	9.52
   1.5	9.12	5.14	5.25	8.93	18.60	6.90
   1.6	6.88	3.75	4.04	6.76	14.81	5.25
   1.7	5.50	2.92	3.13	5.35	12.15	4.38
   1.8	4.42	2.35	2.58	4.38	10.20	3.65
   1.9	3.78	1.91	2.19	3.70	8.74	3.21
   2.0	3.16	1.67	1.89	3.20	7.62	2.87

   Table 5: ARL Performance of M. A., Synthetic, Sign and S2 Chart under Cauchy distribution (n=7, ARL(0) = 128 ).

   1 0	MA Chart			Synthetic Chart	Sign Chart
   	w=2 d=5	w=3 d=4.33	w=4 d=3.5	d=5 L=2	d=7
   1.0	128.00	128.00	128.00	128.00	128.00
   1.1	70.74	59.78	59.53	70.17	84.80
   1.2	43.29	31.73	31.69	42.39	59.58
   1.3	28.21	19.63	18.69	27.71	43.91
   1.4	19.59	12.68	12.42	19.29	33.65
   1.5	14.47	8.79	8.71	14.15	26.64
   1.6	11.11	6.37	6.53	10.84	21.68
   1.7	8.88	4.93	5.10	8.60	18.04
   1.8	7.32	4.02	4.19	7.03	15.31
   1.9	6.00	3.30 /td>	3.43	5.90	13.22
   2.0	5.11	2.72	2.96	5.06	11.57

   Table 6: ARL Performance of M. A., Synthetic, Sign and S2 Chart under Normal distribution (n=10, ARL(0) = 171.88).

   1 0	MA chart			Synthetic Chart	Sign Chart	S2 Chart
   	w=2 d=6	w=3 d=4.33	w=4 d=4	d=6 L=2	d=10	UCL= 2.5748
   1.0	171.88	171.88	171.88	171.88	171.88	171.88
   1.1	69.24	61.38	50.28	66.54	79.97	41.78
   1.2	33.73	29.30	21.92	32.11	43.18	15.39
   1.3	19.47	16.37	11.44	18.16	26.03	7.52
   1.4	12.72	10.79	7.23	11.56	17.05	4.48
   1.5	8.72	7.46	4.98	8.04	11.91	3.06
   1.6	6.60	5.78	3.82	5.98	8.76	2.31
   1.7	5.22	4.68	3.05	4.69	6.71	1.88
   1.8	4.33	3.87	2.58	3.84	5.31	1.61
   1.9	3.64	3.35	2.29	3.25	4.32	1.43
   2.0	3.17	2.96	2.03	2.82	3.60	1.32

   Table 7: ARL Performance of M. A., Synthetic, Sign and S2 Chart under double exponential distribution (n=10, ARL(0) = 171.88 ).

   1 0	MA chart			Synthetic Chart	Sign Chart	S2 Chart
   	w=2 d=6	w=3 d=4.33	w=4 d=4	d=6 L=2	d=10	UCL= 4.022
   1.0	171.88	171.88	171.88	171.88	171.88	171.88
   1.1	80.55	72.22	62.63	78.42	91.53	71.15
   1.2	43.64	38.29	29.79	41.78	54.14	35.73
   1.3	26.53	22.37	16.24	25.03	34.72	20.23
   1.4	17.14	14.97	10.33	16.42	23.72	12.53
   1.5	12.27	10.51	7.31	11.56	17.05	8.49
   1.6	9.55	8.32	5.33	8.61	12.78	6.12
   1.7	7.40	6.60	4.28	6.71	9.90	4.65
   1.8	5.97	5.31	3.55	5.43	7.89	3.76
   1.9	4.97	4.55	2.95	4.53	6.45	3.06
   2.0	4.27	3.93	2.63	3.87	5.37	2.60

   Table 8: ARL Performance of M. A., Synthetic, Sign and S2 Chart under gamma distribution (n=10, ARL(0) = 171.88 ).

   1 0	MA chart			Synthetic Chart	Sign Chart	S2 Chart
   	w=2 d=6	w=3 d=4.33	w=4 d=4	d=6 L=2	d=10	UCL= 3.447
   1.0	171.88	171.88	171.88	171.88	171.88	171.88
   1.1	67.38	60.80	50.45	65.42	78.85	56.41
   1.2	32.70	27.82	20.52	30.97	41.85	24.47
   1.3	18.46	15.71	10.73	17.20	24.76	13.43
   1.4	11.73	10.14	6.61	10.77	15.92	8.42
   1.5	8.09	6.96	4.56	7.38	10.93	5.85
   1.6	5.99	5.41	3.55	5.43	7.89	4.46
   1.7	4.52	4.21	2.85	4.22	5.95	3.56
   1.8	3.77	3.56	2.39	3.43	4.64	3.03
   1.9	3.17	3.04	2.04	2.89	3.72	2.64
   2.0	2.76	2.67	1.84	2.50	3.05	2.34

   Table 9: ARL Performance of M. A., Synthetic, Sign and S2 Chart under Cauchy distribution (n=10, ARL(0) = 171.88 ).

   td>

   3.66

   1 0	MA chart			Synthetic Chart	Sign Chart
   	w=2 d=6	w=3 d=4.33	w=4 d=4	d=6 L=2	d=10
   1.0	171.88	171.88	171.88	171.88	171.88
   1.1	85.29	78.40	66.70	82.55	95.45
   1.2	47.22	41.49	32.10	44.99	57.65
   1.3	28.72	24.84	17.89	27.13	37.28
   1.4	19.31	16.33	11.22	17.75	25.49
   1.5	13.52	11.57	7.81	12.42	18.26
   1.6	10.00	8.62	5.74	9.17	13.60
   1.7	7.69	6.68	4.44	7.08	10.46
   1.8	6.23	5.52	5.67	8.28
   1.9	5.19	4.61	3.09	4.69	6.71
   2.0	4.39	4.04	2.69	3.98	5.55

   CONCLUSIONS

   • The synthetic sign chart performs better than the sign chart for all the process distributions considered here. However the performance of the MA chart is better than the synthetic chart for all the process distributions when the span of the moving average w 3. The ARL performance of the MA chart goes on increasing as the span of the MA chart increases.

   • The MA chart with w=4 performs better than the S2 chart except for the normal distribution. Though the performance of the S2 chart is superior to all the other charts for the normal distribution, it is possible to

   • choose an appropriate w so as to match the ARL performance of MA chart with the S2 chart.

   • The MA chart performs better even for small sample size like n=7 and the performance improves as sample size increases.

VI ILLUSTRATIVE EXAMPLE

To illustrate the construction of the proposed nonparametric moving average control chart for variability, we use the simulated data. Twenty samples of size ten each were randomly generated from normal distribution with mean zero and in-control standard

deviation 0= 1 and then, ten samples of size ten each were randomly generated from normal distribution with mean zero and standard deviation 1= 1.2. Upper control limit of MA chart for i < w(=3) is not shown in

Figure 1. These control limits decrease as the value of w increases. In the present illustration these limits have no effect as far as status of the process is concerned at the beginning.

The values of Moving Average Statistics were calculated with span of moving average w =2, 3, 4 for each sample. In Figure 1, we plotted sign chart using upper control limit d = 8 and moving average chart

with span w = 3 using upper control limit UCL = 4.33. It is observed that the sign chart does not signal at all but the moving average chart signals at sample numbers 28 and 29.

Table 10: Values of Sign and Moving Average Statistics for the samples in the example.

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